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In this paper, we first study the approximate fixed point property for hybrid Caristi type and Mizoguchi–Takahashi type mappings on metric spaces. We give some new generalizations of Mizoguchi–Takahashi's fixed point theorem and Caristi's fixed point theorem under new relaxed conditions which are quite original in the existing literature. We present new generalized Ekeland's variational principle, generalized Takahashi's nonconvex minimization theorem and nonconvex maximal element theorem for uniformly below sequentially lower semicontinuous from above functions and essential distances. Their equivalence relationships are also established. [ABSTRACT FROM AUTHOR]

In this paper we give a quasi-metric version of Caristi's fixed point theorem by using mw-distances. Our theorem generalizes a recent result obtained by Karapinar and Romaguera in [7]. [ABSTRACT FROM AUTHOR]

This paper brings forward the concept of Caristi type hybrid fixed point in MPM-space, by giving two hybrid fixed point theorems and two common hybrid fixed point theorems of sequences of set-valued mappings, the theorems improve and generalize the Caristi's fixed point and correspond to recent important results. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new fixed point theorems for generalized distances and Caristi-type mappings without assuming that the dominated functions possess lower semicontinuity property. As applications of these results, we obtain some new fixed point theorems which generalize and improve the famous Banach contraction principle. [ABSTRACT FROM AUTHOR]

In this paper, we prove several fixed point results for mappings of Caristi type in the setting of G-metric spaces. [ABSTRACT FROM AUTHOR]

We solve a question posed by E. Karapinar, F. Khojasteh and Z.D. Mitrović in their paper "A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces". We also characterize the completeness of b-metric spaces with the help of a variant of the contractivity condition introduced by the authors in the aforementioned article. [ABSTRACT FROM AUTHOR]

In this paper, we attempt to give a unified approach to the existing several versions of Ekeland's variational principle. In the framework of uniform spaces, we introduce p-distances and more generally, q-distances. Then we introduce a new type of completeness for uniform spaces, i.e., sequential completeness with respect to a q-distance (particularly, a p-distance), which is a very extensive concept of completeness. By using q-distances and the new type of completeness, we prove a generalized Takahashi's nonconvex minimization theorem, a generalized Ekeland's variational principle and a generalized Caristi's fixed point theorem. Moreover, we show that the above three theorems are equivalent to each other. From the generalized Ekeland's variational principle, we deduce a number of particular versions of Ekeland's principle, which include many known versions of the principle and their improvements. [ABSTRACT FROM AUTHOR]

In this paper, by applying the abstract maximal element principle of Lin and Du, we present some new existence theorems related with critical point theorem, maximal element theorem, generalized Ekeland's variational principle and common (fuzzy) fixed point theorem for essential distances. [ABSTRACT FROM AUTHOR]

In this paper, we study common fixed point properties for a family of mappings. We present the extensions of Caristi's fixed point theorem and Markov–Kakutani fixed point theorem for a family of set-valued mappings. [ABSTRACT FROM AUTHOR]

We extend Nadler's fixed point theorem to ν-generalized metric spaces. Through the proof of the above extension, we understand more deeply the mathematical structure of a ν-generalized metric space. In particular, we study the completeness of the space. We also improve Caristi's and Subrahmanyam's fixed point theorems in the space. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we first prove some generalizations of Caristi’s fixed point theorem. Then we give some applications to the fixed point theory of weakly contractive set-valued maps and the minimization problem. [Copyright &y& Elsevier]

Abstract: In this paper, we first study existence theorems of solution for quasivariational inclusion problems. We apply existence theorems of solution for quasivariational inclusion problem to study the existence theorems of solution for the Stampacchia generalized vector quasiequilibrium problems and Stampacchia vector quasiequilibrium problems. Our results and approach to the Stampacchia generalized vector quasiequilibrium problem are different from any existence results of this type of problem. Finally, we apply the existence theorems of solution for the Stampacchia vector quasiequilibrium problems to study simultaneous solutions of Caristi’s fixed point theorem and the Stampacchia set-valued vector Ekeland’s variational principle. Our results on Stampacchia set-valued vector Ekeland’s variational principle are different from any existence results of these types of problems. [Copyright &y& Elsevier]

Abstract: In this paper, the famous Banach contraction principle and Caristi''s fixed point theorem are generalized to the case of multi-valued mappings. Our results are extensions of the well-known Nadler''s fixed point theorem [S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475–487], as well as of some Caristi type theorems for multi-valued operators, see [N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177–188; J.P. Aubin, Optima and Equilibria. An Introduction to Nonlinear Analysis, Grad. Texts in Math., Springer-Verlag, Berlin, 1998, p. 17; S.S. Zhang, Q. Luo, Set-valued Caristi fixed point theorem and Ekeland''s variational principle, Appl. Math. Mech. 10 (2) (1989) 111–113 (in Chinese), English translation: Appl. Math. Mech. (English Ed.) 10 (2) (1989) 119–121], etc. [Copyright &y& Elsevier]

Abstract: In this paper, we extend the generalized Caristi''s fixed point theorems proved by Bae [J. Math. Anal. Appl. 284 (2003) 690–697] and others. [Copyright &y& Elsevier]

We solve a question posed by E. Karapinar, F. Khojasteh and Z.D. Mitrović in their paper “A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces”. We also characterize the completeness of b-metric spaces with the help of a variant of the contractivity condition introduced by the authors in the aforementioned article.

In this paper we establish and prove a generalized variational principle for b-metric spaces. As a consequence, we obtain a weak Zhong-type variational principle in b-metric spaces. We show the applicability of the mentioned generalized variational principle by presenting a Caristi-type fixed point theorem and an extension of the main result for bifunctions - both of them stated in b-metric spaces.

Using a pre-order principle in [Qiu JH. A pre-order principle and set-valued Ekeland variational principle. J Math Anal Appl. 2014;419:904-937], we establish a general equilibrium version of set-valued Ekeland variational principle (denoted by EVP), where the objective bimap is defined on the product of left-complete quasi-metric spaces and taking values in a quasi-order linear space, and the perturbation consists of the quasi-metric and a positive vector k0. Here, the ordering is only to be k0-closed, which is strictly weaker than tobe topologically closed. From the general equilibrium version, we deduce a number of particular equilibrium versions of EVP with set-valued bimaps or with vector-valued bimap. As applications of the equilibrium versions of EVP, we present several interesting results on equilibrium problems, vector optimization and fixed point theory in the setting of quasi-metric spaces. These results extend and improve the related known results. Using the obtained EVPs, we further study the existence and the robustness of traps in Behavioural Sciences. [ABSTRACT FROM AUTHOR]

We give generalizations in complete gauge spaces of the following results: Bishop-Phelps' theorem, Ekeland's variational principle, Caristi's fixed point theorem, the drop theorem and the flower petal theorem. We show that our generalizations are equivalent. We apply those results to obtain fixed point theorems for multivalued contractions defined on a closed subset of a complete gauge space and satisfying a generalized inwardness condition. [ABSTRACT FROM AUTHOR]