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An operator theoretical approach to a class of fractional order differential equations. (English) Zbl 1226.47048

Summary: We propose a general method for obtaining the representation of solutions for linear fractional order differential equations based on the theory of (a,k)-regularized families of operators. We illustrate the method for the case of the fractional order differential equation

D t α u ' (t)+μD t α u(t)=Au(t)+t -α Γ(1-α)(u ' (0)+μu(0))+f(t),t>0,0<α1,μ0,

where A is an unbounded closed operator defined on a Banach space X and f is an X-valued function.

MSC:
47D60C-semigroups, regularized semigroups
34A08Fractional differential equations
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