Application Of Critical Point Theory In Delay Differential Cycle Solution

This PH.D.Thesis mainly concerns the applications and generalizations of critical point theory to investigating periodic solutions,multiple periodic solutions and subharmonic solutions for nonlinear differential equations with deviating arguments.It is composed of four chapters.In Chapterâ… ,we introduce the historical background and the recent development of problems to be studied,and main results of this paper are also outlined.In Chapterâ…¡,a new geometrical index is developed,which is a normal index with dimensional property and is a generalization of the Z₂ index and Z_p index in the literature. By using this newly developed index theory,we investigate the multiplicity of subharmonic solutions for nonlinear Hamiltonian systems and second order Hamiltonian systems. Under the same hypotheses,more subharmonic solutions are obtained than those given in the literature,and by these results,a new answer to a conjecture posed by R.Michalek and Liu Jiaquan is given.In this chapter,we also establish an existence theorem of nontrivial periodic solutions to a class of second order Hamiltonian systems with potential changing its sign,and give a solution under centain hypotheses to one of the open problems posed by Antonacci.In Chapterâ…¢,some sufficient conditions verifying variational structure for a given differential system are introduced.According to these results,we develop the variational structures for some first and second order differential difference equations,and study the existence of multiple periodic solutions and subharmonic solutions for these equations by newly developed Z_p geometrical index theory.Some new existence and multiplicity results are obtained for these delay differential systems and even for related ordinary differential systems.We also apply critical point theory to investigate the subharmonic solutions for a class of neutral functional differential equations,and therefore provide a new approach to deal with such problems.In Chapterâ…£,nontrivial periodic solutions for a delay differential equations has been studied.By using saddle point reduction and Morse theory,we obtain an existence theorem of three periodic solutions for this equation.By means of E⁺-Morse theory established by A.Abbondandolo,we also give a lower bound of the number of nontrivial periodic solutions for a differential difference equation.These conclusions generalize the same results on autonomous Hamiltonian systems and second order conservative systems to differential difference equations.In the last two sections of this chapter,we give two applications of the new Z_p geometrical index theory to the existence of periodic solutions and subharmonic solutions for a class of wave equations with deviating arguments.Some new existence and multiplicity theorems are given,which improve the related results on wave equations with normal variables. Due to a substantial generalization of Z_p geometrical index theory,we succeed in applying critical point theory to investigate the periodic solutions of delay differential equations,and solve some important problems.Therefore,this Ph.D.thesis will play an important role in the development of qualitative theory of functional differential equations.