On asymptotic pointwise contractions in modular function spaces

We discuss the existence of fixed points of asymptotic pointwise contractions in modular function spaces.     

A fixed point theorem for pointwise eventually nonexpansive mappings in nearly uniformly convex Banach spaces

The purpose of this paper is to prove the existence of a fixed point for a pointwise eventually nonexpansive mapping in a nearly uniformly convex Banach space. This provides an affirmative answer to a question given by Kirk and Xu [W.A. Kirk, Hong-Kun Xu, Asymptotic pointwise contraction, Nonlinear Anal. 69 (2008), 4706-4712].     

Einige Existenz- und Einschließungssätze für Fixpunkte expandierender Integraloperatoren

Some existence theorems for fixed points of nonlinear integral operators expanding a cone in a Banach space in the sense of Krasnoselskii [5] are given. Using special cones we find pointwise inclusions for the fixed points. The question of finding the best bounds for the fixed points leads to a nonlinear optimization problem with infinitely many restrictions.

     

Best proximity points for asymptotic cyclic contraction mappings

In this paper, we prove the existence and convergence of best proximity points for asymptotic cyclic contractions in metric spaces with the property UC, as well as for asymptotic proximal pointwise contractions in uniformly convex Banach spaces. Moreover, we consider a generalized cyclic contraction mapping and prove the existence of best proximity points for such a mapping in Banach spaces which do not necessarily satisfy the geometric property UC.     

Asymptotic pointwise contractions

A simple and elementary (without using the ultrapower technique) proof for the existence of a fixed point of an asymptotic pointwise contraction is provided. The existence of fixed points for asymptotic pointwise nonexpansive mappings in a uniformly convex Banach space is also investigated.     

On global existence and attractivity results for nonlinear functional integral equations

In this paper two existence results concerning the global attractivity and global asymptotic attractivity for a certain functional nonlinear integral equation are proved. Our existence results include several existence as well as attractivity results obtained earlier by Banas and Dhage (2008) [1], Hu and Yan (2006) [3], Dhage (2009) [15] and Banas and Rzepka (2003) [7] as special cases under some weaker Lipschitz conditions. A measure theoretic fixed point theorem of Dhage (2008) [6] is used in formulating our main results and the characterizations of solutions are obtained in the space of functions defined, continuous and bounded on unbounded intervals.     

Existence theorems for monotone operators in reflexive Banach spaces

In this paper, we obtain an existence theorem for single-valued monotone operators in a reflexive Banach space. Using this result, we prove a fixed point theorem for nonexpansive mappings in a Hilbert space and an existence theorem for maximal monotone operators in a Banach space. Received: 3 July 2006 Revised: 15 January 2007     

Fixed point results for a generalized nonexpansive map in uniformly convex metric spaces

In this paper, the existence of a unique fixed point of a map satisfying a very general contractive condition on a suitable subset of a uniformly convex metric space is proved. This fixed point is approximated by averaging Krasnosel’skii iterations of a generalized nonexpansive map. Our results substantially improve and extend several known results existing in the literature.     

Fixed point theorems for generalized set-contraction maps and their applications

A new generalized set-valued contraction on topological spaces with respect to a measure of noncompactness is introduced. Two fixed point theorems for the KKM type maps which are either generalized set-contraction or condensing ones are given. Furthermore, applications of these results for existence of coincidence points and maximal elements are deduced.     

Proximal pointwise contraction

In this paper, we introduce the notion of proximal pointwise contraction and obtain the existence of a best proximity point on a pair of weakly compact convex subset of a Banach space and generalize a result of [W.A. Kirk, Mappings of generalized contractive type, J. Math. Anal. Appl. 32 (1970) 567-570; W.A. Kirk, H.K. Xu, Asymptotic pointwise contractions, Nonlinear Anal. 69 (2008) 4706-4712].     

On asymptotic pointwise nonexpansive mappings in modular function spaces

We prove the existence of fixed points of asymptotic pointwise nonexpansive mappings in modular function spaces.     

