Fixed Point Theorems for L-Contractions in Generalized...
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Research Article Fixed Point Theorems for L-Contractions in Generalized Metric Spaces Seong-Hoon Cho Department of Mathematics, Hanseo University, Chungnam -, Republic of Korea Correspondence should be addressed to Seong-Hoon Cho; [email protected] Received 21 August 2018; Revised 8 October 2018; Accepted 4 November 2018; Published 2 December 2018 Academic Editor: Aref Jeribi Copyright © 2018 Seong-Hoon Cho. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, the notion of L-contractions is introduced and a new fixed point theorem for such contractions is established. 1. Introduction and Preliminaries Branciari [1] introduced the notion of generalized metric spaces and obtained a generalization of the Banach contrac- tion principle, whereaſter many authors proved various fixed point results in such spaces, for example, [2–8] and references therein. Also, Suzuki et al. [9] and Abtahi et al. [10] studied ]-generalized metric spaces and proved the Banach and Kannan contraction principles in such spaces, and Mitrovi´ c et al. [11] introduced the notion of ] ()-generalized metric spaces and Banach and Reich contraction principles in such spaces. In particular, Jleli and Samet [12] introduced the notion of -contractions and gave a generalization of the Banach contraction principle in generalized metric spaces, where : (0,∞) → (1,∞) is a function satisfying the following conditions: (1) is nondecreasing; (2) ∀{ } ⊂ (0, ∞), lim →∞ ( )=1⇐⇒ lim →∞ =0 + ; (1) (3) ∃ ∈ (0, 1) ∧ ∈ (0, ∞): lim →0 + () − 1 = . (2) Also, Ahmad et al. [13] extended the result of Jleli and Samet [12] to metric spaces by applying the following simple condition (4) instead of (3). (4) is continuous on (0, ∞). Recently, Khojasteh et al. [14] introduced the notion of Z-contractions by defining the concept of simulation functions. ey unified some existing metric fixed point results. Aſterward, many authors ([15–19] and references therein) obtained generalizations of the result of [14]. In the paper, we introduce the concept of a new type of contraction maps, and we establish a new fixed point theorem for such contraction maps in the setting of generalized metric spaces. Let L be the family of all mappings : [1,∞)×[1,∞) → R such that (1) (1, 1) = 1; (2) (, ) < / ∀, > 1; (3) for any sequence { }, { } ⊂ (1, ∞) with ≤ ∀ = 1, 2, 3, ⋅ ⋅ ⋅ lim →∞ = lim →∞ >1⇒ lim →∞ sup ( , ) < 1. (3) We say that ∈ L is a L-simulation function. Note that (, ) < 1 ∀ > 1. Example . Let , , : [1, ∞) × [1, ∞) → R be functions defined as follows, respectively: (1) (, ) = / ∀, ≥ 1, where ∈ (0, 1); Hindawi Abstract and Applied Analysis Volume 2018, Article ID 1327691, 6 pages https://doi.org/10.1155/2018/1327691
Transcript of Fixed Point Theorems for L-Contractions in Generalized...
Research ArticleFixed Point Theorems for L-Contractions inGeneralized Metric Spaces
Seong-Hoon Cho
Department of Mathematics Hanseo University Chungnam 356-706 Republic of Korea
Correspondence should be addressed to Seong-Hoon Cho shchohanseoackr
Received 21 August 2018 Revised 8 October 2018 Accepted 4 November 2018 Published 2 December 2018
Academic Editor Aref Jeribi
Copyright copy 2018 Seong-Hoon Cho This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In this paper the notion ofL-contractions is introduced and a new fixed point theorem for such contractions is established
1 Introduction and Preliminaries
Branciari [1] introduced the notion of generalized metricspaces and obtained a generalization of the Banach contrac-tion principle whereafter many authors proved various fixedpoint results in such spaces for example [2ndash8] and referencestherein Also Suzuki et al [9] and Abtahi et al [10] studied]-generalized metric spaces and proved the Banach andKannan contraction principles in such spaces and Mitrovicet al [11] introduced the notion of 119887](119904)-generalized metricspaces and Banach and Reich contraction principles in suchspaces
In particular Jleli and Samet [12] introduced the notionof 120579-contractions and gave a generalization of the Banachcontraction principle in generalized metric spaces where120579 (0infin) 997888rarr (1infin) is a function satisfying the followingconditions(1205791) 120579 is nondecreasing(1205792) forall119905119899 sub (0infin)
lim119899997888rarrinfin
120579 (119905119899) = 1 lArrrArr lim119899997888rarrinfin
119905119899 = 0+ (1)
(1205793) exist119903 isin (0 1) and 119897 isin (0infin)lim119905997888rarr0+
120579 (119905) minus 1119905119903 = 119897 (2)
Also Ahmad et al [13] extended the result of Jleli andSamet [12] to metric spaces by applying the following simplecondition (1205794) instead of (1205793)
(1205794) 120579 is continuous on (0infin)Recently Khojasteh et al [14] introduced the notion
of Z-contractions by defining the concept of simulationfunctions They unified some existing metric fixed pointresults Afterward many authors ([15ndash19] and referencestherein) obtained generalizations of the result of [14]
In the paper we introduce the concept of a new type ofcontraction maps and we establish a new fixed point theoremfor such contraction maps in the setting of generalized metricspaces
LetL be the family of allmappings 120585 [1infin)times[1infin) 997888rarrR such that
(1205851) 120585(1 1) = 1(1205852) 120585(119905 119904) lt 119904119905 forall119904 119905 gt 1(1205853) for any sequence 119905119899 119904119899 sub (1infin) with 119905119899 le 119904119899 forall119899 =1 2 3 sdot sdot sdot
lim119899997888rarrinfin
119905119899 = lim119899997888rarrinfin
119904119899 gt 1 997904rArr lim119899997888rarrinfin
sup 120585 (119905119899 119904119899) lt 1 (3)
We say that 120585 isin L is aL-simulation functionNote that 120585(119905 119905) lt 1 forall119905 gt 1
Example 1 Let 120585119887 120585119908 120585 [1infin) times [1infin) 997888rarr R be functionsdefined as follows respectively
(1) 120585119887(119905 119904) = 119904119896119905 forall119905 119904 ge 1 where 119896 isin (0 1)
HindawiAbstract and Applied AnalysisVolume 2018 Article ID 1327691 6 pageshttpsdoiorg10115520181327691
2 Abstract and Applied Analysis
(2) 120585119908(119905 119904) = 119904119905120601(119904) forall119905 119904 ge 1 where 120601 [1infin) 997888rarr[1infin) is nondecreasing and lower semicontinuoussuch that 120601minus1(1) = 1
120585 (119905 119904) =
1 if (119904 119905) = (1 1) 1199042119905 if 119904 lt 119905119904120582119905 otherwise
(4)
forall119904 119905 ge 1 where 120582 isin (0 1)Then 120585119887 120585119908 120585 isin L
We recall the following definitions which are in [1]Let 119883 be a nonempty set and let 119889 119883 times 119883 997888rarr [0infin)
be a map such that for all 119909 119910 isin 119883 and for all distinct points119906 V isin 119883 each of them is different from 119909 and 119910(d1) 119889(119909 119910) = 0 if and only if 119909 = 119910(d2) 119889(119909 119910) = 119889(119910 119909)(d3) 119889(119909 119910) le 119889(119909 119906) + 119889(119906 V) + 119889(V 119910)Then 119889 is called a generalized metric on 119883 and (119883 119889) is
called a generalized metric spaceLet (119883 119889) be a generalized metric space let 119909119899 sub 119883 be a
sequence and 119909 isin 119883Then we say that
(1) 119909119899 is convergent to 119909 (denoted by lim119899997888rarrinfin 119909119899 = 119909)if and only if lim119899997888rarrinfin 119889(119909119899 119909) = 0
(2) 119909119899 is Cauchy if and only if lim119899119898997888rarrinfin 119889(119909119899 119909119898) = 0(3) (119883 119889) is complete if and only if every Cauchy
sequence in119883 is convergent to some point in119883
Let (119883 119889) be a generalized metric spaceA map 119879 119883 997888rarr 119883 is called continuous at 119909 isin 119883 if for
any 119881 isin 120591 containing 119879119909 there exists 119880 isin 120591 containing 119909such that 119879119880 sub 119881 where 120591 is the topology on 119883 induced bythe generalized metric 119889 That is
120591 = 119880 sub 119883 forall119909 isin 119880 exist119861 isin 120573 119909 isin 119861 sub 119880 120573 = 119861 (119909 119903) 119909 isin 119883 forall119903 gt 0 119861 (119909 119903) = 119910 isin 119883 119889 (119909 119910) lt 119903
(5)
If 119879 is continuous at each point 119909 isin 119883 then it is calledcontinuous
Note that 119879 is continuous if and only if it is sequentiallycontinuous ie lim119899997888rarrinfin 119889(119879119909119899 119879119909) = 0 for any sequence119909119899 sub 119883 with lim119899997888rarrinfin 119889(119909119899 119909) = 0Remark 2 (see [6]) If 119889 is a generalized metric on 119883 then itis not continuous in each coordinate
Lemma 3 (see [20]) Let (119883 119889) be a generalized metric spacelet 119909119899 sub 119883 be a Cauchy sequence and 119909 119910 isin 119883 If there existsa positive integer119873 such that
(1) 119909119899 = 119909119898 forall119899119898 gt 119873
(2) 119909119899 = 119909 forall119899 gt 119873(3) 119909119899 = 119910 forall119899 gt 119873(4) lim119899997888rarrinfin 119889(119909119899 119909) = lim119899997888rarrinfin 119889(119909119899 119910)then 119909 = 119910
2 Fixed Point Theorems
We denote by Θ the class of all functions 120579 (0infin) 997888rarr(1infin) such that conditions (1205791) and (1205792) holdA mapping 119879 119883 997888rarr 119883 is called L-contraction with
respect to 120585 if there exist 120579 isin Θ and 120585 isin L such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0120585 (120579 (119889 (119879119909 119879119910)) 120579 (119889 (119909 119910))) ge 1 (6)
Note that if T isL-contraction with respect to 120585 then it iscontinuous In fact let 119909 isin 119883 be a point and let 119909119899 sub 119883 beany sequence such that
lim119899997888rarrinfin
119889 (119909119899 119909) = 0+119889 (119879119909119899 119879119909) gt 0
forall119899 = 1 2 3 sdot sdot sdot (7)
Then from (1205792) lim119899997888rarrinfin 120579(119889(119909119899 119909)) = 1It follows from (6) and (1205852) that1 le 120585 (120579 (119889 (119879119909119899 119879119909)) 120579 (119889 (119909119899 119909))) lt 120579 (119889 (119909119899 119909))120579 (119889 (119879119909119899 119879119909)) (8)
which implies
120579 (119889 (119879119909119899 119879119909)) lt 120579 (119889 (119909119899 119909)) (9)
Since 120579 is nondecreasing we have119889 (119879119909119899 119879119909) lt 119889 (119909119899 119909) (10)
and so
lim119899997888rarrinfin
119889 (119879119909119899 119879119909) = 0 (11)
Hence 119879 is continuousNow we prove our main result
Theorem 4 Let (119883 119889) be a complete generalized metric spaceand let 119879 119883 997888rarr 119883 be aL-contraction with respect to 120585
en 119879 has a unique fixed point and for every initial point1199090 isin 119883 the Picard sequence 1198791198991199090 converges to the fixed pointProof Firstly we show uniqueness of fixed point whenever itexists
Assume that 119908 and 119906 are fixed points of 119879If 119906 = 119911 then 119889(119908 119906) gt 0 and so it follows from (6) that
1 le 120585 (120579 (119889 (119879119908 119879119906)) 120579 (119889 (119908 119906)))= 120585 (120579 (119889 (119908 119906)) 120579 (119889 (119908 119906))) lt 120579 (119889 (119908 119906))
120579 (119889 (119908 119906)) (12)
Abstract and Applied Analysis 3
Hence
120579 (119889 (119908 119906)) lt 120579 (119889 (119908 119906)) (13)
which is a contradictionHence 119908 = 119906 and fixed point of 119879 is uniqueSecondly we prove existence of fixed pointLet 1199090 isin 119883 be a point Define a sequence 119909119899 sub 119883 by119909119899 = 119879119909119899minus1 = 1198791198991199090 forall119899 = 1 2 3 sdot sdot sdot If 1199091198990 = 1199091198990+1 for some 1198990 isin N then 1199091198990 is a fixed point of119879 and the proof is finishedAssume that
119909119899minus1 = 119909119899 forall119899 = 1 2 3 sdot sdot sdot (14)
It follows from (6) and (14) that forall119899 = 1 2 3 sdot sdot sdot1 le 120585 (120579 (119889 (119879119909119899minus1 119879119909119899)) 120579 (119889 (119909119899minus1 119909119899)))= 120585 (120579 (119889 (119909119899 119909119899+1)) 120579 (119889 (119909119899minus1 119909119899)))lt 120579 (119889 (119909119899minus1 119909119899))120579 (119889 (119909119899 119909119899+1))
