Nonlinear impulsive problems for fractional differential equations and Ulam stability

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Abstract

In this paper, the first purpose is treating Cauchy problems and boundary value problems for nonlinear impulsive differential equations with Caputo fractional derivative. We introduce the concept of piecewise continuous solutions for impulsive Cauchy problems and impulsive boundary value problems respectively. By using a new fixed point theorem, we obtain many new existence, uniqueness and data dependence results of solutions via some generalized singular Gronwall inequalities. The second purpose is discussing Ulam stability for impulsive fractional differential equations. Some new concepts in stability of impulsive fractional differential equations are offered from different perspectives. Some applications of our results are also provided.

Keywords

Impulsive problems
Fractional differential equations
Cauchy problems
Boundary value problems
Ulam stability

Cited by (0)

The first author acknowledges the support by Key Projects of Science and Technology Research in the Chinese Ministry of Education (211169) and Tianyuan Special Funds of the National Natural Science Foundation of China (11026102). The second author was supported by National Natural Science Foundation of China (10971173). The third author acknowledges the support by Grants VEGA-MS1/0507/11, VEGA-SAV 2/0124/10 and APVV-0134-10.