Existence of continuous solutions for nonlinear functional differential and integral inclusions

A. M. A. El-Sayed; Al-Issa, Sh. M

doi:10.26637/MJM0703/0028

Abstract

In this article, we establish the existence of a positive continuous solution of the functional integral inclusion of fractional order
$$
x(t) \in p(t)+I^\alpha F_1\left(t, I^\beta f_2(t, x(\varphi(t))), t \in[0,1], \alpha, \beta \in(0,1) .\right.
$$
The study holds in the case when the set-valued function has Lipschitz selections.
As an application, we study the initial-value problem of the arbitrary fractional order differential inclusion
$$
\frac{d x}{d t} \in F_1\left(t, D^\gamma x(t)\right) \text {, a.e, } t \in[0,1], \quad \gamma>0
$$
where $F_1(t, x(t))$ is a Lipschitz set-valued function defined on $[0,1] \times R^{+}$.

Keywords:

Set-valued function, functional Integral inclusion, fixed point theorem, Lipschitz selections

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

A. M. A. El-Sayed Faculty of Science, Alexandria University, Alexandria, Egypt. https://orcid.org/0000-0001-7092-7950
Al-Issa, Sh. M Faculty of Science, Lebanese International University, Saida, Lebanon. https://orcid.org/0000-0002-1488-0817

Pages: 541-544
Date Published: 01-07-2019
Vol. 7 No. 03 (2019): Malaya Journal of Matematik (MJM)

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