Caristi's Fixed Point Theorem Research Articles

Caristi type and meir-keeler type fixed point theorems

We generalize the Caristi fixed point theorem by employing a weaker form of continuity and show that contractive type mappings that satisfy the conditions of our theorem provide new solutions to the Rhoades? problem on continuity at fixed point. We also obtain a Meir-Keeler type fixed point theorem which gives a new solution to the Rhoades? problem on the existence of contractive mappings that admit discontinuity at the fixed point. We prove that our theorems characterize completeness of the metric space as well as Cantor?s intersection property.

"A Caristi fixed point theorem for complete quasi-metric spaces by using mw-distances "

The authors thank the support of the Ministry of Economy and Competitiveness of Spain, Grants MTM2012-37894-C02-01 and MTM2015-64373-P (MINECO, FEDER,UE).

Two kinds of fixed point theorems and reverse mathematics

AbstractIn this paper, we investigate the logical strength of two types of fixed point theorems in the context of reverse mathematics. One is concerned with extensions of the Banach contraction principle. Among theorems in this type, we mainly show that the Caristi fixed point theorem is equivalent to over . The other is dedicated to topological fixed point theorems such as the Brouwer fixed point theorem. We introduce some variants of the Fan‐Browder fixed point theorem and the Kakutani fixed point theorem, which we call and , respectively. Then we show that is equivalent to and is equivalent to , over . In addition, we also study the application of the Fan‐Browder fixed point theorem to game systems.

Common fixed point properties for a family of set-valued mappings

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.

Caristi type fixed point theorems in fuzzy metric spaces

In this paper, we extend the generalized Caristi's fixed point theorem proved by Bollenbacher and Hicks to $p$-orbitally complete fuzzy metric spaces by considering the fuzzy metric spaces in the sense of George and Veeramani. We also give some illustrative examples that support our results.

Brondsted order in a metric space and generalizations of Caristi theorem

Fixed point and coincidence theorems for mappings of ordered sets and their metric counterparts generalizing the well-known Caristi fixed point theorem are presented.

Caristi's fixed point theorem and its equivalences in fuzzy metric spaces

Caristi's fixed point theorem and its equivalences in fuzzy metric spaces

Pseudo-metric space and fixed point theorem

The aim of this paper is to give a generalized version of Caristi fixed point theorems in pseudo-metric spaces. Our results generalize and improve many of well-known theorems. As an application of our results, we give a new existence theorem to the generalized nonlinear complementarity problem and a solution of differential inclusion in the distributions setting.

A direct proof of Caristi's fixed point theorem

A direct proof of Caristi's fixed point theorem

On generalized Caristi's fixed point theorem and its equivalence

On generalized Caristi's fixed point theorem and its equivalence

A NEW APPROACH TO CARISTI'S FIXED POINT THEOREM ON NON-ARCHIMEDEAN FUZZY METRIC SPACES

In the present paper, we give a new approach to Caristi's fixed pointtheorem on non-Archimedean fuzzy metric spaces. For this we define anordinary metric $d$ using the non-Archimedean fuzzy metric $M$ on a nonemptyset $X$ and we establish some relationship between $(X,d)$ and $(X,M,ast )$%. Hence, we prove our result by considering the original Caristi's fixedpoint theorem.

Caristi Fixed Point Theorem in Metric Spaces with a Graph

We discuss Caristi’s fixed point theorem for mappings defined on a metric space endowed with a graph. This work should be seen as a generalization of the classical Caristi’s fixed point theorem. It extends some recent works on the extension of Banach contraction principle to metric spaces with graph.

Set-valued Ekeland variational principles in fuzzy metric spaces

In this paper, we establish a general set-valued Ekeland's variational principle in fuzzy metric spaces, where the objective function is a set-valued map defined on a fuzzy metric space and taking values in a pre-ordered locally convex space, and the perturbation involves a quasi-metric family generating the fuzzy topology of the domain space. Moreover, the direction of the perturbation is a convex subset of the positive cone instead of a single positive vector. In our general version, the assumption that the objective function is lower semi-continuous and one that the range of the function is lower bounded are both weakened. From the general Ekeland's variational principle, we obtain several particular set-valued Ekeland's variational principles in fuzzy metric spaces, which generalize and improve some related known results. From these, we deduce the corresponding Caristi's fixed point theorems for set-valued maps and the corresponding Takahashi's non-convex minimization theorems in set-valued optimization. Finally, we extend the obtained results to F-type topological spaces.

Fixed point results on subgraphs of directed graphs

Abstract In this paper, we obtain some fixed point results on subgraphs of directed graphs. We show that the Caristi fixed point theorem and a version of Knaster-Tarski fixed point theorem are special cases of our results. 2010 MSC 47H10; 05C20; 54H25

On the Additivity of the Minkowski Functionals

In this work, we investigate the additivity of the Minkowski functionals associated to a cone in a linear vector space. As an application we discuss the equivalence of the classical Caristi fixed point theorem in metric spaces and its vectorial version in cone metric spaces.

Some generalizations of Caristi's fixed point theorem with applications

Some generalizations of Caristi's fixed point theorem with applications

Vectorial Ekeland's Variational Principle with a W-Distance and its Equivalent Theorems

By using the properties of w-distances and Gerstewitz's functions, we first give a vectorial Takahashi's nonconvex minimization theorem with a w-distance. From this, we deduce a general vectorial Ekeland's variational principle, where the objective function is from a complete metric space into a pre-ordered topological vector space and the perturbation contains a w-distance and a non-decreasing function of the objective function value. From the general vectorial variational principle, we deduce a vectorial Caristi's fixed point theorem with a w-distance. Finally we show that the above three theorems are equivalent to each other. The related known results are generalized and improved. In particular, some conditions in the theorems of [Y. Araya, Ekeland's variational principle and its equivalent theorems in vector optimization, J. Math. Anal. Appl. 346(2008), 9–16[ are weakened or even completely relieved.

Variational principles and fixed point theorems

The paper concerns fundamental variational principles and the Caristi fixed point theorem. The Brézis–Browder theorem is extended and Altmanʼs theorem is investigated. The notion of istance, an extension of ω-distance, is defined. The new concept enables us to prove some elegant and general variational principles which imply a much stronger form of the Caristi and the Takahashi fixed point theorems. Another consequence is an advanced version of the Ekeland variational principle.

Fixed point results for multivalued contractive maps

Abstract Using the concept of u-distance, we prove a fixed point theorem for multivalued contractive maps. We also establish a multivalued version of the Caristi's fixed point theorem and common fixed point result. Consequently, several known fixed point results are either improved or generalized including the corresponding fixed point results of Caristi, Mizoguchi-Takahashi, Kada et al., Suzuki-Takahashi, Suzuki, and Ume. Mathematics Subject Classification (2000): 47H10; 54H25.

Maximal and minimal point theorems and Caristi's fixed point theorem

This study is concerned with the existence of fixed points of Caristi-type mappings motivated by a problem stated by Kirk. First, several existence theorems of maximal and minimal points are established. By using them, some generalized Caristi's fixed point theorems are proved, which improve Caristi's fixed point theorem and the results in the studies of Jachymski, Feng and Liu, Khamsi, and Li. MSC 2010: 06A06; 47H10.