(15)
Consequently we obtain that
120579 (119889 (119909119899 119909119899+1)) lt 120579 (119889 (119909119899minus1 119909119899)) forall119899 = 1 2 3 sdot sdot sdot (16)
which implies
119889 (119909119899 119909119899+1) lt 119889 (119909119899minus1 119909119899) forall119899 = 1 2 3 sdot sdot sdot (17)
Hence 119889(119909119899minus1 119909119899) is a decreasing sequence and so thereexists 119903 ge 0 such that
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899) = 119903 (18)
We now show that 119903 = 0Assume that 119903 = 0Then it follows from (1205792) that
lim119899997888rarrinfin
120579 (119889 (119909119899minus1 119909119899)) = 1 (19)
and so
lim119899997888rarrinfin
120579 (119889 (119909119899minus1 119909119899)) gt 1 (20)
Let 119904119899 = 120579(119889(119909119899minus1 119909119899)) and 119905119899 = 120579(119889(119909119899 119909119899+1)) forall119899 =1 2 3 sdot sdot sdot From (1205853) we obtain
1 le lim119899997888rarrinfin
sup 120585 (119905119899 119904119899) lt 1 (21)
which is a contradictionThus we have
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899) = 0 (22)
and so
lim119899997888rarrinfin
120579 (119889 (119909119899minus1 119909119899)) = 1 (23)
We show that
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899+1) = 0 (24)
We consider three cases
Case 1 119909119899 = 119909119899+2 forall119899 = 1 2 3 sdot sdot sdot From (6) and (14) we obtain that forall119899 = 1 2 3 sdot sdot sdot1 le 120585 (120579 (119889 (119879119909119899minus1 119879119909119899+1)) 120579 (119889 (119909119899minus1 119909119899+1)))= 120585 (120579 (119889 (119909119899 119909119899+2)) 120579 (119889 (119909119899minus1 119909119899+1)))lt 120579 (119889 (119909119899minus1 119909119899+1))120579 (119889 (119909119899 119909119899+2))
(25)
and so
120579 (119889 (119909119899 119909119899+2)) lt 120579 (119889 (119909119899minus1 119909119899+1)) forall119899 = 1 2 3 sdot sdot sdot (26)
which implies
119889 (119909119899 119909119899+2) lt 119889 (119909119899minus1 119909119899+1) forall119899 = 1 2 3 sdot sdot sdot (27)
Hence 119889(119909119899minus1 119909119899+1) is decreasingIn a manner similar to that which proved (22) we have
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899+1) = 0 (28)
Case 2 There exists 1198990 ge 1 such that 1199091198990 = 1199091198990+2From the first term to the 1198990 th term shall be removed
and let 119909119899 = 1199091198990+119899 forall119899 = 1 2 3 sdot sdot sdot Then 119909119899 = 119909119899+2 forall119899 = 1 2 3 sdot sdot sdot By Case 1 we have
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899+1) = 0 (29)
Case 3 119909119899 = 119909119899+2 forall119899 = 0 1 2 sdot sdot sdot We have
119889 (119909119899minus1 119909119899+1) = 0 forall119899 = 1 2 3 sdot sdot sdot (30)
Hence
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899+1) = 0 (31)
In all cases (24) is satisfiedNow we show that 119909119899 is boundedIf 119909119899 is not bounded then there exists a subsequence119909119899(119896) of 119909119899 such that 1198991 = 1 and forall119896 = 1 2 3 119899(119896 + 1)
is the minimum integer greater than 119899(119896) with119889 (119909119899(119896+1) 119909119899(119896)) gt 1
119889 (119909119897 119909119899(119896)) le 1 (32)
for 119899(119896) le 119897 le 119899(119896 + 1) minus 1Then we have
1 lt 119889 (119909119899(119896+1) 119909119899(119896))le 119889 (119909119899(119896+1) 119909119899(119896+1)minus2) + 119889 (119909119899(119896+1)minus2 119909119899(119896+1)minus1)+ 119889 (119909119899(119896+1)minus1 119909119899(119896))
le 119889 (119909119899(119896+1) 119909119899(119896+1)minus2) + 119889 (119909119899(119896+1)minus2 119909119899(119896+1)minus1) + 1(33)
4 Abstract and Applied Analysis
By letting 119896 997888rarr infin in the above we obtain
lim119896997888rarrinfin
119889 (119909119899(119896+1) 119909119899(119896)) = 1 (34)
By using (22) (34) and condition (d3) we deduce that
lim119896997888rarrinfin
119889 (119909119899(119896+1)minus1 119909119899(119896)minus1) = 1 (35)
It follows from (1205792) (34) and (35) that
lim119896997888rarrinfin
120579 (119889 (119909119899(119896+1) 119909119899(119896))) gt 1 (36)
lim119899997888rarrinfin
120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1)) gt 1 (37)
From (6) and (32) we infer that
1 le 120585 (120579 (119889 (119879119909119899(119896+1)minus1 119879119909119899(119896)minus1)) 120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1))) = 120585 (120579 (119889 (119909119899(119896+1) 119909119899(119896))) 120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1))) lt 120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1))
120579 (119889 (119909119899(119896+1) 119909119899(119896)))(38)
which implies
120579 (119889 (119909119899(119896+1) 119909119899(119896))) lt 120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1)) (39)
Let
119904119899 = 120579(119889 (119909119899(119896+1)minus1 119909119899(119896)minus1) 119905119899 = 120579 (119889 (119909119899(119896+1) 119909119899(119896))) forall119899 = 1 2 3 sdot sdot sdot (40)
Then 119905119896 lt 119904119896 forall119899 = 1 2 3 sdot sdot sdot and lim119896997888rarrinfin 119905119896 =lim119896997888rarrinfin 119904119896 gt 1
It follows from (1205853) that1 le lim119896997888rarrinfin
sup 120585 (119905119896 119904119896) lt 1 (41)
which is a contradictionThus 119909119899 is boundedNow we show that 119909119899 is a Cauchy sequenceLet
119872119899 = sup 119889 (119909119894 119909119895) 119894 119895 ge 119899 (42)
Clearly
0 le 119872119899+1 le 119872119899 lt infin forall119899 = 1 2 3 sdot sdot sdot (43)
and so there exists119872 ge 0 such that
lim119899997888rarrinfin
119872119899 = 119872 (44)
Assume that119872 gt 0It follows from (42) that forall119896 = 1 2 3 sdot sdot sdot there exist119899(119896)119898(119896) ge 119896 with
119872119896 minus 1119896 lt 119889 (119909119898(119896) 119909119899(119896)) le 119872119896 (45)
Solim119896997888rarrinfin
119889 (119909119898(119896) 119909119899(119896)) = lim119896997888rarrinfin
119872119896 = 119872 (46)
It follows from (6) and (14) that
120585 (120579 (119889 (119879119909119898(119896)minus1 119879119909119899(119896)minus1)) 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1)))= 120585 (120579 (119889 (119909119898(119896) 119909119899(119896))) 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1)))lt 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1))
120579 (119889 (119909119898(119896) 119909119899(119896)))(47)
which implies
120579 (119889 (119909119898(119896) 119909119899(119896))) lt 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1)) (48)
Hence we have119889 (119909119898(119896) 119909119899(119896)) lt 119889 (119909119898(119896)minus1 119909119899(119896)minus1)
le 119889 (119909119898(119896)minus1 119909119898(119896)) + 119889 (119909119898(119896) 119909119899(119896))+ 119889 (119909119899(119896) 119909119899(119896)minus1)
(49)
Letting 119896 997888rarr infin in the above inequality we obtain
lim119899997888rarrinfin
119889 (119909119898(119896)minus1119909119899(119896)minus1) = 119872 (50)
Let119904119896 = 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1)) 119905119896 = 120579 (119889 (119909119898(119896) 119909119899(119896))) forall119896 = 1 2 3 sdot sdot sdot (51)
Then 119905119896 lt 119904119896 forall119896 = 1 2 3 sdot sdot sdot Since119872 gt 0lim119896997888rarrinfin
119905119896 = lim119896997888rarrinfin
119904119896 gt 1 (52)
Thus we have1 le lim119896997888rarrinfin
sup 120585 (119905119896 119904119896) lt 1 (53)
which is a contradictionHence119872 = 0 and hence 119909119899 is a Cauchy sequenceSince 119883 is complete there exists 119911 isin 119883 such that
lim119899997888rarrinfin
119889 (119909119899 119911) = 0 (54)
Because 119879 is continuouslim119899997888rarrinfin
119889 (119909119899 119879119911) = lim119899997888rarrinfin
119889 (119879119909119899minus1 119879119911) = 0 (55)
By Lemma 3 119911 = 119879119911We give an example to illustrate Theorem 4
Example 5 Let 119883 = 1 2 3 4 and define 119889 119883 times 119883 997888rarr[0infin) as follows119889 (1 2) = 119889 (2 1) = 3119889 (2 3) = 119889 (3 2) = 119889 (1 3) = 119889 (3 1) = 1119889 (1 4) = 119889 (4 1) = 119889 (2 4) = 119889 (4 2) = 119889 (3 4)
= 119889 (4 3) = 4119889 (119909 119909) = 0 forall119909 isin 119883
(56)
Abstract and Applied Analysis 5
Then (119883 119889) is a complete generalized metric space butnot a metric space (see [21])
Define a map 119879 119883 997888rarr 119883 by
119879119909 = 3 (119909 = 4) 1 (119909 = 4) (57)
And define a function 120579 (0infin) 997888rarr (1infin) by120579 (119905) = 119890119905 (58)
We now show that 119879 is a L-contraction with respect to120585119887 where 120585119887(119905 119904) = 119904119896119905 forall119905 119904 ge 1 119896 = 12
We have
119889 (119879119909 119879119910) =
119889 (1 3) = 1 (119909 = 4 119910 = 4) 119889 (1 1) = 0 (119909 = 4 119910 = 4) 119889 (3 3) = 0 (119909 = 4 119910 = 4)
(59)
so
119889 (119879119909 119879119910) gt 0 lArrrArr 119909 = 4 119910 = 4 (60)
We have for 119909 = 4 and 119910 = 4119889 (119909 119910) = 4
119889 (119879119909 119879119910) = 1 (61)
We deduce that for all 119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0120585119887 (120579 (119889 (119879119909 119879119910)) 120579 (119889 (119909 119910))) = [120579 (119889 (119909 119910))]119896
120579 (119889 (119879119909 119879119910))
= [1198904]121198901 = 119890 gt 1
(62)
Thus all hypotheses of Theorem 4 are satisfied and 119879 hasa fixed point 119909lowast = 3
Note that Banachrsquos contraction principle is not satisfiedwith the usual metric 120588(119909 119910) = |119909 minus 119910| forall119909 119910 isin 119883 In fact if119909 = 2 119910 = 4 then
120588 (1198792 1198794) le 119896120588 (2 4) 119896 isin (0 1) (63)
which implies
119896 ge 1 (64)
Also note that the 120579-contraction condition [13] does nothold
Let 120579(119905) = 119890119905 forall119905 gt 0Then 120579(119905) satisfies conditions (1205791) (1205792) and (1205794)If
120579 (120588 (1198792 1198794)) le [120579 (120588 (2 4))]119896 where 119896 isin (0 1) (65)
then
1198902 le [1198902]119896 (66)
and so 119896 ge 1 Hence 119879 is not 120579-contraction mapBy taking 120585 = 120585119887 in Theorem 4 we obtain Corollary 6
Corollary 6 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
120579 (119889 (119879119909 119879119910)) le 120579 (119889 (119909 119910))119896 (67)
where 120579 isin Θ and 119896 isin (0 1)en 119879 has a unique fixed point
Remark 7 Corollary 6 is a generalization of Theorem 21 of[12] without condition (1205793) and Theorem 22 of [13] withoutcondition (1205794)
By taking 120585 = 120585119908 in Theorem 4 we obtain Corollary 8
Corollary 8 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
120579 (119889 (119879119909 119879119910)) le 120579 (119889 (119909 119910))120601 (120579 (119889 (119909 119910))) (68)
where 120579 isin Θ and 120601 [1infin) 997888rarr [1infin) is nondecreasing andlower semicontinuous such that 120601minus1(1) = 1
en 119879 has a unique fixed point
Corollary 9 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
119889 (119879119909 119879119910) le 119889 (119909 119910) minus 120593 (119889 (119909 119910)) (69)
where 120593 [0infin) 997888rarr [0infin) is nondecreasing and lowersemicontinuous such that 120593minus1(0) = 0
en 119879 has a unique fixed point
Proof Condition (69) implies 119879 is continuousLet 120579(119905) = 119890119905 forall119905 gt 0From (69) we have that for all 119909 119910 isin 119883 with 119889(119879119909 119879119910) gt0120579 (119889 (119879119909 119879119910)) = 119890119889(119879119909119879119910) le 119890119889(119909119910)minus120593(119889(119909119910)) = 119890119889(119909119910)
119890120593(119889(119909119910)) (70)
Let 120593(119905) = ln(120601(120579(119905))) forall119905 ge 0 where 120601 [1infin) 997888rarr[1infin) is nondecreasing and lower semicontinuous such that120601minus1(1) = 1Then 120593 is nondecreasing and lower semicontinuous and
