tailieunhanh - Fixed point theory in WC-Banach algebras

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). | Turk J Math (2016) 40: 283 – 291 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Fixed point theory in WC–Banach algebras Aref JERIBI1 , Bilel KRICHEN2,∗, Bilel MEFTEH1 Department of Mathematics, Faculty of Sciences of Sfax, Sfax, Tunisia 2 Department of Mathematics, Preparatory Engineering Institute, Sfax, Tunisia 1 Received: • Accepted/Published Online: • Final Version: 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) . Key words: WC–Banach algebra, fixed point theorems, weak topology, measure of weak noncompactness 1. Introduction In 1998 , Dhage [10] proved the following fixed point theorem involving three operators in a Banach algebra by combining Schauder’s fixed point theorem and Banach’s contraction principle. Theorem Let S be a nonempty, bounded, closed, and convex subset of a Banach algebra X and let A , B , C : S → S be three operators, such that: (i ) A and C are Lipschitzian with Lipschitz constants α and β , respectively, (ii ) ( I−C )−1 A ( I−C ) A exists on B(S), I being the identity operator on X , and the operator ( ) : X → X is defined by I−C x := x−Cx A Ax , (iii ) B is completely continuous, and (iv ) AxBy + Cx ∈ S, ∀ x, y ∈ S . Then the operator equation x = AxBx + Cx has, at least, a solution in S , whenever αM + β 0 , Br denotes the closed ball in X centered at 0X with radius r and D(A) denotes the domain of an operator A . ΩX is the collection of all nonempty bounded subsets of X and Kw is the subset of ΩX consisting of all weakly compact subsets of X . In the remainder, ⇀ denotes the weak convergence and → denotes the strong convergence in X . Recall .