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.
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 Krasnosel'skii type theorem involving the weak topology of Banach spaces.
In this paper, we will prove some fixed point theorems for the sum and the product of nonlinear weakly sequentially continuous operators acting on a WC-Banach algebra. Our results improve and correct some recent results given by Banas and Taoudi, and extend several earlier works using the condition (P) .
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