Caristi's Fixed Point Theorem Research Articles

A note on the Caristi fixed point theorem

A note on the Caristi fixed point theorem

Variants of the New Caristi Theorem

The well-known Caristi fixed point theorem has numerous generalizations and modifications. Recently there
 have appeared its equivalent dual forms and generalizations based on new concept of lower semicontinuity
 from above by several authors. In the present article, we give new proofs of such new versions and their
 equivalent formulations by applying our Metatheorem in the ordered fixed point theory.

Ekeland Variational Principle and Some of Its Equivalents on a Weighted Graph, Completeness and the OSC Property

We prove a version of the Ekeland Variational Principle (EkVP) in a weighted graph G and its equivalence to Caristi fixed point theorem and to the Takahashi minimization principle. The usual completeness and topological notions are replaced with some weaker versions expressed in terms of the graph G. The main tool used in the proof is the OSC property for sequences in a graph. Converse results, meaning the completeness of weighted graphs for which one of these principles holds, are also considered.

Equivalents of various maximum principles

Certain maximum principles can be reformulated to various types of fixed point theorems and conversely, based on Metatheorem due to ourselves. Such principles are Zorn's lemma, Banach contraction principle, Nadler's fixed point theorem, Brézis-Browder principle, Caristi's fixed point theorem, Ekeland's variational principle, Takahashi's nonconvex minimization theorem, some others, and their variants, generalizations, or equivalent formulations. Consequently, we have many new theorems equivalent to known results on fixed point, common fixed point, stationary point, common stationary point, and others. We show that such points are all maximal elements of certain ordered sets. Further, we introduce our earlier related works as a history of our Metatheorem.

On Caristi fixed point theorem for set-valued mappings

On Caristi fixed point theorem for set-valued mappings

Ekeland's variational principle in fuzzy quasi-normed spaces

<abstract> <p>Fuzzy quasi-normed space provides an ideal mathematical framework for studying asymmetric phenomena. In this paper, we prove a version of the Ekeland variational principle in fuzzy quasi-normed spaces and apply it to Caristi's fixed point theorem and Takahashi minimization theorem. Moreover, we prove the equivalence relations among these theorems.</p> </abstract>

Caristi's fixed point theorem, Ekeland's variational principle and Takahashi's maximization theorem in fuzzy quasi-metric spaces

In this paper, a partial order and its basic properties are exploited in a fuzzy quasi-metric space in the sense of George and Veeramani (1994) [17]. Based on this partial order, Caristi's fixed point theorem, Ekeland's variational principle and Takahashi's maximization theorem are extended to fuzzy quasi-metric spaces by using the Brézis and Browder principle on ordered sets. Moreover, an equivalence chain among these theorems is provided.

A Common Coincidence of Fixed Point for Generalized Caristi Fixed Point Theorem

In this paper, the interpolative Caristi type weakly compatible contractive in a complete metric space is applied to show some common fixed points results related to such mappings. Our application shows that the function which is used to prove the obtained results is a bounded map. An example is provided to show the useability of the acquired results.

On the complexity of finding a Caristi's fixed point

We consider the complexity of finding a fixed point whose existence is guaranteed by an order-theoretic fixed point theorem, e.g., Caristi's fixed point theorem or Brøndsted's fixed point theorem. The problem of finding a fixed point guaranteed by some fixed point theorem is often used to characterize a complexity class. The best well-known such a problem is the problem of finding a Brouwer's fixed point. In contrast to this problem, there has been relatively little computational complexity analysis of computing a Caristi's fixed point and computing a Brøndsted's fixed point. In this paper, we show that the problem of finding a Caristi's fixed point is PLS-complete. Furthermore, we prove that the problem of finding a Brøndsted's fixed point is CLS-hard and belongs to PPAD.

Caristi's fixed point theorem in probabilistic metric spaces

In this work, we define a partial order on probabilistic metric spaces and establish some new Caristi's fixed point theorems and Ekeland's variational principle for the class of (right) continuous and Archimedean t-norms. As an application, a partial answer to Kirk's problem in metric spaces is given.

