| It is well known that boundary value problems for nonlinear high-order differential equations have extremely abundant source and wide applications in the. field of physics. If is significant to study the existence and multiplicity of solutions to these problems not only in theory but in practice. This thesis is mainly composed of three chapters in which we discuss some fourth-order and sixth-order 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 a kind of fourth-order two-point boundary value problem. We establish some sufficient conditions on nonlinear term f which arc 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 Nonlinear Anal, the core magazine embodied by SCI. In chapter II, we use the fixed-point, index thoery to consider some fourth-order two-point boundary value problem. The model we discuss is similar to the one discussed in chapter I. We mainly study the existence of sign-changing solution to the problem. Now the research on the existence of sign-changing solution has been attached great importance. In chapter III, we discuss the existence of three positive concave solutions to a kind of sixth-order two-point boundary value problem by using Leggett-Williams' fixed point theorem. This chapter has been published on ,/. Shanxi University.In the following, we state the main results of this thesis concretely .In chapter I, we mainly use the strongly monotone operator principle and the critical point theory to consider the following fourth-order two-point boundary value problem (BVP):where f : [0,1] is continuous.Firstly, we use the strongly monotone operator principle to discuss the existence and uniqueness of solution to BVP (1.1.1). The main results can be stated as the following.Theorem 1.3.1. Suppose that for each t ∈ [0,1], f{t,u) is a nonincreasing function in u, i.e., f(t, u1) ≥ f{t,u2) for all u1 and u2 in R1 with u1 < u2. Then BVP (1.1.1) has a unique solution in C~4[0,l].Theorem 1.3.2. If there exists α ∈ [0,4) such that [f(t,u) - f(t,v)}[u - v]≤ a|u-v2| for all t ∈ [0, 1], and u,v ∈ R~1, then BVP (1.1.1) has a unique solution inC4[O,1].Consequently, the critical point theory is employed to discuss the existence and multiplicity of solutions to BVP (1.1.1). The main results are the following.Theorem 1.4.1. Suppose thatpu/ f(t,v)dv ^ -u2 + b(t)\u\2~1 + c(t), te[0,1], Jo *■where a G [0,tt4), 7 G (0,2), b G L2^[0,1], and c G L[0,1]. Then BVP (1.1.1) has at least one solution in C4[0,1].The famous mountain pass lemma is employed to prove the following theorem.Theorem 1.5.1. Suppose that(Hi) there exist /i G [0,1/2) and M > 0 such that F(t,u) = /?" f(t,v)dv ^ fiuf[t, u) for all \u\ ^ M and t G [0,1];(H2) limsup^Q /(￡, u)/u < ir4, liminfu_>+oo /(￡, u)/u > n4 uniformly for t G[0,1]-Then BVP (1.1.1) has at least one nonzero solution in C4[0,1].We use the even functional critical point theorem to prove the following theorem. Theorem 1.6.1. Suppose that f(t,u) is odd in u, i.e., f(t,—u) = —f(t,-u) for all t G [0,1] and u G R1. And suppose that (Hi) in Theorem 1.5.1 is satisfied andlimsup/(￡, u)/u < 7r4, liminf f(t, u)/u = +00 uniformly for t G [0,1].Then BVP (1.1.1) has infinitely many solutions in C4[0,1].In chapter two, we mainly discuss the following fourth-order two-point boundary value problem (BVP):), te[o,i],u(O)=u(l) = O, (2.1.1)u"(0) = u"(l) = 0,where / : R1 —> R1 is continuous.We can see that the problem is simliar to BVP (1.1.1) discussed in chapter I. We mainly use the fixed-point index theory to discuss the existence of sign-changing solution to BVP (2.1.1). The main result is the following.Theorem 2.1.1. Suppose the following conditions hold(Hi) / : R1 —> R1 is continuous, strictly increasing, and /(0) = 0;(H2) limu^of (u)/u = fa and p0 G ((2￡;07r)4, ((2k0 4- 1)tt)4), where k0 is a positive integer.
|
PDF Full Text Request