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.
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 AnsariMohamed Amine KhamsiAbdul LatifJen-Chih Yao
AnsariQ. H.LalithaC. S.MehtaM.2014Boca Raton, Fla, USACRC Press, Taylor & Francis GroupAnsariQ. H.2010New Delhi, IndiaNarosa Publishing HouseBianchiM.KassayG.PiniR.Existence of equilibria via Ekeland's principle2005305250251210.1016/j.jmaa.2004.11.042MR2130718ZBL1061.49005de FigueiredoD. G.198981Mumbai, IndiaTata Institute of Fundamental Researchvi+96Tata Institute of Fundamental Research Lectures on Mathematics and PhysicsMR1019559EkelandI.Sur les problèmes variationnels197227510571059MR0310670ZBL0249.49004EkelandI.On the variational principle197447324353MR034661910.1016/0022-247X(74)90025-0ZBL0286.49015EkelandI.Nonconvex minimization problems19791344347410.1090/S0273-0979-1979-14595-6MR526967ZBL0441.49011KadaO.SuzukiT.TakahashiW.Nonconvex minimization theorems and fixed point theorems in complete metric spaces1996442381391MR1416281ZBL0897.54029KhamsiM. A.KirkW. A.2001New York, NY, USAWiley-Intersciencex+302Pure and Applied Mathematics10.1002/9781118033074MR1818603CaristiJ.Fixed point theorems for mappings satisfying inwardness conditions1976215241251MR039432910.1090/S0002-9947-1976-0394329-4ZBL0305.47029CaristiJ.KirkW. A.Geometric fixed point theory and inwardness conditions1975490Berlin, GermanySpringer7483Lecture Notes in MathematicsMR0399968ZBL0315.54052