Bilel Krichen

Fixed point theorems for block operator matrix and an application to a structured problem under boundary conditions of Rotenberg’s model type

In this manuscript, we introduce and study the existence of solutions for a coupled system of differential equations under abstract boundary conditions of Rotenberg’s model type, this last arises in growing cell populations. The entries of block operator matrix associated to this system are nonlinear and act on the Banach space Xp:= Lp([0, 1] × [a, b]; dµ dv), where 0 ≤ a < b < ∞; 1 < p < ∞.

Existence of solutions of a two-dimensional boundary value problem for a system of nonlinear equations arising in growing cell populations

In the paper [A. Ben Amar, A. Jeribi, and B. Krichen, Fixed point theorems for block operator matrix and an application to a structured problem under boundary conditions of Rotenbergs model type, to appear in Math. Slovaca. (2014)], the existence of solutions of the two-dimensional boundary value problem (1) and (2) was discussed in the product Banach space Lp×Lp for p∈(1, ∞). Due to the lack of compactness on L1 spaces, the analysis did not cover the case p=1. The purpose of this work is to extend the results of Ben Amar et al. to the case p=1 by establishing new variants of fixed-point theorems for a 2×2 operator matrix, involving weakly compact operators.

Existence of Solutions of a Nonlinear Hammerstein Integral Equation

In this article, we are concerned with existence results for a nonlinear Hammerstein integral equation in L 1 spaces. Making use of the De Blasi measure of weak noncompactness, we establish variants of some Krasnoselskii type theorem involving the weak topology of Banach spaces.

Essential spectra of some matrix operators involving γ-relatively bounded entries and an application

In this paper, we introduce the concept of relative boundedness with respect to an axiomatic measure of noncompactness , as a generalization of the notion of relative boundedness. Then, some spectral properties of a unbounded block operator matrix involving -relatively bounded inputs are given. The obtained results are applied to investigate the essential spectra of a two-dimensional transport operator in with abstract boundary conditions relating the incoming and the outgoing fluxes.

Some Fixed Point Theorems for Orbitally-(p, q)-Quasi-contraction Mappings

In this paper, we provide some existence and uniqueness results for a (p, q)-quasi-contraction mapping acting on an orbitally-complete cone metric space. These results generalize several fixed point theorems, in particular those due to Ilic and Rakocevic’s for quasi-contraction mappings (Ilic and Rakocevic, Appl Math Lett 22(5):728–731, 2009), convex contraction mapping, and two-sided convex contraction of order 2.