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

Abstract

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 Rotenberg's 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 L_p×L_p 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.

Keywords:

• operator matrix
• fixed-point theory
• weak compactness
• growing cell populations