Caristi's Theorem Research Articles

On Modular b-Metrics

The notions of modular b-metric and modular b-metric space were introduced by Ege and Alaca as natural generalizations of the well-known and featured concepts of modular metric and modular metric space presented and discussed by Chistyakov. In particular, they stated generalized forms of Banach’s contraction principle for this new class of spaces thus initiating the study of the fixed point theory for these structures, where other authors have also made extensive contributions. In this paper we endow the modular b-metrics with a metrizable topology that supplies a firm endorsement of the idea of convergence proposed by Ege and Alaca in their article. Moreover, for a large class of modular b-metric spaces, we formulate this topology in terms of an explicitly defined b-metric, which extends both an important metrization theorem due to Chistyakov as well as the so-called topology of metric convergence. This approach allows us to characterize the completeness for this class of modular b-metric spaces that may be viewed as an offsetting of the celebrated Caristi–Kirk theorem to our context. We also include some examples that endorse our results.

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.

A generalization of Caristi's theorem on two Menger spaces

A generalization of Caristi's theorem on two Menger spaces

A Study of Caristi’s Fixed Point Theorem on Normed Space and Its Applications

In this work, we will discuss Caristi’s fixed point theorem for mapping results introduced in the setting of normed spaces. This work is a generalization of the classical Caristi’s fixed point theorem. Also, Caristi’s type of fixed points theorem was partial discussed in Reich, Mizoguchi and Takahashi’s and Amini-Harandi’s results, we developed ideas that many known fixed point theorems can easily be derived from the Caristi theorem.

The Caristi\u2013Kirk Fixed Point Theorem from the point of view of ball spaces

We take a fresh look at the important Caristi–Kirk Fixed Point Theorem and link it to the recently developed theory of ball spaces, which provides generic fixed point theorems for contracting functions in a number of applications including, but not limited to, metric spaces. The connection becomes clear from a proof of the Caristi–Kirk Theorem given by J.-P. Penot in 1976. We define Caristi–Kirk ball spaces and use a generic fixed point theorem to reprove the Caristi–Kirk Theorem. Further, we show that a metric space is complete if and only if all of its Caristi–Kirk ball spaces are spherically complete.

Generalizations of Caristi-Kirk theorem in partial metric spaces and applications

Generalizations of Caristi-Kirk theorem in partial metric spaces and applications

Functional type Caristi-Kirk theorem on two metric spaces and applications

In this paper, we give some generalizations of the functional type Caristi-Kirk theorem (see Functional Type Caristi-Kirk Theorems, 2005) for two mappings on metric spaces. We investigate the existence of some fixed points for two simultaneous projections to find the optimal solutions of the proximity two functions via Caristi-Kirk fixed point theorem.

Some applications of Caristi’s fixed point theorem in metric spaces

AbstractIn this work, partial answers to Reich, Mizoguchi and Takahashi’s and Amini-Harandi’s conjectures are presented via a light version of Caristi’s fixed point theorem. Moreover, we introduce the idea that many of known fixed point theorems can easily be derived from the Caristi theorem. Finally, the existence of bounded solutions of a functional equation is studied.

Generalizations of Caristi Kirk's Theorem on Partial Metric Spaces

AbstractIn this article, lower semi-continuous maps are used to generalize Cristi-Kirk's fixed point theorem on partial metric spaces. First, we prove such a type of fixed point theorem in compact partial metric spaces, and then generalize to complete partial metric spaces. Some more general results are also obtained in partial metric spaces.2000 Mathematics Subject Classification 47H10,54H25

Ordered Non-Archimedean Fuzzy Metric Spaces and Some Fixed Point Results

AbstractIn the present paper we provide two different kinds of fixed point theorems on ordered nonArchimedean fuzzy metric spaces. First, two fixed point theorems are proved for fuzzy order "Equation missing"-contractive type mappings. Then a common fixed point theorem is given for noncontractive type mappings. Kirk's problem on an extension of Caristi's theorem is also discussed.

Quasicone Metric Spaces and Generalizations of Caristi Kirk's Theorem

AbstractCone-valued lower semicontinuous maps are used to generalize Cristi-Kirik's fixed point theorem to Cone metric spaces. The cone under consideration is assumed to be strongly minihedral and normal. First we prove such a type of fixed point theorem in compact cone metric spaces and then generalize to complete cone metric spaces. Some more general results are also obtained in quasicone metric spaces.

