Caristi-Kirk Fixed Point Theorem Research Articles

Ekeland variational principle and its equivalents in T 1-quasi-uniform spaces

The present paper is concerned with the Ekeland Variational Principle (EkVP) and its equivalents (Caristi–Kirk fixed point theorem, Takahashi minimization principle, Oettli-Théra equilibrium version of EkVP) in quasi-uniform spaces. These extend some results proved by Hamel and Löhne [A minimal point theorem in uniform spaces. In: Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday. Vols 1, 2. Dordrecht: Kluwer Academic Publishers; 2003. p. 577–593] and Hamel [Equivalents to Ekeland's variational principle in uniform spaces. Nonlinear Anal. 2005;62:913–924] in uniform spaces, as well as those proved in quasi-metric spaces by various authors. The case of F-quasi-gauge spaces, a non-symmetric version of F-gauge spaces introduced by Fang [The variational principle and fixed point theorems in certain topological spaces. J Math Anal Appl. 1996;202:398–412], is also considered. The paper ends with the quasi-uniform versions of some minimization principles proved by Arutyunov and Gel'man [The minimum of a functional in a metric space, and fixed points. Zh Vychisl Mat Mat Fiz. 2009;49:1167–1174] and Arutyunov [Caristi's condition and existence of a minimum of a lower bounded function in a metric space. Applications to the theory of coincidence points. Proc Steklov Inst Math. 2015;291(1):24–37] in complete metric spaces.

Caristi’s Fixed Point Theorem in Cone Metric Space

In this paper, we provide a short, comprehensive, and brief proof for Caristi-Kirk fixed point result for single and set-valued mappings in cone metric spaces. In addition, we partially addressed an open problem in which Caristi-Kirk fixed point result in cone metric spaces reduces to a classical result in metric spaces and provided a brief justification for a partial positive answer to this open problem using Caristi-Kirk fixed point theorem on uniform space. The proofs given to Caristi-Kirk’s result vary and use different techniques.

Ekeland variational principle in complete weakly symmetric (1, q 2)-quasimetric spaces and applications

In this paper, Ekeland variational principle for bifunctions is given in complete weakly symmetric -quasimetric spaces and it is shown to be equivalent to Oettli–Théra theorem, Caristi–Kirk fixed point theorem and Takahashi's nonconvex minimization principle. Moreover, Ekeland variational principle for the system of bifunctions is obtained. Finally, by applying obtained results, the existence of solutions of quasi-equilibrium problems (QEP) and uncountable system of QEP is provided under the compactness condition and the generalized Caristi-like condition, respectively.

Vectorial Ekeland variational principle for cyclically antimonotone vector equilibrium problems

ABSTRACTIn this paper, we first obtain some results for the vectorial Ekeland variational principle with a W-distance, in which the final space is a real linear space not necessarily endowed with a topology. We introduce a cyclically antimonotone bifunction taking values in a real linear space, show its properties, and then apply the results to deduce a general vector equilibrium version of Ekeland variational principle. Moreover, we apply the properties of the cyclically antimonotone bifunction to study the existence of solutions for vector equilibrium problems defined on compact or noncompact sets. We also prove the existence of solutions for uncountable systems of vector equilibrium problems. Finally, our main results are applied to establish two new general versions of Caristi-Kirk fixed point theorem for set-valued mappings and some new existence theorems for vector variational inequalities under mild conditions.

Caristi\u2013Kirk and Oettli\u2013Th\xe9ra ball spaces and applications

Based on the theory of ball spaces introduced by Kuhlmann and Kuhlmann, we introduce and study Caristi–Kirk and Oettli–Théra ball spaces. We show that if the underlying metric space is complete, then these have a very strong property: every ball contains a singleton ball. This fact provides quick proofs for several results which are equivalent to the Caristi–Kirk fixed point theorem, namely Ekeland’s variational principles, the Oettli–Théra theorem, Takahashi’s theorem and the flower petal 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.

Ekeland’s Variational Principle and Minimization Takahashi’s Theorem in Generalized Metric Spaces

We consider a distance function on generalized metric spaces and we get a generalization of Ekeland Variational Principle (EVP). Next, we prove that EVP is equivalent to Caristi–Kirk fixed point theorem and minimization Takahashi’s theorem.

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 surjectivity theorems with applications

In this paper a new class of mappings, known as locally $\lambda $-strongly $\phi $-accretive mappings, where $\lambda $ and $\phi $ have special meanings, is introduced. This class of mappings constitutes a generalization of the well-known monotone mappings, accretive mappings and strongly $\phi $-accretive mappings. Subsequently, the above notion is used to extend the results of Park and Park, Browder and Ray to locally $\lambda $-strongly $\phi $-accretive mappings by using Caristi-Kirk fixed point theorem. In the sequel, we introduce the notion of generalized directional contractor and prove a surjectivity theorem which is used to solve certain functional equations in Banach spaces.

