Abstract
In this paper, the author puts forward a kind of anti-periodic boundary value problems of fractional equations with the Riemann-Liouville fractional derivative. More precisely, the author is concerned with the following fractional equation: with the anti-periodic boundary value conditions where denotes the standard Riemann-Liouville fractional derivative of order , and the nonlinear function may be singular at . By applying the contraction mapping principle and the other fixed point theorem, the author obtains the existence and uniqueness of solutions. MSC:34A08, 34B15.
Highlights
1 Introduction In the present paper, we are concerned with the existence of solutions for the fractional differential equation
Differential equations with fractional order are a generalization of ordinary differential equations to non-integer order
There has been a significant development in the study of fractional differential equations in recent years; see, for example, [ – ]
Summary
We are concerned with the existence of solutions for the fractional differential equation. With anti-periodic boundary value conditions t –αu(t)|t→ + = –t –αu(t)|t= , t –αu(t) |t→ + = – t –αu(t) |t= , where Dα + denotes the standard Riemann-Liouville fractional derivative of order α ∈ ( , ), and the nonlinear function f (t, ·, ·) may be singular at t =. There appeared a paper dealing with anti-periodic boundary value problems of a fractional equation with the Riemann-Liouville fractional derivative (see [ ]), which will be formulated later. Comparing with the recent article (see [ ]) dealing with the following periodic boundary value problems with Riemann-Liouville fractional derivative. F (t) fn exists by applying the theorem to the limit convergence of function sequences again. We definite an operator T on X as follows: Th(t) = –tα– G (s)h(s) ds + tα– G (s)h(s) ds + I α+h(t), t ∈
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