Weak and pseudo-solutions of an arbitrary (fractional) orders differential equation in nonreflexive Banach space

• Published in: AIMS mathematics Vol. 6; no. 1; pp. 52 - 65
• Main Authors: Hashem, H. H. G., El-Sayed, A. M. A., Alenizi, Maha A.
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2021
• Subjects: fractional pettis integral; measure of weak noncompactness; pseudo-solution; weakly continuous solution; weakly relatively compact
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2021004

In this paper, we establish some existence results of weak solutions and pseudo-solutions for the initial value problem of the arbitrary (fractional) orders differential equation <disp-formula> <tex-math id="FE1"> $ \frac{dx}{dt}~ = ~ f(t, D^\gamma x(t)), ~\gamma \in (0, 1), ~~t~\in [0, T]=\mathbb{I}\\ x(0) = x_0. $ </tex-math> </disp-formula> in nonreflexive Banach spaces $~E, ~$ where $~D^\gamma x(\cdot)~$ is a fractional %pseudo- derivative of the function $~x(\cdot):\mathbb{I} \rightarrow E~$ of order $~\gamma.~$ The function $~f(t, x):\mathbb{I}\times E \rightarrow E~$ will be assumed to be weakly sequentially continuous in $x~$ for each $~t\in \mathbb{I}~$ and Pettis integrable in $~t~$ on $~\mathbb{I}~$ for each $~x\in C[\mathbb{I}, E].~$ Also, a weak noncompactness type condition (expressed in terms of measure of noncompactness) will be imposed.