Fixed point theory in WC--Banach algebras

Abstract

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) .

On the existence of solutions for an infinite system of integral equations

“…Notice that it was shown in [10] that if the operator A involved in (1.2) is weakly compact and satisfy condition (H ) (this condition allows us to distinguish some classes of operators which transform each weakly convergent sequence in a Banach space X into a sequence containing a subsequence being strongly convergent in X) and B and C are w-contractive then the equation (1.2) has at least one solution (see [10,Theorem 3.2]). However, the condition (ii) in Theorem 3.2 in [10] is too restrictive and can be relaxed just by supposing the operator A is w-contractive. This result is developed in Theorem 4 (see section 3).…”

Measure of Weak Noncompactness and Fixed Point Theorems in Banach Algebras with Applications

“…2. In Theorem 5, we do not require the conditions "A satisfies condition (H1)" and "A(Ω) is relatively weakly compact", but in Theorem 3.2 in [23], these conditions are necessary. 3.…”

Fixed Points for Sum or Product of Operators in Weak Topology With Applications (A Survey) H. H. G. Hashem

“…In 2015 Aref Jeribi , Bilel Krichen , Bilel Mefteh [32] proved some fixed point theorems for the sum and the product of nonlinear weakly sequentially continuous operators acting on a WC-Banach algebra. their results improve some recent results and extend some several earlier works using the condition (P).…”