Editorial Variational Analysis, Optimization, and Fixed...
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Editorial Variational Analysis, Optimization, and Fixed Point Theory 2014 Jen-Chih Yao, 1 Abdul Latif, 2 Chong Li, 3 and Adrian Petrusel 4 1 Center of General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan 2 Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia 3 Department of Mathematics, Zhejiang University, Hangzhou 3110027, China 4 Department of Mathematics, Babes-Bolyai University, 400084 Cluj-Napoca, Romania Correspondence should be addressed to Jen-Chih Yao; [email protected] Received 14 September 2014; Accepted 14 September 2014 Copyright © 2015 Jen-Chih Yao et al. 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 the last two decades, the theory of variational analysis including variational inequalities (VI) emerged as a rapidly growing area of research because of its applications in nonlinear analysis, optimization, economics, game theory, and so forth; see, for example, [1] and the references therein. In the recent past, many authors devoted their attention to studying the VI defined on the set of fixed points of a mapping, called hierarchical variational inequalities. Very recently, several iterative methods have been investigated to solve VI, hierarchical variational inequalities, and triple hierarchical variational inequalities. Since the origin of the VI, a tool has been used to study optimization prob- lems. Hierarchical variational inequalities are used to study the bilevel mathematical programming problems. A triple level mathematical programming problem can be studied by using triple hierarchical variational inequalities. Several abstract results in nonlinear analysis are of special interest and applicability in the theory of variational problems, optimization, and mathematical economics. We point out here three of them (we refer to [2] for other methods or approaches). Ekeland’s variational principle provides the existence of an approximate minimizer of a bounded below and lower semicontinuous function. It is one of the most important results from nonlinear analysis and it has applications in different areas of mathematics and mathematical sciences, namely, fixed point theory, optimization, optimal control theory, game theory, nonlinear equations, dynamical systems, and so forth, for example, [3–8] and the references therein. During the last decade, it has been used to study the existence of solutions of equilibrium problems in the setting of metric spaces, for example, [3, 4] and the references therein. Banach’s contraction principle is remarkable in its sim- plicity, yet it is perhaps the most widely applied fixed point theory in all of the analyses. is is because the contractive condition on the mapping is simple and easy to verify and because it requires only completeness of the metric space. Although, the basic idea was known to others earlier, the principle first appeared in explicit form in Banach’s 1922 thesis where it was used to establish the existence of a solution to an integral equation. Caristi’s fixed point theorem [9, 10] has found many applications in nonlinear analysis. It is shown, for example, that this theorem yields essentially all the known inwardness results of geometric fixed point theory in Banach spaces. Recall that inwardness conditions are the ones which assert that, in some sense, points from the domain are mapped toward the domain. is theorem is an amazing equivalent to Ekeland’s variational principle. We refer to the recent monograph [11]. is special issue is concerned with the most recent devel- opment on the topic. Jen-Chih Yao Abdul Latif Chong Li Adrian Petrusel Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2015, Article ID 312823, 2 pages http://dx.doi.org/10.1155/2015/312823
Transcript of Editorial Variational Analysis, Optimization, and Fixed...
EditorialVariational Analysis, Optimization, and Fixed Point Theory 2014
Jen-Chih Yao,1 Abdul Latif,2 Chong Li,3 and Adrian Petrusel4
1Center of General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan2Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia3Department of Mathematics, Zhejiang University, Hangzhou 3110027, China4Department of Mathematics, Babes-Bolyai University, 400084 Cluj-Napoca, Romania
Correspondence should be addressed to Jen-Chih Yao; [email protected]
Received 14 September 2014; Accepted 14 September 2014
Copyright © 2015 Jen-Chih Yao et al. This 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 the last two decades, the theory of variational analysisincluding variational inequalities (VI) emerged as a rapidlygrowing area of research because of its applications innonlinear analysis, optimization, economics, game theory,and so forth; see, for example, [1] and the references therein.In the recent past, many authors devoted their attentionto studying the VI defined on the set of fixed points of amapping, called hierarchical variational inequalities. Veryrecently, several iterative methods have been investigatedto solve VI, hierarchical variational inequalities, and triplehierarchical variational inequalities. Since the origin of theVI, a tool has been used to study optimization prob-lems. Hierarchical variational inequalities are used to studythe bilevel mathematical programming problems. A triplelevel mathematical programming problem can be studiedby using triple hierarchical variational inequalities. Severalabstract results in nonlinear analysis are of special interestand applicability in the theory of variational problems,optimization, and mathematical economics. We point outhere three of them (we refer to [2] for other methods orapproaches).