On Bishop-Phelps partial order, variation mappings and Caristi's fixed point theorem in quasi-metric spaces

On Bishop-Phelps partial order, variation mappings and Caristi's fixed point theorem in quasi-metric spaces

Some Caristi type fixed point theorems

In 1976, Caristi fixed point theorem was proved for ordinary metric spaces. After that, it was shown that this result is equivalent to Ekeland variational principle which has a great number of applications in many branches of mathematics. In this paper, we first introduce a new concept so called strong $$M_{b}$$ -metric to remedy lackness of continuity of $$M_{b}$$ -metric. Then, we investigate whether Caristi fixed point theorem can be extended to this space. Next, we obtain Caristi type fixed point theorem and some generalizations on strong $$M_{b}$$ -metric spaces. Also, we provide some illustrative and interesting examples showing that our theorems extend the results existing in the literature. Finally, we present some applications of our results to ordinary metric spaces.

On Best Proximity Point Results for Some Type of Mappings

In this paper, we give new conditions for existence and uniqueness of a best proximity point for Geraghty- and Caristi-type mappings. The presented results are most valuable generalizations of the Geraghty and Caristi fixed point theorems.

Caristi Fixed Point Theorem in Mv-Metric Spaces

Caristi Fixed Point Theorem in Mv-Metric Spaces

On Caristi’s fixed point theorem in metric spaces with a graph

We generalize the Caristi’s fixed point theorem for single valued as well as multivalued mappings defined on ametric space endowed with a graph andw-distance. Particularly, we modify the concept of the (OSC)-propertydue to Alfuraidan and Khamsi (Alfuraidan M. R. and Khamsi, M. A.,Caristi fixed point theorem in metric spaceswith graph, Abstr. Appl. Anal., (2014) Art. ID 303484, 5.) which enable us to reformulated their stated graphtheory version theorem (Theorem 3.2 in Alfuraidan M. R. and Khamsi, M. A.,Caristi fixed point theorem in metricspaces with graph, Abstr. Appl. Anal., (2014) Art. ID 303484, 5. ) to the case ofw-distance. Consequently,we extend and improve some recent works concerning extension of Banach Contraction Theorem tow-distancewith graph e.g. (Jachymski, J.,The contraction principle for mappings on a metric space with graph, Proc. Amer. Math.Soc.,136(2008), No. 4, 1359–1373; Nieto, J. J., Pouso, R. L. and Rodriguez-Lopez R.,Fixed point theorems in orderedabstract spaces, Proc. Amer. Math. Soc.,135(2007), 2505–2517 and Petrusel, A. and Rus, I.,Fixed point theorems inorderedL−spaces endowed with graph, Proc. Amer, Math. Soc.,134(2006), 411–418.

Fixed point theorems, multi-valued contractive mappings and multi-valuedcaristi type mappings

In this paper, a fixed point theorem is established for multi-valued contractive-mappings in FMS and Caristi fixed point theorem is summed up to the instance of multi-valued mappings in FMS. Illustrations are given to support fundamental outcomes. These outcomes are the expansions of the outcomesacquired by Feng and Liu (J. Math. Anal. Appl. 317:103-112, 2006) to the case of FMS

Caristi type fixed point theorems using Száz principle in quasi-metric spaces

In this paper, we use Szaz maximum principle to prove generalizations of Caristi fixed point theorem in a ´ preordered K-complete quasi metric space. Examples are given to support our results.

Ekeland variational principle on weighted graphs

In this work, we give a graphical version of the Ekeland variational principle which enables us to discover a new version of the Caristi fixed point theorem in weighted digraphs not necessarily generated by a partial order. Then we show that both graphical versions of the Ekeland variational principle and Caristi’s fixed point theorem are equivalent. In addition, we applied our main result on a differential structure Banach space.

Fixed points and coincidences of families of mappings between ordered sets and some metrical consequences

We present theorems on common fixed points and coincidences for families of multi-valued mappings of ordered sets. This generalizes some recent results of the author and Podoprikhin as well as the classical theorems of Knaster– Tarski, Smithson and Zermelo. We also consider connections with the Caristi fixed-point theorem and other metrical results.

完备随机度量空间上Ekeland变分原理与Caristi不动点定理的等价性

完备随机度量空间上Ekeland变分原理与Caristi不动点定理的等价性