Treatment of a Coupled System for Quadratic Functional Integral Equation on the Real Half-Line via Measure of Noncompactness
Abstract
:1. Preliminaries and Introduction
2. Main Result
- (i)
- , are continuous and .
- (ii)
- There exists a continuous function such that
- (iii)
- are continuous and satisfy the Lipschitz condition
- (iv)
- are continuous and there exists a continuous function , such that and
- (v)
- Without loss of generality, we can write , and Now there exists a positive constant C, such that
- ()
- ()
- . In this instance, let us take a look at the function w = given by
3. Asymptotic Stability
4. Particular Cases and Example
- Let ; then, the coupled system (1) takes the formBased on conditions (iii)–(v) of Theorem 2, then(1) has at least one asymptotically stable solution .Moreover, when Then, we have a coupled system of Urysohn integral equations
- Let , in (1), we have
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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El-Sayed, A.M.A.; Omar, Y.M.Y.; Hashem, H.H.G.; Al-Issa, S.M. Treatment of a Coupled System for Quadratic Functional Integral Equation on the Real Half-Line via Measure of Noncompactness. Foundations 2023, 3, 37-48. https://doi.org/10.3390/foundations3010004
El-Sayed AMA, Omar YMY, Hashem HHG, Al-Issa SM. Treatment of a Coupled System for Quadratic Functional Integral Equation on the Real Half-Line via Measure of Noncompactness. Foundations. 2023; 3(1):37-48. https://doi.org/10.3390/foundations3010004
Chicago/Turabian StyleEl-Sayed, Ahmed M. A., Yasmin M. Y. Omar, Hind H. G. Hashem, and Shorouk M. Al-Issa. 2023. "Treatment of a Coupled System for Quadratic Functional Integral Equation on the Real Half-Line via Measure of Noncompactness" Foundations 3, no. 1: 37-48. https://doi.org/10.3390/foundations3010004