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 L 1 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.

Fixed Point Theory for 1-Set Contractions: a Survey

“…Further applications to nonlinear Volterra and Hammerstein equations can be found, e.g., in [2,45,62,63]. …”

Leray–Schauder Fixed Point Theorems for Block Operator Matrix with an Application

“…Also, the theory of block operator matrix is a subject of great interest thanks to the useful applications for studying some systems of integral equations as well as systems of partial or ordinary differential equations. Recent work has employed the fixed point technique for the operator matrix with nonlinear entries acting on Banach spaces or Banach algebras for studying the existence of solutions for several classes of systems of nonlinear integral equations, see, for example, [1][2][3][4][5]. ese operators are defined by a 2 × 2 block operator matrix:…”

“…Due to the lack of compactness on L 1 spaces, the study in [6] did not cover the case p � 1. Later, Jeribi et al [1] proposed to extend the results of Amar et al [6] to the case p � 1 by establishing new variants of fixed point theorems for (1), and their analysis was carried out via arguments of weak topology and, particularly, the technique of measures of weak noncompactness. In the above quoted works, the assumptions that I − A or I − D are invertible play a fundamental role in the arguments.…”

New Types of F-Contraction for Multivalued Mappings and Related Fixed Point Results in Abstract Spaces

“…A point V is said to be a fixed point of a multivalued/selfmapping , if V ∈ V/V = V. Fixed point theory has a large number of applications, for example, [1][2][3][4]. Czewick [5] initiated the study of fixed point in b-metric spaces.…”