Claudio H. Morales

Convergence of paths for pseudo-contractive mappings in Banach spaces

Let X be a real Banach space, let K be a closed convex subset of X, and let T , from K into X, be a pseudo-contractive mapping (i.e. (λ − 1) ‖u−v‖ ≤ ‖(λI−T )(u)−(λI−T )(v)‖ for all u, v ∈ K and λ > 1). Suppose the space X has a uniformly Gâteaux differentiable norm, such that every closed bounded convex subset of K enjoys the Fixed Point Property for nonexpansive self-mappings. Then the path t → xt ∈ K, t ∈ [0, 1), defined by the equation xt = tTxt + (1− t)x0 is continuous and strongly converges to a fixed point of T as t→ 1−, provided that T satisfies the weakly inward condition.

The Mann process for perturbed m-accretive operators in Banach spaces

This work was carried out while this author was visiting the University ofAlabama in Huntsville underthe nancial support by the LG Yonam Foundation, 1998, and he would like to thank Professor Claudio H.Morales for his hospitality in the Department of Mathematics.0362-546X/01/

Convergence of the steepest descent method for accretive operators

-see front matter ? 2001 Elsevier Science Ltd. All rights reserved.PII: S0362-546X(00)00115-2

On linear ordinary differential equations with functionally commutative coefficient matrices

Let X be a uniformly smooth Banach space and let A : X → X be a bounded demicontinuous mapping, which is also α-strongly accretive on X. Let z ∈ X and let x0 be an arbitrary initial value in X. Then the approximating scheme xn+1 = xn − cn(Axn − z), n = 0, 1, 2, . . . , converges strongly to the unique solution of the equation Ax = z, provided that the sequence {cn} fulfills suitable conditions.

Strong convergence of path for continuous pseudo-contractive mappings

A system of linear ordinary differential equations x=A(t)x, t ∈ I, is called semiproper if A(t)A(τ)=A(τ)A(t), t,τ ∈ I [also known as functional commutativity of A(t)]. It is known that a semiproper system has a closed-form fundamental solution matrix XA(t)=exp∫tA(τ)dτ, where the matrix exponential is defined by the power series exp(·)=Σ∞k=1(·)kk!. Therefore the problem of solving a semiproper system amounts to that of finding a finite-form expression for the matrix exponential. Based on some recent results obtained by the authors for decomposing semiproper matrix functions, a systematic approach is developed for finding a finite-form analytical solution for the entire family of semiproper systems. This solution is then used to derive a number of important and practical stability criteria for semiproper systems. Applications of the new results are also discussed.

Fixed Point Theorems for Random Lowersemi-continuous Mappings

The purpose of this paper is to study the convergence of a path that begins at the unique fixed point of a strongly pseudo-contractive operator defined on a closed and convex subset of a reflexive Banach space and converges to a fixed point of a pseudo-contractive mapping. Primarily, it is proven that a convex combination of these two operators is indeed strongly pseudo-contractive under the weakly inward condition. This fact generalizes a result of Barbu for accretive operators.

Zeros for accretive operators satisfying certain boundary conditions

We prove a general principle in Random Fixed Point Theory by introducing a condition named () which was inspired by some of Petryshyns work, and then we apply our result to prove some random fixed points theorems, including generalizations of some Bharucha-Reid theorems.

A Bolzano's theorem in the new millennium

Abstract Let X be a Banach space with the dual space X ∗ to be uniformly convex, let D ⊂ X be open, and let T: D → X be strongly accretive (i.e., for some k λ − k )∥ u − v ∥ ⩽ ∥( λ − 1)( u − v )+ T ( u ) − T ( v )∥ for all u, v ϵ D and λ > k ). Suppose T is demicontinuous and strongly accretive and suppose there exists z ϵ D satisfying: T ( x ) t ( x − z ) for all x ϵ ∂D and t T has a unique zero in D . This result is then applied to the study of existence of zeros of accretive mappings under apparently different types of boundary conditions on T .

The Leray-Schauder condition for continuous pseudo-contractive mappings

1. IntroductionOurmainobjectiveinthispaperistoestablishacloseconnectionbetweenaclassicaltheorem from real analysis (discovered over two centuries ago) and recent works inMonotone Operator Theory for reexive Banach spaces. Nevertheless, throughout thisexposition,wealsogiveabriefdescriptionofhowtheoriginalproblemhasevolvedintime,passingthroughdierentcategoriesofgeneralizationforthelast30years.Whilewe are progressing toward this goal, we are able to obtain some new results such asTheorem4(seeSection3),wheresomeconvexityconditionisnolongerrequired.Weare also able to obtain a new Invariance of Domain Theorem for monotone operators,as can be seen in Theorem 5 below.The study of the existence of zeros for nonlinear functional equations involvingmonotoneoperatorshasbeenextensivelydiscussedtowardtheveryendofthelastmil-lennium. Concerning the study of the existence of zeros under the boundary condition(2) below, we nd among the contributions, the work of Vainberg and Kachurovskii[19],Minty[12,13],Browder[4–6]andShinbrot[17].Forrelatedwork,wealsomen-tion BrBezis et al. [3], Kachurovskii [9], Leray and Lions [11], and Rockafellar [16].However, our main interest here is to attempt to unify some of the work done in thecontourofthiscondition(2),thatwasperhapsrstobservedbythismathematicianofthe XIX century. In spite of the fact that most of the results presented in Section 2 ofthis paper are known, their proofs share some degree of originality.

Spatial decomposition of functionally commutative matrices

Over thirty years ago, Kirk raised the question of whether a nonexpansive mapping, defined on a convex domain with nonempty interior, has a fixed point under the Leray-Schauder condition, provided that its domain enjoys the Fixed Point Property with respect to nonexpansive self-mappings. In the present work we have found the answer to this question to be positive, even for a larger class of mappings. The result, indeed, represents a quite significant extension of a large number of theorems obtained in the last forty years. This also includes new theorems for nonexpansive mappings.