120593 (119905) = 0 lArrrArr 120601 (120579 (119905)) = 1 lArrrArr 120579 (119905) = 119890119905 = 1 lArrrArr 119905 = 0 (71)
It follows from (70) that for all 119909 119910 isin 119883with 119889(119879119909 119879119910) gt0120579 (119889 (119879119909 119879119910)) le 120579 (119889 (119909 119910))
119890ln(120601(120579(119889(119909119910)))) =120579 (119889 (119909 119910))
120601 (120579 (119889 (119909 119910))) (72)
By Corollary 8 119879 has a unique fixed point
By taking 120579(119905) = 2 minus (2120587) arctan(1119905120572) where 120572 isin(0 1) 119905 gt 0 in Corollary 8 we obtain the following result
Corollary 10 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
6 Abstract and Applied Analysis
2 minus 2120587 arctan( 1
[119889 (119879119909 119879119910)]120572) le 2 minus (2120587) arctan (1 [119889 (119909 119910)]120572)120601 (2 minus (2120587) arctan (1 [119889 (119909 119910)]120572)) (73)
where 120572 isin (0 1) and 120601 [1infin) 997888rarr [1infin) is nondecreasingand lower semicontinuous such that 120601minus1(1) = 1
en 119879 has a unique fixed point
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The author express his gratitude to the referees for carefulreading and giving variable comments This research wassupported by Hanseo University
References
[1] A Branciari ldquoA fixed point theorem of Banach-Caccioppolitype on a class of generalized metric spacesrdquo PublicationesMathematicae vol 57 no 1-2 pp 31ndash37 2000
[2] H Aydi E Karapınar and B Samet ldquoFixed points for general-ized (120572120595) contractions on generalized metric spacesrdquo Journalof Inequalities and Applications vol 2014 article 229 2014
[3] C Di Bari and P Vetro ldquoCommon fixed points in generalizedmetric spacesrdquo Applied Mathematics and Computation vol 218no 13 pp 7322ndash7325 2012
[4] P Das ldquoA fixed point theorem on a class of generalized metricspacesrdquo e Korean Journal of Mathematics vol 9 pp 29ndash332002
[5] W A Kirk and N Shahzad ldquoGeneralized metrics and Caristirsquostheoremrdquo Fixed Point eory and Applications vol 2013 article129 2013
[6] B Samet ldquoDiscussion on ldquoA fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spacesrdquo by ABranciarirdquo Publicationes Mathematicae vol 76 no 3-4 pp 493-494 2010
[7] I R Sarma J M Rao and S S Rao ldquoContractions overgeneralizedmetric spacesrdquoe Journal of Nonlinear Science andits Applications vol 2 no 3 pp 180ndash182 2009
[8] W Shatanawi A Al-Rawashdeh Hassen Aydi and H KNashine ldquoOn a Fixed Point for Generalized Contractions inGeneralized Metric Spacesrdquo Abstract and Applied Analysis vol2012 Article ID 246085 13 pages 2012
[9] T Suzuki B Alamri andL A Khan ldquoSome notes on fixed pointtheoems in ]-generalized metric spacesrdquo Bulletin of the KyushuInstitute of TechnologyMathematics Natural Science no 62 pp15ndash23 2015
[10] M Abtahi Z Kadelburg and S Radenovic ldquoFixed points ofCiric-Matakowski-type contractions in ]-generalized metricspacesrdquo RACSAM
[11] Z D Mitrovic and S Radenovic ldquoThe Banach and Reichcontractiond in b](s)-metric spacesrdquo Journal of Fixed Pointeory and Applications vol 19 no 4 pp 3087ndash3095 2017
[12] M Jleli and B Samet ldquoA new generalization of the Banachcontraction principlerdquo Journal of Inequalities and Applicationsvol 2014 no 1 article no 38 8 pages 2014
[13] J Ahmad A E Al-Mazrooei Y J Cho and Y-O YangldquoFixed point results for generalized 120579-contractionsrdquo Journal ofNonlinear Sciences and Applications JNSA vol 10 no 5 pp2350ndash2358 2017
[14] F Khojasteh S Shukla and S Radenovic ldquoA new approachto the study of fixed point theorems via simulation functionsrdquoFilomat vol 29 no 6 pp 1189ndash1194 2015
[15] H H Alsulami E Karapinar F Khojasteh and A-F Roldan-Lopez-de-Hierro ldquoA proposal to the study of contractions inquasi-metric spacesrdquo Discrete Dynamics in Nature and Societyvol 2014 Article ID 269286 pp 1ndash10 2014
[16] H Isik N Gungor C Park and S Jang ldquoFixed PointTheoremsfor Almost Z-Contractions with an ApplicationrdquoMathematicsvol 6 no 3 p 37 2018
[17] E Karapınar ldquoFixed points results via simulation functionsrdquoFilomat vol 30 no 8 pp 2343ndash2350 2016
[18] A-F Roldan-Lopez-de-Hierro E Karapinar C Roldan-Lopez-de-Hierro and J Martinez-Moreno ldquoCoincidence point the-orems on metric spaces via simulation functionsrdquo Journal ofComputational and Applied Mathematics vol 275 pp 345ndash3552015
[19] S Radenovic F Vetro and J Vujakovic ldquoAn alternative andeasy approach to fixed point results via simulation functionsrdquoDemonstratio Mathematica vol 50 no 1 pp 223ndash230 2017
[20] M Jleli and B Samet ldquoThe Kannanrsquos fixed point theorem in acone rectangular metric spacerdquoe Journal of Nonlinear Scienceand its Applications vol 2 no 3 pp 161ndash167 2009
[21] A Azam and M Arshad ldquoKannan fixed point theorem ongeneralizedmetric spacesrdquoe Journal of Nonlinear Science andits Applications vol 1 no 1 pp 45ndash48 2008
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2 Abstract and Applied Analysis
(2) 120585119908(119905 119904) = 119904119905120601(119904) forall119905 119904 ge 1 where 120601 [1infin) 997888rarr[1infin) is nondecreasing and lower semicontinuoussuch that 120601minus1(1) = 1
120585 (119905 119904) =
1 if (119904 119905) = (1 1) 1199042119905 if 119904 lt 119905119904120582119905 otherwise
(4)
forall119904 119905 ge 1 where 120582 isin (0 1)Then 120585119887 120585119908 120585 isin L
We recall the following definitions which are in [1]Let 119883 be a nonempty set and let 119889 119883 times 119883 997888rarr [0infin)
be a map such that for all 119909 119910 isin 119883 and for all distinct points119906 V isin 119883 each of them is different from 119909 and 119910(d1) 119889(119909 119910) = 0 if and only if 119909 = 119910(d2) 119889(119909 119910) = 119889(119910 119909)(d3) 119889(119909 119910) le 119889(119909 119906) + 119889(119906 V) + 119889(V 119910)Then 119889 is called a generalized metric on 119883 and (119883 119889) is
called a generalized metric spaceLet (119883 119889) be a generalized metric space let 119909119899 sub 119883 be a
sequence and 119909 isin 119883Then we say that
(1) 119909119899 is convergent to 119909 (denoted by lim119899997888rarrinfin 119909119899 = 119909)if and only if lim119899997888rarrinfin 119889(119909119899 119909) = 0
(2) 119909119899 is Cauchy if and only if lim119899119898997888rarrinfin 119889(119909119899 119909119898) = 0(3) (119883 119889) is complete if and only if every Cauchy
sequence in119883 is convergent to some point in119883
Let (119883 119889) be a generalized metric spaceA map 119879 119883 997888rarr 119883 is called continuous at 119909 isin 119883 if for
any 119881 isin 120591 containing 119879119909 there exists 119880 isin 120591 containing 119909such that 119879119880 sub 119881 where 120591 is the topology on 119883 induced bythe generalized metric 119889 That is
120591 = 119880 sub 119883 forall119909 isin 119880 exist119861 isin 120573 119909 isin 119861 sub 119880 120573 = 119861 (119909 119903) 119909 isin 119883 forall119903 gt 0 119861 (119909 119903) = 119910 isin 119883 119889 (119909 119910) lt 119903
(5)
If 119879 is continuous at each point 119909 isin 119883 then it is calledcontinuous
Note that 119879 is continuous if and only if it is sequentiallycontinuous ie lim119899997888rarrinfin 119889(119879119909119899 119879119909) = 0 for any sequence119909119899 sub 119883 with lim119899997888rarrinfin 119889(119909119899 119909) = 0Remark 2 (see [6]) If 119889 is a generalized metric on 119883 then itis not continuous in each coordinate
Lemma 3 (see [20]) Let (119883 119889) be a generalized metric spacelet 119909119899 sub 119883 be a Cauchy sequence and 119909 119910 isin 119883 If there existsa positive integer119873 such that
(1) 119909119899 = 119909119898 forall119899119898 gt 119873
(2) 119909119899 = 119909 forall119899 gt 119873(3) 119909119899 = 119910 forall119899 gt 119873(4) lim119899997888rarrinfin 119889(119909119899 119909) = lim119899997888rarrinfin 119889(119909119899 119910)then 119909 = 119910
2 Fixed Point Theorems
We denote by Θ the class of all functions 120579 (0infin) 997888rarr(1infin) such that conditions (1205791) and (1205792) holdA mapping 119879 119883 997888rarr 119883 is called L-contraction with
respect to 120585 if there exist 120579 isin Θ and 120585 isin L such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0120585 (120579 (119889 (119879119909 119879119910)) 120579 (119889 (119909 119910))) ge 1 (6)
Note that if T isL-contraction with respect to 120585 then it iscontinuous In fact let 119909 isin 119883 be a point and let 119909119899 sub 119883 beany sequence such that
lim119899997888rarrinfin
119889 (119909119899 119909) = 0+119889 (119879119909119899 119879119909) gt 0
forall119899 = 1 2 3 sdot sdot sdot (7)
Then from (1205792) lim119899997888rarrinfin 120579(119889(119909119899 119909)) = 1It follows from (6) and (1205852) that1 le 120585 (120579 (119889 (119879119909119899 119879119909)) 120579 (119889 (119909119899 119909))) lt 120579 (119889 (119909119899 119909))120579 (119889 (119879119909119899 119879119909)) (8)
which implies
120579 (119889 (119879119909119899 119879119909)) lt 120579 (119889 (119909119899 119909)) (9)
Since 120579 is nondecreasing we have119889 (119879119909119899 119879119909) lt 119889 (119909119899 119909) (10)
and so
lim119899997888rarrinfin
119889 (119879119909119899 119879119909) = 0 (11)
Hence 119879 is continuousNow we prove our main result
Theorem 4 Let (119883 119889) be a complete generalized metric spaceand let 119879 119883 997888rarr 119883 be aL-contraction with respect to 120585
en 119879 has a unique fixed point and for every initial point1199090 isin 119883 the Picard sequence 1198791198991199090 converges to the fixed pointProof Firstly we show uniqueness of fixed point whenever itexists
Assume that 119908 and 119906 are fixed points of 119879If 119906 = 119911 then 119889(119908 119906) gt 0 and so it follows from (6) that
1 le 120585 (120579 (119889 (119879119908 119879119906)) 120579 (119889 (119908 119906)))= 120585 (120579 (119889 (119908 119906)) 120579 (119889 (119908 119906))) lt 120579 (119889 (119908 119906))
120579 (119889 (119908 119906)) (12)
Abstract and Applied Analysis 3
Hence
120579 (119889 (119908 119906)) lt 120579 (119889 (119908 119906)) (13)
which is a contradictionHence 119908 = 119906 and fixed point of 119879 is uniqueSecondly we prove existence of fixed pointLet 1199090 isin 119883 be a point Define a sequence 119909119899 sub 119883 by119909119899 = 119879119909119899minus1 = 1198791198991199090 forall119899 = 1 2 3 sdot sdot sdot If 1199091198990 = 1199091198990+1 for some 1198990 isin N then 1199091198990 is a fixed point of119879 and the proof is finishedAssume that
119909119899minus1 = 119909119899 forall119899 = 1 2 3 sdot sdot sdot (14)
It follows from (6) and (14) that forall119899 = 1 2 3 sdot sdot sdot1 le 120585 (120579 (119889 (119879119909119899minus1 119879119909119899)) 120579 (119889 (119909119899minus1 119909119899)))= 120585 (120579 (119889 (119909119899 119909119899+1)) 120579 (119889 (119909119899minus1 119909119899)))lt 120579 (119889 (119909119899minus1 119909119899))120579 (119889 (119909119899 119909119899+1))
(15)
Consequently we obtain that
120579 (119889 (119909119899 119909119899+1)) lt 120579 (119889 (119909119899minus1 119909119899)) forall119899 = 1 2 3 sdot sdot sdot (16)
which implies
119889 (119909119899 119909119899+1) lt 119889 (119909119899minus1 119909119899) forall119899 = 1 2 3 sdot sdot sdot (17)
Hence 119889(119909119899minus1 119909119899) is a decreasing sequence and so thereexists 119903 ge 0 such that
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899) = 119903 (18)
We now show that 119903 = 0Assume that 119903 = 0Then it follows from (1205792) that
lim119899997888rarrinfin
120579 (119889 (119909119899minus1 119909119899)) = 1 (19)
and so
lim119899997888rarrinfin
120579 (119889 (119909119899minus1 119909119899)) gt 1 (20)
Let 119904119899 = 120579(119889(119909119899minus1 119909119899)) and 119905119899 = 120579(119889(119909119899 119909119899+1)) forall119899 =1 2 3 sdot sdot sdot From (1205853) we obtain
