Based on the monotone iteration method and Leray–Schauder degree theory, we obtain the Ambrosetti-Prodi type results for multiparameter Neumann systems with mean curvature operator in Minkowski space {[ -(u'/√(1-u'^2))'=f(x,u,v)+r+h(x), x∈ (0,1),; -(v'/√(1-v'^2))'=g(x,u,v)+s+l(x), x∈ (0,1),; u'(0)=u'(1)=0, v'(0)=v'(1)=0, ]. where f,g∈ C([0,1]×ℝ×ℝ), r,s∈ℝ are parameters, h,l∈ C[0,1], and ∫ _0^1h(x)dx=0,∫ _0^1l(x)dx=0.

Abstract In this paper, we focus on the existence of positive solutions for nonlinear fourth-order Neumann boundary value problem where k 1 and k 2 are constants, λ > 0 is the bifurcation parameter, f ∈ C([0, 1] × ℝ+, ℝ), ℝ+ := [0, ∞). We first discuss the sign properties of Green’s function for the elastic beam boundary value problem, and then we show that there exists a global branch of solutions emanating from infinity under some different growth conditions. In addition, we prove that for λ near the bifurcation points, solutions of large norm are indeed positive. The technique for dealing with this paper relies on the global bifurcation theory.

In this paper, the authors obtain an Ambrosetti–Prodi type result of T-periodic solutions of the first-order differential equations u'(t)=a(t)u(t)-f(t,u(t))+s, t∈ R by topological degree method, where T>0 is a constant, a:ℝ→ [0,∞ ) are T-periodic functions with ∫ _0^Ta(t)dt=0 , s∈ℝ is a parameter; f is a Carathéodory function and T-periodic for the first variable, which f(t, u) satisfies locally coercive at infinity.

In this work, we investigate the continuum of one‐sign solutions of the nonlinear one‐dimensional Minkowski‐curvature equation with nonlinear boundary conditions by using unilateral global bifurcation techniques. We obtain the existence and multiplicity of one‐sign solutions for this problem and give the global structure of one‐sign solutions set according to different asymptotic behaviors of nonlinearity near zero.

In this work, we study the multiplicity of solutions for non-homogeneous Dirichlet problem with one-dimension Minkowski-curvature operator $$\begin{aligned} \Bigg (\frac{u'}{\sqrt{1-\kappa u'^2}}\Bigg )'+f(u)=0,\ t\in (0,1),\ \ \qquad u(0)=sA,\quad u(1)=sB, \end{aligned}$$ where $$\kappa >0$$ is a constant, $$A,B\in {\mathbb {R}}$$ are constants, $$s\in {\mathbb {R}}$$ is a parameter and $$f:[0,\infty )\rightarrow {\mathbb {R}}$$ is continuous. The results depend on the values of the real numbers s, A, B and on the behaviour of f(u)/u for u near zero.

用Krasnoselskii不动点定理给出带非线性边界条件的一类离散梁方程{△4u(t-2)=λh(t)f(u(t)),t ∈[2,T]z,u(0)=△u(0)=0,Δ2u(T)=0,△3u(T-1)+c(u(T))u(T)=0正解的存在性结果,其中:λ＞0为参数;h:[2,T]z→[0,∞)为函数;f:(0,∞)→R连续且在u=0处允许有奇性,在u=∞处超线性增长.

In this article,we established the structure of all eigenvalues and the oscillation property of corresponding eigenfunctions for discrete clamped beam equation △4u(k-2) =λm(k)u(k),k ∈[2,N+ 1]z,u(0) =△u(0) =0 =u(N+2) =△u(N+2) with the weight function m :[2,N + 1]z → (0,∞),[2,N + 1]z ={2,3,…,N + 1}.As an application,we obtain the global structure of nodal solutions of the corresponding nonlinear problems based on the nonlinearity satisfying suitable growth conditions at zero and infinity.

基于锥上的不动点定理,获得二阶变系数离散Neumann超线性半正边值问题({Δ2u(t-1)+q(t)u(t)=λf(t,u(t)),t∈[1,T]Z,Δu(0)=0,Δu(T)=0)正解的存在性与多解性,其中,0≤q(t)<2(1-cosπ/2T),f:[1,T]Z×[0,+∞)→[-M,+∞)连续,[1,T]Z:={1,2,…,T},M>0为常数,λ>0为参数.

