Stability and dependency of the solution of a nonlinear quadratic functional integral inclusion with distributed delay
Authors
A. M. A. El-Sayed
- Faculty of Science, Alexandria University, Alexandria, Egypt.
N. F. M. El-Haddad
- Faculty of Science, Damanhour University, Behera, Egypt.
Abstract
Let \(\mathcal{E}\) be a reflexive Banach space. In this paper, we are interested in the solvability of the nonlinear quadratic functional integral inclusion with distributed delay
\[ {\mathrm{x(t)}} \in {\mathcal{F}}({\mathrm{t}},{\mathrm{g(t,x(t))}}\int_{0}^{\varphi(t)}\mathrm{u(t,s,x(s))ds}),~~~\mathrm{t}\in [0,\infty),\]
on the real half-axis. Our thought is found within the space \( {\mathcal{BC(R_+,E)}}\) of bounded continuous functions on the real half-axis \(\mathcal{R_+}\) and taking values in a reflexive Banach space \(\mathcal{E}\) beneath the presumption that the multi-valued function \(\mathcal{F}\) fulfill Lipschitz condition in \(\mathcal{E}\). The main tool applied in this work is the procedure associated with measures of non-compactness in the space \({\mathcal{BC(R_+,E)}}\) by a given norm of continuity and using Darbo's fixed point theorem.
The asymptotic stability and the asymptotic dependency of the solution will be studied. We additionally provide an example to demonstrate the effectiveness and value of our results.
Share and Cite
ISRP Style
A. M. A. El-Sayed, N. F. M. El-Haddad, Stability and dependency of the solution of a nonlinear quadratic functional integral inclusion with distributed delay, Journal of Mathematics and Computer Science, 35 (2024), no. 1, 35--51
AMA Style
El-Sayed A. M. A., El-Haddad N. F. M., Stability and dependency of the solution of a nonlinear quadratic functional integral inclusion with distributed delay. J Math Comput SCI-JM. (2024); 35(1):35--51
Chicago/Turabian Style
El-Sayed, A. M. A., El-Haddad, N. F. M.. "Stability and dependency of the solution of a nonlinear quadratic functional integral inclusion with distributed delay." Journal of Mathematics and Computer Science, 35, no. 1 (2024): 35--51
Keywords
- Multi-valued function
- quadratic functional integral inclusion
- measure of non-compactness
- Lipshitz condition
- reflexive Banach space
- asymptotic stability
- asymptotic dependency
MSC
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