## Abstract and Applied Analysis

### Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem

#### 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 $v(t)={S}_{\alpha ,\beta }(t){x}_{\mathrm{0}}+({g}_{\mathrm{1}}\mathrm{\ast}{S}_{\alpha ,\beta })(t){x}_{\mathrm{1}}+({g}_{\alpha -\mathrm{1}}\mathrm{\ast}{S}_{\alpha ,\beta }\mathrm{\ast}f)(t)$ and mild solution and classical solution of it. Finally, for the problem ${\mathrm{}}^{c}{D}_{t}^{\alpha }v(t)=Av(t)+{g}_{\beta +\mathrm{1}}(t)x$, $t>\mathrm{0}$, $\mathrm{}{v}^{(k)}(\mathrm{0})=\mathrm{0}$, $k=\mathrm{0,1},\dots \!,N-\mathrm{1},$ 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 $A{v}_{x}\in {L}_{\text{l}\text{o}\text{c}}^{\mathrm{1}}({\Bbb R}^{+},X).$ Our results extend and generalize some related results in the literature.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 430418, 9 pages.

Dates
First available in Project Euclid: 26 March 2014

https://projecteuclid.org/euclid.aaa/1395858516

Digital Object Identifier
doi:10.1155/2014/430418

Mathematical Reviews number (MathSciNet)
MR3166612

Zentralblatt MATH identifier
07022377

#### Citation

Li, Ya-Ning; Sun, Hong-Rui. Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem. Abstr. Appl. Anal. 2014 (2014), Article ID 430418, 9 pages. doi:10.1155/2014/430418. https://projecteuclid.org/euclid.aaa/1395858516

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