EXISTENCE OF VECTOR EQUILIBRIA VIA EKELAND’S VARIATIONAL PRINCIPLE

In this paper, we prove Ekeland’s type of variational principle for a vector equilibrium problem, and present a Caristi-Kirk type fixed point theorem and an existence result for vector equilibrium solution.

Converses to fixed point theorems of Zermelo and Caristi

Let X be an abstract nonempty set and T be a self-map of X. Let Per T and Fix T denote the sets of all periodic points and all fixed points of T, respectively. Our main theorem says that if PerT= Fix T≠∅ , then there exists a partial ordering ≼ such that every chain in ( X,≼) has a supremum and for all x∈ X, x≼ Tx. This result is a converse to Zermelo's fixed point theorem. We also show that, from a purely set-theoretical point of view, fixed point theorems of Zermelo and Caristi are equivalent. Finally, we discuss relations between Caristi's theorem and its restriction to mappings satisfying Caristi's condition with a continuous real function ϕ.

NOTE ON THE RESULTS WITH LOWER SEMI-CONTINUITY

In this paper, we introduce the concept of lower semi-continuous from above functions and show that many well-known results, such as Ekland's and Caristi's theorems, remain also true under lower semi-continuous from above functions.

A minimum condition and some realted fixed-point theorems

AbstractThe classic Banach Contraction Principle assumes that the self-map is a contraction. Rather than requiring that a single operator be a contraction, we weaken this hypothesis by considering a minimum involving a set of iterates of that operator. This idea is a central motif for many of the results of this paper, in which we also study how this weakended hypothesis may be applied in Caristi's theorem, and how combinatorial arguments may be used in proving fixed-point theorems.

Caristi's Fixed Point Theorem and Selections of Set-Valued Contractions

We show (without the axiom of choice) that the Zermelo theorem implies directly a restriction of the Caristi fixed point theorem to continuous functions. Under the axiom of choice, this restriction is proved to be equivalent to Caristi's theorem. We also discuss Kirk's problem on an extension of the Caristi theorem and we establish two selection theorems for set-valued contractions.

Completeness and the Contraction Principle

We prove (something more general than) the result that a subset of a Banach space is closed if and only if every contraction of the space leaving the set invariant has a fixed point in that subset. This implies that a normed space is complete if and only if every contraction on the space has a fixed point. We also show that these results fail if convex is replaced by or starshaped. In the last few years a metric space variational principle due to Ekeland [3] and an equivalent fixed point theorem due to Caristi have been profitably exploited in a variety of contexts ([2,4, 12], and the references therein). It is easy to deduce the contraction principle [1] from either of these results [2, 12]; and it is equally apparent from the proof of Ekeland's principle given in [3] that the later result is of contractive type. This leads naturally to the question as to whether Ekeland's principle may be derived directly from the contraction principle? Moreover, in [5] Kirk showed that Caristi's theorem holds only in a complete metric setting and Sullivan [11] showed the same for Ekeland's theorem. Thus we are led to ask under what circumstances a metric space with the contraction propertv (every metric contraction of the space has a fixed point) is complete? Janos [8] calls such spaces pseudocomplete. Many other authors ([5,8,9, 10] included) have considered the contraction property in some form or other. Our positive result requires the following definition. Let us say that a subset C of metric space X endowed with a metric d is uniformlv Lipschitz-connected if there exists a positive constant L such that given any xo and xl in C there exists an arc g: [0, 1 -C with g(0) = xo and g(l) = xt such that ( 1 ) dd(g(s), g(t)) ? L I s-t I d(g(0). g(l)) for 0 s, t 1. The prototype of a uniformly Lipschitz-connected set is a subset of a normed space. In this case g(s) := sxl + (1 s)xo satisfies (1) with L: = 1. This is also true in certain translation invariant linear metric spaces. It will be apparent that there are many other types of uniformly Lipschitz-connected sets. These include continuously differentiable finite dimensional surfaces and bi-Lipschitz images of uniformly Lipschitz-connected sets. Received by the editors January 20, 1982 and, in revised form, April 9. 1982. 1980 Mathemnatics Subject Classification. Primary 47H 10, 54H25.