Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations

Abstract

We investigate mild solutions of the fractional order nonhomogeneous Cauchy problem $D_t^{\alpha }u(t)=Au(t)+f(t),\mathrm{}\mathrm{}t>\mathrm{0}$, where When $A$ is the generator of a $C_{\mathrm{0}}$-semigroup $(T(t){)}_{t\ge \mathrm{0}}$ on a Banach space $X$, we obtain an explicit representation of mild solutions of the above problem in terms of the semigroup. We then prove that this problem under the boundary condition $u(\mathrm{0})=u(\mathrm{1})$ admits a unique mild solution for each $f\in C([\mathrm{0,1}];X)$ if and only if the operator $I-S_{\alpha }(\mathrm{1})$ is invertible. Here, we use the representation $S_{\alpha }(t)x={\int }_{\mathrm{0}}^{\mathrm{\infty }}\mathrm{‍}{\mathrm{\Phi }}_{\alpha }(s)T(s{t}^{\alpha })\mathrm{x\hspace{0.17em}}ds,\mathrm{}\mathrm{}t>\mathrm{0}$ in which ${\mathrm{\Phi }}_{\alpha }$ is a Wright type function. For the first order case, that is, $\alpha =\mathrm{1}$, the corresponding result was proved by Prüss in 1984. In case $X$ is a Banach lattice and the semigroup $(T(t){)}_{t\ge \mathrm{0}}$ is positive, we obtain existence of solutions of the semilinear problem