1 le lim119899997888rarrinfin
sup 120585 (119905119899 119904119899) lt 1 (21)
which is a contradictionThus we have
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899) = 0 (22)
and so
lim119899997888rarrinfin
120579 (119889 (119909119899minus1 119909119899)) = 1 (23)
We show that
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899+1) = 0 (24)
We consider three cases
Case 1 119909119899 = 119909119899+2 forall119899 = 1 2 3 sdot sdot sdot From (6) and (14) we obtain that forall119899 = 1 2 3 sdot sdot sdot1 le 120585 (120579 (119889 (119879119909119899minus1 119879119909119899+1)) 120579 (119889 (119909119899minus1 119909119899+1)))= 120585 (120579 (119889 (119909119899 119909119899+2)) 120579 (119889 (119909119899minus1 119909119899+1)))lt 120579 (119889 (119909119899minus1 119909119899+1))120579 (119889 (119909119899 119909119899+2))
(25)
and so
120579 (119889 (119909119899 119909119899+2)) lt 120579 (119889 (119909119899minus1 119909119899+1)) forall119899 = 1 2 3 sdot sdot sdot (26)
which implies
119889 (119909119899 119909119899+2) lt 119889 (119909119899minus1 119909119899+1) forall119899 = 1 2 3 sdot sdot sdot (27)
Hence 119889(119909119899minus1 119909119899+1) is decreasingIn a manner similar to that which proved (22) we have
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899+1) = 0 (28)
Case 2 There exists 1198990 ge 1 such that 1199091198990 = 1199091198990+2From the first term to the 1198990 th term shall be removed
and let 119909119899 = 1199091198990+119899 forall119899 = 1 2 3 sdot sdot sdot Then 119909119899 = 119909119899+2 forall119899 = 1 2 3 sdot sdot sdot By Case 1 we have
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899+1) = 0 (29)
Case 3 119909119899 = 119909119899+2 forall119899 = 0 1 2 sdot sdot sdot We have
119889 (119909119899minus1 119909119899+1) = 0 forall119899 = 1 2 3 sdot sdot sdot (30)
Hence
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899+1) = 0 (31)
In all cases (24) is satisfiedNow we show that 119909119899 is boundedIf 119909119899 is not bounded then there exists a subsequence119909119899(119896) of 119909119899 such that 1198991 = 1 and forall119896 = 1 2 3 119899(119896 + 1)
is the minimum integer greater than 119899(119896) with119889 (119909119899(119896+1) 119909119899(119896)) gt 1
119889 (119909119897 119909119899(119896)) le 1 (32)
for 119899(119896) le 119897 le 119899(119896 + 1) minus 1Then we have
1 lt 119889 (119909119899(119896+1) 119909119899(119896))le 119889 (119909119899(119896+1) 119909119899(119896+1)minus2) + 119889 (119909119899(119896+1)minus2 119909119899(119896+1)minus1)+ 119889 (119909119899(119896+1)minus1 119909119899(119896))
le 119889 (119909119899(119896+1) 119909119899(119896+1)minus2) + 119889 (119909119899(119896+1)minus2 119909119899(119896+1)minus1) + 1(33)
4 Abstract and Applied Analysis
By letting 119896 997888rarr infin in the above we obtain
lim119896997888rarrinfin
119889 (119909119899(119896+1) 119909119899(119896)) = 1 (34)
By using (22) (34) and condition (d3) we deduce that
lim119896997888rarrinfin
119889 (119909119899(119896+1)minus1 119909119899(119896)minus1) = 1 (35)
It follows from (1205792) (34) and (35) that
lim119896997888rarrinfin
120579 (119889 (119909119899(119896+1) 119909119899(119896))) gt 1 (36)
lim119899997888rarrinfin
120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1)) gt 1 (37)
From (6) and (32) we infer that
1 le 120585 (120579 (119889 (119879119909119899(119896+1)minus1 119879119909119899(119896)minus1)) 120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1))) = 120585 (120579 (119889 (119909119899(119896+1) 119909119899(119896))) 120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1))) lt 120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1))
120579 (119889 (119909119899(119896+1) 119909119899(119896)))(38)
which implies
120579 (119889 (119909119899(119896+1) 119909119899(119896))) lt 120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1)) (39)
Let
119904119899 = 120579(119889 (119909119899(119896+1)minus1 119909119899(119896)minus1) 119905119899 = 120579 (119889 (119909119899(119896+1) 119909119899(119896))) forall119899 = 1 2 3 sdot sdot sdot (40)
Then 119905119896 lt 119904119896 forall119899 = 1 2 3 sdot sdot sdot and lim119896997888rarrinfin 119905119896 =lim119896997888rarrinfin 119904119896 gt 1
It follows from (1205853) that1 le lim119896997888rarrinfin
sup 120585 (119905119896 119904119896) lt 1 (41)
which is a contradictionThus 119909119899 is boundedNow we show that 119909119899 is a Cauchy sequenceLet
119872119899 = sup 119889 (119909119894 119909119895) 119894 119895 ge 119899 (42)
Clearly
0 le 119872119899+1 le 119872119899 lt infin forall119899 = 1 2 3 sdot sdot sdot (43)
and so there exists119872 ge 0 such that
lim119899997888rarrinfin
119872119899 = 119872 (44)
Assume that119872 gt 0It follows from (42) that forall119896 = 1 2 3 sdot sdot sdot there exist119899(119896)119898(119896) ge 119896 with
119872119896 minus 1119896 lt 119889 (119909119898(119896) 119909119899(119896)) le 119872119896 (45)
Solim119896997888rarrinfin
119889 (119909119898(119896) 119909119899(119896)) = lim119896997888rarrinfin
119872119896 = 119872 (46)
It follows from (6) and (14) that
120585 (120579 (119889 (119879119909119898(119896)minus1 119879119909119899(119896)minus1)) 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1)))= 120585 (120579 (119889 (119909119898(119896) 119909119899(119896))) 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1)))lt 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1))
120579 (119889 (119909119898(119896) 119909119899(119896)))(47)
which implies
120579 (119889 (119909119898(119896) 119909119899(119896))) lt 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1)) (48)
Hence we have119889 (119909119898(119896) 119909119899(119896)) lt 119889 (119909119898(119896)minus1 119909119899(119896)minus1)
le 119889 (119909119898(119896)minus1 119909119898(119896)) + 119889 (119909119898(119896) 119909119899(119896))+ 119889 (119909119899(119896) 119909119899(119896)minus1)
(49)
Letting 119896 997888rarr infin in the above inequality we obtain
lim119899997888rarrinfin
119889 (119909119898(119896)minus1119909119899(119896)minus1) = 119872 (50)
Let119904119896 = 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1)) 119905119896 = 120579 (119889 (119909119898(119896) 119909119899(119896))) forall119896 = 1 2 3 sdot sdot sdot (51)
Then 119905119896 lt 119904119896 forall119896 = 1 2 3 sdot sdot sdot Since119872 gt 0lim119896997888rarrinfin
119905119896 = lim119896997888rarrinfin
119904119896 gt 1 (52)
Thus we have1 le lim119896997888rarrinfin
sup 120585 (119905119896 119904119896) lt 1 (53)
which is a contradictionHence119872 = 0 and hence 119909119899 is a Cauchy sequenceSince 119883 is complete there exists 119911 isin 119883 such that
lim119899997888rarrinfin
119889 (119909119899 119911) = 0 (54)
Because 119879 is continuouslim119899997888rarrinfin
119889 (119909119899 119879119911) = lim119899997888rarrinfin
119889 (119879119909119899minus1 119879119911) = 0 (55)
By Lemma 3 119911 = 119879119911We give an example to illustrate Theorem 4
Example 5 Let 119883 = 1 2 3 4 and define 119889 119883 times 119883 997888rarr[0infin) as follows119889 (1 2) = 119889 (2 1) = 3119889 (2 3) = 119889 (3 2) = 119889 (1 3) = 119889 (3 1) = 1119889 (1 4) = 119889 (4 1) = 119889 (2 4) = 119889 (4 2) = 119889 (3 4)
= 119889 (4 3) = 4119889 (119909 119909) = 0 forall119909 isin 119883
(56)
Abstract and Applied Analysis 5
Then (119883 119889) is a complete generalized metric space butnot a metric space (see [21])
Define a map 119879 119883 997888rarr 119883 by
119879119909 = 3 (119909 = 4) 1 (119909 = 4) (57)
And define a function 120579 (0infin) 997888rarr (1infin) by120579 (119905) = 119890119905 (58)
We now show that 119879 is a L-contraction with respect to120585119887 where 120585119887(119905 119904) = 119904119896119905 forall119905 119904 ge 1 119896 = 12
We have
119889 (119879119909 119879119910) =
119889 (1 3) = 1 (119909 = 4 119910 = 4) 119889 (1 1) = 0 (119909 = 4 119910 = 4) 119889 (3 3) = 0 (119909 = 4 119910 = 4)
(59)
so
119889 (119879119909 119879119910) gt 0 lArrrArr 119909 = 4 119910 = 4 (60)
We have for 119909 = 4 and 119910 = 4119889 (119909 119910) = 4
119889 (119879119909 119879119910) = 1 (61)
We deduce that for all 119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0120585119887 (120579 (119889 (119879119909 119879119910)) 120579 (119889 (119909 119910))) = [120579 (119889 (119909 119910))]119896
120579 (119889 (119879119909 119879119910))
= [1198904]121198901 = 119890 gt 1
(62)
Thus all hypotheses of Theorem 4 are satisfied and 119879 hasa fixed point 119909lowast = 3
Note that Banachrsquos contraction principle is not satisfiedwith the usual metric 120588(119909 119910) = |119909 minus 119910| forall119909 119910 isin 119883 In fact if119909 = 2 119910 = 4 then
120588 (1198792 1198794) le 119896120588 (2 4) 119896 isin (0 1) (63)
which implies
119896 ge 1 (64)
Also note that the 120579-contraction condition [13] does nothold
Let 120579(119905) = 119890119905 forall119905 gt 0Then 120579(119905) satisfies conditions (1205791) (1205792) and (1205794)If
120579 (120588 (1198792 1198794)) le [120579 (120588 (2 4))]119896 where 119896 isin (0 1) (65)
then
1198902 le [1198902]119896 (66)
and so 119896 ge 1 Hence 119879 is not 120579-contraction mapBy taking 120585 = 120585119887 in Theorem 4 we obtain Corollary 6
Corollary 6 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
120579 (119889 (119879119909 119879119910)) le 120579 (119889 (119909 119910))119896 (67)
where 120579 isin Θ and 119896 isin (0 1)en 119879 has a unique fixed point
Remark 7 Corollary 6 is a generalization of Theorem 21 of[12] without condition (1205793) and Theorem 22 of [13] withoutcondition (1205794)
By taking 120585 = 120585119908 in Theorem 4 we obtain Corollary 8
Corollary 8 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
120579 (119889 (119879119909 119879119910)) le 120579 (119889 (119909 119910))120601 (120579 (119889 (119909 119910))) (68)
where 120579 isin Θ and 120601 [1infin) 997888rarr [1infin) is nondecreasing andlower semicontinuous such that 120601minus1(1) = 1
en 119879 has a unique fixed point
Corollary 9 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
119889 (119879119909 119879119910) le 119889 (119909 119910) minus 120593 (119889 (119909 119910)) (69)
where 120593 [0infin) 997888rarr [0infin) is nondecreasing and lowersemicontinuous such that 120593minus1(0) = 0
en 119879 has a unique fixed point
Proof Condition (69) implies 119879 is continuousLet 120579(119905) = 119890119905 forall119905 gt 0From (69) we have that for all 119909 119910 isin 119883 with 119889(119879119909 119879119910) gt0120579 (119889 (119879119909 119879119910)) = 119890119889(119879119909119879119910) le 119890119889(119909119910)minus120593(119889(119909119910)) = 119890119889(119909119910)
119890120593(119889(119909119910)) (70)
Let 120593(119905) = ln(120601(120579(119905))) forall119905 ge 0 where 120601 [1infin) 997888rarr[1infin) is nondecreasing and lower semicontinuous such that120601minus1(1) = 1Then 120593 is nondecreasing and lower semicontinuous and
120593 (119905) = 0 lArrrArr 120601 (120579 (119905)) = 1 lArrrArr 120579 (119905) = 119890119905 = 1 lArrrArr 119905 = 0 (71)
It follows from (70) that for all 119909 119910 isin 119883with 119889(119879119909 119879119910) gt0120579 (119889 (119879119909 119879119910)) le 120579 (119889 (119909 119910))
119890ln(120601(120579(119889(119909119910)))) =120579 (119889 (119909 119910))
120601 (120579 (119889 (119909 119910))) (72)
By Corollary 8 119879 has a unique fixed point
By taking 120579(119905) = 2 minus (2120587) arctan(1119905120572) where 120572 isin(0 1) 119905 gt 0 in Corollary 8 we obtain the following result
Corollary 10 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
6 Abstract and Applied Analysis
2 minus 2120587 arctan( 1
[119889 (119879119909 119879119910)]120572) le 2 minus (2120587) arctan (1 [119889 (119909 119910)]120572)120601 (2 minus (2120587) arctan (1 [119889 (119909 119910)]120572)) (73)
where 120572 isin (0 1) and 120601 [1infin) 997888rarr [1infin) is nondecreasingand lower semicontinuous such that 120601minus1(1) = 1
