Existence of solution for impulsive hybrid differential equation

Purpose: Many problems in science and engineering can be modeled by nonlinear differential equations with initial conditions. The logistic equation is used widely in the study of population growth and decay. The logistic equation also has applications in ecology, statistics, medicine, chemistry, and physics, to name a few. Over time, the population growth or decay tends toward the stable equilibrium. However, in reality, some populations approach extinction or grow away from the equilibrium, and a new equilibrium will be formed. In addition, due to natural causes or unpredictable circumstances like explosions in an oil rig, hurricanes, or flooding at some time, t, the population will jump down or up in a small interval of time. These combined situations can be modeled as impulsive hybrid differential equations. In general, solutions to the nonlinear differential equations cannot be computed analytically.Methods: It is known that a fruitful method,which is both theoretical and computational, is the generalized monotone method with coupled lower and upper solutions. Furthermore, under uniqueness assumption, we proved the existence of a unique solution.Results: In this work, we extend the generalized monotone method with coupled upper and lower solutions to impulsive hybrid nonlinear differential equation.Conclusions: Finally, we provide some numerical examples.