Periodic Solutions For Superlinear Impulsive Equations With Time-dependent Potential

In this paper,we consider the existence,multiplicity of periodic solutions for super-linear impulsive equations with time-dependent potential by using fixed point theorems The thesis consists of the following two parts1.The existence and multiplicity of periodic solutions for super linear time-dependent Hamiltonian equations with impulsive terms2.The existence of periodic solutions for non-conservative superlinear time-dependent equations with impulsive termsConsider superlinear second order differential equations with impulsive terms,when impulsive terms are bounded or oriented,it is relatively easy to deal with the influence of the impulsive terms on the rotation,the existence of periodic solutions for the equation is discussed by analyzing rotation of the solution for angular in the phase plane.When the impulsive terms is linear or superlinear and the equation satisfies the global existence of the solution,by analyzing the rotation angle of the global homomorphism represented by the rectangular coordinate on the phase plane,the influence of the impulsive terms on the rotation is dealt with,and the existence of the periodic solution of the equation is discussed in previous researches.How to discuss the existence and multiplicity of periodic solutions of the equation under the absence of global existence conditions and the condition that impulsive terms are unbounded or degenerate has not been achieved before.The motivation of this thesis is to explore and study these problemsWe introduce the concept of spiral curves,and analyze the spiral property of the solution on the phase plane under the influence of the jump map brought by the impulsive term.Then a priori estimate of some specific solutions is obtained by the spiral property of the solution,which inspires a method for dealing with the problem without global existence conditionIn the first part of the thesis,we consider the existence and multiplicity of peri-odic solutions for conservative impulsive equations with time-dependent potential,the equation is partial superlinear,and the jump maps are global orientation-preserving and area-preserving homomorphisms with the finite twist property.We use the Poincare-Birkhoff twist theorem,combined with the analysis of the spiral property of the solution and the analysis of the rotation angle of the global homeomorphism represented by the rectangular coordinate on the phase plane,the existence and multiplicity of periodic so-lutions for conservative impulsive equations with time-dependent potential are proved At the same time,the finite twist properties of asymptotic homeomorphisms of global homeomorphisms with finite twist properties are also givenIn the second part of the thesis,we consider the existence of periodic solutions for non-conservative time-dependent equations with impulsive terms,where the equations has the property of spiral and rapid rotation,the jump maps are linear maps,which may be degraded,and the analysis on the phase plane is not limited to infinity.When the jump maps are non-degenerate linear maps,we use the twist theorem for non area-preserving continuous mappings.When the jump maps are degenerate linear maps,using the theory of topological degree,we establish a partial fixed point theorem with an angular description,this theorem is suitable for dealing with degenerate situations.Combined with the technique of spiral curve and the angle estimation of linear maps,we prove the existence of periodic solutions for non-conservative superlinear time-dependent equations with impulsive terms.