Research Article New Results in the Startpoint Theory for...
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Research Article New Results in the Startpoint Theory for Quasipseudometric Spaces Yaé Ulrich Gaba Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, Cape Town 7701, South Africa Correspondence should be addressed to Ya´ e Ulrich Gaba; [email protected] Received 22 July 2014; Revised 24 October 2014; Accepted 27 October 2014; Published 17 November 2014 Academic Editor: Aref Jeribi Copyright © 2014 Ya´ e Ulrich Gaba. 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. We give two generalizations of eorem 35 proved by Gaba (2014). More precisely, we change the structure of the contractive condition; namely, we introduce a function Φ instead of a simple constant . Dedicated to Professor Guy A. Degla for his mentorship 1. Introduction and Preliminaries In [1], we introduced the concept of startpoint and end- point for set-valued mappings defined on quasipseudometric spaces. As mentioned there, the purpose of this theory is to study fixed point like related properties. In the present, we give more results from the theory. More precisely, we generalize eorem 35 of [1] by changing the structure of the contractive condition; namely, we introduce a function Φ instead of a simple constant (as it appears in the original statement). is new condition is interesting in the sense that it allows us to have a condition involving a functional of the variables and not just the variables themselves. For the convenience of the reader, we will recall some necessary definitions but for a detailed expos´ e of the definition and examples, the interested reader is referred to [1]. Definition 1. Let be a nonempty set. A function :× → [0, ∞) is called quasipseudometric on if (i) (, ) = 0 ∀ ∈ ; (ii) (, ) ≤ (, ) + (, ) ∀, , ∈ . Moreover, if (, ) = 0 = (, ) ⇒ = , then is said to be a 0 -quasipseudometric. e latter condition is referred to as the 0 -condition. Remark 2. (i) Let be quasipseudometric on ; then the map −1 defined by −1 (, ) = (, ) whenever , ∈ is also a quasipseudometric on , called the conjugate of . In the literature, −1 is also denoted as or . (ii) It is easy to verify that the function defined by := ∨ −1 , that is, (, ) = max{(, ), (, )}, defines a metric on whenever is a 0 -quasipseudometric on . e quasipseudemetric induces a topology () on . Definition 3. Let (, ) be a quasipseudometric space. e -convergence of a sequence ( ) to a point , also called leſt-convergence and denoted by →, is defined in the following way: → ⇐⇒ (, ) → 0. (1) Similarly, we define the −1 -convergence of a sequence ( ) to a point or right convergence and denote it by −1 → , in the following way: −1 → ⇐⇒ ( ,) → 0. (2) Finally, in a quasipseudometric space (, ), we will say that a sequence ( ) -converges to if it is both leſt and right Hindawi Publishing Corporation Journal of Operators Volume 2014, Article ID 741818, 5 pages http://dx.doi.org/10.1155/2014/741818
Transcript of Research Article New Results in the Startpoint Theory for...
Research ArticleNew Results in the Startpoint Theory forQuasipseudometric Spaces
Yaeacute Ulrich Gaba
Department of Mathematics and Applied Mathematics University of Cape Town Rondebosch Cape Town 7701 South Africa
Correspondence should be addressed to Yae Ulrich Gaba gabayae2gmailcom
Received 22 July 2014 Revised 24 October 2014 Accepted 27 October 2014 Published 17 November 2014
Academic Editor Aref Jeribi
Copyright copy 2014 Yae Ulrich Gaba 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
We give two generalizations of Theorem 35 proved by Gaba (2014) More precisely we change the structure of the contractivecondition namely we introduce a functionΦ instead of a simple constant 119888
Dedicated to Professor Guy A Degla for his mentorship
1 Introduction and Preliminaries
In [1] we introduced the concept of startpoint and end-point for set-valued mappings defined on quasipseudometricspaces As mentioned there the purpose of this theory isto study fixed point like related properties In the presentwe give more results from the theory More precisely wegeneralize Theorem 35 of [1] by changing the structure ofthe contractive condition namely we introduce a functionΦinstead of a simple constant 119888 (as it appears in the originalstatement) This new condition is interesting in the sensethat it allows us to have a condition involving a functionalof the variables and not just the variables themselves Forthe convenience of the reader we will recall some necessarydefinitions but for a detailed expose of the definition andexamples the interested reader is referred to [1]
Definition 1 Let 119883 be a nonempty set A function 119889 119883 times119883 rarr [0infin) is called quasipseudometric on119883 if
(i) 119889(119909 119909) = 0 forall119909 isin 119883(ii) 119889(119909 119911) le 119889(119909 119910) + 119889(119910 119911) forall119909 119910 119911 isin 119883
Moreover if 119889(119909 119910) = 0 = 119889(119910 119909) rArr 119909 = 119910 then 119889 is saidto be a 119879
0-quasipseudometric The latter condition is referred
to as the 1198790-condition
Remark 2 (i) Let 119889 be quasipseudometric on119883 then themap119889minus1 defined by 119889minus1(119909 119910) = 119889(119910 119909) whenever 119909 119910 isin 119883 is also
a quasipseudometric on 119883 called the conjugate of 119889 In theliterature 119889minus1 is also denoted as 119889119905 or 119889
(ii) It is easy to verify that the function 119889119904 defined by 119889119904 =119889 or 119889
minus1 that is 119889119904(119909 119910) = max119889(119909 119910) 119889(119910 119909) defines ametric on119883 whenever 119889 is a 119879
0-quasipseudometric on119883
The quasipseudemetric 119889 induces a topology 120591(119889) on119883
Definition 3 Let (119883 119889) be a quasipseudometric space The119889-convergence of a sequence (119909
119899) to a point 119909 also called
left-convergence and denoted by 119909119899
119889
997888rarr 119909 is defined in thefollowing way
119909119899
119889
997888997888rarr 119909 lArrrArr 119889 (119909 119909119899) 997888rarr 0 (1)
Similarly we define the 119889minus1-convergence of a sequence
(119909119899) to a point 119909 or right convergence and denote it by 119909
119899
119889minus1
997888997888997888rarr
119909 in the following way
119909119899
119889minus1
997888997888997888rarr 119909 lArrrArr 119889 (119909119899 119909) 997888rarr 0 (2)
Finally in a quasipseudometric space (119883 119889) we will saythat a sequence (119909
119899) 119889119904-converges to119909 if it is both left and right
Hindawi Publishing CorporationJournal of OperatorsVolume 2014 Article ID 741818 5 pageshttpdxdoiorg1011552014741818
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convergent to119909 andwedenote it as119909119899
119889119904
997888rarr 119909 or119909119899rarr 119909 when
there is no confusion Hence
119909119899
119889119904
997888997888rarr 119909 lArrrArr 119909119899
119889
997888997888rarr 119909 119909119899
119889minus1
997888997888rarr 119909 (3)
Definition 4 A sequence (119909119899) in quasipseudometric (119883 119889) is
called
(a) left 119889-Cauchy if for every 120598 gt 0 there exist 119909 isin 119883 and1198990isin N such that
forall119899 ge 1198990119889 (119909 119909
119899) lt 120598 (4)
(b) left 119870-Cauchy if for every 120598 gt 0 there exists 1198990isin N
such that
forall119899 119896 1198990le 119896 le 119899 119889 (119909
119896 119909119899) lt 120598 (5)
(c) 119889119904-Cauchy if for every 120598 gt 0 there exists 1198990isin N such
that
forall119899 119896 ge 1198990119889 (119909119899 119909119896) lt 120598 (6)
Dually we define right 119889-Cauchy and right 119870-Cauchysequences
Definition 5 A quasipseudometric space (119883 119889) is called
(i) left 119870-complete provided that any left 119870-Cauchysequence is 119889-convergent
(ii) left Smyth sequentially complete if any left 119870-Cauchysequence is 119889119904-convergent
Definition 6 A 1198790-quasipseudometric space (119883 119889) is called
bicomplete provided that the metric 119889119904 on119883 is complete
As usual a subset 119860 of a quasipseudometric space (119883 119889)will be called bounded provided that there exists a positivereal constant119872 such that 119889(119909 119910) lt 119872 whenever 119909 119910 isin 119860
Let (119883 119889) be a quasipseudometric spaceWe setP0(119883) =
2119883 0 where 2119883 denotes the power set of119883 For 119909 isin 119883 and
119860 119861 isin P0(119883) we define
119889 (119909 119860) = inf 119889 (119909 119886) 119886 isin 119860
119889 (119860 119909) = inf 119889 (119886 119909) 119886 isin 119860 (7)
and119867(119860 119861) by
119867(119860 119861) = maxsup119886isin119860
119889 (119886 119861) sup119887isin119861
119889 (119860 119887) (8)
Then 119867 is an extended quasipseudometric on P0(119883)
Moreover we know from [2] that the restriction of 119867 to119878119888119897(119883) = 119860 sube 119883 119860 = (119888119897
120591(119889)119860) cap (119888119897
120591(119889minus1
)119860) is an
extended 1198790-quasipseudometric We will denote by 119862119861(119883)