Vectorial form of Ekeland-type variational principle

In this paper, we deal with a vectorial form of Ekeland-type variational principle for multivalued bioperator whose domain is a complete metric space and its range is a subset of a locally convex Hausdorff topological space. From this theorem, Caristi-Kirk fixed point theorem for multivalued maps is established in a more general setting and our techniques allow us to improve and to extend their results in (Ansari in J. Math. Anal. Appl. 335: 561-575, 2007; Bednarczuk and Zagrodny in Arch. Math. 93: 577-586, 2009; Bianchi, Kassay and Pini in J. Math. Anal. Appl. 305: 502-512, 2005).MSC:49K30, 90C29.

Vectorial Form of Ekeland-Type Variational Principle in Locally Convex Spaces and Its Applications

AbstractBy using a Dane"Equation missing"' drop theorem in locally convex spaces we obtain a vectorial form of Ekeland-type variational principle in locally convex spaces. From this theorem, we derive some versions of vectorial Caristi-Kirk's fixed-point theorem, Takahashi's nonconvex minimization theorem, and Oettli-Théra's theorem. Furthermore, we show that these results are equivalent to each other. Also, the existence of solution of vector equilibrium problem is given.

Equivalent Extensions to Caristi-Kirk's Fixed Point Theorem, Ekeland's Variational Principle, and Takahashi's Minimization Theorem

With a recent result of Suzuki (2001) we extend Caristi-Kirk's fixed point theorem, Ekeland's variational principle, and Takahashi's minimization theorem in a complete metric space by replacing the distance with a -distance. In addition, these extensions are shown to be equivalent. When the -distance is l.s.c. in its second variable, they are applicable to establish more equivalent results about the generalized weak sharp minima and error bounds, which are in turn useful for extending some existing results such as the petal theorem.

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.

Vectorial form of Ekeland-type variational principle with applications to vector equilibrium problems and fixed point theory

This paper introduces a vectorial form of equilibrium version of Ekeland-type variational principle. Some equivalent results to our variational principle are given. As applications, we derive the existence of solutions of a vector equilibrium problem in the setting of complete quasi-metric spaces with a W-distance. Caristi–Kirk fixed point theorem for multivalued maps is also established in a more general setting.

Equivalent formulations of Ekeland's variational principle

We prove that for some 0< α and 0< ε⩽+∞ a proper lower semicontinuous and bounded below function f on a metric space ( X, d) satisfies that for each x∈ X with inf X f<f(x)< inf X f+ε there exists y∈ X such that 0< αd( x, y)⩽ f( x)− f( y) iff for each such x this inequality holds for some minimizer z of f. Similar conditions are shown to be sufficient for f to possess minimizers, weak sharp minima and error bounds. A fixed point theorem is also established. Moreover, these results all turn out to be equivalent to the Ekeland variational principle, the Caristi–Kirk fixed point theorem and the Takahashi theorem.

Some existence results of efficiency in vector optimization

In this paper, we present several existence results for efficient solutions and efficient points in vector optimization problems. Firstly, we apply a corollary of a recently obtained Caristi-Kirk fixed point theorem ([3]) to obtain existence results for efficient solutions of a vector optimization problem, which generalize the existence theorems of efficient solutions in [2] (Theorem 9 and its Corollary). Secondly, we generalize Theorem 10 in [2] to the vector case, obtaining an existence result for efficient points of a vector optimization problem. As a result, an open problem following the Corollary of Theorem 10 in [2] is solved in some way. Finally, the concept of nuclear cones introduced in [5] is extended, somehow answering another open question in [2] (in the Remark following the Corollary of Theorem 9). Applying this concept of generalized nuclear cones, we derive another existence theorem of efficient points.

Surjectivity of 𝜑-accretive operators

Let X X and Y Y be Banach spaces, ϕ : X → Y ∗ \phi :X \to {Y^ * } and P : X → Y : P:X \to Y: ; P P is said to be strongly ϕ \phi -accretive if ⟨ P x − P y , ϕ ( x − y ) ⟩ ⩾ c | | x − y | | 2 \langle Px - Py,\;\phi \left ( {x - y} \right )\rangle \geqslant c{|| {x - y} ||^2} for some c &gt; 0 c &gt; 0 and each x x , y ∈ X y \in X . These maps constitute a generalization simultaneously of monotone maps (when Y = X ∗ Y = {X^ * } ) and accretive maps (when Y = X Y = X ). By applying the Caristi-Kirk fixed point theorem, W. O. Ray showed that a localized class of these maps must be surjective under appropriate geometric assumptions on Y ∗ {Y^ * } and continuity assumptions on the duality map. In this paper we show that such geometric assumptions can be removed without affecting the conclusion of Ray.