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

Abstract

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 \[ %\frac{dx}{dt}~=~ f(t,D^\gamma x(t)),~\gamma \in (0,1), ~~t~\in~I=[0,T] %\] \begin{eqnarray*}\label{2} \hspace{-3cm}\frac{dx}{dt}&=& f(t,D^\gamma x(t)),~\gamma \in (0,1), ~~t~\in~[0,T]=\mathbb{I}\nonumber\\ &&\\ x(0)&=&x_0. \nonumber \end{eqnarray*} 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.

Keywords

• measure of weak noncompactness
• weakly continuous solution
• pseudo-solution
• weakly relatively compact
• fractional pettis integral