Maher Mnif

Some Fixed Point Theorems and Application to Biological Model

In this paper, we establish some results regarding the existence of solution on L 1 spaces to a nonlinear boundary value problem originally proposed by Rotenberg (J. Theo. Biol. 1983; 103:181–199) to model the growth of cell population. Our strategy consists in establishing new variants of fixed point theorems in general Banach spaces. These topological results can be used to resolve some open problems posed by Latrach and Jeribi (Nonlinear Anal. T.M.A. 1999; 36:843–862).

Perturbation theory of lower semi-Browder multivalued linear operators

In the present paper we introduce the notion of lower semi-Browder linear relation and we study the perturbation problem under compact operator perturbations.

Some new properties in Fredholm theory, Schechter essential spectrum, and application to transport theory.

The theory of measures of noncompactness has many applications on topology, functional analysis, and operator theory. In this paper, we consider one axiomatic approach to this notion which includes the most important classical definitions. We give some results concerning a certain class of semi-Fredholm and Fredholm operators via the concept of measures of noncompactness. Moreover, we establish a fine description of the Schechter essential spectrum of a closed densely defined operators. These results are exploited to investigate the Schechter essential spectrum of a multidimensional neutron transport operator.

SPECTRAL MAPPING THEOREM FOR ASCENT, ESSENTIAL ASCENT, DESCENT AND ESSENTIAL DESCENT SPECTRUM OF LINEAR RELATIONS ∗

Abstract In [7], Cross showed that the spectrum of a linear relation T on a normed space satisfies the spectral mapping theorem. In this paper, we extend the notion of essential ascent and descent for an operator acting on a vector space to linear relations acting on Banach spaces. We focus to define and study the descent, essential descent, ascent and essential ascent spectrum of a linear relation everywhere defined on a Banach space X . In particular, we show that the corresponding spectrum satisfy the polynomial version of the spectral mapping theorem.

Browder spectra of upper triangular matrix linear relations

In this paper, we define a matrix linear relation and present some properties of this one. When A ∈ BCR(H) and B ∈ BCR(K) are given, we denote by MC the matrix linear relation acting on the infinite dimensional separable Hilbert space H ⊕K, of the form MC = ( A C 0 B ) . It is shown that MC is Browder relation for some operator C ∈ B(K,H) if and only if A is upper semi Fredholm relation with finite ascent, B is lower semi Fredholm relation with finite descent and n(A) + n(B) = d(A) + d(B).

Coperturbation function and lower semi-Browder multivalued linear operators

In this article, we investigate the perturbation theory of lower semi-Browder and Browder linear relations. Our approach is based on the concept of a coperturbation function for linear relations in order to establish some perturbation theorems and deduce the stability under strictly cosingular operator perturbations. Furthermore, we apply the obtained results to study the invariance and the characterization of Browders essential defect spectrum and Browders essential spectrum.

BROWDER AND SEMI-BROWDER OPERATORS＊

Abstract In this article, we study characterization, stability, and spectral mapping theorem for Browders essential spectrum, Browders essential defect spectrum and Browders essential approximate point spectrum of closed densely defined linear operators on Banach spaces.

ESSENTIAL APPROXIMATE POINT SPECTRA FOR UPPER TRIANGULAR MATRIX OF LINEAR RELATIONS

Abstract When A ∈ ℒ ℛ ( H ) and B ∈ ℒ ℛ ( K ) are given, for B ∈ ℒ ℛ ( K ) we denote by MC the linear relation acting on the infinite dimensional separable Hilbert space H ⊕ K of the form M C = ( A C 0 B ) . In this paper, we give the necessary and sufficient conditions on A and B for which MC is upper semi-Fredholm with negative index or Weyl for some C ∈ ℒ ℛ ( K , H ) .

On graph measures in banach spaces and description of essential spectra of multidimensional transport equation

Abstract In this paper, we define new measures called respectively graph measure of noncompactness and graph measure of weak noncompactness. Moreover, we apply the obtained results to discuss the incidence of some perturbation results realized in [2] on the behavior of essential spectra of such closed densely defined linear operators on Banach spaces. These results are exploited to investigate the essential spectra of a multidimensional neutron transport operator on L 1 spaces.

Fixed-Point Theory on a Frechet Topological Vector Space

We establish some versions of fixed-point theorem in a Frechet topological vector space . The main result is that every map (where is a continuous map and is a continuous linear weakly compact operator) from a closed convex subset of a Frechet topological vector space having the Dunford-Pettis property into itself has fixed-point. Based on this result, we present two versions of the Krasnoselskii fixed-point theorem. Our first result extend the well-known Krasnoselskiis fixed-point theorem for U-contractions and weakly compact mappings, while the second one, by assuming that the family where and a compact is nonlinear equicontractive, we give a fixed-point theorem for the operator of the form .