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

It is shown that if the modulus X of nearly uniform smoothness of a reflexive Banach space satisfies 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.

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 problemwhere 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 < ∞. ∆ 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).  2005 Elsevier Inc. All rights reserved.

The 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.

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Key words Fixed points, sum of operators, dissipative operators, integral equations, Dirichlet problems MSC (2000) 47H10, 47H17, 47B44The 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.

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.