Abstract
In this paper, the author is concerned with the fractional equation with the anti-periodic boundary value conditions where denotes the Caputo fractional derivative of order γ, the constants α, , , , satisfy the conditions , , . Different from the recent studies, the function f involves the Caputo fractional derivative and . In addition, the author put forward new anti-periodic boundary value conditions, which are more suitable than those studied in the recent literature. By applying the Banach contraction mapping principle and the Leray-Schauder degree theory, some existence results of solutions are obtained. MSC:34A08, 34B15.
Highlights
In the present paper, we are concerned with the existence of solutions for the fractional differential equationCDα +u(t) = f t, u(t), CDα + u(t), CDα + u(t), t ∈ (, ), ( . )with anti-periodic boundary value conditions⎧ ⎨u( ) = –u( ), tβ – C Dβ + u(t)|t→ = –tβ – C Dβ + u(t)|t=,⎩tβ – C Dβ + u(t)|t→ = –tβ – C Dβ + u(t)|t=, Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering
There has been a significant development in the study of fractional differential equations and inclusions in recent years; see the monographs of Podlubny [ ], Kilbas et al [ ], Lakshmikantham et al [ ], Samko et al [ ], Diethelm [ ], and the survey by Agarwal et al [ ]
Anti-periodic boundary value problems occur in the mathematic modeling of a variety of physical processes and have recently received considerable attention
Summary
Where CDγ + denotes the Caputo fractional derivative of order γ , the constants α, α , α , β , β satisfy conditions < α ≤ , < α ≤ < α ≤ , < β < < β < , and f is a given continuous function. For examples and details of anti-periodic fractional boundary conditions, see [ – ]. In [ ], Agarwal and Ahmad studied the solvability of the following anti-periodic boundary value problem for nonlinear fractional differential equation: In [ ], Wang, Ahmad, Zhang investigated the following impulsive anti-periodic fractional boundary value problem:
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