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Existence of periodic solution for a nonlinear fractional differential equation. (English) Zbl 1181.34006

Summary: We consider the following nonlinear fractional differential equation of the form

D δ u(t)-λu(t)=f(t,u(t)),tJ:=(0,1],0<δ<1,(1·1)

where D δ is the standard Riemann-Liouville fractional derivative, f is continuous, and λ.

Due to the singularity of the possible solutions, we introduce a new and proper concept of periodic boundary value conditions. We present Green’s function and give some existence results for the linear case and then we study the nonlinear problem.

MSC:
34A08Fractional differential equations
34C25Periodic solutions of ODE
34B15Nonlinear boundary value problems for ODE
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