Solutions To 2m-order Neumann Boundary Value Problems

In this paper, we discuss the existence and multiplicity of solutions to the following 2mth-order differential equation Neumann boundary value problem where Lu= is an 2mth-order linear differential operator, ai∈R1,i= 1,2,…,m,f∈C1([0,1]×R1,R1).This paper is mainly composed of three chapters:Chapter 1 is Introduction. Chapter 2 gives preliminaries and some necessary lemmas and theorems. In Chapter 3, we obtain some existence results of solutions to problem (1.1) when f satisfies specific conditions.In paper [1], the authors obtained all kinds of existence results of solutions to problem (1.1) by using the strongly monotone operator principle and the critical point theory. In paper [2], the authors studied the existence and multiplicity of positive solutions and sign-changing solutions to fourth-order Neumann boundary value prob-lem by using fixed-point index theory and Morse theory. At the same time, the paper [3] studied fourth-order Neumann boundary value problems with two parameters and with variable coefficient by using Morse theory, the critical point theory, fixed-point in-dex theory and Leggett-Williams three solutions theorem. Moreover, motivated by [4], some papers studied existence results of solutions to higher order differential equation Neuamnn boundary value problem with parameters by applying local linking theorem and the critical point theory. On this basis, we further study the problem (1.1).In this paper, Neumann boundary value problem (1.1) is transformed into the in-tegral equation, and then by using the square root operator and the Morse theory, we obtained the existence and multiplicity of solutions to the problem (1.1) under some conditions of f. Compared with the results of this aspect, theorem 3.1 gives the result that the problem (1.1) possesses at least one nontrivial solution under weaken condi-tions, and theorem 3.2 proves that the problem (1.1) possesses at least one nontrivial solution directly using Morse theory,improving the methods. Theorem 3.3,theorem 3.4 give the results that the problem (1.1) possesses at least three nontrivial solutions and at least n pairs of solutions. These results are new. At the end,theorem 3.5 proves that the problem (1.1) possesses infinitely many solutions under weaken conditions directly using Morse theory. These are the main difference from other literatures.We realign the eigenvalues of the linear differential operator L and denoteλ0<λ1<λ2<…, where is the eigenpolynomial of the differential operator L. The following assumption holds throughout the paper:The following theorems are the main results of this paper:Theorem 3.1 Assume that f satisfies the conditions:(f1)f (t, 0)＝0, t∈[0,1];(f2) limx→0 F(t, x)/x2< (1-c)/2 for all t∈[0,1], where c>-p(k2π2),k∈N0,F(t,x)＝∫0x f(t,y)dy;(f3) there existμ∈(0,1/2), R> 0 such that 0< F(t, x)≤μxf(t, x)+(μ-1/2)cx2 for all t∈[0,1],|x|≥R.Then the problem (1.1) possesses at least one nontrivial solution.Compared with the conditions of [1, Theorem 4.13, p.973], the condition (f2) of theorem 3.1 is weaken than (A2). Moreover it is simpler.Theorem 3.2 Assume that f satisfies the conditions (f1), (f3) and(f4) there exists n∈No such thatλn< fx'(t,0)<λn+1, t∈[0,1]. Then the problem (1.1) possesses at least one nontrivial solution.Theorem 3.2 proves directly the result with Morse theory, improving the method that we proved results indirectly using the conclusions proved.Theorem 3.3 Assume that f satisfies the conditions (f1), (f4) and(f5) there exist a, b∈R1 withα<λ0/2 such that F(t, x)≤ax2+b, (t, x)∈[0,1]×R1.Then the problem (1.1) possesses at least three nontrivial solutions.Theorem 3.4. Assume that f satisfies the conditions (f5) and(f6)/(t,-x)=-f(t,x),(t,x)∈[0,1]×R1; (f7) there existε> 0 such that F(t, x)≥((λn+ε)/2)x2, (t, x)∈[0,1]×[-δ,δ]. Then the problem (1.1) possesses at least n pairs of solutions in C2m[0,1].Theorem 3.5. Assume that f satisfies the conditions (f3) and (f6). Then the problem (1.1) possesses infinitely many solutions.Compared with the conditions of [1, Theorem 4.14, p.974], theorem 3.5 removes the condition that f needs to satisfy (?)sup and +∞uniformly for t∈[0,1]. Then It proves directly the result with Morse theory.