An Investigation of a Nonlinear Delay Functional Equation with a Quadratic Functional Integral Constraint

Ahmed M. A. El-Sayed, Malak M. S. Ba-Ali () and Eman M. A. Hamdallah
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Ahmed M. A. El-Sayed: Faculty of Science, Alexandria University, Alexandria 21521, Egypt
Malak M. S. Ba-Ali: Faculty of Science, Princess Nourah Bint Abdul Rahman University, Riyadh 11671, Saudi Arabia
Eman M. A. Hamdallah: Faculty of Science, Alexandria University, Alexandria 21521, Egypt

Mathematics, 2023, vol. 11, issue 21, 1-24

Abstract: This research paper focuses on investigating the solvability of a constrained problem involving a nonlinear delay functional equation subject to a quadratic functional integral constraint, in two significant cases: firstly, the existence of nondecreasing solutions in a bounded interval L 1 [ 0 , T ] and, secondly, the existence of nonincreasing solutions in unbounded interval L 1 ( R + ) . Moreover, the paper explores various qualitative properties associated with these solutions for the given problem. To establish the validity of our claims, we employ the De Blasi measure of noncompactness (MNC) technique as a basic tool for our proofs. In the first case, we provide sufficient conditions for the uniqueness of the solution ψ ∈ L 1 [ 0 , T ] and rigorously demonstrate its continuous dependence on some parameters. Additionally, we establish the equivalence between the constrained problem and an implicit hybrid functional integral equation (IHFIE). Furthermore, we delve into the study of Hyers–Ulam stability. In the second case, we examine both the asymptotic stability and continuous dependence of the solution ψ ∈ L 1 ( R + ) on some parameters. Finally, some examples are provided to verify our investigation.

Keywords: constrained problem; Hyers–Ulam stability; measure of noncompactness; implicit hybrid functional integral equation; asymptotic stability and dependency (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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