Existence of fixed points for the sum of two operators

The purpose of this paper is to study the existence of fixed points for the sum of two nonlinear operators in the framework of real Banach spaces. Later on, we give some examples of applications of this type of results (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Isomorphically expansive mappings in $l_2$

Iterative approximation to a coincidence point of two mappings

In this article two methods for approximating the coincidence point of two mappings are studied and moreover, rates of convergence for both methods are given. These results are illustrated by several examples, in particular we apply such results to study the convergence and their rate of convergence of these methods to the solution of a nonlinear integral equation and of a nonlinear differential equation.

Fixed point theory for a class of generalized nonexpansive mappings

AbstractIn this paper we introduce two new classes of generalized nonexpansive mapping and we study both the existence of fixed points and their asymptotic behavior.

Wardowski conditions to the coincidence problem

In this article we first discuss the existence and uniqueness of a solution for the coincidence problem: Find p ∈ X such that Tp = Sp, where X is a nonempty set, Y is a complete metric space, and T, S:X → Y are two mappings satisfying a Wardowski type condition of contractivity. Later on, we will state the convergence of the Picard-Juncgk iteration process to the above coincidence problem as well as a rate of convergence for this iteration scheme. Finally, we shall apply our results to study the existence and uniqueness of a solution as well as the convergence of the Picard-Juncgk iteration process toward the solution of a second order differential equation. Ministerio de Economía y Competi…

Existence theorems for m-accretive operators in Banach spaces

Abstract In 1985, the second author proved a surjective result for m -accretive and ϕ -expansive mappings for uniformly smooth Banach spaces. However, in this case, we have been able to remove the uniform smoothness of the Banach space, without any additional assumption.

Fixed point theory for multivalued generalized nonexpansive mappings

A very general class of multivalued generalized nonexpansive mappings is defined. We also give some fixed point results for these mappings, and finally we compare and separate this class from the other multivalued generalized nonexpansive mappings introduced in the recent literature.

Existence of fixed points and measures of weak noncompactness

Abstract The purpose of this paper is to study the existence of fixed points by using measures of weak noncompactness. Later on, we provide an existence principle for solutions for a nonlinear integral equation.

Fixed Point Theorems with Applications to the Solvability of Operator Equations and Inclusions on Function Spaces

1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2Department of Mathematical Analysis, University of Valencia, Spain 3Centre Universitaire Polydisciplinaire, Kelaa des Sraghna, Morocco 4Universite Cadi Ayyad, Laboratoire de Mathematiques et de Dynamique de Populations, Marrakech, Morocco 5Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy

Fixed point theory for almost convex functions

Traditionally, metric fixed point theory has sought classes of spaces in which a given type of mapping (nonexpansive, assymptotically or generalized nonexpansive, uniformly Lipschitz, etc.) from a nonempty weakly compact convex set into itself always has a fixed point. In some situations the class of space is determined by the application while there is some degree of freedom in constructing the map to be used. With this in mind we seek to relax the conditions on the space by considering more restrictive types of mappings.

Schaefer–Krasnoselskii fixed point theorems using a usual measure of weak noncompactness

Abstract We present some extension of a well-known fixed point theorem due to Burton and Kirk [T.A. Burton, C. Kirk, A fixed point theorem of Krasnoselskii–Schaefer type, Math. Nachr. 189 (1998) 423–431] for the sum of two nonlinear operators one of them compact and the other one a strict contraction. The novelty of our results is that the involved operators need not to be weakly continuous. Finally, an example is given to illustrate our results.

The fixed point property for mappings admitting a center

Abstract We introduce a class of nonlinear continuous mappings in Banach spaces which allow us to characterize the Banach spaces without noncompact flat parts in their spheres as those that have the fixed point property for this type of mapping. Later on, we give an application to the existence of zeroes for certain kinds of accretive operators.

The asymptotic behavior of the solutions of the Cauchy problem generated by ϕ-accretive operators

Abstract The purpose of this paper is to study the asymptotic behavior of the solutions of certain type of differential inclusions posed in Banach spaces. In particular, we obtain the abstract result on the asymptotic behavior of the solution of the boundary value problem { u t − Δ p ( u ) + | u | γ − 1 u = f on  ] 0 , ∞ [ × Ω , − ∂ u ∂ η ∈ β ( u ) on  [ 0 , ∞ [ × ∂ Ω , u ( 0 , x ) = u 0 ( x ) in  Ω , where Ω is a bounded open domain in R n with smooth boundary ∂Ω, f ( t , x ) is a given L 1 -function on ] 0 , ∞ [ × Ω , γ ⩾ 1 and 1 ⩽ p ∞ . Δ p represents the p-Laplacian operator, ∂ ∂ η is the associated Neumann boundary operator and β a maximal monotone graph in R × R with 0 ∈ β ( 0 ) .

Domains of accretive operators in Banach spaces

LetD(A)be the domain of anm-accretive operatorAon a Banach spaceE. We provide sufficient conditions for the closure ofD(A)to be convex and forD(A)to coincide withEitself. Several related results and pertinent examples are also included.

