Weak and pseudo-solutions of an arbitrary (fractional) orders differential equation in nonreflexive Banach space
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}\f...
Main Authors: | , , |
---|---|
Format: | Article |
Language: | English |
Published: |
AIMS Press
2021-10-01
|
Series: | AIMS Mathematics |
Subjects: | |
Online Access: | https://www.aimspress.com/article/10.3934/math.2021004/fulltext.html |
id |
doaj-10c0713307bd45e2866d956448251136 |
---|---|
record_format |
Article |
spelling |
doaj-10c0713307bd45e2866d9564482511362020-11-25T04:00:46ZengAIMS PressAIMS Mathematics2473-69882021-10-0161526510.3934/math.2021004Weak and pseudo-solutions of an arbitrary (fractional) orders differential equation in nonreflexive Banach spaceH. H. G. Hashem0A. M. A. El-Sayed1Maha A. Alenizi21 Department of mathematics, College of Science, Qassim University, P.O. Box 6644 Buraidah 51452, Saudi Arabia2 Faculty of Science, Alexandria University, Alexandria, Egypt1 Department of mathematics, College of Science, Qassim University, P.O. Box 6644 Buraidah 51452, Saudi ArabiaIn 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.https://www.aimspress.com/article/10.3934/math.2021004/fulltext.htmlmeasure of weak noncompactnessweakly continuous solutionpseudo-solutionweakly relatively compactfractional pettis integral |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
H. H. G. Hashem A. M. A. El-Sayed Maha A. Alenizi |
spellingShingle |
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 AIMS Mathematics measure of weak noncompactness weakly continuous solution pseudo-solution weakly relatively compact fractional pettis integral |
author_facet |
H. H. G. Hashem A. M. A. El-Sayed Maha A. Alenizi |
author_sort |
H. H. G. Hashem |
title |
Weak and pseudo-solutions of an arbitrary (fractional) orders differential equation in nonreflexive Banach space |
title_short |
Weak and pseudo-solutions of an arbitrary (fractional) orders differential equation in nonreflexive Banach space |
title_full |
Weak and pseudo-solutions of an arbitrary (fractional) orders differential equation in nonreflexive Banach space |
title_fullStr |
Weak and pseudo-solutions of an arbitrary (fractional) orders differential equation in nonreflexive Banach space |
title_full_unstemmed |
Weak and pseudo-solutions of an arbitrary (fractional) orders differential equation in nonreflexive Banach space |
title_sort |
weak and pseudo-solutions of an arbitrary (fractional) orders differential equation in nonreflexive banach space |
publisher |
AIMS Press |
series |
AIMS Mathematics |
issn |
2473-6988 |
publishDate |
2021-10-01 |
description |
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. |
topic |
measure of weak noncompactness weakly continuous solution pseudo-solution weakly relatively compact fractional pettis integral |
url |
https://www.aimspress.com/article/10.3934/math.2021004/fulltext.html |
work_keys_str_mv |
AT hhghashem weakandpseudosolutionsofanarbitraryfractionalordersdifferentialequationinnonreflexivebanachspace AT amaelsayed weakandpseudosolutionsofanarbitraryfractionalordersdifferentialequationinnonreflexivebanachspace AT mahaaalenizi weakandpseudosolutionsofanarbitraryfractionalordersdifferentialequationinnonreflexivebanachspace |
_version_ |
1724449309926096896 |