Weak solutions to boundary value problems for fractional differential equations via variational methods

Volume 9, Issue 5, pp 2971--2981 http://dx.doi.org/10.22436/jnsa.009.05.89

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Authors

Peiluan Li - School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023, P. R. China. Changjin Xu - Guizhou Key Laboratory of Economics System Simulation, Guizhou College of Finance and Economics, Guiyang, 550004, P. R. China. Hui Wang - College of Information Engineering, Henan University of Science and Technology, Luoyang, 471003, P. R. China.

Abstract

Using variational methods, we investigate the solutions to the boundary value problems for fractional order differential equations. First, we consider the eigenvalue problem associated with it. Then, we obtain the existence of at least two weak solutions for every real number via Brezis and Nirenberg's Linking Theorem. Furthermore, for every positive integer k, the existence criteria of k pairs of weak solutions are established by using Clark Theorem. At last, some examples are also given to illustrate the results.

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Keywords

• Fractional differential equations
• critical points
• variational method
• eigenvalue problem.

MSC

•  34B37
•  34K10
•  26A33

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