Fixed Points for Pseudocontractive Mappings on Unbounded Domains

We give some fixed point results for pseudocontractive mappings on nonbounded domains which allow us to obtain generalizations of recent fixed point theorems of Penot, Isac, and Németh. An application to integral equations is given.

Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings

Abstract It is shown that if the modulus Γ X of nearly uniform smoothness of a reflexive Banach space satisfies Γ X ′ ( 0 ) 1 , then every bounded closed convex subset of X has the fixed point property for nonexpansive mappings. In particular, uniformly nonsquare Banach spaces have this property since they are properly included in this class of spaces. This answers a long-standing question in the theory.

Fixed point theory for 1-set contractive and pseudocontractive mappings

The purpose of this paper is to study the existence and uniqueness of fixed point for a class of nonlinear mappings defined on a real Banach space, which, among others, contains the class of separate contractive mappings, as well as to see that an important class of 1-set contractions and of pseudocontractions falls into this type of nonlinear mappings. As a particular case, we give an iterative method to approach the fixed point of a nonexpansive mapping. Later on, we establish some fixed point results of Krasnoselskii type for the sum of two nonlinear mappings where one of them is either a 1-set contraction or a pseudocontraction and the another one is completely continuous, which extend …

Existence and uniqueness of solution to several kinds of differential equations using the coincidence theory

The purpose of this article is to study the existence of a coincidence point for two mappings defined on a nonempty set and taking values on a Banach space using the fixed point theory for nonexpansive mappings. Moreover, this type of results will be applied to obtain the existence of solutions for some classes of ordinary differential equations. Ministerio de Economía y Competitividad Junta de Andalucía

Fixed point methods and accretivity for perturbed nonlinear equations in Banach spaces

Abstract In this paper we use fixed point theorems to guarantee the existence of solutions for inclusions of the form A u + λ u + F u ∋ g , where A is a quasi-m-accretive operator defined in a Banach space, λ > 0 , and the nonlinear perturbation F satisfies some suitable conditions. We apply the obtained results, among other things, to guarantee the existence of solutions of boundary value problems of the type − Δ ρ ( u ( x ) ) + λ u ( x ) + F u ( x ) = g ( x ) , x ∈ Ω , and ρ ( u ) = 0 on ∂Ω, where the Laplace operator Δ should be understood in the sense of distributions over Ω and to study the existence and uniqueness of solution for a nonlinear integro-differential equation posed in L 1 …

Fixed point property in Banach lattices with Banach-Saks property

Existence results and asymptotic behavior for nonlocal abstract Cauchy problems

AbstractThe purpose of this paper is to study the existence and asymptotic behavior of solutions for Cauchy problems with nonlocal initial datum generated by accretive operators in Banach spaces.

Existence and uniqueness results for a nonlinear evolution equation arising in growing cell populations

Abstract The present paper is concerned with a nonlinear initial–boundary value problem derived from a model introduced by Rotenberg (1983) describing the growth of a cell population. Each cell of this population is distinguished by two parameters: its degree of maturity μ and its maturation velocity v . At mitosis, the daughter cells and mother cells are related by a general reproduction rule. We prove existence and uniqueness results in the case where the total cross-section and the boundary conditions are depending on the total density of population. Local and nonlocal reproduction rules are discussed.

Banach spaces which are somewhat uniformly noncreasy

AbstractWe consider a family of spaces wider than r-UNC spaces and we give some fixed point results in the setting of these spaces.

Well-posedness of a nonlinear evolution equation arising in growing cell population

We prove that a nonlinear evolution equation which comes from a model of an age-structured cell population endowed with general reproduction laws is well-posed. Copyright © 2011 John Wiley & Sons, Ltd.

An existence and uniqueness principle for a nonlinear version of the Lebowitz-Rubinow model with infinite maximum cycle length

The present article deals with existence and uniqueness results for a nonlinear evolution initial-boundary value problem, which originates in an age-structured cell population model introduced by Lebowitz and Rubinow (1974) describing the growth of a cell population. Cells of this population are distinguished by age a and cycle length l. In our framework, daughter and mother cells are related by a general reproduction rule that covers all known biological ones. In this paper, the cycle length l is allowed to be infinite. This hypothesis introduces some mathematical difficulties. We consider both local and nonlocal boundary conditions.

INTEGRAL SOLUTIONS TO A CLASS OF NONLOCAL EVOLUTION EQUATIONS

We study the existence of integral solutions to a class of nonlinear evolution equations of the form [Formula: see text] where A : D(A) ⊆ X → 2X is an m-accretive operator on a Banach space X, and f : [0, T] × X → X and [Formula: see text] are given functions. We obtain sufficient conditions for this problem to have a unique integral solution.

Banach spaces which are r-uniformly noncreasy

Abstract We consider a family of spaces wider than UNC spaces introduced by Prus, and we give some fixed point results in the setting of these spaces.

Stability of the Fixed Point Property for Nonexpansive Mappings

In 1971 Zidler [Zi 71] showed that every separable Banach space (X, ‖·‖) admits an equivalent renorming, (X, ‖·‖0), which is uniformly convex in every direction (UCED), and consequently it has weak normal structure and so the weak fixed point property (WFPP) [D-J-S 71].