| This thesis is mainly concerned with the existence of positive solutions of singular sec-ond order periodic boundary value problems with sign-changing nonlinearities and extremalsolutions of three-point boundary value problems for second order impulsive differentialequations with deviating arguments. Some results are obtained by virtue of the fixed pointindex theory and the method of lower and upper solutions coupled with monotone itera-tive technique, respectively, which either extend or complement some previously obtainedresults.It consists of three chapters.As the introduction, in the first chapter, the background and history of two-point andthree-point boundary value problems of ordinary and impulsive differential equations arebrie?y addressed and the main work of this paper are given.In Chapter 2, by using the fixed point index theory, we discuss the existence of positivesolutions of singular second order periodic boundary value problems with sign-changingnonlinearities. With the help of a new integral representation of the solution of the associatedlinear periodic boundary value problems, we obtain some sufficient conditions guaranteeingthe existence of at least one positive solution followed by one existence result of positiveperiodic solutions. Finally, we give some examples to illustrate the validity of our results.Chapter 3 deals with the existence of extremal solutions of three-point boundary valueproblem for second order impulsive differential equations with deviating arguments andnonlinear boundary conditions. Some new comparison results under weaker conditions aregiven, which allow us to introduce new concept of lower and upper solutions. Meanwhile,the monotone iterative technique is employed to establish some sufficient conditions on theexistence of solutions of this problem. Our results extend part of previous results of thecorresponding ordinary differential equations. Moreover, some examples with nonlinearboundary conditions are presented.
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