en 119879 has a unique fixed point
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The author express his gratitude to the referees for carefulreading and giving variable comments This research wassupported by Hanseo University
References
[1] A Branciari ldquoA fixed point theorem of Banach-Caccioppolitype on a class of generalized metric spacesrdquo PublicationesMathematicae vol 57 no 1-2 pp 31ndash37 2000
[2] H Aydi E Karapınar and B Samet ldquoFixed points for general-ized (120572120595) contractions on generalized metric spacesrdquo Journalof Inequalities and Applications vol 2014 article 229 2014
[3] C Di Bari and P Vetro ldquoCommon fixed points in generalizedmetric spacesrdquo Applied Mathematics and Computation vol 218no 13 pp 7322ndash7325 2012
[4] P Das ldquoA fixed point theorem on a class of generalized metricspacesrdquo e Korean Journal of Mathematics vol 9 pp 29ndash332002
[5] W A Kirk and N Shahzad ldquoGeneralized metrics and Caristirsquostheoremrdquo Fixed Point eory and Applications vol 2013 article129 2013
[6] B Samet ldquoDiscussion on ldquoA fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spacesrdquo by ABranciarirdquo Publicationes Mathematicae vol 76 no 3-4 pp 493-494 2010
[7] I R Sarma J M Rao and S S Rao ldquoContractions overgeneralizedmetric spacesrdquoe Journal of Nonlinear Science andits Applications vol 2 no 3 pp 180ndash182 2009
[8] W Shatanawi A Al-Rawashdeh Hassen Aydi and H KNashine ldquoOn a Fixed Point for Generalized Contractions inGeneralized Metric Spacesrdquo Abstract and Applied Analysis vol2012 Article ID 246085 13 pages 2012
[9] T Suzuki B Alamri andL A Khan ldquoSome notes on fixed pointtheoems in ]-generalized metric spacesrdquo Bulletin of the KyushuInstitute of TechnologyMathematics Natural Science no 62 pp15ndash23 2015
[10] M Abtahi Z Kadelburg and S Radenovic ldquoFixed points ofCiric-Matakowski-type contractions in ]-generalized metricspacesrdquo RACSAM
[11] Z D Mitrovic and S Radenovic ldquoThe Banach and Reichcontractiond in b](s)-metric spacesrdquo Journal of Fixed Pointeory and Applications vol 19 no 4 pp 3087ndash3095 2017
[12] M Jleli and B Samet ldquoA new generalization of the Banachcontraction principlerdquo Journal of Inequalities and Applicationsvol 2014 no 1 article no 38 8 pages 2014
[13] J Ahmad A E Al-Mazrooei Y J Cho and Y-O YangldquoFixed point results for generalized 120579-contractionsrdquo Journal ofNonlinear Sciences and Applications JNSA vol 10 no 5 pp2350ndash2358 2017
[14] F Khojasteh S Shukla and S Radenovic ldquoA new approachto the study of fixed point theorems via simulation functionsrdquoFilomat vol 29 no 6 pp 1189ndash1194 2015
[15] H H Alsulami E Karapinar F Khojasteh and A-F Roldan-Lopez-de-Hierro ldquoA proposal to the study of contractions inquasi-metric spacesrdquo Discrete Dynamics in Nature and Societyvol 2014 Article ID 269286 pp 1ndash10 2014
[16] H Isik N Gungor C Park and S Jang ldquoFixed PointTheoremsfor Almost Z-Contractions with an ApplicationrdquoMathematicsvol 6 no 3 p 37 2018
[17] E Karapınar ldquoFixed points results via simulation functionsrdquoFilomat vol 30 no 8 pp 2343ndash2350 2016
[18] A-F Roldan-Lopez-de-Hierro E Karapinar C Roldan-Lopez-de-Hierro and J Martinez-Moreno ldquoCoincidence point the-orems on metric spaces via simulation functionsrdquo Journal ofComputational and Applied Mathematics vol 275 pp 345ndash3552015
[19] S Radenovic F Vetro and J Vujakovic ldquoAn alternative andeasy approach to fixed point results via simulation functionsrdquoDemonstratio Mathematica vol 50 no 1 pp 223ndash230 2017
[20] M Jleli and B Samet ldquoThe Kannanrsquos fixed point theorem in acone rectangular metric spacerdquoe Journal of Nonlinear Scienceand its Applications vol 2 no 3 pp 161ndash167 2009
[21] A Azam and M Arshad ldquoKannan fixed point theorem ongeneralizedmetric spacesrdquoe Journal of Nonlinear Science andits Applications vol 1 no 1 pp 45ndash48 2008
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Abstract and Applied Analysis 3
Hence
120579 (119889 (119908 119906)) lt 120579 (119889 (119908 119906)) (13)
which is a contradictionHence 119908 = 119906 and fixed point of 119879 is uniqueSecondly we prove existence of fixed pointLet 1199090 isin 119883 be a point Define a sequence 119909119899 sub 119883 by119909119899 = 119879119909119899minus1 = 1198791198991199090 forall119899 = 1 2 3 sdot sdot sdot If 1199091198990 = 1199091198990+1 for some 1198990 isin N then 1199091198990 is a fixed point of119879 and the proof is finishedAssume that
119909119899minus1 = 119909119899 forall119899 = 1 2 3 sdot sdot sdot (14)
It follows from (6) and (14) that forall119899 = 1 2 3 sdot sdot sdot1 le 120585 (120579 (119889 (119879119909119899minus1 119879119909119899)) 120579 (119889 (119909119899minus1 119909119899)))= 120585 (120579 (119889 (119909119899 119909119899+1)) 120579 (119889 (119909119899minus1 119909119899)))lt 120579 (119889 (119909119899minus1 119909119899))120579 (119889 (119909119899 119909119899+1))
(15)
Consequently we obtain that
120579 (119889 (119909119899 119909119899+1)) lt 120579 (119889 (119909119899minus1 119909119899)) forall119899 = 1 2 3 sdot sdot sdot (16)
which implies
119889 (119909119899 119909119899+1) lt 119889 (119909119899minus1 119909119899) forall119899 = 1 2 3 sdot sdot sdot (17)
Hence 119889(119909119899minus1 119909119899) is a decreasing sequence and so thereexists 119903 ge 0 such that
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899) = 119903 (18)
We now show that 119903 = 0Assume that 119903 = 0Then it follows from (1205792) that
lim119899997888rarrinfin
120579 (119889 (119909119899minus1 119909119899)) = 1 (19)
and so
lim119899997888rarrinfin
120579 (119889 (119909119899minus1 119909119899)) gt 1 (20)
Let 119904119899 = 120579(119889(119909119899minus1 119909119899)) and 119905119899 = 120579(119889(119909119899 119909119899+1)) forall119899 =1 2 3 sdot sdot sdot From (1205853) we obtain
1 le lim119899997888rarrinfin
sup 120585 (119905119899 119904119899) lt 1 (21)
which is a contradictionThus we have
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899) = 0 (22)
and so
lim119899997888rarrinfin
120579 (119889 (119909119899minus1 119909119899)) = 1 (23)
We show that
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899+1) = 0 (24)
We consider three cases
Case 1 119909119899 = 119909119899+2 forall119899 = 1 2 3 sdot sdot sdot From (6) and (14) we obtain that forall119899 = 1 2 3 sdot sdot sdot1 le 120585 (120579 (119889 (119879119909119899minus1 119879119909119899+1)) 120579 (119889 (119909119899minus1 119909119899+1)))= 120585 (120579 (119889 (119909119899 119909119899+2)) 120579 (119889 (119909119899minus1 119909119899+1)))lt 120579 (119889 (119909119899minus1 119909119899+1))120579 (119889 (119909119899 119909119899+2))
(25)
and so
120579 (119889 (119909119899 119909119899+2)) lt 120579 (119889 (119909119899minus1 119909119899+1)) forall119899 = 1 2 3 sdot sdot sdot (26)
which implies
119889 (119909119899 119909119899+2) lt 119889 (119909119899minus1 119909119899+1) forall119899 = 1 2 3 sdot sdot sdot (27)
Hence 119889(119909119899minus1 119909119899+1) is decreasingIn a manner similar to that which proved (22) we have
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899+1) = 0 (28)
Case 2 There exists 1198990 ge 1 such that 1199091198990 = 1199091198990+2From the first term to the 1198990 th term shall be removed
and let 119909119899 = 1199091198990+119899 forall119899 = 1 2 3 sdot sdot sdot Then 119909119899 = 119909119899+2 forall119899 = 1 2 3 sdot sdot sdot By Case 1 we have
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899+1) = 0 (29)
Case 3 119909119899 = 119909119899+2 forall119899 = 0 1 2 sdot sdot sdot We have
119889 (119909119899minus1 119909119899+1) = 0 forall119899 = 1 2 3 sdot sdot sdot (30)
Hence
lim119899997888rarrinfin
119889 (119909119899minus1 119909119899+1) = 0 (31)
In all cases (24) is satisfiedNow we show that 119909119899 is boundedIf 119909119899 is not bounded then there exists a subsequence119909119899(119896) of 119909119899 such that 1198991 = 1 and forall119896 = 1 2 3 119899(119896 + 1)
is the minimum integer greater than 119899(119896) with119889 (119909119899(119896+1) 119909119899(119896)) gt 1
119889 (119909119897 119909119899(119896)) le 1 (32)
for 119899(119896) le 119897 le 119899(119896 + 1) minus 1Then we have
1 lt 119889 (119909119899(119896+1) 119909119899(119896))le 119889 (119909119899(119896+1) 119909119899(119896+1)minus2) + 119889 (119909119899(119896+1)minus2 119909119899(119896+1)minus1)+ 119889 (119909119899(119896+1)minus1 119909119899(119896))
le 119889 (119909119899(119896+1) 119909119899(119896+1)minus2) + 119889 (119909119899(119896+1)minus2 119909119899(119896+1)minus1) + 1(33)
4 Abstract and Applied Analysis
By letting 119896 997888rarr infin in the above we obtain
lim119896997888rarrinfin
119889 (119909119899(119896+1) 119909119899(119896)) = 1 (34)
By using (22) (34) and condition (d3) we deduce that
lim119896997888rarrinfin
119889 (119909119899(119896+1)minus1 119909119899(119896)minus1) = 1 (35)
It follows from (1205792) (34) and (35) that
lim119896997888rarrinfin
120579 (119889 (119909119899(119896+1) 119909119899(119896))) gt 1 (36)
lim119899997888rarrinfin
120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1)) gt 1 (37)
From (6) and (32) we infer that
1 le 120585 (120579 (119889 (119879119909119899(119896+1)minus1 119879119909119899(119896)minus1)) 120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1))) = 120585 (120579 (119889 (119909119899(119896+1) 119909119899(119896))) 120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1))) lt 120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1))
120579 (119889 (119909119899(119896+1) 119909119899(119896)))(38)
which implies
120579 (119889 (119909119899(119896+1) 119909119899(119896))) lt 120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1)) (39)
Let
119904119899 = 120579(119889 (119909119899(119896+1)minus1 119909119899(119896)minus1) 119905119899 = 120579 (119889 (119909119899(119896+1) 119909119899(119896))) forall119899 = 1 2 3 sdot sdot sdot (40)
Then 119905119896 lt 119904119896 forall119899 = 1 2 3 sdot sdot sdot and lim119896997888rarrinfin 119905119896 =lim119896997888rarrinfin 119904119896 gt 1
It follows from (1205853) that1 le lim119896997888rarrinfin
sup 120585 (119905119896 119904119896) lt 1 (41)
which is a contradictionThus 119909119899 is boundedNow we show that 119909119899 is a Cauchy sequenceLet
119872119899 = sup 119889 (119909119894 119909119895) 119894 119895 ge 119899 (42)
Clearly
0 le 119872119899+1 le 119872119899 lt infin forall119899 = 1 2 3 sdot sdot sdot (43)
and so there exists119872 ge 0 such that
lim119899997888rarrinfin
119872119899 = 119872 (44)
Assume that119872 gt 0It follows from (42) that forall119896 = 1 2 3 sdot sdot sdot there exist119899(119896)119898(119896) ge 119896 with
119872119896 minus 1119896 lt 119889 (119909119898(119896) 119909119899(119896)) le 119872119896 (45)
Solim119896997888rarrinfin
119889 (119909119898(119896) 119909119899(119896)) = lim119896997888rarrinfin
119872119896 = 119872 (46)
It follows from (6) and (14) that
120585 (120579 (119889 (119879119909119898(119896)minus1 119879119909119899(119896)minus1)) 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1)))= 120585 (120579 (119889 (119909119898(119896) 119909119899(119896))) 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1)))lt 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1))
120579 (119889 (119909119898(119896) 119909119899(119896)))(47)
which implies
120579 (119889 (119909119898(119896) 119909119899(119896))) lt 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1)) (48)
Hence we have119889 (119909119898(119896) 119909119899(119896)) lt 119889 (119909119898(119896)minus1 119909119899(119896)minus1)
le 119889 (119909119898(119896)minus1 119909119898(119896)) + 119889 (119909119898(119896) 119909119899(119896))+ 119889 (119909119899(119896) 119909119899(119896)minus1)
(49)
Letting 119896 997888rarr infin in the above inequality we obtain
lim119899997888rarrinfin
119889 (119909119898(119896)minus1119909119899(119896)minus1) = 119872 (50)
