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

Phan Dinh Phung, Le Xuan Truong,
Fractional Calculus and Applied Analysis
September 2013, Volume 16, Issue 3, pp 538-558

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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

Dαu(t)F(t,u(t),Dα1u(t)),a.e.,t[0,1],Iβu(t)|t=0=0,u(1)=01u(t)dt,(*)

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 E α,1 (I). An application in control theory is also provided by using the Young measures.

Keywords

  • 26A33
  • 34A60
  • 34B10
  • 34A08
  • 47N70
  • fractional differential inclusion
  • boundary value problem
  • Green’s function
  • contractive set valued-map
  • retract
  • Young measures