On the Solvability of Quadratic Multi-Term Hybrid Equation with Nonlocal Hybrid Condition in Banach Algebra Spaces

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DOI: https://doi.org/10.28924/2291-8639-24-2026-25
Published: Jan 29, 2026
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Sh. M Al-Issa, A. M. A. El-Sayed, I. H. Kaddoura, R. M. Al-Sehly, On the Solvability of Quadratic Multi-Term Hybrid Equation with Nonlocal Hybrid Condition in Banach Algebra Spaces, Int. J. Anal. Appl., 24 (2026), 25.

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Sh. M Al-Issa, A. M. A. El-Sayed, I. H. Kaddoura, R. M. Al-Sehly

Abstract

This paper investigates the existence of solutions for a class of multi-term hybrid functional equations subject to nonlocal and fractional conditions. We first establish sufficient conditions to ensure the existence of at least one continuous solution by applying Dhage’s fixed-point theorem within an appropriate Banach algebra framework. Subsequently, we extended the analysis to integrable solutions in the Lebesgue space L1(J,R) under Carathéodory-type growth conditions. The uniqueness of solutions is then addressed by imposing Lipschitz-type constraints on nonlinear and hybrid terms. Furthermore, we examine the continuous dependence of solutions on initial data and parameters. Several illustrative examples are presented to demonstrate the applicability and validity of the obtained results. The theoretical framework developed here unifies and generalizes various existing results for hybrid and nonlocal fractional differential equations.

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