Let119904119896 = 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1)) 119905119896 = 120579 (119889 (119909119898(119896) 119909119899(119896))) forall119896 = 1 2 3 sdot sdot sdot (51)
Then 119905119896 lt 119904119896 forall119896 = 1 2 3 sdot sdot sdot Since119872 gt 0lim119896997888rarrinfin
119905119896 = lim119896997888rarrinfin
119904119896 gt 1 (52)
Thus we have1 le lim119896997888rarrinfin
sup 120585 (119905119896 119904119896) lt 1 (53)
which is a contradictionHence119872 = 0 and hence 119909119899 is a Cauchy sequenceSince 119883 is complete there exists 119911 isin 119883 such that
lim119899997888rarrinfin
119889 (119909119899 119911) = 0 (54)
Because 119879 is continuouslim119899997888rarrinfin
119889 (119909119899 119879119911) = lim119899997888rarrinfin
119889 (119879119909119899minus1 119879119911) = 0 (55)
By Lemma 3 119911 = 119879119911We give an example to illustrate Theorem 4
Example 5 Let 119883 = 1 2 3 4 and define 119889 119883 times 119883 997888rarr[0infin) as follows119889 (1 2) = 119889 (2 1) = 3119889 (2 3) = 119889 (3 2) = 119889 (1 3) = 119889 (3 1) = 1119889 (1 4) = 119889 (4 1) = 119889 (2 4) = 119889 (4 2) = 119889 (3 4)
= 119889 (4 3) = 4119889 (119909 119909) = 0 forall119909 isin 119883
(56)
Abstract and Applied Analysis 5
Then (119883 119889) is a complete generalized metric space butnot a metric space (see [21])
Define a map 119879 119883 997888rarr 119883 by
119879119909 = 3 (119909 = 4) 1 (119909 = 4) (57)
And define a function 120579 (0infin) 997888rarr (1infin) by120579 (119905) = 119890119905 (58)
We now show that 119879 is a L-contraction with respect to120585119887 where 120585119887(119905 119904) = 119904119896119905 forall119905 119904 ge 1 119896 = 12
We have
119889 (119879119909 119879119910) =
119889 (1 3) = 1 (119909 = 4 119910 = 4) 119889 (1 1) = 0 (119909 = 4 119910 = 4) 119889 (3 3) = 0 (119909 = 4 119910 = 4)
(59)
so
119889 (119879119909 119879119910) gt 0 lArrrArr 119909 = 4 119910 = 4 (60)
We have for 119909 = 4 and 119910 = 4119889 (119909 119910) = 4
119889 (119879119909 119879119910) = 1 (61)
We deduce that for all 119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0120585119887 (120579 (119889 (119879119909 119879119910)) 120579 (119889 (119909 119910))) = [120579 (119889 (119909 119910))]119896
120579 (119889 (119879119909 119879119910))
= [1198904]121198901 = 119890 gt 1
(62)
Thus all hypotheses of Theorem 4 are satisfied and 119879 hasa fixed point 119909lowast = 3
Note that Banachrsquos contraction principle is not satisfiedwith the usual metric 120588(119909 119910) = |119909 minus 119910| forall119909 119910 isin 119883 In fact if119909 = 2 119910 = 4 then
120588 (1198792 1198794) le 119896120588 (2 4) 119896 isin (0 1) (63)
which implies
119896 ge 1 (64)
Also note that the 120579-contraction condition [13] does nothold
Let 120579(119905) = 119890119905 forall119905 gt 0Then 120579(119905) satisfies conditions (1205791) (1205792) and (1205794)If
120579 (120588 (1198792 1198794)) le [120579 (120588 (2 4))]119896 where 119896 isin (0 1) (65)
then
1198902 le [1198902]119896 (66)
and so 119896 ge 1 Hence 119879 is not 120579-contraction mapBy taking 120585 = 120585119887 in Theorem 4 we obtain Corollary 6
Corollary 6 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
120579 (119889 (119879119909 119879119910)) le 120579 (119889 (119909 119910))119896 (67)
where 120579 isin Θ and 119896 isin (0 1)en 119879 has a unique fixed point
Remark 7 Corollary 6 is a generalization of Theorem 21 of[12] without condition (1205793) and Theorem 22 of [13] withoutcondition (1205794)
By taking 120585 = 120585119908 in Theorem 4 we obtain Corollary 8
Corollary 8 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
120579 (119889 (119879119909 119879119910)) le 120579 (119889 (119909 119910))120601 (120579 (119889 (119909 119910))) (68)
where 120579 isin Θ and 120601 [1infin) 997888rarr [1infin) is nondecreasing andlower semicontinuous such that 120601minus1(1) = 1
en 119879 has a unique fixed point
Corollary 9 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
119889 (119879119909 119879119910) le 119889 (119909 119910) minus 120593 (119889 (119909 119910)) (69)
where 120593 [0infin) 997888rarr [0infin) is nondecreasing and lowersemicontinuous such that 120593minus1(0) = 0
en 119879 has a unique fixed point
Proof Condition (69) implies 119879 is continuousLet 120579(119905) = 119890119905 forall119905 gt 0From (69) we have that for all 119909 119910 isin 119883 with 119889(119879119909 119879119910) gt0120579 (119889 (119879119909 119879119910)) = 119890119889(119879119909119879119910) le 119890119889(119909119910)minus120593(119889(119909119910)) = 119890119889(119909119910)
119890120593(119889(119909119910)) (70)
Let 120593(119905) = ln(120601(120579(119905))) forall119905 ge 0 where 120601 [1infin) 997888rarr[1infin) is nondecreasing and lower semicontinuous such that120601minus1(1) = 1Then 120593 is nondecreasing and lower semicontinuous and
120593 (119905) = 0 lArrrArr 120601 (120579 (119905)) = 1 lArrrArr 120579 (119905) = 119890119905 = 1 lArrrArr 119905 = 0 (71)
It follows from (70) that for all 119909 119910 isin 119883with 119889(119879119909 119879119910) gt0120579 (119889 (119879119909 119879119910)) le 120579 (119889 (119909 119910))
119890ln(120601(120579(119889(119909119910)))) =120579 (119889 (119909 119910))
120601 (120579 (119889 (119909 119910))) (72)
By Corollary 8 119879 has a unique fixed point
By taking 120579(119905) = 2 minus (2120587) arctan(1119905120572) where 120572 isin(0 1) 119905 gt 0 in Corollary 8 we obtain the following result
Corollary 10 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
6 Abstract and Applied Analysis
2 minus 2120587 arctan( 1
[119889 (119879119909 119879119910)]120572) le 2 minus (2120587) arctan (1 [119889 (119909 119910)]120572)120601 (2 minus (2120587) arctan (1 [119889 (119909 119910)]120572)) (73)
where 120572 isin (0 1) and 120601 [1infin) 997888rarr [1infin) is nondecreasingand lower semicontinuous such that 120601minus1(1) = 1
en 119879 has a unique fixed point
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The author express his gratitude to the referees for carefulreading and giving variable comments This research wassupported by Hanseo University
References
[1] A Branciari ldquoA fixed point theorem of Banach-Caccioppolitype on a class of generalized metric spacesrdquo PublicationesMathematicae vol 57 no 1-2 pp 31ndash37 2000
[2] H Aydi E Karapınar and B Samet ldquoFixed points for general-ized (120572120595) contractions on generalized metric spacesrdquo Journalof Inequalities and Applications vol 2014 article 229 2014
[3] C Di Bari and P Vetro ldquoCommon fixed points in generalizedmetric spacesrdquo Applied Mathematics and Computation vol 218no 13 pp 7322ndash7325 2012
[4] P Das ldquoA fixed point theorem on a class of generalized metricspacesrdquo e Korean Journal of Mathematics vol 9 pp 29ndash332002
[5] W A Kirk and N Shahzad ldquoGeneralized metrics and Caristirsquostheoremrdquo Fixed Point eory and Applications vol 2013 article129 2013
[6] B Samet ldquoDiscussion on ldquoA fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spacesrdquo by ABranciarirdquo Publicationes Mathematicae vol 76 no 3-4 pp 493-494 2010
[7] I R Sarma J M Rao and S S Rao ldquoContractions overgeneralizedmetric spacesrdquoe Journal of Nonlinear Science andits Applications vol 2 no 3 pp 180ndash182 2009
[8] W Shatanawi A Al-Rawashdeh Hassen Aydi and H KNashine ldquoOn a Fixed Point for Generalized Contractions inGeneralized Metric Spacesrdquo Abstract and Applied Analysis vol2012 Article ID 246085 13 pages 2012
[9] T Suzuki B Alamri andL A Khan ldquoSome notes on fixed pointtheoems in ]-generalized metric spacesrdquo Bulletin of the KyushuInstitute of TechnologyMathematics Natural Science no 62 pp15ndash23 2015
[10] M Abtahi Z Kadelburg and S Radenovic ldquoFixed points ofCiric-Matakowski-type contractions in ]-generalized metricspacesrdquo RACSAM
[11] Z D Mitrovic and S Radenovic ldquoThe Banach and Reichcontractiond in b](s)-metric spacesrdquo Journal of Fixed Pointeory and Applications vol 19 no 4 pp 3087ndash3095 2017
[12] M Jleli and B Samet ldquoA new generalization of the Banachcontraction principlerdquo Journal of Inequalities and Applicationsvol 2014 no 1 article no 38 8 pages 2014
[13] J Ahmad A E Al-Mazrooei Y J Cho and Y-O YangldquoFixed point results for generalized 120579-contractionsrdquo Journal ofNonlinear Sciences and Applications JNSA vol 10 no 5 pp2350ndash2358 2017
[14] F Khojasteh S Shukla and S Radenovic ldquoA new approachto the study of fixed point theorems via simulation functionsrdquoFilomat vol 29 no 6 pp 1189ndash1194 2015
[15] H H Alsulami E Karapinar F Khojasteh and A-F Roldan-Lopez-de-Hierro ldquoA proposal to the study of contractions inquasi-metric spacesrdquo Discrete Dynamics in Nature and Societyvol 2014 Article ID 269286 pp 1ndash10 2014
[16] H Isik N Gungor C Park and S Jang ldquoFixed PointTheoremsfor Almost Z-Contractions with an ApplicationrdquoMathematicsvol 6 no 3 p 37 2018
[17] E Karapınar ldquoFixed points results via simulation functionsrdquoFilomat vol 30 no 8 pp 2343ndash2350 2016
[18] A-F Roldan-Lopez-de-Hierro E Karapinar C Roldan-Lopez-de-Hierro and J Martinez-Moreno ldquoCoincidence point the-orems on metric spaces via simulation functionsrdquo Journal ofComputational and Applied Mathematics vol 275 pp 345ndash3552015
[19] S Radenovic F Vetro and J Vujakovic ldquoAn alternative andeasy approach to fixed point results via simulation functionsrdquoDemonstratio Mathematica vol 50 no 1 pp 223ndash230 2017
[20] M Jleli and B Samet ldquoThe Kannanrsquos fixed point theorem in acone rectangular metric spacerdquoe Journal of Nonlinear Scienceand its Applications vol 2 no 3 pp 161ndash167 2009
[21] A Azam and M Arshad ldquoKannan fixed point theorem ongeneralizedmetric spacesrdquoe Journal of Nonlinear Science andits Applications vol 1 no 1 pp 45ndash48 2008
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4 Abstract and Applied Analysis
By letting 119896 997888rarr infin in the above we obtain
lim119896997888rarrinfin
119889 (119909119899(119896+1) 119909119899(119896)) = 1 (34)
By using (22) (34) and condition (d3) we deduce that
lim119896997888rarrinfin
119889 (119909119899(119896+1)minus1 119909119899(119896)minus1) = 1 (35)
It follows from (1205792) (34) and (35) that
lim119896997888rarrinfin
120579 (119889 (119909119899(119896+1) 119909119899(119896))) gt 1 (36)
lim119899997888rarrinfin
120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1)) gt 1 (37)
From (6) and (32) we infer that
1 le 120585 (120579 (119889 (119879119909119899(119896+1)minus1 119879119909119899(119896)minus1)) 120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1))) = 120585 (120579 (119889 (119909119899(119896+1) 119909119899(119896))) 120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1))) lt 120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1))
120579 (119889 (119909119899(119896+1) 119909119899(119896)))(38)
which implies
120579 (119889 (119909119899(119896+1) 119909119899(119896))) lt 120579 (119889 (119909119899(119896+1)minus1 119909119899(119896)minus1)) (39)
Let
119904119899 = 120579(119889 (119909119899(119896+1)minus1 119909119899(119896)minus1) 119905119899 = 120579 (119889 (119909119899(119896+1) 119909119899(119896))) forall119899 = 1 2 3 sdot sdot sdot (40)
Then 119905119896 lt 119904119896 forall119899 = 1 2 3 sdot sdot sdot and lim119896997888rarrinfin 119905119896 =lim119896997888rarrinfin 119904119896 gt 1
It follows from (1205853) that1 le lim119896997888rarrinfin
sup 120585 (119905119896 119904119896) lt 1 (41)
which is a contradictionThus 119909119899 is boundedNow we show that 119909119899 is a Cauchy sequenceLet
119872119899 = sup 119889 (119909119894 119909119895) 119894 119895 ge 119899 (42)
Clearly
0 le 119872119899+1 le 119872119899 lt infin forall119899 = 1 2 3 sdot sdot sdot (43)
and so there exists119872 ge 0 such that
lim119899997888rarrinfin
119872119899 = 119872 (44)
Assume that119872 gt 0It follows from (42) that forall119896 = 1 2 3 sdot sdot sdot there exist119899(119896)119898(119896) ge 119896 with
119872119896 minus 1119896 lt 119889 (119909119898(119896) 119909119899(119896)) le 119872119896 (45)
Solim119896997888rarrinfin
