Existence And Multiplicity Results Of Positive Solutions Of Boundary Value Problems For Nonlinear Difference Equations

This dissertation deals with the positive solutions of boundary value problemsfor second order self-adjoint difference equations, resonant difference equations, p-Laplacian difference equations and semipositone difference equations with param-eters by applying critical point theory and its generalization-nonsmooth criticalpoint theory. In short, we reduce the problem of finding the positive solutions ofboundary value problems for these difference equations to the problems of seek-ing the critical points of corresponding functional on suitable function space. Weobtain some new results. This dissertation is composed of five chatpers.Chapter 1 concentrates on the brief introduction of the historic backgroundand up-to-date progress, main results in this dissertation and some primary toolson the investigated problems.Chapter 2 is devoted to the study of positive solutions of second order self-adjoint difference equations with two classes of different boundary conditions.When the nonlinearity is an odd function, multiple positive solutions for the sec-ond order self-adjoint difference equation with the Neumann boundary conditionΔu(0) = 0,Δu(T) = 0 are obtained by applying Clark's theorem. For the mixedboundary condition u(0) = 0,Δu(T) = 0, the existence of positive solutions for sec-ond order self-adjoint difference equation is also studied via Mountain Pass Lemmawhen the nonlinearity is nonnegative and asymptotically linear at both 0+ and +∞.It's the first time in the literature to consider the existence of multiple positivesolutions of resonant difference equations in Chapter 3. The corresponding func-tional of the problem is constructed. When the nonlinearity is an odd function,we obtain some suffcient conditions which guarantee the existence of the multi-ple positive solutions of the problem by using a theorem given by Goeleven andMotreanu.In Chapter 4, we investigate positive solutions of boundary value problems fora class of p-Laplacian difference equations. In the case that p = 2, multiple positivesolutions of the problem are obtained by applying Clark's theorem. When p≠2,by using three critical point theorem, we prove that the problem has at least twopositive solutions.It's also the first time for us to discuss the existece of positive solutions ofboundary value probems for semipositone difference equation with parameters in Chapter 5. For semipositone problem with one parameter, we prove that the semi-positone problem has at least two positive solutions by using nonsmooth threecritical points theorem. For two-parameter-semipositone problem, we apply nons-mooth Mountain Pass Theorem and super-subsolutions method to study the exis-tence, multiplicity and nonexistence of the semipositone problem.