the collection of all nonempty bounded and 119889119904-closed subsetsof119883
Definition 7 (compare [1]) Let 119865 119883 rarr 2119883 be a set-valued
map An element 119909 isin 119883 is said to be
(i) a fixed point of 119865 if 119909 isin 119865119909(ii) a startpoint of 119865 if119867(119909 119865119909) = 0(iii) an endpoint of 119865 if119867(119865119909 119909) = 0
We complete this section by recalling the followinglemma
Lemma 8 (compare [1]) Let (119883 119889) be a quasipseudometricspace For every fixed 119909 isin 119883 the mapping 119910 997891rarr 119889(119909 119910) is 120591(119889)upper semicontinuous (120591(119889)-119906119904119888 in short) and 120591(119889minus1) lowersemicontinuous (120591(119889minus1)-lsc in short) For every fixed 119910 isin 119883the mapping 119909 997891rarr 119889(119909 119910) is 120591(119889)-lsc and 120591(119889minus1)-usc
2 Main Results
We commence this section with the main result of this paper
Theorem9 Let (119883 119889) be a left119870-complete quasipseudometricspace Let 119879 119883 rarr 119862119861(119883) be a set-valued map and define119891 119883 rarr R as119891(119909) = 119867(119909 119879119909) LetΦ [0infin) rarr [0 1) bea function such that lim sup
119903rarr 119905+Φ(119903) lt 1 for each 119905 isin [0infin)
Moreover assume that for any 119909 isin 119883 there exists 119910 isin 119879119909satisfying
119867(119910 119879119910) le Φ (119867 (119909 119910))119867 (119909 119910) (9)
and then 119879 has a startpoint
Proof First observe that since Φ(119867(119909 119910)) lt 1 for any119909 119910 isin 119883 it follows that 2 minus Φ(119867(119909 119910)) gt 1 for any 119909 119910 isin119883 and hence
119867(119909 119910) le [2 minus Φ (119867 (119909 119910))]119867 (119909 119879119909) (10)
for any 119909 isin 119883 and 119910 isin 119879119909For any initial 119909
0isin 119883 there exists (for all actually) 119909
1isin
1198791199090sube 119883 such that
119867(1199090 1199091) le [2 minus Φ (119867 (119909
0 1199091))]119867 (119909
0 1198791199090)
(11)
From (9) we get
119867(1199091 1198791199091) le Φ (119867 (119909
0 1199091))119867 (119909
0 1199091)
le Φ (119867 (1199090 1199091))
times [2 minus Φ (119867 (1199090 1199091))]119867 (119909
0 1198791199090)
= Ψ (119867 (1199090 1199091))119867 (119909
0 1198791199090)
(12)
where Ψ [0infin) rarr [0 1) is defined by
Ψ (119905) = Φ (119905) (2 minus Φ (119905)) (13)
Now choosing 1199092isin 1198651199091sube 119883 we have that
119867(1199091 1199092) le [2 minus Φ (119867 (119909
1 1199092))]119867 (119909
1 1198791199091)
(14)
Journal of Operators 3
and from (9) we get
119867(1199092 1198791199092) le Ψ (119867 (119909
1 1199092))119867 (119909
1 1198791199091) (15)
Continuing this process we obtain a sequence (119909119899)where
119909119899+1isin 119879119909119899sube 119883 with
119867(119909119899 119909119899+1)
le [2 minus Φ (119867 (119909119899 119909119899+1))]119867 (119909
119899 119879119909119899)
(16)
119867(119909119899+1 119879119909119899+1) le Ψ (119867 (119909
119899 119909119899+1))119867 (119909
119899 119879119909119899)
119899 = 1 2
(17)
For simplicity denote 119889119899= 119867(119909
119899 119909119899+1) and 119863
119899=
119867(119909119899 119879119909119899) for all 119899 ge 0 So from (17) we can write
119863119899+1le Ψ (119889
119899)119863119899le 119863119899 (18)
for all 119899 ge 0 Hence (119863119899) is a strictly decreasing sequence and
hence there exists 120575 ge 0 such that
lim119899rarrinfin
119863119899= 120575 (19)
From (16) it is easy to see that
119889119899lt 2119863119899 (20)
Thus the sequence (119889119899) is bounded and so there is 119889 ge 0
such that lim sup119899rarrinfin
119889119899= 119889 and hence a subsequence (119889
119899119896
)
of (119889119899) such that lim
119896rarrinfin119889119899119896
= 119889+ From (17)we have119863
119899119896
+1le
Ψ(119889119899119896
)119863119899119896
and thus
120575 = lim sup119896rarrinfin
119863119899119896
+1
le (lim sup119896rarrinfin
Ψ(119889119899119896
))(lim sup119896rarrinfin
119863119899119896
)
le lim sup119889119899
119896
rarr119889+
Ψ(119889119899119896
) 120575
(21)
This together with the fact lim sup119903rarr 119905+Φ(119903) lt 1 for each
119905 isin [0infin) implies that 120575 = 0 Then from (19) and (20) wederive that lim
119899rarrinfin119889119899= 0
Claim 1 (119909119899) is a left 119870-Cauchy sequence
Now let 120572 = lim sup119889119899
rarr0+Ψ(119889119899) and 119902 such that 120572 lt 119902 lt
1 This choice of 119902 is always possible since 120572 lt 1 Then thereis 1198990such that Ψ(119889
119899) lt 119902 for all 119899 ge 119899
0 So from (18) we have
119863119899+1le 119902119863
119899for all 119899 ge 119899
0 Then by induction we get 119863
119899le
119902119899minus11989901198631198990
for all 119899 ge 1198990+ 1 Combining this and inequality
(20) we get
119898
sum
119896=1198990
119889 (119909119896 119909119896+1) le 2
119898
sum
119896=1198990
119902119896minus11989901198631198990
le 21
1 minus 1199021198631198990
(22)
for all119898 gt 119899 ge 1198990+1 Hence (119909
119899) is a left119870-Cauchy sequence
According to the left 119870-completeness of (119883 119889) thereexists 119909lowast isin 119883 such that 119909
119899
119889
997888rarr 119909lowast
Claim 2 119909lowast is a startpoint of 119879Observe that the sequence 119863
119899= (119891119909
119899) = (119867(119909
119899 119879119909119899))
converges to 0 Since 119891 is 120591(119889)-lower semicontinuous (assupremumof 120591(119889)-lower semicontinuous functions) we have
0 le 119891 (119909lowast) le lim inf119899rarrinfin
119891 (119909119899) = 0 (23)
Hence 119891(119909lowast) = 0 that is119867(119909lowast 119879119909lowast) = 0This completes the proof
We give below an example to illustrate the theorem
Example 10 Let 119883 = [0 1] and 119889 119883 times 119883 rarr R be themapping defined by 119889(119886 119887) = max119886 minus 119887 0 Then 119889 is a1198790-quasipseudometric on119883 Observe that any left119870-Cauchy
sequence in (119883 119889) is 119889-convergent to 0 Indeed if (119909119899) is a left
119870-Cauchy sequence for every 120598 gt 0 there exists 1198990isin N such
that
forall119899 119896 1198990le 119896 le 119899 119889 (119909
119896 119909119899) lt 120598 (24)
This entails that forall119899 1198990lt 119899
119889 (0 119909119899) le 119889 (0 119909
119899minus1) + 119889 (119909
119899minus1 119909119899) = 0 + 119889 (119909
119899minus1 119909119899) lt 120598
(25)
Hence 119889(0 119909119899) rarr 0 that is 119909
119899
119889
997888rarr 0 Therefore (119883 119889) isleft 119870-complete Let 119879 119883 rarr 119862119861(119883) be such that
119879 (119909) =
1
21199092 for 119909 isin [0 15
32) cup (
15
32 1]
17
961
4 for 119909 = 15
32
(26)
Let Φ [0infin) rarr [0 34) sub [0 1) be defined by
Φ (119903) =
3
2119903 for 119903 isin [0 7
24) cup (
7
241
2)
425
768 for 119903 = 7
24
1
2 for 119903 isin [1
2infin)
(27)
An explicit computation of 119891(119909) = 119867(119909 119879119909) gives
119891 (119909) =
119909 minus1
21199092 for 119909 isin [0 15
32) cup (
15
32 1]
28
96 for 119909 = 15
32
(28)
Moreover for each 119909 isin [0 1532) cup (1532 1] 119910 =
(12)1199092 and we have
119867(119910 119879119910) le Φ (119867 (119909 119910))119867 (119909 119910) (29)
Of course inequality (29) also holds in the case of119909 = 1532 and 119910 = 1796 Therefore all assumptions ofTheorem 9 are satisfied and the endpoint of 119879 is 119909 = 0
4 Journal of Operators
Remark 11 In fact every sequence in (119883 119889) 119889-converges to0
Corollary 12 Let (119883 119889) be a right 119870-complete quasipseudo-metric space Let 119879 119883 rarr 119862119861(119883) be a set-valued map anddefine 119891 119883 rarr R as 119891(119909) = 119867(119879119909 119909) Let Φ [0infin) rarr[0 1) be a function such that lim sup
119903rarr 119905+Φ(119903) lt 1 for each
119905 isin [0infin) Moreover assume that for any 119909 isin 119883 there exists119910 isin 119879119909 satisfying
119867(119879119910 119910) le Φ (119867 (119910 119909))119867 (119910 119909) (30)
and then 119879 has an endpoint
Corollary 13 Let (119883 119889) be a bicomplete quasipseudometricspace Let 119879 119883 rarr 119862119861(119883) be a set-valued map and119891 119883 rarr R defined by 119891(119909) = 119867
119904(119879119909 119909) =
max119867(119879119909 119909)119867(119909 119879119909) If there exists 119888 isin (0 1) suchthat for all 119909 isin 119883 there exists 119910 isin 119865119909 satisfying
119867119904(119910 119865119910) le min Φ (119886) 119886 Φ (119887) 119887 (31)
where 119886 = 119867(119910 119909) and 119887 = 119867(119909 119910) then 119879 has a fixedpoint
Proof We give here the main idea of the proof Observe thatinequality (31) guarantees that the sequence (119909
119899) constructed
in the proof ofTheorem 9 is a 119889119904-Cauchy sequence and hence119889119904-converges to some 119909lowast Using the fact that 119891 is 120591(119889119904)-
lower semicontinuous (as supremum of 120591(119889119904)-continuousfunctions) we have
0 le 119891 (119909lowast) le lim inf119899rarrinfin