Integrable solutions of a mixed type operator equation

In this paper we prove the existence of integrable solutions for a generalized mixed type operator equation, which contains many key integral and functional equations appearing frequently in Mathematical literature. Our main tool is a Krasnosel’skii type fixed point theorem recently proved by Latrach and Taoudi, the first author. An existence theory for a class of nonlinear transport equations is also developed.     

KKM mappings in metric type spaces

In this work we discuss some recent results about KKM mappings in cone metric spaces. We also discuss the fixed point existence results of multivalued mappings defined on such metric spaces. In particular we show that most of the new results are merely copies of the classical ones and do not necessitate the underlying Banach space nor the associated cone.     

A variational inequality theory for demicontinuous S-contractive maps with applications to semilinear elliptic inequalities

A variational inequality theory for demicontinuous S-contractive maps in Hilbert spaces is established by employing the ideas of Granas' topological transversality. Such a variational inequality theory has many properties similar to those of fixed point theory for demicontinuous weakly inward S-contractive maps and to those of fixed point index for condensing maps. The variational inequality theory will be applied to study the existence of positive weak solutions and eigenvalue problems for semilinear second-order elliptic inequalities with nonlinearities which satisfy suitable lower bound conditions involving the critical Sobolev exponent. There has been little discussion for such elliptic inequalities involving the critical Sobolev exponent in the literature.     

Characterizations of reproducing cones and uniqueness of fixed points

The purpose of this paper is to discuss the existence and uniqueness of fixed point in a partially ordered Banach space. Based on the characterizations of reproducing cones, some fixed point theorems for nonlinear operators are proved. As an application, the existence and uniqueness of periodical solution for a first order differential equation is discussed.     

Local asymptotic attractivity for nonlinear quadratic functional integral equations

In this paper, an existence result for local asymptotic attractivity of the solutions is proved for a nonlinear quadratic functional integral equation under certain growth conditions which in turn gives the existence as well as asymptotic stability of solutions. A couple of examples are provided for indicating the natural realizations of abstract theory presented in the paper.     

Endpoints of set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and strict contractions in uniform spaces

In this paper, we introduce the concepts of the set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and set-valued dynamical systems of strict contractions in uniform spaces and we present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions. The result, concerning the investigations of problems of the set-valued asymptotic fixed point theory, include some well-known results of Meir and Keeler, Kirk and Suzuki concerning the asymptotic fixed point theory of single-valued maps in metric spaces. The result, concerning set-valued strict contractions (in which the contractive coefficient is not constant), is different from the result of Yuan concerning the existence of endpoints of Tarafdar–Vyborny generalized contractions (in which the contractive coefficient is constant) in bounded metric spaces and provides some examples of Tarafdar–Yuan topological contractions in compact uniform spaces. Definitions and results presented here are new for set-valued dynamical systems in uniform, locally convex and metric spaces and even for single-valued maps. Examples show a fundamental difference between our results and the well-known ones.     

Existence and uniqueness of fixed points for mixed monotone operators with applications

In this paper, we study the uniqueness and existence of fixed points of mixed monotone operators in the partially ordered Banach space. Our conclusions essentially improve the relevant results obtained by Liang and others. Moreover, as an application of our results, we prove the existence and uniqueness of a positive solution for a class of integral equations which cannot be solved by using previously available methods.     

Common fixed points for Banach operator pairs with applications

The existence of common fixed point results for a Banach operator pair under certain generalized contractions is established. The invariant best approximation results are proved as applications and the existence of solutions of variational inequalities is obtained. We also study the solution of functional equations arising from dynamic programming.     

A fixed point theorem for weakly Zamfirescu mappings

In [5], Zamfirescu (1972) gave a fixed point theorem that generalizes the classical fixed point theorems by Banach, Kannan, and Chatterjea. In this paper, we follow the ideas of Dugundji and Granas to extend Zamfirescu’s fixed point theorem to the class of weakly Zamfirescu maps. A continuation method for this class of maps is also given.