| Nonlinear phenomenon is the general phenomenon in natural world. Nonlinear problems are the general problems in the natural science and engineering areas, which determined the importance of the research of nonlinear differential equation systems.To the differential equation Dirichlet boundary value problems or periodic boundary value problems, predecessors already have fruitful articles([1]-[8]). In [1,2,3], Li studies the existence of positive solutions to the fourth or higher-order ordinary differential equation with the Dirichlet boundary condition or periodic boundary condition. However, as the difficulties of calculations and researches, people rarely study the Neumann boundary value problems. The ordinary differential equation Neumann boundary value problem still be worth to study. Thus, our topic is fresh.This paper mainly investigates the existence of the positive, negative and sign-changing solutions to the2mth-order ordinary differential equation Neumann boundary value problems (BVP) where is a2mth-order linear differential operator,α0≠0, αj∈=1,2,...,m-1, and m∈N.In [4], Li, Zhang and Li just investigates the existence sign-changing solutions to problem (1-1.1) in the case of m=2,α0=α1=1. This paper is the extension of [4], thus the content is creativeness.We first study the existence of positive solutions to the Neumann boundary value prob-lems (1.1.1) via Fixed point index theory ([9,10]). Then, we investigate the existence of multiple and sign-changing solutions to the Neumann boundary value problems (1.1.1) via Fixed point index theory, Critical group and Morse theory, while the nonlinearity is in reso-nant case and non-resonant case, respectively. Thus, the process of our research is innovative.Let p2m be the characteristic polynomial of L2m, N(p2m) be the set of all null points of p2m in the complex plane C Denote f∞=(?)f (t,x)/x,f∞=(?)f(t,x)/x.Our main results are as follows:Theorem1.3.1.If α0>0and f∈C([0,1]×R+,R+).Assume N(p2m)(?){z∈C:|Imz|<Ï€/2).(H0) If one of the conditions holds:(i)f0<α0<f∞；(ii)f∞<α0<f0. Then the Neumann boundary value problems(1.1.1)has at least one positive solution.Theorem2.2.1.Assume(H0)and if f satisfies:(f1)f∈C1([0,l]×R,R)and f(t,x)x≥0for all(t,x)∈[0,1]×Rï¼›(f2)there exist two positive integers n0,n∞.such that μ2n0-1<α0<μ2n0,μ2n∞-1<α∞<μ2n∞, where α0=limxâ†'0f(t,x)/x,α∞=lim|x|â†'∞,(t,z)/x uniformly for t∈[0,1]ï¼›(f3)there exists a constant T>0such that|f(t,x)|<α0T,t∈[0,1],|x|≤T. Then the BVP(1.1.1)has at least six different nontrivial solutions:two positive solutions, two negative solutions and two sign-changing solutions.Theorem2.2.2.Suppose that(H0),(f1)-(f3)hold and(f4)f is odd in x,that is,f(t,-x)=-f(t,x),(t,x)∈[0,1]×R. Then the BVP(1.1.1)has at least eight different nontrivial solutions:two positive solutions, two negative solutions and four sign-changing solutions.Theorem3.2.1.Suppose the conditions(H0),(f1),(f3)hold.And f satisfies:(f5)fx’(t,0)=μ2n0for some no≥1with μ2n0-1<μ2n0and for all t∈[0,1].Moreover, there exists δ>0such that f(t,x)x≤μ2n0x2for all(t,x)∈[0,1]×[-δ,δ].(f6)there exist n∞≥1with μ2n∞-1<μ2n∞,C1>0,and α∈(0,1)such that lim|x|â†'∞f(t,x)/x=μ2n∞exists uniformly for t∈[0,1]and|f(t,x)-μ2n∞x|≤Cl(1+|x|α),(t,x)∈[0,1]×R, uniformly for t∈[0,1], where F(t,x)=(?)f(t,y)dy. Then the BVP (1.1.1) has at least six nontrivial solutions. Moreover, if BVP (1.1.1) has only finitely many solutions then, of these solutions, there are two positive solutions, two negative solutions and two sign-changing solutions.Theorem3.2.2. Suppose that the conditions (H0),(f1) and (f3)-(f6) hold. Then BVP (1.1.1) has at least eight different nontrivial solutions. Moreover, if BVP (1.1.1) has only finitely many solutions then, of these solutions, there are two positive solutions, two negative solutions and two sign-changing solutions. |
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