119891 (119909119899) = 0 (32)
Hence 119891(119909lowast) = 0 that is 119867(119909lowast 119879119909lowast) = 0 =
119867(119879119909lowast 119909lowast) and we are done
The following theorem is the second generalization thatwe propose
Theorem 14 Let (119883 119889) be a left119870-complete quasipseudomet-ric space Let 119879 119883 rarr 119862119861(119883) be a set-valued map anddefine 119891 119883 rarr R as 119891(119909) = 119867(119909 119879119909) Let 119887 [0infin) rarr[119886 1) 119886 gt 0 be a nondecreasing function Let Φ [0infin) rarr[0 1) be a function such that Φ(119905) lt 119887(119905) for each 119905 isin [0infin)and lim sup
119903rarr 119905+Φ(119903) lt lim sup
119903rarr 119905+119887(119903) for each 119905 isin [0infin)
Moreover assume that for any 119909 isin 119883 there exists 119910 isin 119879119909satisfying
119867(119910 119879119910) le Φ (119867 (119909 119910))119867 (119909 119910) (33)
and then 119879 has a startpoint
Proof Observe that because 119887(119867(119909 119910)) lt 1 for all 119909 119910 isin119883
119887 (119867 (119909 119910))119867 (119909 119910) le 119867 (119909 119879119909) (34)
for any 119910 isin 119879119909 Let 1199090isin 119883 be arbitrary Then we can choose
1199091isin 1198791199090such that
119887 (119867 (1199090 1199091))119867 (119909
0 1199091) le 119867 (119909
0 1198791199090) (35)
119867(1199091 1198791199091) le Φ (119867 (119909
0 1199091))119867 (119909
0 1199091) (36)
Define the function Ψ [0infin) rarr [0 1) by Ψ(119905) =Φ(119905)119887(119905) Hence (35) and (36) together sum to
119867(1199091 1198791199091) le Ψ (119867 (119909
0 1199091))119867 (119909
0 1198791199090) (37)
Now we choose 1199092isin 1198791199091such that
119887 (119867 (1199091 1199092))119867 (119909
1 1199092) le 119867 (119909
1 1198791199091)
119867 (1199092 1198791199092) le Φ (119867 (119909
1 1199092))119867 (119909
1 1199092)
(38)
which lead to
119867(1199092 1198791199092) le Ψ (119867 (119909
1 1199092))119867 (119909
1 1198791199091) (39)
Continuing this process we get an iterative sequence (119909119899)
where 119909119899+1isin 119879119909119899sube 119883 and denoting 119889
119899= 119867(119909
119899 119909119899+1)
and119863119899= 119867(119909
119899 119879119909119899) for all 119899 ge 0 we can write that
119887 (119889119899) 119889119899le 119863119899 119863
119899+1le Φ (119889
119899) 119889119899 (40)
for all 119899 ge 0 Hence
119863119899+1le Ψ (119889
119899)119863119899 (41)
If 119863119896= 0 for some 119896 ge 0 then we trivially have
lim119899rarrinfin
119863119899= 0 and the conclusion is immediate So without
loss of generality we can assume that 119863119899gt 0 for all 119899 ge 0
and from (41) we have119863119899+1lt 119863119899for all 119899 ge 0
Observe also that if for some 119899 119889119899le 119889119899+1
we are led toa contradiction Indeed from (40) and using the fact that thefunction 119887 is nondecreasing we have
119889119899le 119889119899+1leΦ (119889119899)
119887 (119889119899+1)119889119899leΦ (119889119899)
119887 (119889119899)119889119899= Ψ (119889
119899) 119889119899lt 119889119899
(42)
Hence 119889119899+1lt 119889119899for all 119899 ge 0 Thus there exist 120575 ge 0 and
119889 ge 0 such that lim119899rarrinfin
119863119899= 120575 and lim
119899rarrinfin119889119899= 119889 From
(41) we get
120575 le (lim sup119899rarrinfin
Ψ (119889119899)) 120575 = lim sup
119889119899
rarr119889+
Ψ (119889119899) 120575 (43)
and hence 120575 = 0 (since lim sup119903rarr 119905+Φ(119903) lt 1 for all 119905 isin
[0infin)) Moreover since
119886119889119899le 119887 (119889
119899) 119889119899le 119863119899le Ψ (119889
119899minus1)119863119899minus1 (44)
this forces 119889 to be 0 that is lim119899rarrinfin
119889119899= 0
Furthermore setting 120572 = lim sup119889119899
rarr0+Ψ(119889119899) and letting
119902 gt 0 be a positive number such that 120572 lt 119902 lt 1 there is 1198990
such that Ψ(119889119899) lt 119902 for all 119899 ge 119899
0 Hence from (41) and (44)
we get
119886119889119899le Ψ (119889
119899minus1)119863119899minus1le 119902119863119899minus1le sdot sdot sdot le 119902
119899minus11989901198631198990
(45)
for all 119899 ge 1198990 So 119889
119899le (1119886)119902
119899minus11989901198631198990
for all 119899 ge 1198990 Using a
similar argument as the one used in the proof of Theorem 9we conclude that (119909
119899) is a left119870-Cauchy sequence and that its
limit point is a starpoint of 119879
Journal of Operators 5
Example 15 Let 119883 = [0 1) and 119889 119883 times 119883 rarr R be themapping defined by 119889(119886 119887) = max119886 minus 119887 0 Then 119889 is a1198790-quasipseudometric on 119883 We know that (119883 119889) is left 119870-
complete Let 119879 119883 rarr 119862119861(119883) be such that
119879 (119909) = 1
21199092 (46)
Let Φ [0infin) rarr [0 34) sub [0 1) be defined by
Φ (119903) =
3
2119903 for 119903 isin [0 1
2)
0 for 119903 isin [12infin)
(47)
Let 119887 [0infin) rarr [23 1) be defined by
119887 (119905) =
5
6119905 for 119905 isin [0 1
2)
1
2 for 119905 isin [1
2infin)
(48)
An explicit computation of 119891(119909) = 119867(119909 119879119909) gives
119891 (119909) = 119909 minus1
21199092 for 119909 isin [0 1) (49)
Moreover for each 119909 isin [0 1) there exists 119910 = (12)1199092and we have
119867(119910 119879119910) le3
2119867 (119909 119910)119867 (119909 119910)
= Φ (119867 (119909 119910))119867 (119909 119910)
(50)
Therefore all assumptions ofTheorem 14 are satisfied andthe endpoint of 119879 is 119909 = 0
Corollary 16 Let (119883 119889) be a right 119870-complete quasipseudo-metric space Let 119879 119883 rarr 119862119861(119883) be a set-valued mapand define 119891 119883 rarr R as 119891(119909) = 119867
minus1(119909 119879119909) Let
119887 [0infin) rarr [119886 1) 119886 gt 0 be a nondecreasing function LetΦ [0infin) rarr [0 1) be a function such that Φ(119905) lt 119887(119905) foreach 119905 isin [0infin) and lim sup
119903rarr 119905+Φ(119903) lt lim sup
119903rarr 119905+119887(119903) for
each 119905 isin [0infin) Moreover assume that for any 119909 isin 119883 thereexists 119910 isin 119879119909 satisfying
119867minus1(119910 119879119910) le Φ (119867
minus1(119909 119910))119867
minus1(119909 119910) (51)
Φ(119905) lt 119887(119905) and lim sup119903rarr 119905+Φ(119903) lt lim sup
119903rarr 119905+119887(119903) for each
119905 isin [0infin) Moreover assume that for any 119909 isin 119883 there exists119910 isin 119879119909 satisfying and then 119879 has an endpoint
Corollary 17 Let (119883 119889) be a bicomplete quasipseudometricspace Let 119879 119883 rarr 119862119861(119883) be a set-valued map and define119891 119883 rarr R as 119891(119909) = 119867119904(119909 119879119909) Let 119887 [0infin) rarr [119886 1)119886 gt 0 be a nondecreasing function Let Φ [0infin) rarr [0 1)
be a function such that Φ(119905) lt 119887(119905) and lim sup119903rarr 119905+Φ(119903) lt
lim sup119903rarr 119905+119887(119903) for each 119905 isin [0infin) Moreover assume that
for any 119909 isin 119883 there exists 119910 isin 119879119909 satisfying
119867119904(119910 119879119910) le min Φ (119886) 119886 Φ (119887) 119887 (52)
where 119886 = 119867(119909 119910) and 119887 = 119867minus1(119909 119910) Then 119879 has afixed point
Remark 18 All the results given remain true when we replaceaccordingly the bicomplete quasipseudometric space (119883 119889)by a left Smyth sequentially completeleft 119870-complete or aright Smyth sequentially completeright 119870-complete space
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y U Gaba ldquoStartpoints and (120572 120574)-contractions in quasi-pseudometric spacesrdquo Journal of Mathematics vol 2014 ArticleID 709253 8 pages 2014
[2] H-P Kunzi and C Ryser ldquoThe Bourbaki quasi-uniformityrdquoTopology Proceedings vol 20 pp 161ndash183 1995
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Stochastic AnalysisInternational Journal of
2 Journal of Operators
convergent to119909 andwedenote it as119909119899
119889119904
997888rarr 119909 or119909119899rarr 119909 when
there is no confusion Hence
119909119899
119889119904
997888997888rarr 119909 lArrrArr 119909119899
119889
997888997888rarr 119909 119909119899
119889minus1
997888997888rarr 119909 (3)
Definition 4 A sequence (119909119899) in quasipseudometric (119883 119889) is
called
(a) left 119889-Cauchy if for every 120598 gt 0 there exist 119909 isin 119883 and1198990isin N such that
forall119899 ge 1198990119889 (119909 119909
119899) lt 120598 (4)
(b) left 119870-Cauchy if for every 120598 gt 0 there exists 1198990isin N
such that
forall119899 119896 1198990le 119896 le 119899 119889 (119909
119896 119909119899) lt 120598 (5)
(c) 119889119904-Cauchy if for every 120598 gt 0 there exists 1198990isin N such
that
forall119899 119896 ge 1198990119889 (119909119899 119909119896) lt 120598 (6)
Dually we define right 119889-Cauchy and right 119870-Cauchysequences
Definition 5 A quasipseudometric space (119883 119889) is called
(i) left 119870-complete provided that any left 119870-Cauchysequence is 119889-convergent
(ii) left Smyth sequentially complete if any left 119870-Cauchysequence is 119889119904-convergent
