Equivalence of some properties in the theory of Banach algebras and applications

Leray Schauder Type Fixed Point Theorems in RWC-Banach Algebras and Application to Chandrasekhar Integral Equations

“…the space C(K, X) of all continuous functions from a Hausdorff compact space K to X is a RWC-Banach algebra whenever X is reflexive (resp. X is a RWC-Banach algebra), see [13].…”

“…Theorem 2.11. [13] A Banach algebra X is a RWC-Banach algebra, if and only, if X satisfies the condition P . Lemma 2.12.…”

“…Define the operators A, C : E → E and B : B r → E by: Proof. In view of Theorem 2.2 in [13], X is a RWC-Banach algebra. We shall prove that A, B and C satisfy all conditions of Corollary 3.4.…”

“…In [19], a mixed method combines the contraction principal and the Schauder fixed point theorem was used in order to resolve (1.1). Subsequently, many extensions and development of this method in the strong and the weak topologies are developped, see [8,12,13,24]. Because the weak topology represent a natural framework of different applied problems, and since the sequential weak continuity property of the product of two sequentially weakly continuous operators is not always conserved, Ben Amar et al in [8] have introduced a new class of Banach algebras satisfying certain sequential condition, called condition (P).…”