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On global solutions to fractional functional differential equations with infinite delay in Fréchet spaces. (English) Zbl 1228.34015
Summary: We investigate global uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. We shall rely on a nonlinear alternative of Leray-Schauder type in Fréchet spaces due to M. Frigon and A. Granas [Ann. Sci. Math. Qué. 22, No. 2, 161–168 (1998; Zbl 1100.47514)]. The results are obtained by using the \(\alpha \)-resolvent family \((S_{\alpha }(t))_{t\geq 0}\) on a complex Banach space \(X\) combined with the above-mentioned fixed point theorem. As an application, a controllability result with one parameter is also provided to illustrate the theory.

MSC:
34A08 Fractional ordinary differential equations and fractional differential inclusions
47H10 Fixed-point theorems
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
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