## Abstract and Applied Analysis

### 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 ${D}_{t}^{\alpha }u\left(t\right)=Au\left(t\right)+f\left(t\right), \mathrm{}\mathrm{}t>\mathrm{0}$, where $\mathrm{0}<\alpha <\mathrm{1}.$ When $A$ is the generator of a ${C}_{\mathrm{0}}$-semigroup $\left(T\left(t\right){\right)}_{t\ge \mathrm{0}}$ 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 $u\left(\mathrm{0}\right)=u\left(\mathrm{1}\right)$ admits a unique mild solution for each $f\in C\left(\left[\mathrm{0,1}\right];X\right)$ if and only if the operator $I-{S}_{\alpha }\left(\mathrm{1}\right)$ is invertible. Here, we use the representation ${S}_{\alpha }\left(t\right)x={\int }_{\mathrm{0}}^{\mathrm{\infty }}\mathrm{‍}{\mathrm{\Phi }}_{\alpha }\left(s\right)T\left(s{t}^{\alpha }\right)\mathrm{x }ds, \mathrm{}\mathrm{}t>\mathrm{0}$ 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 $\left(T\left(t\right){\right)}_{t\ge \mathrm{0}}$ is positive, we obtain existence of solutions of the semilinear problem ${D}_{t}^{\alpha }u\left(t\right)=Au\left(t\right)+f\left(t,u\left(t\right)\right),\mathrm{}\mathrm{}\mathrm{}\mathrm{}t>\mathrm{0},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{0}<\alpha <\mathrm{1}.$

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 614328, 11 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393512131

Digital Object Identifier
doi:10.1155/2013/614328

Mathematical Reviews number (MathSciNet)
MR3129328

Zentralblatt MATH identifier
07095167

#### Citation

Keyantuo, Valentin; Lizama, Carlos; Warma, Mahamadi. Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations. Abstr. Appl. Anal. 2013 (2013), Article ID 614328, 11 pages. doi:10.1155/2013/614328. https://projecteuclid.org/euclid.aaa/1393512131

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