Applications Of Variational Methods In Difference Equations And Impulsive Differential Equations

This dissertation consists of two parts. The first part (Chapter 2 to Chapter 6) isdevoted to the study of the existence of periodic, homoclinic, heteroclinic solutions, het-eroclinic chains and chaos for a class of difference equations. The numerical heteroclinicsolutions for a class of second order Hamiltonian system are also studied in this part. Thesecond part (Chapter 7 to Chapter 9) is devoted to the study of the existence of periodic,homoclinic and heteroclinic solutions for a class of impulsive differential equations.In the first part, we first proved the existence of periodic solutions, heteroclinic solu-tions and heteroclinic chains (a heteroclinic chain is formed by a number of heteroclinicsolutions which are connected to each other) for a class of difference equations. Then ifthere exists an isolated heteroclinic chain, we obtained the existence of infinitely manyheteroclinic, homoclinic and periodic solutions. Based on these results, we proved thatthese equations possess an approximate Bernoulli shift structure, and its time?T map istopologically chaotic, that is, it has a positive entropy. Specially, if these difference equa-tionssatisfysomeperiodicconditions, weshowedthatthesesystemsareLi-Yorkechaotic.Afterthat, westudiedthenumericalheteroclinicsolutionsforaclassofcontinuousHamil-tonian system, and proved that the heteroclinic solutions of the difference equation whichobtained from the finite difference scheme of the Hamiltonian system are convergent tothe heteroclinic chains of the Hamiltonian system. In the last section of this part, we stud-ied the numerical heteroclinic solutions of the pendulum equation to illustrate the resultswe obtained. We also proved that when the step length of the difference is not very small,discrete pendulum equation is Li-Yorke chaotic.In the second part, we first studied the existence of impulse-generated periodic solu-tions for a class of second order impulsive differential equations. Here a periodic solutionis said to be impulse-generated if its existence depends on the existence of the non-zeroimpulses (the formal definition will be given in the dissertation). Via variational methods,we not only gave the existence of such periodic solutions, but also gave a lower estimationof the number of such solutions. This estimation is totally determined by the number ofthe impulses occurring in a period of the system. Next we proved that when the period ofthe solutions just obtained goes to infinity, these periodic solutions possess a subsequence whichisconvergenttoanimpulse-generatedhomoclinicsolutionoftheimpulsivesystem.In the last section of this part, we proved the existence of heteroclinic solutions for a classof impulsive differential equations. Several results obtained in this part can be applied toordinary differential equations and difference equation as well as impulsive differentialequations, and some new and interesting results were obtained in these applications.