Abstract
We establish sufficient criteria for the existence of solutions for a nonlinear generalized Langevin-type nonlocal fractional-order integral multivalued problem. The convex and non-convex cases for the multivalued map involved in the given problem are considered. Our results rely on Leray–Schauder nonlinear alternative for multivalued maps and Covitz and Nadler’s fixed point theorem. Illustrative examples for the main results are included.
Highlights
Fractional calculus is the extension of classical calculus which deals with differential and integral operators of fractional order
We study the existence of solutions for a nonlinear generalized Langevin type nonlocal fractional-order integral multivalued problem given by Mathematics 2019, 7, 1015; doi:10.3390/math7111015
We have introduced a new class of multivalued boundary value problems on an arbitrary domain containing Caputo-type generalized fractional differential operators of different orders and a generalized integral operator
Summary
Fractional calculus is the extension of classical calculus which deals with differential and integral operators of fractional order. In [8], existence and uniqueness results for a nonlinear Langevin equation involving two fractional orders supplemented with three-point boundary conditions were obtained. In [11], the authors proved the existence of and uniqueness results for an anti-periodic boundary value problem of a system of Langevin fractional differential equations. In [12], the authors investigated a nonlinear fractional Langevin equation with anti-periodic boundary conditions by applying coupled fixed point theorems. In a recent work [13], the authors obtained some existence results for a fractional Langevin equation with nonlinearity depending on Riemann–Liouville fractional integral, and complemented with nonlocal multi-point and multi-strip boundary conditions. We study the existence of solutions for a nonlinear generalized Langevin type nonlocal fractional-order integral multivalued problem given by Mathematics 2019, 7, 1015; doi:10.3390/math7111015 www.mdpi.com/journal/mathematics. We summarize the work established in this paper, and its implications, in the last section
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