Variational Analysis, Optimization, and Fixed Point TheoryView this Special Issue
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Variational Analysis, Optimization, and Fixed Point Theory
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,  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.
The 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 simplicity, yet it is perhaps the most widely applied fixed point theory in all of analysis. This 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. This theorem is amazing equivalent to Ekeland’s variational principle.
Qamrul Hasan Ansari
Mohamed Amine Khamsi
- 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.
- Q. H. Ansari, Metric Spaces: Including Fixed Point Theory and Set-Valued Maps, Narosa Publishing House, New Delhi, India, 2010.
- M. Bianchi, G. Kassay, and R. Pini, “Existence of equilibria via Ekeland's principle,” Journal of Mathematical Analysis and Applications, vol. 305, no. 2, pp. 502–512, 2005.
- D. G. de Figueiredo, Lectures on the Ekeland Variational Principle with Applications and Detours, vol. 81 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Tata Institute of Fundamental Research, Mumbai, India, 1989.
- I. Ekeland, “Sur les problèmes variationnels,” Comptes Rendus de l'Académie des Sciences, vol. 275, pp. 1057–1059, 1972.
- I. Ekeland, “On the variational principle,” Journal of Mathematical Analysis and Applications, vol. 47, pp. 324–353, 1974.
- I. Ekeland, “Nonconvex minimization problems,” Bulletin of the American Mathematical Society, vol. 1, no. 3, pp. 443–474, 1979.
- O. Kada, T. Suzuki, and W. Takahashi, “Nonconvex minimization theorems and fixed point theorems in complete metric spaces,” Mathematica Japonica, vol. 44, no. 2, pp. 381–391, 1996.
- M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics, Wiley-Interscience, New York, NY, USA, 2001.
- J. Caristi, “Fixed point theorems for mappings satisfying inwardness conditions,” Transactions of the American Mathematical Society, vol. 215, pp. 241–251, 1976.
- J. Caristi and W. A. Kirk, “Geometric fixed point theory and inwardness conditions,” in The Geometry of Metric and Linear Spaces, vol. 490 of Lecture Notes in Mathematics, pp. 74–83, Springer, Berlin, Germany, 1975.
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