119889 (119909119898(119896) 119909119899(119896)) = lim119896997888rarrinfin
119872119896 = 119872 (46)
It follows from (6) and (14) that
120585 (120579 (119889 (119879119909119898(119896)minus1 119879119909119899(119896)minus1)) 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1)))= 120585 (120579 (119889 (119909119898(119896) 119909119899(119896))) 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1)))lt 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1))
120579 (119889 (119909119898(119896) 119909119899(119896)))(47)
which implies
120579 (119889 (119909119898(119896) 119909119899(119896))) lt 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1)) (48)
Hence we have119889 (119909119898(119896) 119909119899(119896)) lt 119889 (119909119898(119896)minus1 119909119899(119896)minus1)
le 119889 (119909119898(119896)minus1 119909119898(119896)) + 119889 (119909119898(119896) 119909119899(119896))+ 119889 (119909119899(119896) 119909119899(119896)minus1)
(49)
Letting 119896 997888rarr infin in the above inequality we obtain
lim119899997888rarrinfin
119889 (119909119898(119896)minus1119909119899(119896)minus1) = 119872 (50)
Let119904119896 = 120579 (119889 (119909119898(119896)minus1 119909119899(119896)minus1)) 119905119896 = 120579 (119889 (119909119898(119896) 119909119899(119896))) forall119896 = 1 2 3 sdot sdot sdot (51)
Then 119905119896 lt 119904119896 forall119896 = 1 2 3 sdot sdot sdot Since119872 gt 0lim119896997888rarrinfin
119905119896 = lim119896997888rarrinfin
119904119896 gt 1 (52)
Thus we have1 le lim119896997888rarrinfin
sup 120585 (119905119896 119904119896) lt 1 (53)
which is a contradictionHence119872 = 0 and hence 119909119899 is a Cauchy sequenceSince 119883 is complete there exists 119911 isin 119883 such that
lim119899997888rarrinfin
119889 (119909119899 119911) = 0 (54)
Because 119879 is continuouslim119899997888rarrinfin
119889 (119909119899 119879119911) = lim119899997888rarrinfin
119889 (119879119909119899minus1 119879119911) = 0 (55)
By Lemma 3 119911 = 119879119911We give an example to illustrate Theorem 4
Example 5 Let 119883 = 1 2 3 4 and define 119889 119883 times 119883 997888rarr[0infin) as follows119889 (1 2) = 119889 (2 1) = 3119889 (2 3) = 119889 (3 2) = 119889 (1 3) = 119889 (3 1) = 1119889 (1 4) = 119889 (4 1) = 119889 (2 4) = 119889 (4 2) = 119889 (3 4)
= 119889 (4 3) = 4119889 (119909 119909) = 0 forall119909 isin 119883
(56)
Abstract and Applied Analysis 5
Then (119883 119889) is a complete generalized metric space butnot a metric space (see [21])
Define a map 119879 119883 997888rarr 119883 by
119879119909 = 3 (119909 = 4) 1 (119909 = 4) (57)
And define a function 120579 (0infin) 997888rarr (1infin) by120579 (119905) = 119890119905 (58)
We now show that 119879 is a L-contraction with respect to120585119887 where 120585119887(119905 119904) = 119904119896119905 forall119905 119904 ge 1 119896 = 12
We have
119889 (119879119909 119879119910) =
119889 (1 3) = 1 (119909 = 4 119910 = 4) 119889 (1 1) = 0 (119909 = 4 119910 = 4) 119889 (3 3) = 0 (119909 = 4 119910 = 4)
(59)
so
119889 (119879119909 119879119910) gt 0 lArrrArr 119909 = 4 119910 = 4 (60)
We have for 119909 = 4 and 119910 = 4119889 (119909 119910) = 4
119889 (119879119909 119879119910) = 1 (61)
We deduce that for all 119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0120585119887 (120579 (119889 (119879119909 119879119910)) 120579 (119889 (119909 119910))) = [120579 (119889 (119909 119910))]119896
120579 (119889 (119879119909 119879119910))
= [1198904]121198901 = 119890 gt 1
(62)
Thus all hypotheses of Theorem 4 are satisfied and 119879 hasa fixed point 119909lowast = 3
Note that Banachrsquos contraction principle is not satisfiedwith the usual metric 120588(119909 119910) = |119909 minus 119910| forall119909 119910 isin 119883 In fact if119909 = 2 119910 = 4 then
120588 (1198792 1198794) le 119896120588 (2 4) 119896 isin (0 1) (63)
which implies
119896 ge 1 (64)
Also note that the 120579-contraction condition [13] does nothold
Let 120579(119905) = 119890119905 forall119905 gt 0Then 120579(119905) satisfies conditions (1205791) (1205792) and (1205794)If
120579 (120588 (1198792 1198794)) le [120579 (120588 (2 4))]119896 where 119896 isin (0 1) (65)
then
1198902 le [1198902]119896 (66)
and so 119896 ge 1 Hence 119879 is not 120579-contraction mapBy taking 120585 = 120585119887 in Theorem 4 we obtain Corollary 6
Corollary 6 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
120579 (119889 (119879119909 119879119910)) le 120579 (119889 (119909 119910))119896 (67)
where 120579 isin Θ and 119896 isin (0 1)en 119879 has a unique fixed point
Remark 7 Corollary 6 is a generalization of Theorem 21 of[12] without condition (1205793) and Theorem 22 of [13] withoutcondition (1205794)
By taking 120585 = 120585119908 in Theorem 4 we obtain Corollary 8
Corollary 8 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
120579 (119889 (119879119909 119879119910)) le 120579 (119889 (119909 119910))120601 (120579 (119889 (119909 119910))) (68)
where 120579 isin Θ and 120601 [1infin) 997888rarr [1infin) is nondecreasing andlower semicontinuous such that 120601minus1(1) = 1
en 119879 has a unique fixed point
Corollary 9 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
119889 (119879119909 119879119910) le 119889 (119909 119910) minus 120593 (119889 (119909 119910)) (69)
where 120593 [0infin) 997888rarr [0infin) is nondecreasing and lowersemicontinuous such that 120593minus1(0) = 0
en 119879 has a unique fixed point
Proof Condition (69) implies 119879 is continuousLet 120579(119905) = 119890119905 forall119905 gt 0From (69) we have that for all 119909 119910 isin 119883 with 119889(119879119909 119879119910) gt0120579 (119889 (119879119909 119879119910)) = 119890119889(119879119909119879119910) le 119890119889(119909119910)minus120593(119889(119909119910)) = 119890119889(119909119910)
119890120593(119889(119909119910)) (70)
Let 120593(119905) = ln(120601(120579(119905))) forall119905 ge 0 where 120601 [1infin) 997888rarr[1infin) is nondecreasing and lower semicontinuous such that120601minus1(1) = 1Then 120593 is nondecreasing and lower semicontinuous and
120593 (119905) = 0 lArrrArr 120601 (120579 (119905)) = 1 lArrrArr 120579 (119905) = 119890119905 = 1 lArrrArr 119905 = 0 (71)
It follows from (70) that for all 119909 119910 isin 119883with 119889(119879119909 119879119910) gt0120579 (119889 (119879119909 119879119910)) le 120579 (119889 (119909 119910))
119890ln(120601(120579(119889(119909119910)))) =120579 (119889 (119909 119910))
120601 (120579 (119889 (119909 119910))) (72)
By Corollary 8 119879 has a unique fixed point
By taking 120579(119905) = 2 minus (2120587) arctan(1119905120572) where 120572 isin(0 1) 119905 gt 0 in Corollary 8 we obtain the following result
Corollary 10 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
6 Abstract and Applied Analysis
2 minus 2120587 arctan( 1
[119889 (119879119909 119879119910)]120572) le 2 minus (2120587) arctan (1 [119889 (119909 119910)]120572)120601 (2 minus (2120587) arctan (1 [119889 (119909 119910)]120572)) (73)
where 120572 isin (0 1) and 120601 [1infin) 997888rarr [1infin) is nondecreasingand lower semicontinuous such that 120601minus1(1) = 1
en 119879 has a unique fixed point
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The author express his gratitude to the referees for carefulreading and giving variable comments This research wassupported by Hanseo University
References
[1] A Branciari ldquoA fixed point theorem of Banach-Caccioppolitype on a class of generalized metric spacesrdquo PublicationesMathematicae vol 57 no 1-2 pp 31ndash37 2000
[2] H Aydi E Karapınar and B Samet ldquoFixed points for general-ized (120572120595) contractions on generalized metric spacesrdquo Journalof Inequalities and Applications vol 2014 article 229 2014
[3] C Di Bari and P Vetro ldquoCommon fixed points in generalizedmetric spacesrdquo Applied Mathematics and Computation vol 218no 13 pp 7322ndash7325 2012
[4] P Das ldquoA fixed point theorem on a class of generalized metricspacesrdquo e Korean Journal of Mathematics vol 9 pp 29ndash332002
[5] W A Kirk and N Shahzad ldquoGeneralized metrics and Caristirsquostheoremrdquo Fixed Point eory and Applications vol 2013 article129 2013
[6] B Samet ldquoDiscussion on ldquoA fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spacesrdquo by ABranciarirdquo Publicationes Mathematicae vol 76 no 3-4 pp 493-494 2010
[7] I R Sarma J M Rao and S S Rao ldquoContractions overgeneralizedmetric spacesrdquoe Journal of Nonlinear Science andits Applications vol 2 no 3 pp 180ndash182 2009
[8] W Shatanawi A Al-Rawashdeh Hassen Aydi and H KNashine ldquoOn a Fixed Point for Generalized Contractions inGeneralized Metric Spacesrdquo Abstract and Applied Analysis vol2012 Article ID 246085 13 pages 2012
[9] T Suzuki B Alamri andL A Khan ldquoSome notes on fixed pointtheoems in ]-generalized metric spacesrdquo Bulletin of the KyushuInstitute of TechnologyMathematics Natural Science no 62 pp15ndash23 2015
[10] M Abtahi Z Kadelburg and S Radenovic ldquoFixed points ofCiric-Matakowski-type contractions in ]-generalized metricspacesrdquo RACSAM
[11] Z D Mitrovic and S Radenovic ldquoThe Banach and Reichcontractiond in b](s)-metric spacesrdquo Journal of Fixed Pointeory and Applications vol 19 no 4 pp 3087ndash3095 2017
[12] M Jleli and B Samet ldquoA new generalization of the Banachcontraction principlerdquo Journal of Inequalities and Applicationsvol 2014 no 1 article no 38 8 pages 2014
[13] J Ahmad A E Al-Mazrooei Y J Cho and Y-O YangldquoFixed point results for generalized 120579-contractionsrdquo Journal ofNonlinear Sciences and Applications JNSA vol 10 no 5 pp2350ndash2358 2017
[14] F Khojasteh S Shukla and S Radenovic ldquoA new approachto the study of fixed point theorems via simulation functionsrdquoFilomat vol 29 no 6 pp 1189ndash1194 2015
[15] H H Alsulami E Karapinar F Khojasteh and A-F Roldan-Lopez-de-Hierro ldquoA proposal to the study of contractions inquasi-metric spacesrdquo Discrete Dynamics in Nature and Societyvol 2014 Article ID 269286 pp 1ndash10 2014
[16] H Isik N Gungor C Park and S Jang ldquoFixed PointTheoremsfor Almost Z-Contractions with an ApplicationrdquoMathematicsvol 6 no 3 p 37 2018
[17] E Karapınar ldquoFixed points results via simulation functionsrdquoFilomat vol 30 no 8 pp 2343ndash2350 2016
[18] A-F Roldan-Lopez-de-Hierro E Karapinar C Roldan-Lopez-de-Hierro and J Martinez-Moreno ldquoCoincidence point the-orems on metric spaces via simulation functionsrdquo Journal ofComputational and Applied Mathematics vol 275 pp 345ndash3552015
[19] S Radenovic F Vetro and J Vujakovic ldquoAn alternative andeasy approach to fixed point results via simulation functionsrdquoDemonstratio Mathematica vol 50 no 1 pp 223ndash230 2017
[20] M Jleli and B Samet ldquoThe Kannanrsquos fixed point theorem in acone rectangular metric spacerdquoe Journal of Nonlinear Scienceand its Applications vol 2 no 3 pp 161ndash167 2009
[21] A Azam and M Arshad ldquoKannan fixed point theorem ongeneralizedmetric spacesrdquoe Journal of Nonlinear Science andits Applications vol 1 no 1 pp 45ndash48 2008
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Abstract and Applied Analysis 5
Then (119883 119889) is a complete generalized metric space butnot a metric space (see [21])
Define a map 119879 119883 997888rarr 119883 by
119879119909 = 3 (119909 = 4) 1 (119909 = 4) (57)
And define a function 120579 (0infin) 997888rarr (1infin) by120579 (119905) = 119890119905 (58)
We now show that 119879 is a L-contraction with respect to120585119887 where 120585119887(119905 119904) = 119904119896119905 forall119905 119904 ge 1 119896 = 12
We have
119889 (119879119909 119879119910) =
119889 (1 3) = 1 (119909 = 4 119910 = 4) 119889 (1 1) = 0 (119909 = 4 119910 = 4) 119889 (3 3) = 0 (119909 = 4 119910 = 4)
(59)
so
119889 (119879119909 119879119910) gt 0 lArrrArr 119909 = 4 119910 = 4 (60)
We have for 119909 = 4 and 119910 = 4119889 (119909 119910) = 4
119889 (119879119909 119879119910) = 1 (61)
We deduce that for all 119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0120585119887 (120579 (119889 (119879119909 119879119910)) 120579 (119889 (119909 119910))) = [120579 (119889 (119909 119910))]119896
120579 (119889 (119879119909 119879119910))
