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On type of periodicity and ergodicity to a class of fractional order differential equations. (English) Zbl 1194.34007

In this paper, the authors study some sufficient conditions for the existence and uniqueness of: a) pseudo almost periodic (in the sense of Zhang) mild solutions to some semilinear fractional differential equations, and b) asymptotically almost automorphic (in the sense of N’Guérékata) mild solutions to some semilinear fractional integro-differential equations; in all cases, the derivative \(D^{\alpha}_{t}\) is considered in the sense of Riemann-Liouville with \(1<\alpha<2\) and the operator \(A\) is sectorial of negative type. The authors reach their goals using a theoretical operator theory approach and fixed point techniques. The results extend and complete several recent works by the authors and others (including C. Lizama, G. N’Guérékata, G. Mophou). An application to some fractional relaxation-oscillation equation is also given.

MSC:

34A08 Fractional ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
47N20 Applications of operator theory to differential and integral equations
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