An Implicit Hybrid Delay Functional Integral Equation: Existence of Integrable Solutions and Continuous Dependence

Ahmed M A El-Sayed,Shorouk M Al-Issa,Hind H G Hashem

doi:10.3390/math9243234

Ahmed M A El-Sayed, Shorouk M Al-Issa + Show 1 more

Open Access

https://doi.org/10.3390/math9243234

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Abstract

In this work, we are discussing the solvability of an implicit hybrid delay nonlinear functional integral equation. We prove the existence of integrable solutions by using the well known technique of measure of noncompactnes. Next, we give the sufficient conditions of the uniqueness of the solution and continuous dependence of the solution on the delay function and on some functions. Finally, we present some examples to illustrate our results.

Highlights

The study of implicit differential and integral equations has received much attention over the last 30 years or so
IFDEs have recently been studied by several researchers; Dhage and Lakshmikantham [3] have proposed and studied hybrid differential equations
Zhao et al [4] have worked at hybrid fractional differential equations and expanded Dhage’s approach to fractional order

Summary

Introduction

The study of implicit differential and integral equations has received much attention over the last 30 years or so. Allowing Qr to be a subset of Br containing all functions that are nonnegative and a.e. nondecreasing on I, we may conclude that Qr is nonempty, closed, convex, bounded, and compact in measure Since l1 + M l2 < 1, it follows, from fixed point theorem [18], that A is a contraction with regard to the measure of noncompactness χ and has at least one fixed point in Qr which show that Equation (7) has at least one positive nondecreasing a.e. solution x ∈ L1. Continuous Dependence Here, we investigate the continuous dependence of the unique solution x ∈ L1 on the delay function φ and on the two functions f1 and f2

Uniqueness of the Solution Consider the assumptions:

Continuous Dependence on the Functions f1 and f2

Conclusions