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 $v(t)=S_{\alpha ,\beta }(t)x_{\mathrm{0}}+(g_{\mathrm{1}}\mathrm{*}{S}_{\alpha ,\beta })(t)x_{\mathrm{1}}+(g_{\alpha -\mathrm{1}}\mathrm{*}{S}_{\alpha ,\beta }\mathrm{*}f)(t)$ and mild solution and classical solution of it. Finally, for the problem ${\mathrm{}}^c{D}_t^{\alpha }v(t)=Av(t)+g_{\beta +\mathrm{1}}(t)x$, $t>\mathrm{0}$, $\mathrm{}{v}^{(k)}(\mathrm{0})=\mathrm{0}$, $k=\mathrm{0,1},\dots ,N-\mathrm{1},$ 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 $A{v}_x\in L_{\text{l}\text{o}\text{c}}^{\mathrm{1}}({ℝ}^{+},X).$ Our results extend and generalize some related results in the literature.