A new method for converting boundary value problems for impulsive fractional differential equations to integral equations and its applications

作者：Liu, Yuji^*

来源：Advances in Nonlinear Analysis, 2019, 8(1): 386-454.

DOI：10.1515/anona-2016-0064

• Impulsive fractional differential equation
• fractional Langevin equation
• anti-periodic boundary value problem
• Riemann-Liouville fractional derivative
• Caputo fractional derivative
• Hadamard fractional derivative
• Caputo-type Hadamard fractional derivative
• Picard iterative technique
• fixed point theorem

In this paper, we present a new method for converting boundary value problems of impulsive fractional differential equations to integral equations. Applications of this method are given to solve some types of anti-periodic boundary value problems for impulsive fractional differential equations. Firstly by using iterative method, we prove existence and uniqueness of solutions of Cauchy problems of differential equations involving Caputo fractional derivative, Riemann-Liouville and Hadamard fractional derivatives with order q is an element of(0, 1), see Theorem 3.1.2, Theorem 3.1.4, Theorem 3.1.6 and Theorem 3.1.8. Then we obtain exact expression of piecewise continuous solutions of these fractional differential equations see Theorem 3.2.1, Theorem 3.2.2, Theorem 3.2.3 and Theorem 3.2.4. Finally, four classes of integral type anti-periodic boundary value problems of singular fractional differential equations with impulse effects are proposed. Sufficient conditions are given for the existence of solutions of these problems. See Theorems 4.1-4.4. We allow the nonlinearity p(t)f(t, x) in fractional differential equations to be singular at t = 0, 1 and be involved a super-linear and sub-linear term. The analysis relies on Schaefer's fixed point theorem. In order to avoid misleading readers, we correct the results in [28] and [65]. We establish sufficient conditions for the existence of solutions of an anti-periodic boundary value problem for impulsive fractional differential equation. The results in [68] are complemented. The results in [81] are corrected. See Lemma 5.1, Lemma 5.7, Lemma 5.10 and Lemma 5.13.

• 出版日期2019-1
• 单位广东财经大学