On the existence of zeros of monotone operators in reflexive Banach spaces

This paper is devoted to the investigation on the existence of zeros of monotone operators in reflexive Banach spaces. We first present a sufficient condition under which single-valued monotone operators have zeros. The obtained theorem includes a previous result as a special case. A necessary and sufficient condition for the existence of zeros of maximal monotone operators is presented.     

Strong convergence studied by a hybrid type method for monotone operators in a Banach space

In this paper, we study a strong convergence for monotone operators. We first introduce the hybrid type algorithm for monotone operators. Next, we obtain a strong convergence theorem (Theorem 3.3) for finding a zero point of an inverse-strongly monotone operator in a Banach space. Finally, we apply our convergence theorem to the problem of finding a minimizer of a convex function.     

Maximality of sums of two maximal monotone operators in general Banach space

We combine methods from convex analysis, based on a function of Simon Fitzpatrick, with a fine recent idea due to Voisei, to prove maximality of the sum of two maximal monotone operators in Banach space under various natural domain and transversality conditions.

     

On the connection between the existence of zeros and the asymptotic behavior of resolvents of maximal monotone operators in reflexive Banach spaces

A more systematic approach is introduced in the theory of zeros of maximal monotone operators , where is a real Banach space. A basic pair of necessary and sufficient boundary conditions is given for the existence of a zero of such an operator . These conditions are then shown to be equivalent to a certain asymptotic behavior of the resolvents or the Yosida resolvents of . Furthermore, several interesting corollaries are given, and the extendability of the necessary and sufficient conditions to the existence of zeros of locally defined, demicontinuous, monotone mappings is demonstrated. A result of Guan, about a pathwise connected set lying in the range of a monotone operator, is improved by including non-convex domains. A partial answer to Nirenberg's problem is also given. Namely, it is shown that a continuous, expansive mapping on a real Hilbert space is surjective if there exists a constant such that The methods for these results do not involve explicit use of any degree theory.

     

On iterations methods for zeros of accretive operators in Banach spaces

In this paper, three iterations are designed to approach zeros of set-valued accretive operators in Banach spaces. The first one is the continuous Picard type iteration involving the resolvent, the second one is the approximate Picard type iteration involving the resolvent and the third one is the Halpern type iteration involving the resolvent. Some strong convergence theorems for three iterations are proved.     

On the convergence of sequence of maximal monotone operators of type (D) in Banach spaces

Positivity - Within the setting of general real Banach spaces, we prove that the sequence of maximal monotone operators of type (D) graphically converges provided, their corresponding class of...     

On the range of the sum of monotone operators in general Banach spaces

The purpose of this paper is to generalize the Brézis-Haraux theorem on the range of the sum of monotone operators from a Hilbert space to general Banach spaces. The result obtained provides that the range is topologically almost equal to the sum where is a compatible topology in as proposed by Gossez. To illustrate the main result we consider some basic properties of densely maximal monotone operators.

     

On polynomially bounded operators acting on a Banach space

By the Von Neumann inequality every contraction on a Hilbert space is polynomially bounded. A simple example shows that this result does not extend to Banach space contractions. In this paper, we give general conditions under which an arbitrary Banach space contraction is polynomially bounded. These conditions concern the thinness of the spectrum and the behaviour of the resolvent or the sequence of negative powers. To do this we use techniques from harmonic analysis, in particular, results concerning thin sets such as Helson sets, Kronecker sets and sets that satisfy spectral synthesis.     

On the solution of nonlinear equations with monotone opera tors in a Banach space

Siberian Mathematical Journal -     

On some functional calculus of closed operators in a Banach space

We develop a functional calculus of closed operators in a Banach space based on the class of functions in the form 1/g, where g belongs to the class R[a, b] introduced by M. G. Krein. We prove continuity, stability, uniqueness, spectral mapping, and inverse operator theorems and describe some other properties of the considered calculus.     

Approximating zeros of monotone operators by proximal point algorithms

In this paper, we introduce two kinds of iterative algorithms for the problem of finding zeros of maximal monotone operators. Weak and strong convergence theorems are established in a real Hilbert space. As applications, we consider a problem of finding a minimizer of a convex function.     

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     

On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces

Let be a real reflexive Banach space with dual and open and bounded and such that  Let be maximal monotone with and and with and A general and more unified eigenvalue theory is developed for the pair of operators  Further conditions are given for the existence of a pair such that

The ``implicit" eigenvalue problem, with in place of is also considered.  The existence of continuous branches of eigenvectors of infinite length is investigated, and a Fredholm alternative in the spirit of Necas is given for a pair of homogeneous operators No compactness assumptions have been made in most of the results.  The degree theories of Browder and Skrypnik are used, as well as the degree theories of the authors involving densely defined perturbations of maximal monotone operators.  Applications to nonlinear partial differential equations are included.