Editorial Variational Analysis, Optimization, and...
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Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 823070, 2 pages http://dx.doi.org/10.1155/2013/823070 Editorial Variational Analysis, Optimization, and Fixed Point Theory Qamrul Hasan Ansari, 1 Mohamed Amine Khamsi, 2 Abdul Latif, 3 and Jen-Chih Yao 4 1 Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India 2 Department of Mathematical Sciences, e University of Texas at El Paso, El Paso, TX 79968, USA 3 Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia 4 Center of General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan Correspondence should be addressed to Qamrul Hasan Ansari; [email protected] Received 5 September 2013; Accepted 5 September 2013 Copyright © 2013 Qamrul Hasan Ansari 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 study 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, it has been used as a tool to study optimization problems. 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. e 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, [2–9] 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, [2, 3] 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 analysis. 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 [10, 11] 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 amazing equivalent to Ekeland’s variational principle. Qamrul Hasan Ansari Mohamed Amine Khamsi Abdul Latif Jen-Chih Yao References [1] Q. H. Ansari, C. S. Lalitha, and M. Mehta, Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization, CRC Press, Taylor & Francis Group, Boca Raton, Fla, USA, 2014. [2] Q. H. Ansari, Metric Spaces: Including Fixed Point eory and Set-Valued Maps, Narosa Publishing House, New Delhi, India, 2010.
Transcript of Editorial Variational Analysis, Optimization, and...
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 823070, 2 pageshttp://dx.doi.org/10.1155/2013/823070
EditorialVariational Analysis, Optimization, and Fixed Point Theory
Qamrul Hasan Ansari,1 Mohamed Amine Khamsi,2 Abdul Latif,3 and Jen-Chih Yao4
1 Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India2Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79968, USA3Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia4Center of General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan
Correspondence should be addressed to Qamrul Hasan Ansari; [email protected]
Received 5 September 2013; Accepted 5 September 2013
Copyright © 2013 Qamrul Hasan Ansari et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.
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 study 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, it has been used as a tool to study optimization problems.Hierarchical variational inequalities are used to study thebilevel mathematical programming problems. A triple levelmathematical programming problem can be studied by usingtriple hierarchical variational inequalities.
The Ekeland’s variational principle provides the existenceof an 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, [2–9] 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, [2, 3] 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 analysis. 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 [10, 11] 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 amazing equivalent toEkeland’s variational principle.
Qamrul Hasan AnsariMohamed Amine Khamsi
Abdul LatifJen-Chih Yao
References
[1] Q. H. Ansari, C. S. Lalitha, and M. Mehta, GeneralizedConvexity, Nonsmooth Variational Inequalities, and NonsmoothOptimization, CRC Press, Taylor & Francis Group, Boca Raton,Fla, USA, 2014.
[2] Q. H. Ansari, Metric Spaces: Including Fixed Point Theory andSet-Valued Maps, Narosa Publishing House, New Delhi, India,2010.
2 Abstract and Applied Analysis
[3] 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.
[4] D. G. de Figueiredo, Lectures on the Ekeland VariationalPrinciple with Applications and Detours, vol. 81 of Tata Instituteof Fundamental Research Lectures on Mathematics and Physics,Tata Institute of Fundamental Research, Mumbai, India, 1989.
[5] I. Ekeland, “Sur les problemes variationnels,” Comptes Rendusde l’Academie des Sciences, vol. 275, pp. 1057–1059, 1972.
[6] I. Ekeland, “On the variational principle,” Journal of Mathemat-ical Analysis and Applications, vol. 47, pp. 324–353, 1974.
[7] I. Ekeland, “Nonconvex minimization problems,” Bulletin of theAmerican Mathematical Society, vol. 1, no. 3, pp. 443–474, 1979.
[8] O. Kada, T. Suzuki, and W. Takahashi, “Nonconvex minimiza-tion theorems and fixed point theorems in complete metricspaces,”Mathematica Japonica, vol. 44, no. 2, pp. 381–391, 1996.
[9] M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spacesand Fixed PointTheory, Pure and Applied Mathematics, Wiley-Interscience, New York, NY, USA, 2001.
[10] J. Caristi, “Fixed point theorems for mappings satisfyinginwardness conditions,” Transactions of the American Mathe-matical Society, vol. 215, pp. 241–251, 1976.
[11] 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.
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