Abstract
In this paper we establish the existence of solutions for the nonlinear abstract Cauchy problem of order α ∈ (1, 2), where the fractional derivative is considered in the sense of Caputo. The autonomous and nonautonomous cases are studied. We assume the existence of an α-resolvent family for the homogeneous linear problem. By using this α-resolvent family and appropriate conditions on the forcing function, we study the existence of classical solutions of the nonhomogeneus semilinear problem. The non-autonomous problem is discussed as a perturbation of the autonomous case. We establish a variation of the constants formula for the nonautonomous and nonhomogeneous equation.
Acknowledgements
The authors are very grateful to the editor and the anonymous reviewers for their careful reading of the manuscript, comments and suggestions, which allowed to significantly improve the original version of the text.
H. R. Henríquez was partially supported by Vicerrectoría de Investigación, Desarrollo e Innovación de la Universidad de Santiago under Grant DICYT-USACH 041733HM; V. Poblete was partially supported by project Fondecyt 1191137, and J. C. Pozo was partially supported by project Fondecyt 1181084.
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