Existence results for a system of nonlinear integral equations in Banach algebras under weak topology

Abstract

Abstract. This paper is devoted to the study of a coupled system of nonlinear functional integral equations in suitable Banach algebras. This system is reduced to a fixed point problem for a 2 × 2 block operator matrix with nonlinear inputs. Hence, certain assumptions on its entries are given under a weak topology setting. These assumptions involve in particular the De Blasi measure of weak noncompactness in order to ensure the existence of solutions. Key Words and Phrases: integral equation, Banach algebra, weakly sequentially continuous, measure of weak noncompactness, fixed point theory. 2010 Mathematics Subject Classification: 47H10, 45G15. References

Existence of solutions for a system of Chandrasekhar’s equations in Banach algebras underweak topology

“…♦ If V ∈ B(X) and K ∈ W(X), then β(V • K) ≤ K β(V).♦ If F is Lipschitzian with constant α and is weakly sequentially continuous on X, thenβ(F(V)) ≤ αβ(V), for all V ∈ B(X). ♦In[18], A. Jeribi, N. Kaddachi and B. Krichen give a proof of the next result in case of Lipschitzians mappings. Now, we give a proof for the case of D-Lipschitizians maps.Theorem 2.9.…”

“…In this direction, the authors A. Jeribi, N. Kaddachi and B. Krichen in [18] have established some fixed point for a 2 × 2 operator matrix (2), when X is a Banach algebra satisfying certain condition. An application to a system of nonlinear integral equation occurring in some physical and biological problem.…”

“…In the next section, we give some preliminaries and results needed in the sequel. In addition, we give a reformulation of Theorem 3.4 in [18]. In Section 3, we apply Theorem 2.9 to discuss the existence of solutions to equations of system (1).…”

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:…”

Generalized form of fixed point theorems in Banach algebras under weak topology with an application

“…In [20,22], the authors have established some fixed point results for the block operator matrix (2), where the inputs are nonlinear mappings based on the convexity of the bounded domain, on the well-known Schauder's fixed point theorem, and also on the properties of the inputs (cf. completely continuous [22,25], weakly sequentially continuous [20], etc, . .…”