The Existence Of Periodic Solutions To The Second Order Impulsive Differential Equations

The periodic solutions of differential equations arouse the interests of people not only for their demonstration of the periodic movements, but also for their universality. This universality may approxima- tely portray some non-periodic movement, especially the advancing of the issue of resonates which has affected the development of the qualitative theory of differential equations since seventies of twenty century.The Duffing equation is an important equation belongs to the non-linear vibration mechanics, and many mechanical problems can be reduced to corresponding problems of the Duffing equation. Some fast changing phenomena which arise in a certain stage during the process of tackling the problems was named impulsive phenomena. The mode of impulsive phenomena can be categorized into impulsive differential system. A large quantity of remarkable academic researches concerning Duffing equation with impulses have been done in last two and three decades.Some researchers made a research about the existence of positive periodic solutions of differential equations with impulses and delays by using the Krasnoselskii fixed point theorem. Besides, by utilizing a completely continuous operator on cone, they provided some sufficient conditions about positive periodic solutions[29]. They discussed the periodic boundary value problems and provide some existence results by using Schaeffer's fixed point theorem and the method of upper and lower solutions combined with the monotone iterative technique[30]. They studied the existence of periodic solutions for the Duffing equation with impulses and delays by the help of Mawhin's coincidence degree theory and established some sufficient conditions[31].The paper, on the basis of reference [31] and by means of the coincidence degree theory, studies the following differential equation with impulse and delay the author tries to prove the above equation has at least one periodic solution. Theorem A [35] Let X, Y be a Banach space, L: DomLΩX→Y be a Fredholm mapping of index zero,Ωis an open bounded subset of X ,and let N :Ω×[0,1]→Xbe called L-compact onΩ,suppose and deg{ JQN(x,0),Ω∩KerL,0}≠0,Q : Y→Y be a mapping operator, and ImL = KerQ,J:ImQ→KerLcan be the identity mapping, then the equation Lx = N( x,1),has at least on solution lying inΩ.Definition operator thus equation (2.1)has periodic solution and corresponding to the operator equationLemma 2.1[31] Let L be a Fredholm mapping of index zero, and and Here x′( t₀ )=x₀′is original value.Lemma 2.2[31] Let N be L-compact onΩ₀.Theorem Suppose (H₆) there exist a positive constant K such that fxkxR6≤(?)∈Then equation (2.1) has at least one T- periodic solution.