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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
\[ D^\delta u(t)-\lambda u(t)=f(t,u(t)),\quad t\in J:=(0,1], \;0<\delta<1,\tag{1.1} \]
where \(D^\delta\) is the standard Riemann-Liouville fractional derivative, \(f\) is continuous, and \(\lambda\in\mathbb R\).
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:

34A08 Fractional ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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