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\(S\)-asymptotically \(\omega \)-periodic solutions of semilinear fractional integro-differential equations. (English) Zbl 1176.47035

The authors consider the following semilinear fractional integro-differential equation
\[ v'(t) = \int_0^t \frac{(t - s)^{\alpha - 2}}{\Gamma (\alpha - 1)} Av(s)\,ds + f(t,v(t)), \quad t \geq 0, \quad v(0) = u_0 \in X, \]
where \(1 < \alpha < 2\), \(A:D(A) \subset X \to X\) is a linear densely defined operator of sectorial type on a complex Banach space \(X\), and \(f:[0,\infty )\times X \to X\) is a continuous function satisfying a suitable Lipschitz type condition. The existence and uniqueness of an \(S\)-asymptotically \(\omega\)-periodic mild solution to this problem is obtained.

MSC:

47G20 Integro-differential operators
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
45K05 Integro-partial differential equations
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