Abstract
In this paper, we study boundary value problems, involving the Hilfer fractional derivative, for pantograph fractional differential equations and inclusions supplemented by nonlocal integral boundary conditions. Existence and uniqueness results are obtained by using well-known fixed point theorems for single and multi-valued functions. Examples illustrating our results are also presented.
Highlights
In recent years, the theory of fractional differential equations has played a very important role in a new branch of applied mathematics, which has been utilized for mathematical models in engineering, physics, chemistry, signal analysis, etc
A generalization of derivatives of both Riemann–Liouville and Caputo was given by Hilfer in [8], known as the Hilfer fractional derivative of order α and a type β ∈ [0, 1], which can be reduced to the Riemann–Liouville and Caputo fractional derivatives when β = 0 and β = 1, respectively
For the lower semicontinuous case the existence result is based in nonlinear alternative of Leray–Schauder type together with a selection theorem for lower semicontinuous maps with decomposable values
Summary
The theory of fractional differential equations has played a very important role in a new branch of applied mathematics, which has been utilized for mathematical models in engineering, physics, chemistry, signal analysis, etc. We prove existence as well as existence and uniqueness results, for the boundary value problem (1.1)–(1.2) by using well-known fixed point theorems. We deduce by the Banach contraction mapping principle that A has a fixed point which is the unique solution of the boundary value problem (1.1)–(1.2).
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