Definition 6 A 1198790-quasipseudometric space (119883 119889) is called
bicomplete provided that the metric 119889119904 on119883 is complete
As usual a subset 119860 of a quasipseudometric space (119883 119889)will be called bounded provided that there exists a positivereal constant119872 such that 119889(119909 119910) lt 119872 whenever 119909 119910 isin 119860
Let (119883 119889) be a quasipseudometric spaceWe setP0(119883) =
2119883 0 where 2119883 denotes the power set of119883 For 119909 isin 119883 and
119860 119861 isin P0(119883) we define
119889 (119909 119860) = inf 119889 (119909 119886) 119886 isin 119860
119889 (119860 119909) = inf 119889 (119886 119909) 119886 isin 119860 (7)
and119867(119860 119861) by
119867(119860 119861) = maxsup119886isin119860
119889 (119886 119861) sup119887isin119861
119889 (119860 119887) (8)
Then 119867 is an extended quasipseudometric on P0(119883)
Moreover we know from [2] that the restriction of 119867 to119878119888119897(119883) = 119860 sube 119883 119860 = (119888119897
120591(119889)119860) cap (119888119897
120591(119889minus1
)119860) is an
extended 1198790-quasipseudometric We will denote by 119862119861(119883)
the collection of all nonempty bounded and 119889119904-closed subsetsof119883
Definition 7 (compare [1]) Let 119865 119883 rarr 2119883 be a set-valued
map An element 119909 isin 119883 is said to be
(i) a fixed point of 119865 if 119909 isin 119865119909(ii) a startpoint of 119865 if119867(119909 119865119909) = 0(iii) an endpoint of 119865 if119867(119865119909 119909) = 0
We complete this section by recalling the followinglemma
Lemma 8 (compare [1]) Let (119883 119889) be a quasipseudometricspace For every fixed 119909 isin 119883 the mapping 119910 997891rarr 119889(119909 119910) is 120591(119889)upper semicontinuous (120591(119889)-119906119904119888 in short) and 120591(119889minus1) lowersemicontinuous (120591(119889minus1)-lsc in short) For every fixed 119910 isin 119883the mapping 119909 997891rarr 119889(119909 119910) is 120591(119889)-lsc and 120591(119889minus1)-usc
2 Main Results
We commence this section with the main result of this paper
Theorem9 Let (119883 119889) be a left119870-complete quasipseudometricspace Let 119879 119883 rarr 119862119861(119883) be a set-valued map and define119891 119883 rarr R as119891(119909) = 119867(119909 119879119909) LetΦ [0infin) rarr [0 1) bea function such that lim sup
119903rarr 119905+Φ(119903) lt 1 for each 119905 isin [0infin)
Moreover assume that for any 119909 isin 119883 there exists 119910 isin 119879119909satisfying
119867(119910 119879119910) le Φ (119867 (119909 119910))119867 (119909 119910) (9)
and then 119879 has a startpoint
Proof First observe that since Φ(119867(119909 119910)) lt 1 for any119909 119910 isin 119883 it follows that 2 minus Φ(119867(119909 119910)) gt 1 for any 119909 119910 isin119883 and hence
119867(119909 119910) le [2 minus Φ (119867 (119909 119910))]119867 (119909 119879119909) (10)
for any 119909 isin 119883 and 119910 isin 119879119909For any initial 119909
0isin 119883 there exists (for all actually) 119909
1isin
1198791199090sube 119883 such that
119867(1199090 1199091) le [2 minus Φ (119867 (119909
0 1199091))]119867 (119909
0 1198791199090)
(11)
From (9) we get
119867(1199091 1198791199091) le Φ (119867 (119909
0 1199091))119867 (119909
0 1199091)
le Φ (119867 (1199090 1199091))
times [2 minus Φ (119867 (1199090 1199091))]119867 (119909
0 1198791199090)
= Ψ (119867 (1199090 1199091))119867 (119909
0 1198791199090)
(12)
where Ψ [0infin) rarr [0 1) is defined by
Ψ (119905) = Φ (119905) (2 minus Φ (119905)) (13)
Now choosing 1199092isin 1198651199091sube 119883 we have that
119867(1199091 1199092) le [2 minus Φ (119867 (119909
1 1199092))]119867 (119909
1 1198791199091)
(14)
Journal of Operators 3
and from (9) we get
119867(1199092 1198791199092) le Ψ (119867 (119909
1 1199092))119867 (119909
1 1198791199091) (15)
Continuing this process we obtain a sequence (119909119899)where
119909119899+1isin 119879119909119899sube 119883 with
119867(119909119899 119909119899+1)
le [2 minus Φ (119867 (119909119899 119909119899+1))]119867 (119909
119899 119879119909119899)
(16)
119867(119909119899+1 119879119909119899+1) le Ψ (119867 (119909
119899 119909119899+1))119867 (119909
119899 119879119909119899)
119899 = 1 2
(17)
For simplicity denote 119889119899= 119867(119909
119899 119909119899+1) and 119863
119899=
119867(119909119899 119879119909119899) for all 119899 ge 0 So from (17) we can write
119863119899+1le Ψ (119889
119899)119863119899le 119863119899 (18)
for all 119899 ge 0 Hence (119863119899) is a strictly decreasing sequence and
hence there exists 120575 ge 0 such that
lim119899rarrinfin
119863119899= 120575 (19)
From (16) it is easy to see that
119889119899lt 2119863119899 (20)
Thus the sequence (119889119899) is bounded and so there is 119889 ge 0
such that lim sup119899rarrinfin
119889119899= 119889 and hence a subsequence (119889
119899119896
)
of (119889119899) such that lim
119896rarrinfin119889119899119896
= 119889+ From (17)we have119863
119899119896
+1le
Ψ(119889119899119896
)119863119899119896
and thus
120575 = lim sup119896rarrinfin
119863119899119896
+1
le (lim sup119896rarrinfin
Ψ(119889119899119896
))(lim sup119896rarrinfin
119863119899119896
)
le lim sup119889119899
119896
rarr119889+
Ψ(119889119899119896
) 120575
(21)
This together with the fact lim sup119903rarr 119905+Φ(119903) lt 1 for each
119905 isin [0infin) implies that 120575 = 0 Then from (19) and (20) wederive that lim
119899rarrinfin119889119899= 0
Claim 1 (119909119899) is a left 119870-Cauchy sequence
Now let 120572 = lim sup119889119899
rarr0+Ψ(119889119899) and 119902 such that 120572 lt 119902 lt
1 This choice of 119902 is always possible since 120572 lt 1 Then thereis 1198990such that Ψ(119889
119899) lt 119902 for all 119899 ge 119899
0 So from (18) we have
119863119899+1le 119902119863
119899for all 119899 ge 119899
0 Then by induction we get 119863
119899le
119902119899minus11989901198631198990
for all 119899 ge 1198990+ 1 Combining this and inequality
(20) we get
119898
sum
119896=1198990
119889 (119909119896 119909119896+1) le 2
119898
sum
119896=1198990
119902119896minus11989901198631198990
le 21
1 minus 1199021198631198990
(22)
for all119898 gt 119899 ge 1198990+1 Hence (119909
119899) is a left119870-Cauchy sequence
According to the left 119870-completeness of (119883 119889) thereexists 119909lowast isin 119883 such that 119909
119899
119889
997888rarr 119909lowast
Claim 2 119909lowast is a startpoint of 119879Observe that the sequence 119863
119899= (119891119909
119899) = (119867(119909
119899 119879119909119899))
converges to 0 Since 119891 is 120591(119889)-lower semicontinuous (assupremumof 120591(119889)-lower semicontinuous functions) we have
0 le 119891 (119909lowast) le lim inf119899rarrinfin
119891 (119909119899) = 0 (23)
Hence 119891(119909lowast) = 0 that is119867(119909lowast 119879119909lowast) = 0This completes the proof
We give below an example to illustrate the theorem
Example 10 Let 119883 = [0 1] and 119889 119883 times 119883 rarr R be themapping defined by 119889(119886 119887) = max119886 minus 119887 0 Then 119889 is a1198790-quasipseudometric on119883 Observe that any left119870-Cauchy
sequence in (119883 119889) is 119889-convergent to 0 Indeed if (119909119899) is a left
119870-Cauchy sequence for every 120598 gt 0 there exists 1198990isin N such
that
forall119899 119896 1198990le 119896 le 119899 119889 (119909
119896 119909119899) lt 120598 (24)
This entails that forall119899 1198990lt 119899
119889 (0 119909119899) le 119889 (0 119909
119899minus1) + 119889 (119909
119899minus1 119909119899) = 0 + 119889 (119909
119899minus1 119909119899) lt 120598
(25)
Hence 119889(0 119909119899) rarr 0 that is 119909
119899
119889
997888rarr 0 Therefore (119883 119889) isleft 119870-complete Let 119879 119883 rarr 119862119861(119883) be such that
119879 (119909) =
1
21199092 for 119909 isin [0 15
32) cup (
15
32 1]
17
961
4 for 119909 = 15
32
(26)
Let Φ [0infin) rarr [0 34) sub [0 1) be defined by
Φ (119903) =
3
2119903 for 119903 isin [0 7
24) cup (
7
241
2)
425
768 for 119903 = 7
24
1
2 for 119903 isin [1
2infin)
(27)
An explicit computation of 119891(119909) = 119867(119909 119879119909) gives
119891 (119909) =
119909 minus1
21199092 for 119909 isin [0 15
32) cup (
15
32 1]
28
96 for 119909 = 15
32
(28)
Moreover for each 119909 isin [0 1532) cup (1532 1] 119910 =
(12)1199092 and we have
119867(119910 119879119910) le Φ (119867 (119909 119910))119867 (119909 119910) (29)
Of course inequality (29) also holds in the case of119909 = 1532 and 119910 = 1796 Therefore all assumptions ofTheorem 9 are satisfied and the endpoint of 119879 is 119909 = 0
4 Journal of Operators
Remark 11 In fact every sequence in (119883 119889) 119889-converges to0
Corollary 12 Let (119883 119889) be a right 119870-complete quasipseudo-metric space Let 119879 119883 rarr 119862119861(119883) be a set-valued map anddefine 119891 119883 rarr R as 119891(119909) = 119867(119879119909 119909) Let Φ [0infin) rarr[0 1) be a function such that lim sup
119903rarr 119905+Φ(119903) lt 1 for each
