Abstract
In this paper, we study the existence of solutions for a boundary value problem involving Hadamard type fractional differential inclusions and integral boundary conditions. Some new existence results for convex as well as non-convex multivalued maps are obtained by using standard fixed point theorems for multivalued maps. The paper concludes with an illustrative example.MSC:34A60, 34A08.
Highlights
1 Introduction The intensive development of fractional calculus and its widespread applications in several disciplines clearly indicate the interest of researchers and modelers in the subject
For a detailed account of applications and recent results on initial and boundary value problems of fractional differential equations and inclusions, we refer the reader to a series of books and papers [ – ]
We study the following boundary value problem with an integral nonlocal boundary condition: Dαx(t) ∈ F(t, x(t)), < t < e, < α ≤, x( ) =, AIγ x(η) + Bx(e) = c, < η < e, ( . )
Summary
The intensive development of fractional calculus and its widespread applications in several disciplines clearly indicate the interest of researchers and modelers in the subject. If the multivalued map G is completely continuous with nonempty compact values, G is u.s.c. if and only if G has a closed graph, i.e., xn → x∗, yn → y∗, yn ∈ G(xn) imply y∗ ∈ G(x∗). A multivalued map F : [ , e] × R → P(R) is said to be Carathéodory if (i) t −→ F(t, x) is measurable for each x ∈ R; (ii) x −→ F(t, x) is upper semi-continuous for almost all t ∈ [ , e]; Further a Carathéodory function F is called L -Carathéodory if (iii) for each ρ > , there exists φρ ∈ L ([ , e], R+) such that
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