Abstract
Differential and integral equations in reflexive Banach spaces have gained great attention and hve been investigated in many studies and monographs. Inspired by those, we study the existence of the solution to a delay functional integral equation of Volterra-Stieltjes type and its corresponding delay-functional integro-differential equation in reflexive Banach space E. Sufficient conditions for the uniqueness of the solutions are given. The continuous dependence of the solutions on the delay function, the initial data, and some others parameters are proved.
Highlights
The existence of weak solutions for ordinary differential equations in Banach spaces has been investigated in many papers, for example, in Cichoń [1,2], Cramer et al [3], Knight [4], Kubiaczyk and Szufla [5], and [6,7,8,9,10,11] and the references therein for fractionalorder differential equations in Banach spaces, and [12,13,14] for quadratic integral equations in reflexive Banach algebra
It is clear that the functions f 1, f 2, g satisfy the assumptions of Theorem 5; the functional integral Equation (7) has one weak solution x ∈ C ( I, E)
The theory of differential equations in abstract Banach spaces has been established by some authors from different viewpoints, for example, [24,25,26,27]
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. He obtained sufficient conditions for the existence of a Kantorovich principal solution of the nonlinear Volterra integral equation of the second kind (4) on the half-line [0, ∞) and on a finite interval I. He suggested a method for computing the boundary of an interval outside which the solution can blow up. (O’Regan fixed point theorem) Let E be a Banach space, and let Q be a nonempty, bounded, closed and convex subset of C ( I, E) and let F : Q → Q be weakly sequentially continuous and assume that { FQ(t)} is relatively weakly compact in E for each t ∈ I. F has a fixed point in the set Q (see [22])
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