119905 isin [0infin) Moreover assume that for any 119909 isin 119883 there exists119910 isin 119879119909 satisfying
119867(119879119910 119910) le Φ (119867 (119910 119909))119867 (119910 119909) (30)
and then 119879 has an endpoint
Corollary 13 Let (119883 119889) be a bicomplete quasipseudometricspace Let 119879 119883 rarr 119862119861(119883) be a set-valued map and119891 119883 rarr R defined by 119891(119909) = 119867
119904(119879119909 119909) =
max119867(119879119909 119909)119867(119909 119879119909) If there exists 119888 isin (0 1) suchthat for all 119909 isin 119883 there exists 119910 isin 119865119909 satisfying
119867119904(119910 119865119910) le min Φ (119886) 119886 Φ (119887) 119887 (31)
where 119886 = 119867(119910 119909) and 119887 = 119867(119909 119910) then 119879 has a fixedpoint
Proof We give here the main idea of the proof Observe thatinequality (31) guarantees that the sequence (119909
119899) constructed
in the proof ofTheorem 9 is a 119889119904-Cauchy sequence and hence119889119904-converges to some 119909lowast Using the fact that 119891 is 120591(119889119904)-
lower semicontinuous (as supremum of 120591(119889119904)-continuousfunctions) we have
0 le 119891 (119909lowast) le lim inf119899rarrinfin
119891 (119909119899) = 0 (32)
Hence 119891(119909lowast) = 0 that is 119867(119909lowast 119879119909lowast) = 0 =
119867(119879119909lowast 119909lowast) and we are done
The following theorem is the second generalization thatwe propose
Theorem 14 Let (119883 119889) be a left119870-complete quasipseudomet-ric space Let 119879 119883 rarr 119862119861(119883) be a set-valued map anddefine 119891 119883 rarr R as 119891(119909) = 119867(119909 119879119909) Let 119887 [0infin) rarr[119886 1) 119886 gt 0 be a nondecreasing function Let Φ [0infin) rarr[0 1) be a function such that Φ(119905) lt 119887(119905) for each 119905 isin [0infin)and lim sup
119903rarr 119905+Φ(119903) lt lim sup
119903rarr 119905+119887(119903) for each 119905 isin [0infin)
Moreover assume that for any 119909 isin 119883 there exists 119910 isin 119879119909satisfying
119867(119910 119879119910) le Φ (119867 (119909 119910))119867 (119909 119910) (33)
and then 119879 has a startpoint
Proof Observe that because 119887(119867(119909 119910)) lt 1 for all 119909 119910 isin119883
119887 (119867 (119909 119910))119867 (119909 119910) le 119867 (119909 119879119909) (34)
for any 119910 isin 119879119909 Let 1199090isin 119883 be arbitrary Then we can choose
1199091isin 1198791199090such that
119887 (119867 (1199090 1199091))119867 (119909
0 1199091) le 119867 (119909
0 1198791199090) (35)
119867(1199091 1198791199091) le Φ (119867 (119909
0 1199091))119867 (119909
0 1199091) (36)
Define the function Ψ [0infin) rarr [0 1) by Ψ(119905) =Φ(119905)119887(119905) Hence (35) and (36) together sum to
119867(1199091 1198791199091) le Ψ (119867 (119909
0 1199091))119867 (119909
0 1198791199090) (37)
Now we choose 1199092isin 1198791199091such that
119887 (119867 (1199091 1199092))119867 (119909
1 1199092) le 119867 (119909
1 1198791199091)
119867 (1199092 1198791199092) le Φ (119867 (119909
1 1199092))119867 (119909
1 1199092)
(38)
which lead to
119867(1199092 1198791199092) le Ψ (119867 (119909
1 1199092))119867 (119909
1 1198791199091) (39)
Continuing this process we get an iterative sequence (119909119899)
where 119909119899+1isin 119879119909119899sube 119883 and denoting 119889
119899= 119867(119909
119899 119909119899+1)
and119863119899= 119867(119909
119899 119879119909119899) for all 119899 ge 0 we can write that
119887 (119889119899) 119889119899le 119863119899 119863
119899+1le Φ (119889
119899) 119889119899 (40)
for all 119899 ge 0 Hence
119863119899+1le Ψ (119889
119899)119863119899 (41)
If 119863119896= 0 for some 119896 ge 0 then we trivially have
lim119899rarrinfin
119863119899= 0 and the conclusion is immediate So without
loss of generality we can assume that 119863119899gt 0 for all 119899 ge 0
and from (41) we have119863119899+1lt 119863119899for all 119899 ge 0
Observe also that if for some 119899 119889119899le 119889119899+1
we are led toa contradiction Indeed from (40) and using the fact that thefunction 119887 is nondecreasing we have
119889119899le 119889119899+1leΦ (119889119899)
119887 (119889119899+1)119889119899leΦ (119889119899)
119887 (119889119899)119889119899= Ψ (119889
119899) 119889119899lt 119889119899
(42)
Hence 119889119899+1lt 119889119899for all 119899 ge 0 Thus there exist 120575 ge 0 and
119889 ge 0 such that lim119899rarrinfin
119863119899= 120575 and lim
119899rarrinfin119889119899= 119889 From
(41) we get
120575 le (lim sup119899rarrinfin
Ψ (119889119899)) 120575 = lim sup
119889119899
rarr119889+
Ψ (119889119899) 120575 (43)
and hence 120575 = 0 (since lim sup119903rarr 119905+Φ(119903) lt 1 for all 119905 isin
[0infin)) Moreover since
119886119889119899le 119887 (119889
119899) 119889119899le 119863119899le Ψ (119889
119899minus1)119863119899minus1 (44)
this forces 119889 to be 0 that is lim119899rarrinfin
119889119899= 0
Furthermore setting 120572 = lim sup119889119899
rarr0+Ψ(119889119899) and letting
119902 gt 0 be a positive number such that 120572 lt 119902 lt 1 there is 1198990
such that Ψ(119889119899) lt 119902 for all 119899 ge 119899
0 Hence from (41) and (44)
we get
119886119889119899le Ψ (119889
119899minus1)119863119899minus1le 119902119863119899minus1le sdot sdot sdot le 119902
119899minus11989901198631198990
(45)
for all 119899 ge 1198990 So 119889
119899le (1119886)119902
119899minus11989901198631198990
for all 119899 ge 1198990 Using a
similar argument as the one used in the proof of Theorem 9we conclude that (119909
119899) is a left119870-Cauchy sequence and that its
limit point is a starpoint of 119879
Journal of Operators 5
Example 15 Let 119883 = [0 1) and 119889 119883 times 119883 rarr R be themapping defined by 119889(119886 119887) = max119886 minus 119887 0 Then 119889 is a1198790-quasipseudometric on 119883 We know that (119883 119889) is left 119870-
complete Let 119879 119883 rarr 119862119861(119883) be such that
119879 (119909) = 1
21199092 (46)
Let Φ [0infin) rarr [0 34) sub [0 1) be defined by
Φ (119903) =
3
2119903 for 119903 isin [0 1
2)
0 for 119903 isin [12infin)
(47)
Let 119887 [0infin) rarr [23 1) be defined by
119887 (119905) =
5
6119905 for 119905 isin [0 1
2)
1
2 for 119905 isin [1
2infin)
(48)
An explicit computation of 119891(119909) = 119867(119909 119879119909) gives
119891 (119909) = 119909 minus1
21199092 for 119909 isin [0 1) (49)
Moreover for each 119909 isin [0 1) there exists 119910 = (12)1199092and we have
119867(119910 119879119910) le3
2119867 (119909 119910)119867 (119909 119910)
= Φ (119867 (119909 119910))119867 (119909 119910)
(50)
Therefore all assumptions ofTheorem 14 are satisfied andthe endpoint of 119879 is 119909 = 0
Corollary 16 Let (119883 119889) be a right 119870-complete quasipseudo-metric space Let 119879 119883 rarr 119862119861(119883) be a set-valued mapand define 119891 119883 rarr R as 119891(119909) = 119867
minus1(119909 119879119909) Let
119887 [0infin) rarr [119886 1) 119886 gt 0 be a nondecreasing function LetΦ [0infin) rarr [0 1) be a function such that Φ(119905) lt 119887(119905) foreach 119905 isin [0infin) and lim sup
119903rarr 119905+Φ(119903) lt lim sup
119903rarr 119905+119887(119903) for
each 119905 isin [0infin) Moreover assume that for any 119909 isin 119883 thereexists 119910 isin 119879119909 satisfying
119867minus1(119910 119879119910) le Φ (119867
minus1(119909 119910))119867
minus1(119909 119910) (51)
Φ(119905) lt 119887(119905) and lim sup119903rarr 119905+Φ(119903) lt lim sup
119903rarr 119905+119887(119903) for each
119905 isin [0infin) Moreover assume that for any 119909 isin 119883 there exists119910 isin 119879119909 satisfying and then 119879 has an endpoint
Corollary 17 Let (119883 119889) be a bicomplete quasipseudometricspace Let 119879 119883 rarr 119862119861(119883) be a set-valued map and define119891 119883 rarr R as 119891(119909) = 119867119904(119909 119879119909) Let 119887 [0infin) rarr [119886 1)119886 gt 0 be a nondecreasing function Let Φ [0infin) rarr [0 1)
be a function such that Φ(119905) lt 119887(119905) and lim sup119903rarr 119905+Φ(119903) lt
lim sup119903rarr 119905+119887(119903) for each 119905 isin [0infin) Moreover assume that
for any 119909 isin 119883 there exists 119910 isin 119879119909 satisfying
119867119904(119910 119879119910) le min Φ (119886) 119886 Φ (119887) 119887 (52)
where 119886 = 119867(119909 119910) and 119887 = 119867minus1(119909 119910) Then 119879 has afixed point
Remark 18 All the results given remain true when we replaceaccordingly the bicomplete quasipseudometric space (119883 119889)by a left Smyth sequentially completeleft 119870-complete or aright Smyth sequentially completeright 119870-complete space
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y U Gaba ldquoStartpoints and (120572 120574)-contractions in quasi-pseudometric spacesrdquo Journal of Mathematics vol 2014 ArticleID 709253 8 pages 2014
[2] H-P Kunzi and C Ryser ldquoThe Bourbaki quasi-uniformityrdquoTopology Proceedings vol 20 pp 161ndash183 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Operators 3
and from (9) we get
