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# Fractional Calculus and Applied Analysis

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Volume 16, Issue 3

# On a fractional differential inclusion with integral boundary conditions in Banach space

Phan Phung
/ Le Truong
• Department of Mathematics and Statistics, University of Economics HoChiMinh City, 59C, Nguyen Dinh Chieu Str, District 3, HoChiMinh City, Vietnam
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Published Online: 2013-06-26 | DOI: https://doi.org/10.2478/s13540-013-0035-6

## Abstract

We consider a class of boundary value problem in a separable Banach space E, involving a nonlinear differential inclusion of fractional order with integral boundary conditions, of the form (*)$\left\{ \begin{gathered} D^\alpha u(t) \in F(t,u(t),D^{\alpha - 1} u(t)),a.e.,t \in [0,1], \hfill \\ I^\beta u(t)|_{t = 0} = 0,u(1) = \int\limits_0^1 {u(t)dt,} \hfill \\ \end{gathered} \right.$ where D α is the standard Riemann-Liouville fractional derivative, F is a closed valued mapping. Under suitable conditions we prove that the solutions set of (*) is nonempty and is a retract in W Eα,1(I). An application in control theory is also provided by using the Young measures.

MSC: 26A33; 34A60; 34B10; 34A08; 47N70

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Published Online: 2013-06-26

Published in Print: 2013-09-01

Citation Information: Fractional Calculus and Applied Analysis, Volume 16, Issue 3, Pages 538–558, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454,

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