Existence And Multiplicity Of Solutions To High Order Boundary Value Problems

We mainly study the existence and multiplicity of solutions to high order differential equations in this paper. This thesis is mainly composed of three chapters in which we discuss some fourth-order with two parameters and fourth-order singular boundary value problems for nonlinear ordinary differential equations. In Chapter I, we mainly use the strongly monotone operator principle and the critical point theory to discuss the existence and multiplicity of solutions to a kind of nonlinear operator equations. As an application, we investigate the existence and multiplicity of solutions to fourth-order boundary value problems for ordinary differential equations with two parameters, and give some new existence results of solutions. We establish some sufficient conditions on nonlinearity f which are able to guarantee that the problem has a unique solution, at least one nonzero solution, and infinitely many solutions. This part has been accepted by J. Math. Anal. Appl, see [17], the core magazine embodied by SCI. In Chapter II, we use the abstract results about nonlinear operator equations to study fourth-order singular boundary value problems. Under certain conditions on nonlinearity f, we establish conditions such that the problem has at least two positive solutions, at least one positive solution and no solutions, respectively. This chapter has been published on J. Shanxi Univ., see [24]. In Chapter III, the model we discuss is similar to the one discussed in Chapter I. Firstly, we mainly study the existence of sign-changing solution to nonlinear operator equations by using the topological degree and fixed point index theory. Furthermore, the theoretical results are successfully applied to fourth-order boundary value problems with two parameters. The main theorems, in fact, are some new three-solution theorems which involve in sign-changing solutions and generalize Amann's three-solution theorems. This part has been accepted by J. Math. Anal. Appl., see [18], the core magazine embodied by SCI.In the following, we state the main results of this thesis concretely.In Chapter I, we mainly consider the following fourth-order boundary value problem with two paremeters (BVP):The main results can be stated as the following.