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Extremal solutions of Cauchy problems for abstract fractional differential equations

1Guizhou Normal College Guiyang

2Xiangtan University Xiangtan

3Comenius University Bratislava

© 2013 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

Citation Information: Mathematica Slovaca. Volume 63, Issue 4, Pages 769–792, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: 10.2478/s12175-013-0134-1, August 2013

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Published Online:


In this paper, we study the extremal solutions of Cauchy problems for abstract fractional differential equations. Some definitions such as L 1-Lipschitz-like, L 1-Carathéodory-like and L 1-Chandrabhan-like are introduced. By virtue of the singular integral inequalities with several nonlinearities due to Medved’, the properties of solutions are given. By using a hybrid fixed point theorem due to Dhage, existence results for extremal solutions are established. Finally, we present an example to illustrate our main results.

MSC: Primary 26A33, 06D35, 34A40.

Keywords: fractional differential equations; extremal solutions; existence; fixed point method

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