Research article

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

  • Received: 19 July 2020 Accepted: 17 September 2020 Published: 28 September 2020
  • MSC : 34A12, 47G10, 28B05

  • 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 [0, T]=\mathbb{I}\\ x(0) = x_0. $ 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.

    Citation: H. H. G. Hashem, A. M. A. El-Sayed, Maha A. Alenizi. Weak and pseudo-solutions of an arbitrary (fractional) orders differential equation in nonreflexive Banach space[J]. AIMS Mathematics, 2021, 6(1): 52-65. doi: 10.3934/math.2021004

    Related Papers:

  • 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 [0, T]=\mathbb{I}\\ x(0) = x_0. $ 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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