Non-instantaneous impulsive fractional-order implicit differential equations with random effects

ABSTRACT

In this article, we study existence and stability of a class of non-instantaneous impulsive fractional-order implicit differential equations with random effects. First, we establish a framework to study impulsive fractional sample path associated with impulsive fractional L^p-problem, and present the relationship between them. We also derive the formula of the solution for inhomogeneous impulsive fractional L^p-problem and sample path. Second, we construct a sequence of Picard functions, which admits us to apply successive approximations method to seek the solution of impulsive fractional sample path. Further, we derive the existence of solutions to impulsive fractional L^p-problem. Third, the concepts of Ulam's type stability are introduced and sufficient conditions to guarantee Ulam–Hyers–Rassias stability are derived. Finally, an example is given to illustrate the theoretical results.

KEYWORDS:

• Non-instantaneous impulsive
• fractional-order implicit differential equations
• random effects
• existence
• Ulam–Hyers–Rassias stability

MATHEMATICS SUBJECT CLASSIFICATION:

• 34A08
• 34A37
• 34F05

Acknowledgments

The authors are grateful to the referees and editor for their careful reading of the manuscript and valuable comments.

Funding

The authors acknowledge the National Natural Science Foundation of China (11661016) and Training Object of High Level and Innovative Talents of Guizhou Province ([2016]4006), Unite Foundation of Guizhou Province ([2015]7640) and Graduate Course of Guizhou University (ZDKC[2015]003).