Existence, Uniqueness and UHR Stability of Solutions to Nonlinear Ordinary Differential Equations with Noninstantaneous Impulses

Xuping Zhang 1  and Zhen Xin 2
  • 1 Department of Mathematics, Northwest Normal University, #967 anning east road, Lanzhou, China
  • 2 Department of Mathematics, Northwest Normal University, Gansu 730070, Lanzhou, China
Xuping Zhang
  • Corresponding author
  • Department of Mathematics, Northwest Normal University, #967 anning east road, Lanzhou, 730070, China
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and Zhen Xin
  • Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, China
  • Search for other articles:
  • degruyter.comGoogle Scholar

Abstract

We consider the existence, uniqueness and Ulam–Hyers–Rassias stability of solutions to the initial value problem with noninstantaneous impulses on ordered Banach spaces. The existence and uniqueness of solutions for nonlinear ordinary differential equation with noninstantaneous impulses is obtained by using perturbation technique, monotone iterative method and a new estimation technique of the measure of noncompactness under the situation that the corresponding noninstantaneous impulsive functions gi are compact and not compact, respectively. Furthermore, the UHR stability of solutions is also obtained, which provides an approach to find approximate solution to noninstantaneous impulsive equations in the sense of UHR stability.

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