# Existence of solutions for nonlinear Caputo-Hadamard fractional differential equations via the method of upper and lower solutions

Volume 10, Issue 11, pp 5744--5752 Publication Date: November 15, 2017       Article History
• 880 Views ### Authors

Yunru Bai - Institute of Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Lojasiewicza 6, 30-348 Krakow, Poland. - Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, China. Hua Kong - Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, China.

### Abstract

The purpose of this paper is devoted to consider the existence of solutions for a class of nonlinear Caputo-Hadamard fractional differential equations with integral terms ((CHFDE), for short). Firstly, by applying the semi-group property of Hadamard fractional integral operator, a necessary condition of solvability for (CHFDE) is established. Then, under the suitable conditions, we prove the solution set of (CHFDE) is nonempty by using the method of upper and lower solutions, and Arzel\`{a}-Ascoli theorem. Finally, we present several numerical examples to explicate the main results.

### Keywords

• fractional differential equations
• upper and lower solutions
• monotone sequences
• Arzela-Ascoli theorem

•  49J53
•  49K40
•  90C33
•  90C46

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