Resonance Of Impulsive Differential Equations

In this paper,we consider the resonance of impulsive differential equations,including the following three problems:1.The resonate phenomenon of linear impulsive equation and the existence of peri-odic solution under the Landesman-Lazer condition2.The coexistence of periodic solution and unbounded solutions of weakly nonlinear impulsive equations3.The multiplicity of periodic solutions of impulsive equations with oscillatory po-tential.The necessary and suffcient conditions for the existence of periodic solutions,the coexistence of periodic solutions and unbounded solutions,and the multiple solutions of periodic solutions are important ways to understand the resonance phenomenon of impulsive differential equations.We know little about them.This paper uses phase-plane analysis,topological degree and Poincaré-Birkhoff twist theorem to study related problems and obtains some new resultsIn the first part,we study the resonance of linear impulsive equations,that is,we dis-cuss the sufficient and necessary conditions for the existence of periodic solutions of linear impulsive equations.In the approach of using the variational method and critical point theory to discuss impulsive equations,the periodic solution or the solution of boundary value problems is transformed into the critical point of a functional.Thus one needs that the solutions of the impulsive equations don't have impulse and only the derivatives of the solutions have impulses.We apply topological approach,allowing both the solution and the derivative of the solution to have impulses.We first convert the periodic solu-tion problem of impulsive differential equations into fixed point problems,then analyze a prior condition that homotopy needs in the topological approach,and then we obtain the existence of periodic solutions of impulsive equations.We then proved that all solutions are unbounded when the periodic solution does not exist by the method of variation of constants.At the same time,we also use the degree of topology to discuss the existence of periodic solutions under Landesman-Lazer condition.In the second part,we consider the coexistence of periodic solutions and unbounded solutions of weakly nonlinear impulsive differential equations.This is a typical non-linear phenomenon.We use the successor map and our approach is based on the estimations of the time when the solutions are between those two adjacent zeros and the corresponding differences of the derivative values of the solutions.These estimations are used to get a prior condition required in the topological approach,and then we prove the existence of the periodic solution of the equation using the properties of the topology degree.The proof of the existence of the unbounded solution is mainly using Poincaré map to construct a Liapunov function that makes the solution grow infinitely along the orbit.So that the solution of the equation is unbounded.In the third part,we study the multiplicity of periodic solutions for impulsive dif-ferential equation.First of all,for its autonomous equation,at the impulse point,under the influence of the impulse,the two motions with and without impulses are different.We introduce some assumption on impulsive functions,to control these differences such that the information valid for the equation without impulse can always be used for the impulsive one.Then the phase plane analysis is used to consider the influence of the non-autonomous equation,and the problem is reduced to the small disturbance of the autonomous equation.By estimating the relationship between the solutions of the two equation,the Poincaré-Birkhoff twist theorem is used to obtain an infinite number of periodic solutions of the equation.