Kirk Fixed Point Theorem Research Articles

A generalization of the Goebel–Kirk fixed point theorem for asymptotically nonexpansive mappings

Some results extending the Goebel–Kirk fixed point theorem for asymptotically nonexpansive mappings are presented.

Existence of solutions for Caputo fractional delay differential equations with nonlocal and integral boundary conditions

In this paper, the existence and uniqueness of the solutions of Caputo fractional delay differential equations under nonlocal and integral boundary value conditions are studied. By using the Banach contraction principle and the Burton and Kirk fixed-point theorem, some new conclusions about the existence and uniqueness of solutions are obtained. An example is given to illustrate the main results.

A refinement of the Browder–Göhde–Kirk fixed point theorem and some applications

The following generalization of the Browder–Göhde–Kirk fixed point theorem is proved: ifCis a nonempty bounded closed and convex subset of a uniformly convex normed spaceXandTis a self-mapping ofCsuch thatleft| Tx-Tyright| le beta left( left| x-yright| right) for all x,yin C,xne y,where a functionbeta :left( 0,infty right) rightarrow left[ 0,infty right) is such that lim _{trightarrow 0+}frac{beta left( tright) }{t}=1,thenThas a fixed point. Two modifications of this theorem as well as some accompanying results on Lipschitz-type mappings are given. An application in the theory of L^{p}-solutions of an iterative functional equation, and some refinements of the Radamacher theorem are proposed.

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.

Fixed Point Theory for Study the Controllability of Boundary Control Problems in Reflexive Banach Spaces

In this paper, we extend the work of our proplem in uniformly convex Banach spaces using Kirk fixed point theorem. Thus the existence and sufficient conditions for the controllability to general formulation of nonlinear boundary control problems in reflexive Banach spaces are introduced. The results are obtained by using fixed point theorem that deals with nonexpanisive mapping defined on a set has normal structure and strongly continuous semigroup theory. An application is given to illustrate the importance of the results.

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.

Two fixed point theorems in complete random normed modules and their applications to backward stochastic equations

Let (Ω,F,F,P) be a filtered probability space with a filtration F=(Ft)t∈[0,T] satisfying the usual conditions and T a finite time, L0(F0) the algebra of equivalence classes of F0–measurable real–valued random variables on Ω, Lp(FT,Rd) the usual function space and LF0p(FT,Rd) the L0(F0)–module generated by Lp(FT,Rd). The usual backward stochastic equations (BSEs) are studied for their terminal conditions ξ in Lp(FT,Rd). Motivated by the study of continuous–time conditional mean–conditional convex risk measure portfolio selection, this paper, for the first time, formulates and studies a more general class of BSEs with their terminal conditions in LF0p(FT,Rd). The main results of this paper are to prove two fixed point theorems in complete random normed modules, which are respectively the random generalizations of Banach contraction mapping principle and Browder–Kirk fixed point theorem, and give their applications to the general class of BSEs.

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.

A geometric property in and its applications

AbstractIn this work, we initiate the study of the geometry of the variable exponent sequence space when . In 1931 Orlicz introduced the variable exponent sequence spaces while studying lacunary Fourier series. Since then, much progress has been made in the understanding of these spaces and of their continuous counterpart. In particular, it is well known that is uniformly convex if and only if the exponent is bounded away from 1 and infinity. The geometry of when either or remains largely ill‐understood. We state and prove a modular version of the geometric property of when , known as uniform convexity in every direction. We present specific applications to fixed point theory. In particular we obtain an analogue to the classical Kirk's fixed point theorem in when .

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.

$G$-asymptotic contractions in metric spaces with a graph and fixed point results

In this paper, we discuss the existence and uniqueness of fixed points for $G$-asymptotic contractions in metric spaces endowed with a graph. The result given here is a new version of Kirk's fixed point theorem for asymptotic contractions in metric spaces endowed with a graph. The given result here is a generalization of fixed point theorem for asymptotic contraction from metric s paces to metric spaces endowed with a graph.

Goebel and Kirk fixed point theorem for multivalued asymptotically nonexpansive mappings

We introduce the concept of a multivalued asymptotically nonexpansive mapping and establish Goebel and Kirk fixed point theorem for these mappings in uniformly hyperbolic metric spaces. We also define a modified Mann iteration process for this class of mappings and obtain an extension of some well-known results for singlevalued mappings defined on linear as well as nonlinear domains.

A new approach to the Assad–Kirk fixed point theorem

In the present paper, considering the Wardowski’s technique, we give a new approach to the Assad–Kirk fixed point theorem on metrically convex metric spaces.We also provide a nontrivial example showing that our result is a proper extension of the Assad–Kirk fixed point theorem.

Normal Structure and Common Fixed Point Properties for Semigroups of Nonexpansive Mappings in Banach Spaces

AbstractIn 1965, Kirk proved that if "Equation missing" is a nonempty weakly compact convex subset of a Banach space with normal structure, then every nonexpansive mapping "Equation missing" has a fixed point. The purpose of this paper is to outline various generalizations of Kirk's fixed point theorem to semigroup of nonexpansive mappings and for Banach spaces associated to a locally compact group.

Bead spaces and fixed point theorems

The notion of a bead metric space defined here (see Definition 6) is a nice generalization of that of the uniformly convex normed space. In turn, the idea of a central point for a mapping when combined with the “single central point” property of the bead spaces enables us to obtain strong and elegant extensions of the Browder–Göhde–Kirk fixed point theorem for nonexpansive mappings (see Theorems 14–17). Their proofs are based on a very simple reasoning. We also prove two theorems on continuous selections for metric and Hilbert spaces. They are followed by fixed point theorems of Schauder type. In the final part we obtain a result on nonempty intersection.

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.

A rate of convergence for asymptotic contractions

In [P. Gerhardy, A quantitative version of Kirk's fixed point theorem for asymptotic contractions, J. Math. Anal. Appl. 316 (2006) 339–345], P. Gerhardy gives a quantitative version of Kirk's fixed point theorem for asymptotic contractions. This involves modifying the definition of an asymptotic contraction, subsuming the old definition under the new one, and giving a bound, expressed in the relevant moduli and a bound on the Picard iteration sequence, on how far one must go in the iteration sequence to at least once get close to the fixed point. However, since the convergence to the fixed point needs not be monotone, this theorem does not provide a full rate of convergence. We here give an explicit rate of convergence for the iteration sequence, expressed in the relevant moduli and a bound on the sequence. We furthermore give a characterization of asymptotic contractions on bounded, complete metric spaces, showing that they are exactly the mappings for which every Picard iteration sequence converges to the same point with a rate of convergence which is uniform in the starting point.

A quantitative version of Kirk's fixed point theorem for asymptotic contractions

In [W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645–650], W.A. Kirk introduced the notion of asymptotic contractions and proved a fixed point theorem for such mappings. Using techniques from proof mining, we develop a variant of the notion of asymptotic contractions and prove a quantitative version of the corresponding fixed point theorem.

Applications of W. A. Kirk's fixed-point theorem to generalized nonlinear variational-like inequalities in reflexive Banach spaces

We introduce and study a new class of generalized nonlinear variational-like inequalities, which includes these variational inequalities and variational-like inequalities due to Bose, Cubiotti, Dien, Ding, Ding and Tarafdar, Noor, Parida, Sahoo, and Kumar, and Yao, and others as special cases. By applying Kirk's fixed-point theorem and Ding-Tan minimax inequality, we establish the existence theorems of solutions for the generalized nonlinear variational-like inequalities in reflexive Banach spaces.