119867(1199092 1198791199092) le Ψ (119867 (119909
1 1199092))119867 (119909
1 1198791199091) (15)
Continuing this process we obtain a sequence (119909119899)where
119909119899+1isin 119879119909119899sube 119883 with
119867(119909119899 119909119899+1)
le [2 minus Φ (119867 (119909119899 119909119899+1))]119867 (119909
119899 119879119909119899)
(16)
119867(119909119899+1 119879119909119899+1) le Ψ (119867 (119909
119899 119909119899+1))119867 (119909
119899 119879119909119899)
119899 = 1 2
(17)
For simplicity denote 119889119899= 119867(119909
119899 119909119899+1) and 119863
119899=
119867(119909119899 119879119909119899) for all 119899 ge 0 So from (17) we can write
119863119899+1le Ψ (119889
119899)119863119899le 119863119899 (18)
for all 119899 ge 0 Hence (119863119899) is a strictly decreasing sequence and
hence there exists 120575 ge 0 such that
lim119899rarrinfin
119863119899= 120575 (19)
From (16) it is easy to see that
119889119899lt 2119863119899 (20)
Thus the sequence (119889119899) is bounded and so there is 119889 ge 0
such that lim sup119899rarrinfin
119889119899= 119889 and hence a subsequence (119889
119899119896
)
of (119889119899) such that lim
119896rarrinfin119889119899119896
= 119889+ From (17)we have119863
119899119896
+1le
Ψ(119889119899119896
)119863119899119896
and thus
120575 = lim sup119896rarrinfin
119863119899119896
+1
le (lim sup119896rarrinfin
Ψ(119889119899119896
))(lim sup119896rarrinfin
119863119899119896
)
le lim sup119889119899
119896
rarr119889+
Ψ(119889119899119896
) 120575
(21)
This together with the fact lim sup119903rarr 119905+Φ(119903) lt 1 for each
119905 isin [0infin) implies that 120575 = 0 Then from (19) and (20) wederive that lim
119899rarrinfin119889119899= 0
Claim 1 (119909119899) is a left 119870-Cauchy sequence
Now let 120572 = lim sup119889119899
rarr0+Ψ(119889119899) and 119902 such that 120572 lt 119902 lt
1 This choice of 119902 is always possible since 120572 lt 1 Then thereis 1198990such that Ψ(119889
119899) lt 119902 for all 119899 ge 119899
0 So from (18) we have
119863119899+1le 119902119863
119899for all 119899 ge 119899
0 Then by induction we get 119863
119899le
119902119899minus11989901198631198990
for all 119899 ge 1198990+ 1 Combining this and inequality
(20) we get
119898
sum
119896=1198990
119889 (119909119896 119909119896+1) le 2
119898
sum
119896=1198990
119902119896minus11989901198631198990
le 21
1 minus 1199021198631198990
(22)
for all119898 gt 119899 ge 1198990+1 Hence (119909
119899) is a left119870-Cauchy sequence
According to the left 119870-completeness of (119883 119889) thereexists 119909lowast isin 119883 such that 119909
119899
119889
997888rarr 119909lowast
Claim 2 119909lowast is a startpoint of 119879Observe that the sequence 119863
119899= (119891119909
119899) = (119867(119909
119899 119879119909119899))
converges to 0 Since 119891 is 120591(119889)-lower semicontinuous (assupremumof 120591(119889)-lower semicontinuous functions) we have
0 le 119891 (119909lowast) le lim inf119899rarrinfin
119891 (119909119899) = 0 (23)
Hence 119891(119909lowast) = 0 that is119867(119909lowast 119879119909lowast) = 0This completes the proof
We give below an example to illustrate the theorem
Example 10 Let 119883 = [0 1] and 119889 119883 times 119883 rarr R be themapping defined by 119889(119886 119887) = max119886 minus 119887 0 Then 119889 is a1198790-quasipseudometric on119883 Observe that any left119870-Cauchy
sequence in (119883 119889) is 119889-convergent to 0 Indeed if (119909119899) is a left
119870-Cauchy sequence for every 120598 gt 0 there exists 1198990isin N such
that
forall119899 119896 1198990le 119896 le 119899 119889 (119909
119896 119909119899) lt 120598 (24)
This entails that forall119899 1198990lt 119899
119889 (0 119909119899) le 119889 (0 119909
119899minus1) + 119889 (119909
119899minus1 119909119899) = 0 + 119889 (119909
119899minus1 119909119899) lt 120598
(25)
Hence 119889(0 119909119899) rarr 0 that is 119909
119899
119889
997888rarr 0 Therefore (119883 119889) isleft 119870-complete Let 119879 119883 rarr 119862119861(119883) be such that
119879 (119909) =
1
21199092 for 119909 isin [0 15
32) cup (
15
32 1]
17
961
4 for 119909 = 15
32
(26)
Let Φ [0infin) rarr [0 34) sub [0 1) be defined by
Φ (119903) =
3
2119903 for 119903 isin [0 7
24) cup (
7
241
2)
425
768 for 119903 = 7
24
1
2 for 119903 isin [1
2infin)
(27)
An explicit computation of 119891(119909) = 119867(119909 119879119909) gives
119891 (119909) =
119909 minus1
21199092 for 119909 isin [0 15
32) cup (
15
32 1]
28
96 for 119909 = 15
32
(28)
Moreover for each 119909 isin [0 1532) cup (1532 1] 119910 =
(12)1199092 and we have
119867(119910 119879119910) le Φ (119867 (119909 119910))119867 (119909 119910) (29)
Of course inequality (29) also holds in the case of119909 = 1532 and 119910 = 1796 Therefore all assumptions ofTheorem 9 are satisfied and the endpoint of 119879 is 119909 = 0
4 Journal of Operators
Remark 11 In fact every sequence in (119883 119889) 119889-converges to0
Corollary 12 Let (119883 119889) be a right 119870-complete quasipseudo-metric space Let 119879 119883 rarr 119862119861(119883) be a set-valued map anddefine 119891 119883 rarr R as 119891(119909) = 119867(119879119909 119909) Let Φ [0infin) rarr[0 1) be a function such that lim sup
119903rarr 119905+Φ(119903) lt 1 for each
119905 isin [0infin) Moreover assume that for any 119909 isin 119883 there exists119910 isin 119879119909 satisfying
119867(119879119910 119910) le Φ (119867 (119910 119909))119867 (119910 119909) (30)
and then 119879 has an endpoint
Corollary 13 Let (119883 119889) be a bicomplete quasipseudometricspace Let 119879 119883 rarr 119862119861(119883) be a set-valued map and119891 119883 rarr R defined by 119891(119909) = 119867
119904(119879119909 119909) =
max119867(119879119909 119909)119867(119909 119879119909) If there exists 119888 isin (0 1) suchthat for all 119909 isin 119883 there exists 119910 isin 119865119909 satisfying
119867119904(119910 119865119910) le min Φ (119886) 119886 Φ (119887) 119887 (31)
where 119886 = 119867(119910 119909) and 119887 = 119867(119909 119910) then 119879 has a fixedpoint
Proof We give here the main idea of the proof Observe thatinequality (31) guarantees that the sequence (119909
119899) constructed
in the proof ofTheorem 9 is a 119889119904-Cauchy sequence and hence119889119904-converges to some 119909lowast Using the fact that 119891 is 120591(119889119904)-
lower semicontinuous (as supremum of 120591(119889119904)-continuousfunctions) we have
0 le 119891 (119909lowast) le lim inf119899rarrinfin
119891 (119909119899) = 0 (32)
Hence 119891(119909lowast) = 0 that is 119867(119909lowast 119879119909lowast) = 0 =
119867(119879119909lowast 119909lowast) and we are done
The following theorem is the second generalization thatwe propose
Theorem 14 Let (119883 119889) be a left119870-complete quasipseudomet-ric space Let 119879 119883 rarr 119862119861(119883) be a set-valued map anddefine 119891 119883 rarr R as 119891(119909) = 119867(119909 119879119909) Let 119887 [0infin) rarr[119886 1) 119886 gt 0 be a nondecreasing function Let Φ [0infin) rarr[0 1) be a function such that Φ(119905) lt 119887(119905) for each 119905 isin [0infin)and lim sup
119903rarr 119905+Φ(119903) lt lim sup
119903rarr 119905+119887(119903) for each 119905 isin [0infin)
Moreover assume that for any 119909 isin 119883 there exists 119910 isin 119879119909satisfying
119867(119910 119879119910) le Φ (119867 (119909 119910))119867 (119909 119910) (33)
and then 119879 has a startpoint
Proof Observe that because 119887(119867(119909 119910)) lt 1 for all 119909 119910 isin119883
119887 (119867 (119909 119910))119867 (119909 119910) le 119867 (119909 119879119909) (34)
for any 119910 isin 119879119909 Let 1199090isin 119883 be arbitrary Then we can choose
1199091isin 1198791199090such that
119887 (119867 (1199090 1199091))119867 (119909
0 1199091) le 119867 (119909
0 1198791199090) (35)
119867(1199091 1198791199091) le Φ (119867 (119909
0 1199091))119867 (119909
0 1199091) (36)
Define the function Ψ [0infin) rarr [0 1) by Ψ(119905) =Φ(119905)119887(119905) Hence (35) and (36) together sum to
119867(1199091 1198791199091) le Ψ (119867 (119909
0 1199091))119867 (119909
0 1198791199090) (37)
Now we choose 1199092isin 1198791199091such that
119887 (119867 (1199091 1199092))119867 (119909
1 1199092) le 119867 (119909
1 1198791199091)
119867 (1199092 1198791199092) le Φ (119867 (119909
1 1199092))119867 (119909
1 1199092)
(38)
which lead to
119867(1199092 1198791199092) le Ψ (119867 (119909
1 1199092))119867 (119909
1 1198791199091) (39)
Continuing this process we get an iterative sequence (119909119899)
where 119909119899+1isin 119879119909119899sube 119883 and denoting 119889
119899= 119867(119909
119899 119909119899+1)
and119863119899= 119867(119909
119899 119879119909119899) for all 119899 ge 0 we can write that
119887 (119889119899) 119889119899le 119863119899 119863
119899+1le Φ (119889
119899) 119889119899 (40)
for all 119899 ge 0 Hence
119863119899+1le Ψ (119889
119899)119863119899 (41)
If 119863119896= 0 for some 119896 ge 0 then we trivially have
lim119899rarrinfin
119863119899= 0 and the conclusion is immediate So without
