On the Weak Solutions of a Delay Composite Functional Integral Equation of Volterra-Stieltjes Type in Reflexive Banach Space

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.

Convergence of AA-Iterative Algorithm for Generalized α-Nonexpansive Mappings with an Application

“…In 2022, El-sayed and Omar [32] established the existence and uniqueness of the weak solution of a delay composite functional differential equation of the Volterra-Stieljes type. Many authors solved the delay composite functional differential equation of the Volterra-Stieljes type.…”

“…Finding the solution of (39) and ( 40) is equivalent to finding the solution of the following integral equation [32]:…”

Superconvergent Nyström and Degenerate Kernel Methods for Integro-Differential Equations

“…Theorem 2. Let y S n,1 and y S n,2 be the approximate solutions defined, respectively, by (11) and (13). Let ỹS n,1 and ỹS n,2 be the iterated versions defined respectively by ( 12) and ( 14).…”

“…Among the existing methods in the literature, we cite the Adomian decomposition [5], homotopy analysis [2], Chebyshev and Taylor collocation [6], Taylor series expansion [7,8], integral mean value [9], and decomposition method [10]. For other methods to solve integro-differential equations, see [11][12][13][14].…”

“…. , N h , are obtained by replacing y n,1 given by (40) into the approximate Equation (11) and by identifying coefficients of the family of functions {φ j , k j }, j = 1, . .…”