Ekeland’s variational principle provides the existence ofan approximate minimizer of a bounded below and lowersemicontinuous function. It is one of the most importantresults from nonlinear analysis and it has applications indifferent areas of mathematics and mathematical sciences,namely, fixed point theory, optimization, optimal controltheory, game theory, nonlinear equations, dynamical systems,and so forth, for example, [3–8] and the references therein.
During the last decade, it has been used to study the existenceof solutions of equilibrium problems in the setting of metricspaces, for example, [3, 4] and the references therein.
Banach’s contraction principle is remarkable in its sim-plicity, yet it is perhaps the most widely applied fixed pointtheory in all of the analyses. This is because the contractivecondition on the mapping is simple and easy to verify andbecause it requires only completeness of the metric space.Although, the basic idea was known to others earlier, theprinciple first appeared in explicit form inBanach’s 1922 thesiswhere it was used to establish the existence of a solution to anintegral equation.
Caristi’s fixed point theorem [9, 10] has found manyapplications in nonlinear analysis. It is shown, for example,that this theorem yields essentially all the known inwardnessresults of geometric fixed point theory in Banach spaces.Recall that inwardness conditions are the ones which assertthat, in some sense, points from the domain are mappedtoward the domain. This theorem is an amazing equivalentto Ekeland’s variational principle. We refer to the recentmonograph [11].
This special issue is concernedwith themost recent devel-opment on the topic.
Jen-Chih YaoAbdul Latif
Chong LiAdrian Petrusel
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015, Article ID 312823, 2 pageshttp://dx.doi.org/10.1155/2015/312823
2 Abstract and Applied Analysis
References
[1] Q. H. Ansari, C. S. Lalitha, and M. Mehta, Generalized Con-vexity, Nonsmooth Variational Inequalities, and NonsmoothOptimization, CRC Press, Taylor & Francis, Boca Raton, Fla,USA, 2014.
[2] C. Border Kim, Fixed Point Theorems with Applications to Eco-nomics and Game Theory, Cambridge University Press, Cam-bridge, UK, 1989.
[3] Q. H. Ansari, Metric Spaces: Including Fixed Point Theory andSet-Valued Maps, Narosa Publishing House, New Delhi, India,2010.
[4] M. Bianchi, G. Kassay, and R. Pini, “Existence of equilibriavia Ekeland’s principle,” Journal of Mathematical Analysis andApplications, vol. 305, no. 2, pp. 502–512, 2005.
[5] D. G. De Figueiredo, The Ekeland Variational Principle withApplications and Detours, Tata Institute of FundamentalResearch, Mumbai, India, 1989.
[6] I. Ekeland, “Sur les prolems variationnels,” Comptes Rendus del’Academie des Sciences, vol. 275, pp. 1057–1059, 1972.
[7] I. Ekeland, “On the variational principle,” Journal of Mathemat-ical Analysis and Applications, vol. 47, pp. 324–353, 1974.
[8] I. Ekeland, “Nonconvexminimization problems,”TheAmericanMathematical Society. Bulletin, vol. 1, no. 3, pp. 445–474, 1979.
[9] J. Caristi, “Fixed point theorems for mappings satisfyinginwardness conditions,” Transactions of the American Mathe-matical Society, vol. 215, pp. 241–251, 1976.
[10] J. Caristi and W. A. Kirk, “Geometric fixed point theory andinwardness conditions,” in The Geometry of Metric and LinearSpaces, vol. 490 of Lecture Notes in Mathematics, pp. 74–83,Springer, Berlin, Germany, 1975.
[11] M. A. Khamsi andW. A. Kirk,An Introdunction toMetric Spacesand Fixed Point Theory, Wiley, New York, NY, USA, 2001.
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