## Abstract

We firstly prove that $\beta $-times integrated $\alpha $-resolvent operator function ($(\alpha ,\beta )$-ROF) satisfies a functional equation which extends that of $\beta $-times integrated semigroup and $\alpha $-resolvent operator function. Secondly, for the inhomogeneous $\alpha $-Cauchy problem ${\mathrm{}}^{c}{D}_{t}^{\alpha}u(t)=Au(t)+f(t)$, $\mathrm{}\mathrm{}t\in (\mathrm{0},T)$, $\mathrm{}u(\mathrm{0})={x}_{\mathrm{0}}$, $\mathrm{}\mathrm{}u\mathrm{\text{'}}(\mathrm{0})={x}_{\mathrm{1}},$ if $A$ is the generator of an $(\alpha ,\beta )$-ROF, we give the relation between the function $v(t)={S}_{\alpha ,\beta}(t){x}_{\mathrm{0}}+({g}_{\mathrm{1}}\mathrm{*}{S}_{\alpha ,\beta})(t){x}_{\mathrm{1}}+({g}_{\alpha -\mathrm{1}}\mathrm{*}{S}_{\alpha ,\beta}\mathrm{*}f)(t)$ and mild solution and classical solution of it. Finally, for the problem ${\mathrm{}}^{c}{D}_{t}^{\alpha}v(t)=Av(t)+{g}_{\beta +\mathrm{1}}(t)x$, $t>\mathrm{0}$, $\mathrm{}{v}^{(k)}(\mathrm{0})=\mathrm{0}$, $k=\mathrm{0,1},\dots \ue899,N-\mathrm{1},$ where $A$ is a linear closed operator. We show that $A$ generates an exponentially bounded $(\alpha ,\beta )$-ROF on a Banach space $X$ if and only if the problem has a unique exponentially bounded classical solution ${v}_{x}$ and $A{v}_{x}\in {L}_{\text{l}\text{o}\text{c}}^{\mathrm{1}}({\mathbb{R}}^{+},X).$ Our results extend and generalize some related results in the literature.

## Citation

Ya-Ning Li. Hong-Rui Sun. "Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem." Abstr. Appl. Anal. 2014 1 - 9, 2014. https://doi.org/10.1155/2014/430418