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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 $\left(a,k\right)$-regularized families of operators. We illustrate the method for the case of the fractional order differential equation

${D}_{t}^{\alpha }{u}^{\text{'}}\left(t\right)+\mu {D}_{t}^{\alpha }u\left(t\right)=Au\left(t\right)+\frac{{t}^{-\alpha }}{{\Gamma }\left(1-\alpha \right)}\phantom{\rule{0.166667em}{0ex}}\left({u}^{\text{'}}\left(0\right)+\mu u\left(0\right)\right)+f\left(t\right),\phantom{\rule{1.em}{0ex}}t>0,\phantom{\rule{4pt}{0ex}}0<\alpha \le 1,\phantom{\rule{4pt}{0ex}}\mu \ge 0,$

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

##### MSC:
 47D60 $C$-semigroups, regularized semigroups 34A08 Fractional differential equations
##### References:
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