= [1198904]121198901 = 119890 gt 1
(62)
Thus all hypotheses of Theorem 4 are satisfied and 119879 hasa fixed point 119909lowast = 3
Note that Banachrsquos contraction principle is not satisfiedwith the usual metric 120588(119909 119910) = |119909 minus 119910| forall119909 119910 isin 119883 In fact if119909 = 2 119910 = 4 then
120588 (1198792 1198794) le 119896120588 (2 4) 119896 isin (0 1) (63)
which implies
119896 ge 1 (64)
Also note that the 120579-contraction condition [13] does nothold
Let 120579(119905) = 119890119905 forall119905 gt 0Then 120579(119905) satisfies conditions (1205791) (1205792) and (1205794)If
120579 (120588 (1198792 1198794)) le [120579 (120588 (2 4))]119896 where 119896 isin (0 1) (65)
then
1198902 le [1198902]119896 (66)
and so 119896 ge 1 Hence 119879 is not 120579-contraction mapBy taking 120585 = 120585119887 in Theorem 4 we obtain Corollary 6
Corollary 6 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
120579 (119889 (119879119909 119879119910)) le 120579 (119889 (119909 119910))119896 (67)
where 120579 isin Θ and 119896 isin (0 1)en 119879 has a unique fixed point
Remark 7 Corollary 6 is a generalization of Theorem 21 of[12] without condition (1205793) and Theorem 22 of [13] withoutcondition (1205794)
By taking 120585 = 120585119908 in Theorem 4 we obtain Corollary 8
Corollary 8 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
120579 (119889 (119879119909 119879119910)) le 120579 (119889 (119909 119910))120601 (120579 (119889 (119909 119910))) (68)
where 120579 isin Θ and 120601 [1infin) 997888rarr [1infin) is nondecreasing andlower semicontinuous such that 120601minus1(1) = 1
en 119879 has a unique fixed point
Corollary 9 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
119889 (119879119909 119879119910) le 119889 (119909 119910) minus 120593 (119889 (119909 119910)) (69)
where 120593 [0infin) 997888rarr [0infin) is nondecreasing and lowersemicontinuous such that 120593minus1(0) = 0
en 119879 has a unique fixed point
Proof Condition (69) implies 119879 is continuousLet 120579(119905) = 119890119905 forall119905 gt 0From (69) we have that for all 119909 119910 isin 119883 with 119889(119879119909 119879119910) gt0120579 (119889 (119879119909 119879119910)) = 119890119889(119879119909119879119910) le 119890119889(119909119910)minus120593(119889(119909119910)) = 119890119889(119909119910)
119890120593(119889(119909119910)) (70)
Let 120593(119905) = ln(120601(120579(119905))) forall119905 ge 0 where 120601 [1infin) 997888rarr[1infin) is nondecreasing and lower semicontinuous such that120601minus1(1) = 1Then 120593 is nondecreasing and lower semicontinuous and
120593 (119905) = 0 lArrrArr 120601 (120579 (119905)) = 1 lArrrArr 120579 (119905) = 119890119905 = 1 lArrrArr 119905 = 0 (71)
It follows from (70) that for all 119909 119910 isin 119883with 119889(119879119909 119879119910) gt0120579 (119889 (119879119909 119879119910)) le 120579 (119889 (119909 119910))
119890ln(120601(120579(119889(119909119910)))) =120579 (119889 (119909 119910))
120601 (120579 (119889 (119909 119910))) (72)
By Corollary 8 119879 has a unique fixed point
By taking 120579(119905) = 2 minus (2120587) arctan(1119905120572) where 120572 isin(0 1) 119905 gt 0 in Corollary 8 we obtain the following result
Corollary 10 Let (119883 119889) be a complete generalized metricspace and let 119879 119883 997888rarr 119883 be a mapping such that for all119909 119910 isin 119883 with 119889(119879119909 119879119910) gt 0
6 Abstract and Applied Analysis
2 minus 2120587 arctan( 1
[119889 (119879119909 119879119910)]120572) le 2 minus (2120587) arctan (1 [119889 (119909 119910)]120572)120601 (2 minus (2120587) arctan (1 [119889 (119909 119910)]120572)) (73)
where 120572 isin (0 1) and 120601 [1infin) 997888rarr [1infin) is nondecreasingand lower semicontinuous such that 120601minus1(1) = 1
en 119879 has a unique fixed point
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The author express his gratitude to the referees for carefulreading and giving variable comments This research wassupported by Hanseo University
References
[1] A Branciari ldquoA fixed point theorem of Banach-Caccioppolitype on a class of generalized metric spacesrdquo PublicationesMathematicae vol 57 no 1-2 pp 31ndash37 2000
[2] H Aydi E Karapınar and B Samet ldquoFixed points for general-ized (120572120595) contractions on generalized metric spacesrdquo Journalof Inequalities and Applications vol 2014 article 229 2014
[3] C Di Bari and P Vetro ldquoCommon fixed points in generalizedmetric spacesrdquo Applied Mathematics and Computation vol 218no 13 pp 7322ndash7325 2012
[4] P Das ldquoA fixed point theorem on a class of generalized metricspacesrdquo e Korean Journal of Mathematics vol 9 pp 29ndash332002
[5] W A Kirk and N Shahzad ldquoGeneralized metrics and Caristirsquostheoremrdquo Fixed Point eory and Applications vol 2013 article129 2013
[6] B Samet ldquoDiscussion on ldquoA fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spacesrdquo by ABranciarirdquo Publicationes Mathematicae vol 76 no 3-4 pp 493-494 2010
[7] I R Sarma J M Rao and S S Rao ldquoContractions overgeneralizedmetric spacesrdquoe Journal of Nonlinear Science andits Applications vol 2 no 3 pp 180ndash182 2009
[8] W Shatanawi A Al-Rawashdeh Hassen Aydi and H KNashine ldquoOn a Fixed Point for Generalized Contractions inGeneralized Metric Spacesrdquo Abstract and Applied Analysis vol2012 Article ID 246085 13 pages 2012
[9] T Suzuki B Alamri andL A Khan ldquoSome notes on fixed pointtheoems in ]-generalized metric spacesrdquo Bulletin of the KyushuInstitute of TechnologyMathematics Natural Science no 62 pp15ndash23 2015
[10] M Abtahi Z Kadelburg and S Radenovic ldquoFixed points ofCiric-Matakowski-type contractions in ]-generalized metricspacesrdquo RACSAM
[11] Z D Mitrovic and S Radenovic ldquoThe Banach and Reichcontractiond in b](s)-metric spacesrdquo Journal of Fixed Pointeory and Applications vol 19 no 4 pp 3087ndash3095 2017
[12] M Jleli and B Samet ldquoA new generalization of the Banachcontraction principlerdquo Journal of Inequalities and Applicationsvol 2014 no 1 article no 38 8 pages 2014
[13] J Ahmad A E Al-Mazrooei Y J Cho and Y-O YangldquoFixed point results for generalized 120579-contractionsrdquo Journal ofNonlinear Sciences and Applications JNSA vol 10 no 5 pp2350ndash2358 2017
[14] F Khojasteh S Shukla and S Radenovic ldquoA new approachto the study of fixed point theorems via simulation functionsrdquoFilomat vol 29 no 6 pp 1189ndash1194 2015
[15] H H Alsulami E Karapinar F Khojasteh and A-F Roldan-Lopez-de-Hierro ldquoA proposal to the study of contractions inquasi-metric spacesrdquo Discrete Dynamics in Nature and Societyvol 2014 Article ID 269286 pp 1ndash10 2014
[16] H Isik N Gungor C Park and S Jang ldquoFixed PointTheoremsfor Almost Z-Contractions with an ApplicationrdquoMathematicsvol 6 no 3 p 37 2018
[17] E Karapınar ldquoFixed points results via simulation functionsrdquoFilomat vol 30 no 8 pp 2343ndash2350 2016
[18] A-F Roldan-Lopez-de-Hierro E Karapinar C Roldan-Lopez-de-Hierro and J Martinez-Moreno ldquoCoincidence point the-orems on metric spaces via simulation functionsrdquo Journal ofComputational and Applied Mathematics vol 275 pp 345ndash3552015
[19] S Radenovic F Vetro and J Vujakovic ldquoAn alternative andeasy approach to fixed point results via simulation functionsrdquoDemonstratio Mathematica vol 50 no 1 pp 223ndash230 2017
[20] M Jleli and B Samet ldquoThe Kannanrsquos fixed point theorem in acone rectangular metric spacerdquoe Journal of Nonlinear Scienceand its Applications vol 2 no 3 pp 161ndash167 2009
[21] A Azam and M Arshad ldquoKannan fixed point theorem ongeneralizedmetric spacesrdquoe Journal of Nonlinear Science andits Applications vol 1 no 1 pp 45ndash48 2008
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
6 Abstract and Applied Analysis
2 minus 2120587 arctan( 1
[119889 (119879119909 119879119910)]120572) le 2 minus (2120587) arctan (1 [119889 (119909 119910)]120572)120601 (2 minus (2120587) arctan (1 [119889 (119909 119910)]120572)) (73)
where 120572 isin (0 1) and 120601 [1infin) 997888rarr [1infin) is nondecreasingand lower semicontinuous such that 120601minus1(1) = 1
en 119879 has a unique fixed point
Data Availability
No data were used to support this study
Conflicts of Interest
The author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The author express his gratitude to the referees for carefulreading and giving variable comments This research wassupported by Hanseo University
References
[1] A Branciari ldquoA fixed point theorem of Banach-Caccioppolitype on a class of generalized metric spacesrdquo PublicationesMathematicae vol 57 no 1-2 pp 31ndash37 2000
[2] H Aydi E Karapınar and B Samet ldquoFixed points for general-ized (120572120595) contractions on generalized metric spacesrdquo Journalof Inequalities and Applications vol 2014 article 229 2014
[3] C Di Bari and P Vetro ldquoCommon fixed points in generalizedmetric spacesrdquo Applied Mathematics and Computation vol 218no 13 pp 7322ndash7325 2012
[4] P Das ldquoA fixed point theorem on a class of generalized metricspacesrdquo e Korean Journal of Mathematics vol 9 pp 29ndash332002
[5] W A Kirk and N Shahzad ldquoGeneralized metrics and Caristirsquostheoremrdquo Fixed Point eory and Applications vol 2013 article129 2013
[6] B Samet ldquoDiscussion on ldquoA fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spacesrdquo by ABranciarirdquo Publicationes Mathematicae vol 76 no 3-4 pp 493-494 2010
[7] I R Sarma J M Rao and S S Rao ldquoContractions overgeneralizedmetric spacesrdquoe Journal of Nonlinear Science andits Applications vol 2 no 3 pp 180ndash182 2009
[8] W Shatanawi A Al-Rawashdeh Hassen Aydi and H KNashine ldquoOn a Fixed Point for Generalized Contractions inGeneralized Metric Spacesrdquo Abstract and Applied Analysis vol2012 Article ID 246085 13 pages 2012
[9] T Suzuki B Alamri andL A Khan ldquoSome notes on fixed pointtheoems in ]-generalized metric spacesrdquo Bulletin of the KyushuInstitute of TechnologyMathematics Natural Science no 62 pp15ndash23 2015
[10] M Abtahi Z Kadelburg and S Radenovic ldquoFixed points ofCiric-Matakowski-type contractions in ]-generalized metricspacesrdquo RACSAM
[11] Z D Mitrovic and S Radenovic ldquoThe Banach and Reichcontractiond in b](s)-metric spacesrdquo Journal of Fixed Pointeory and Applications vol 19 no 4 pp 3087ndash3095 2017
[12] M Jleli and B Samet ldquoA new generalization of the Banachcontraction principlerdquo Journal of Inequalities and Applicationsvol 2014 no 1 article no 38 8 pages 2014
[13] J Ahmad A E Al-Mazrooei Y J Cho and Y-O YangldquoFixed point results for generalized 120579-contractionsrdquo Journal ofNonlinear Sciences and Applications JNSA vol 10 no 5 pp2350ndash2358 2017
[14] F Khojasteh S Shukla and S Radenovic ldquoA new approachto the study of fixed point theorems via simulation functionsrdquoFilomat vol 29 no 6 pp 1189ndash1194 2015
[15] H H Alsulami E Karapinar F Khojasteh and A-F Roldan-Lopez-de-Hierro ldquoA proposal to the study of contractions inquasi-metric spacesrdquo Discrete Dynamics in Nature and Societyvol 2014 Article ID 269286 pp 1ndash10 2014
[16] H Isik N Gungor C Park and S Jang ldquoFixed PointTheoremsfor Almost Z-Contractions with an ApplicationrdquoMathematicsvol 6 no 3 p 37 2018
[17] E Karapınar ldquoFixed points results via simulation functionsrdquoFilomat vol 30 no 8 pp 2343ndash2350 2016
[18] A-F Roldan-Lopez-de-Hierro E Karapinar C Roldan-Lopez-de-Hierro and J Martinez-Moreno ldquoCoincidence point the-orems on metric spaces via simulation functionsrdquo Journal ofComputational and Applied Mathematics vol 275 pp 345ndash3552015
[19] S Radenovic F Vetro and J Vujakovic ldquoAn alternative andeasy approach to fixed point results via simulation functionsrdquoDemonstratio Mathematica vol 50 no 1 pp 223ndash230 2017
[20] M Jleli and B Samet ldquoThe Kannanrsquos fixed point theorem in acone rectangular metric spacerdquoe Journal of Nonlinear Scienceand its Applications vol 2 no 3 pp 161ndash167 2009
[21] A Azam and M Arshad ldquoKannan fixed point theorem ongeneralizedmetric spacesrdquoe Journal of Nonlinear Science andits Applications vol 1 no 1 pp 45ndash48 2008
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