loss of generality we can assume that 119863119899gt 0 for all 119899 ge 0
and from (41) we have119863119899+1lt 119863119899for all 119899 ge 0
Observe also that if for some 119899 119889119899le 119889119899+1
we are led toa contradiction Indeed from (40) and using the fact that thefunction 119887 is nondecreasing we have
119889119899le 119889119899+1leΦ (119889119899)
119887 (119889119899+1)119889119899leΦ (119889119899)
119887 (119889119899)119889119899= Ψ (119889
119899) 119889119899lt 119889119899
(42)
Hence 119889119899+1lt 119889119899for all 119899 ge 0 Thus there exist 120575 ge 0 and
119889 ge 0 such that lim119899rarrinfin
119863119899= 120575 and lim
119899rarrinfin119889119899= 119889 From
(41) we get
120575 le (lim sup119899rarrinfin
Ψ (119889119899)) 120575 = lim sup
119889119899
rarr119889+
Ψ (119889119899) 120575 (43)
and hence 120575 = 0 (since lim sup119903rarr 119905+Φ(119903) lt 1 for all 119905 isin
[0infin)) Moreover since
119886119889119899le 119887 (119889
119899) 119889119899le 119863119899le Ψ (119889
119899minus1)119863119899minus1 (44)
this forces 119889 to be 0 that is lim119899rarrinfin
119889119899= 0
Furthermore setting 120572 = lim sup119889119899
rarr0+Ψ(119889119899) and letting
119902 gt 0 be a positive number such that 120572 lt 119902 lt 1 there is 1198990
such that Ψ(119889119899) lt 119902 for all 119899 ge 119899
0 Hence from (41) and (44)
we get
119886119889119899le Ψ (119889
119899minus1)119863119899minus1le 119902119863119899minus1le sdot sdot sdot le 119902
119899minus11989901198631198990
(45)
for all 119899 ge 1198990 So 119889
119899le (1119886)119902
119899minus11989901198631198990
for all 119899 ge 1198990 Using a
similar argument as the one used in the proof of Theorem 9we conclude that (119909
119899) is a left119870-Cauchy sequence and that its
limit point is a starpoint of 119879
Journal of Operators 5
Example 15 Let 119883 = [0 1) and 119889 119883 times 119883 rarr R be themapping defined by 119889(119886 119887) = max119886 minus 119887 0 Then 119889 is a1198790-quasipseudometric on 119883 We know that (119883 119889) is left 119870-
complete Let 119879 119883 rarr 119862119861(119883) be such that
119879 (119909) = 1
21199092 (46)
Let Φ [0infin) rarr [0 34) sub [0 1) be defined by
Φ (119903) =
3
2119903 for 119903 isin [0 1
2)
0 for 119903 isin [12infin)
(47)
Let 119887 [0infin) rarr [23 1) be defined by
119887 (119905) =
5
6119905 for 119905 isin [0 1
2)
1
2 for 119905 isin [1
2infin)
(48)
An explicit computation of 119891(119909) = 119867(119909 119879119909) gives
119891 (119909) = 119909 minus1
21199092 for 119909 isin [0 1) (49)
Moreover for each 119909 isin [0 1) there exists 119910 = (12)1199092and we have
119867(119910 119879119910) le3
2119867 (119909 119910)119867 (119909 119910)
= Φ (119867 (119909 119910))119867 (119909 119910)
(50)
Therefore all assumptions ofTheorem 14 are satisfied andthe endpoint of 119879 is 119909 = 0
Corollary 16 Let (119883 119889) be a right 119870-complete quasipseudo-metric space Let 119879 119883 rarr 119862119861(119883) be a set-valued mapand define 119891 119883 rarr R as 119891(119909) = 119867
minus1(119909 119879119909) Let
119887 [0infin) rarr [119886 1) 119886 gt 0 be a nondecreasing function LetΦ [0infin) rarr [0 1) be a function such that Φ(119905) lt 119887(119905) foreach 119905 isin [0infin) and lim sup
119903rarr 119905+Φ(119903) lt lim sup
119903rarr 119905+119887(119903) for
each 119905 isin [0infin) Moreover assume that for any 119909 isin 119883 thereexists 119910 isin 119879119909 satisfying
119867minus1(119910 119879119910) le Φ (119867
minus1(119909 119910))119867
minus1(119909 119910) (51)
Φ(119905) lt 119887(119905) and lim sup119903rarr 119905+Φ(119903) lt lim sup
119903rarr 119905+119887(119903) for each
119905 isin [0infin) Moreover assume that for any 119909 isin 119883 there exists119910 isin 119879119909 satisfying and then 119879 has an endpoint
Corollary 17 Let (119883 119889) be a bicomplete quasipseudometricspace Let 119879 119883 rarr 119862119861(119883) be a set-valued map and define119891 119883 rarr R as 119891(119909) = 119867119904(119909 119879119909) Let 119887 [0infin) rarr [119886 1)119886 gt 0 be a nondecreasing function Let Φ [0infin) rarr [0 1)
be a function such that Φ(119905) lt 119887(119905) and lim sup119903rarr 119905+Φ(119903) lt
lim sup119903rarr 119905+119887(119903) for each 119905 isin [0infin) Moreover assume that
for any 119909 isin 119883 there exists 119910 isin 119879119909 satisfying
119867119904(119910 119879119910) le min Φ (119886) 119886 Φ (119887) 119887 (52)
where 119886 = 119867(119909 119910) and 119887 = 119867minus1(119909 119910) Then 119879 has afixed point
Remark 18 All the results given remain true when we replaceaccordingly the bicomplete quasipseudometric space (119883 119889)by a left Smyth sequentially completeleft 119870-complete or aright Smyth sequentially completeright 119870-complete space
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y U Gaba ldquoStartpoints and (120572 120574)-contractions in quasi-pseudometric spacesrdquo Journal of Mathematics vol 2014 ArticleID 709253 8 pages 2014
[2] H-P Kunzi and C Ryser ldquoThe Bourbaki quasi-uniformityrdquoTopology Proceedings vol 20 pp 161ndash183 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Operators
Remark 11 In fact every sequence in (119883 119889) 119889-converges to0
Corollary 12 Let (119883 119889) be a right 119870-complete quasipseudo-metric space Let 119879 119883 rarr 119862119861(119883) be a set-valued map anddefine 119891 119883 rarr R as 119891(119909) = 119867(119879119909 119909) Let Φ [0infin) rarr[0 1) be a function such that lim sup
119903rarr 119905+Φ(119903) lt 1 for each
119905 isin [0infin) Moreover assume that for any 119909 isin 119883 there exists119910 isin 119879119909 satisfying
119867(119879119910 119910) le Φ (119867 (119910 119909))119867 (119910 119909) (30)
and then 119879 has an endpoint
Corollary 13 Let (119883 119889) be a bicomplete quasipseudometricspace Let 119879 119883 rarr 119862119861(119883) be a set-valued map and119891 119883 rarr R defined by 119891(119909) = 119867
119904(119879119909 119909) =
max119867(119879119909 119909)119867(119909 119879119909) If there exists 119888 isin (0 1) suchthat for all 119909 isin 119883 there exists 119910 isin 119865119909 satisfying
119867119904(119910 119865119910) le min Φ (119886) 119886 Φ (119887) 119887 (31)
where 119886 = 119867(119910 119909) and 119887 = 119867(119909 119910) then 119879 has a fixedpoint
Proof We give here the main idea of the proof Observe thatinequality (31) guarantees that the sequence (119909
119899) constructed
in the proof ofTheorem 9 is a 119889119904-Cauchy sequence and hence119889119904-converges to some 119909lowast Using the fact that 119891 is 120591(119889119904)-
lower semicontinuous (as supremum of 120591(119889119904)-continuousfunctions) we have
0 le 119891 (119909lowast) le lim inf119899rarrinfin
119891 (119909119899) = 0 (32)
Hence 119891(119909lowast) = 0 that is 119867(119909lowast 119879119909lowast) = 0 =
119867(119879119909lowast 119909lowast) and we are done
The following theorem is the second generalization thatwe propose
Theorem 14 Let (119883 119889) be a left119870-complete quasipseudomet-ric space Let 119879 119883 rarr 119862119861(119883) be a set-valued map anddefine 119891 119883 rarr R as 119891(119909) = 119867(119909 119879119909) Let 119887 [0infin) rarr[119886 1) 119886 gt 0 be a nondecreasing function Let Φ [0infin) rarr[0 1) be a function such that Φ(119905) lt 119887(119905) for each 119905 isin [0infin)and lim sup
119903rarr 119905+Φ(119903) lt lim sup
119903rarr 119905+119887(119903) for each 119905 isin [0infin)
Moreover assume that for any 119909 isin 119883 there exists 119910 isin 119879119909satisfying
119867(119910 119879119910) le Φ (119867 (119909 119910))119867 (119909 119910) (33)
and then 119879 has a startpoint
Proof Observe that because 119887(119867(119909 119910)) lt 1 for all 119909 119910 isin119883
119887 (119867 (119909 119910))119867 (119909 119910) le 119867 (119909 119879119909) (34)
for any 119910 isin 119879119909 Let 1199090isin 119883 be arbitrary Then we can choose
1199091isin 1198791199090such that
119887 (119867 (1199090 1199091))119867 (119909
0 1199091) le 119867 (119909
0 1198791199090) (35)
119867(1199091 1198791199091) le Φ (119867 (119909
0 1199091))119867 (119909
0 1199091) (36)
Define the function Ψ [0infin) rarr [0 1) by Ψ(119905) =Φ(119905)119887(119905) Hence (35) and (36) together sum to
119867(1199091 1198791199091) le Ψ (119867 (119909
0 1199091))119867 (119909
0 1198791199090) (37)
Now we choose 1199092isin 1198791199091such that
119887 (119867 (1199091 1199092))119867 (119909
1 1199092) le 119867 (119909
1 1198791199091)
119867 (1199092 1198791199092) le Φ (119867 (119909
1 1199092))119867 (119909
1 1199092)
(38)
which lead to
119867(1199092 1198791199092) le Ψ (119867 (119909
1 1199092))119867 (119909
1 1198791199091) (39)
Continuing this process we get an iterative sequence (119909119899)
where 119909119899+1isin 119879119909119899sube 119883 and denoting 119889
119899= 119867(119909
119899 119909119899+1)
and119863119899= 119867(119909
