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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 614328, 11 pages
http://dx.doi.org/10.1155/2013/614328
Research Article

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

1Department of Mathematics, Faculty of Natural Sciences, University of Puerto Rico, Rio Piedras Campus, P.O. Box 70377, San Juan, PR 00936-8377, USA
2Departamento de Matemática, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

Received 12 June 2013; Revised 2 October 2013; Accepted 4 October 2013

Academic Editor: Fuding Xie

Copyright © 2013 Valentin Keyantuo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate mild solutions of the fractional order nonhomogeneous Cauchy problem , where When is the generator of a -semigroup on a Banach space , 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 admits a unique mild solution for each if and only if the operator is invertible. Here, we use the representation in which is a Wright type function. For the first order case, that is, , the corresponding result was proved by Prüss in 1984. In case is a Banach lattice and the semigroup is positive, we obtain existence of solutions of the semilinear problem