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.

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非自反Banach空间中任意（分数）阶微分方程的弱解和拟解

本文针对任意（分数阶）微分方程\ [％\ frac {dx} {dt}〜=〜f（t，D ^）的初值问题，建立了弱解和伪解的存在性结果。 \ 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。非自反Banach空间$〜E，〜$中的\ nonumber \ end {eqnarray *}，其中$〜D ^ \ gamma x（\ cdot）〜$是函数$〜x（\ cdot）的分数％伪衍生物： $〜\ gamma。〜$的\ mathbb {I} \ rightarrow E〜$函数$〜f（t，x）：\ mathbb {I} \ times E \ rightarrow E〜$被假定为弱连续的\ mathbb {I}〜$中每个$〜t \在$ x〜$中连续，而C [\ mathbb中每个$〜x \的Petis可在$〜\ mathbb {I}〜$上在$〜t〜$积分{I}，E]。