119899 119879119909119899) for all 119899 ge 0 we can write that
119887 (119889119899) 119889119899le 119863119899 119863
119899+1le Φ (119889
119899) 119889119899 (40)
for all 119899 ge 0 Hence
119863119899+1le Ψ (119889
119899)119863119899 (41)
If 119863119896= 0 for some 119896 ge 0 then we trivially have
lim119899rarrinfin
119863119899= 0 and the conclusion is immediate So without
loss of generality we can assume that 119863119899gt 0 for all 119899 ge 0
and from (41) we have119863119899+1lt 119863119899for all 119899 ge 0
Observe also that if for some 119899 119889119899le 119889119899+1
we are led toa contradiction Indeed from (40) and using the fact that thefunction 119887 is nondecreasing we have
119889119899le 119889119899+1leΦ (119889119899)
119887 (119889119899+1)119889119899leΦ (119889119899)
119887 (119889119899)119889119899= Ψ (119889
119899) 119889119899lt 119889119899
(42)
Hence 119889119899+1lt 119889119899for all 119899 ge 0 Thus there exist 120575 ge 0 and
119889 ge 0 such that lim119899rarrinfin
119863119899= 120575 and lim
119899rarrinfin119889119899= 119889 From
(41) we get
120575 le (lim sup119899rarrinfin
Ψ (119889119899)) 120575 = lim sup
119889119899
rarr119889+
Ψ (119889119899) 120575 (43)
and hence 120575 = 0 (since lim sup119903rarr 119905+Φ(119903) lt 1 for all 119905 isin
[0infin)) Moreover since
119886119889119899le 119887 (119889
119899) 119889119899le 119863119899le Ψ (119889
119899minus1)119863119899minus1 (44)
this forces 119889 to be 0 that is lim119899rarrinfin
119889119899= 0
Furthermore setting 120572 = lim sup119889119899
rarr0+Ψ(119889119899) and letting
119902 gt 0 be a positive number such that 120572 lt 119902 lt 1 there is 1198990
such that Ψ(119889119899) lt 119902 for all 119899 ge 119899
0 Hence from (41) and (44)
we get
119886119889119899le Ψ (119889
119899minus1)119863119899minus1le 119902119863119899minus1le sdot sdot sdot le 119902
119899minus11989901198631198990
(45)
for all 119899 ge 1198990 So 119889
119899le (1119886)119902
119899minus11989901198631198990
for all 119899 ge 1198990 Using a
similar argument as the one used in the proof of Theorem 9we conclude that (119909
119899) is a left119870-Cauchy sequence and that its
limit point is a starpoint of 119879
Journal of Operators 5
Example 15 Let 119883 = [0 1) and 119889 119883 times 119883 rarr R be themapping defined by 119889(119886 119887) = max119886 minus 119887 0 Then 119889 is a1198790-quasipseudometric on 119883 We know that (119883 119889) is left 119870-
complete Let 119879 119883 rarr 119862119861(119883) be such that
119879 (119909) = 1
21199092 (46)
Let Φ [0infin) rarr [0 34) sub [0 1) be defined by
Φ (119903) =
3
2119903 for 119903 isin [0 1
2)
0 for 119903 isin [12infin)
(47)
Let 119887 [0infin) rarr [23 1) be defined by
119887 (119905) =
5
6119905 for 119905 isin [0 1
2)
1
2 for 119905 isin [1
2infin)
(48)
An explicit computation of 119891(119909) = 119867(119909 119879119909) gives
119891 (119909) = 119909 minus1
21199092 for 119909 isin [0 1) (49)
Moreover for each 119909 isin [0 1) there exists 119910 = (12)1199092and we have
119867(119910 119879119910) le3
2119867 (119909 119910)119867 (119909 119910)
= Φ (119867 (119909 119910))119867 (119909 119910)
(50)
Therefore all assumptions ofTheorem 14 are satisfied andthe endpoint of 119879 is 119909 = 0
Corollary 16 Let (119883 119889) be a right 119870-complete quasipseudo-metric space Let 119879 119883 rarr 119862119861(119883) be a set-valued mapand define 119891 119883 rarr R as 119891(119909) = 119867
minus1(119909 119879119909) Let
119887 [0infin) rarr [119886 1) 119886 gt 0 be a nondecreasing function LetΦ [0infin) rarr [0 1) be a function such that Φ(119905) lt 119887(119905) foreach 119905 isin [0infin) and lim sup
119903rarr 119905+Φ(119903) lt lim sup
119903rarr 119905+119887(119903) for
each 119905 isin [0infin) Moreover assume that for any 119909 isin 119883 thereexists 119910 isin 119879119909 satisfying
119867minus1(119910 119879119910) le Φ (119867
minus1(119909 119910))119867
minus1(119909 119910) (51)
Φ(119905) lt 119887(119905) and lim sup119903rarr 119905+Φ(119903) lt lim sup
119903rarr 119905+119887(119903) for each
119905 isin [0infin) Moreover assume that for any 119909 isin 119883 there exists119910 isin 119879119909 satisfying and then 119879 has an endpoint
Corollary 17 Let (119883 119889) be a bicomplete quasipseudometricspace Let 119879 119883 rarr 119862119861(119883) be a set-valued map and define119891 119883 rarr R as 119891(119909) = 119867119904(119909 119879119909) Let 119887 [0infin) rarr [119886 1)119886 gt 0 be a nondecreasing function Let Φ [0infin) rarr [0 1)
be a function such that Φ(119905) lt 119887(119905) and lim sup119903rarr 119905+Φ(119903) lt
lim sup119903rarr 119905+119887(119903) for each 119905 isin [0infin) Moreover assume that
for any 119909 isin 119883 there exists 119910 isin 119879119909 satisfying
119867119904(119910 119879119910) le min Φ (119886) 119886 Φ (119887) 119887 (52)
where 119886 = 119867(119909 119910) and 119887 = 119867minus1(119909 119910) Then 119879 has afixed point
Remark 18 All the results given remain true when we replaceaccordingly the bicomplete quasipseudometric space (119883 119889)by a left Smyth sequentially completeleft 119870-complete or aright Smyth sequentially completeright 119870-complete space
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y U Gaba ldquoStartpoints and (120572 120574)-contractions in quasi-pseudometric spacesrdquo Journal of Mathematics vol 2014 ArticleID 709253 8 pages 2014
[2] H-P Kunzi and C Ryser ldquoThe Bourbaki quasi-uniformityrdquoTopology Proceedings vol 20 pp 161ndash183 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Operators 5
Example 15 Let 119883 = [0 1) and 119889 119883 times 119883 rarr R be themapping defined by 119889(119886 119887) = max119886 minus 119887 0 Then 119889 is a1198790-quasipseudometric on 119883 We know that (119883 119889) is left 119870-
complete Let 119879 119883 rarr 119862119861(119883) be such that
119879 (119909) = 1
21199092 (46)
Let Φ [0infin) rarr [0 34) sub [0 1) be defined by
Φ (119903) =
3
2119903 for 119903 isin [0 1
2)
0 for 119903 isin [12infin)
(47)
Let 119887 [0infin) rarr [23 1) be defined by
119887 (119905) =
5
6119905 for 119905 isin [0 1
2)
1
2 for 119905 isin [1
2infin)
(48)
An explicit computation of 119891(119909) = 119867(119909 119879119909) gives
119891 (119909) = 119909 minus1
21199092 for 119909 isin [0 1) (49)
Moreover for each 119909 isin [0 1) there exists 119910 = (12)1199092and we have
119867(119910 119879119910) le3
2119867 (119909 119910)119867 (119909 119910)
= Φ (119867 (119909 119910))119867 (119909 119910)
(50)
Therefore all assumptions ofTheorem 14 are satisfied andthe endpoint of 119879 is 119909 = 0
Corollary 16 Let (119883 119889) be a right 119870-complete quasipseudo-metric space Let 119879 119883 rarr 119862119861(119883) be a set-valued mapand define 119891 119883 rarr R as 119891(119909) = 119867
minus1(119909 119879119909) Let
119887 [0infin) rarr [119886 1) 119886 gt 0 be a nondecreasing function LetΦ [0infin) rarr [0 1) be a function such that Φ(119905) lt 119887(119905) foreach 119905 isin [0infin) and lim sup
119903rarr 119905+Φ(119903) lt lim sup
119903rarr 119905+119887(119903) for
each 119905 isin [0infin) Moreover assume that for any 119909 isin 119883 thereexists 119910 isin 119879119909 satisfying
119867minus1(119910 119879119910) le Φ (119867
minus1(119909 119910))119867
minus1(119909 119910) (51)
Φ(119905) lt 119887(119905) and lim sup119903rarr 119905+Φ(119903) lt lim sup
119903rarr 119905+119887(119903) for each
119905 isin [0infin) Moreover assume that for any 119909 isin 119883 there exists119910 isin 119879119909 satisfying and then 119879 has an endpoint
Corollary 17 Let (119883 119889) be a bicomplete quasipseudometricspace Let 119879 119883 rarr 119862119861(119883) be a set-valued map and define119891 119883 rarr R as 119891(119909) = 119867119904(119909 119879119909) Let 119887 [0infin) rarr [119886 1)119886 gt 0 be a nondecreasing function Let Φ [0infin) rarr [0 1)
be a function such that Φ(119905) lt 119887(119905) and lim sup119903rarr 119905+Φ(119903) lt
lim sup119903rarr 119905+119887(119903) for each 119905 isin [0infin) Moreover assume that
for any 119909 isin 119883 there exists 119910 isin 119879119909 satisfying
119867119904(119910 119879119910) le min Φ (119886) 119886 Φ (119887) 119887 (52)
where 119886 = 119867(119909 119910) and 119887 = 119867minus1(119909 119910) Then 119879 has afixed point
Remark 18 All the results given remain true when we replaceaccordingly the bicomplete quasipseudometric space (119883 119889)by a left Smyth sequentially completeleft 119870-complete or aright Smyth sequentially completeright 119870-complete space
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] Y U Gaba ldquoStartpoints and (120572 120574)-contractions in quasi-pseudometric spacesrdquo Journal of Mathematics vol 2014 ArticleID 709253 8 pages 2014
[2] H-P Kunzi and C Ryser ldquoThe Bourbaki quasi-uniformityrdquoTopology Proceedings vol 20 pp 161ndash183 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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