Variational Methods For Second Order Sturm-Liouville Boundary Value Problems On Time Scales

In this paper, using variational methods, we consider a class of second or-der Sturm-Liouville boundary value problems with positive parameterλon time scales (?)In Chapter 2, we research the existence of solutions for (Pλ).In Section 1, we con-struct the corresponding variational framework for the problem (Pλ) and present the properties of variational framework. In Section 2, we consider the second or-der Sturm-Liouville eigenvalues problems and give the characterization of the first eigenvalue in Sobolev space, then, we prove the existence interval of A such that the problem (Pa) has one or two solution by direct variational method, Mountain pass lemma and the critical point theorem of Brezis and Nirenberg. In Section 3, we derive the existence of three solutions of (Pλ) whenλlies in a suitable in-terval by using a three critical point theorem Bonanno and Candito established. Finally, we give some examples to illustrate our results in Section 4.In Chapter 3, we study the existence of positive solutions for (Pλ). In Section 1, we prove the existence of positive solutions for (Pλ) is equivalent to existence of solutions for the following boundary value problems when the nonlinear term satisfies that f(t,O) is positive and p(t) is non-decreasing (?) Then, we construct the corresponding variational framework for the problem (P+λ) and present the properties of variational framework. In Section 2, we derive the existence of positive solutions of (P+λ) whenλlies in a suitable interval by direct variational method and saddle point theorem. In the final section, we further discuss the existence of positive solutions for (Pλ) by method of supersolutions and subsolutions.