本文讨论了二阶离散周期边值问题{?2u(t?1)+f?u(t)+g(t,u(t))=s,t∈[1,T]Z,u(0)=u(T?1),?u(0)=?u(T?1)解的个数与参数s的关系,其中g:[1,T]Z×R→R是连续函数,f≥0是常数,T≥2是一个整数,s∈R.本文运用上下解方法及拓扑度理论获得了存在常数s0∈R,当s与s0位置关系变化时该问题没有解、至少有一个解、至少有两个解的结果.

We shall discuss the existence and multiplicity of positive solutions for the discrete Dirichlet problem with one-dimensional prescribed mean curvature operator. Based on the critical point theory, we shall show the existence of either one, or two, or three, or infinity many positive solutions depending on the asymptotic behavior of nonlinearity near zero.

用Guo-Krasnoselskii不动点定理给出半正二阶离散周期边值问题{Δ2 u(t-1)+a(t)u(t)=λf(t,u(t)),t∈[1,T]?,u(0)=u(T),Δu(0)=Δu(T)正解的存在性和多解性结果,其中λ>0为参数,[1,T]?={1,2,…,T},f:[1,T]?×[0,∞)→?连续且存在常数D>0,使得f(t,u)≥-D,(t,u)∈[1,T]? ×[0,∞),a:[1,T]?→(0,∞),0

This paper is concerned with the global behavior of components of radial nodal solutions of semilinear elliptic problems −∆v = λh(x, v) in Ω, v = 0 on ∂Ω, where Ω = {x ∈ RN : r1 < |x| < r2} with 0 < r1 < r2, N ≥ 2. The nonlinear term is continuous and satisfies h(x, 0) = h(x, s1(x)) = h(x, s2(x)) = 0 for suitable positive, concave function s1 and negative, convex function s2, as well as sh(x, s) > 0 for s ∈ R \ {0, s1(x), s2(x)}. Moreover, we give the intervals for the parameter λ which ensure the existence and multiplicity of radial nodal solutions for the above problem. For this, we use global bifurcation techniques to prove our main results.

In this work, we apply the 'disconjugacy theory' and Elias's spectrum theory to study the disconjugacy u((4 ))+/beta u '' -alpha u = 0 with two parameters alpha, beta is an element of R and the spectrum structure of the linear operator u((4))+Bu ''-alpha u coupled with the clamped beam conditions u(0) = u'/(0) = u(1) = u'(1) = 0. As the application of our results, we obtain the global structure of nodal solutions of the corresponding nonlinear analogue based on the bifurcation theory.

运用锥上的不动点指数理论获得了格林函数非负时二阶连续Neumann边值问题()正解存在的最优条件,其中f∈C(R+,R+),a(?)∈C([0,T],(0,+∞))使得相应的齐次线性问题只有平凡解;g∈C((0,T),R+)且在t=0和t=T处g(t)允许有奇性,R+:=[0,∞).

应用锥上的不动点指数理论获得了二阶变系数离散Neumann边值问题{Δ2 u(t-1)+q(t)u(t)=f(t,u(t)),t∈[1,T]ZΔu(0)=0,Δu(T)=0正解存在的条件,其中0≤q(t)<2(1-cosπ/2 T)且q(t)≠0,f:[1,T]Z×R+→R+连续,[1,T]Z:={1,2,?,T},R+:=[0,∞).

运用锥上的不动点定理获得了带Neumann边界条件的半正非线性弹性梁方程边值问题{y(4)(x)+(k1+k2)y"(x)+k1k2y(x)=λf(x,y(x)),x∈[0,1],y'(0)=y'(1)=y?(0)=y?(1)=0在条件0<k1<k2≤π2/4下正解的存在性和多解性,其中λ>0,f∈C([0,1]×[0,∞),(-∞∞))存在正常数X使得f(x,y)≥-X成立.

运用锥上的不动点指数理论,获得了格林函数非负时二阶离散周期边值问题｛△2y(n-1)+a(n)y(n)=g(n)f(y(n)), n∈[1,N]z,y(0)=y(N),△y(0) =△y(N)正解存在的最优条件,其中[1,N]z={1,2,…,N},f:[1,N]z×R+→R+连续,a:[1,N]z→(0,+∞)且n∈max[1,N]z a(n)≤4sin2(π/2N),g∈C([1,N]z,R+),R+:=[0,∞).

运用锥上的不动点指数理论获得了四阶Neumann边值问题{y(4)(x)+(k1 + k2)y''(x) + k1k2y(x) =f(x,y(x)),x∈ [0,1],y'(0)=y'(1)=y'''(0)=y'''(1)=0在条件k1＜k2＜0下正解存在的最优条件,其中f∈C([0,1]×[0,∞),[0,∞)).

We establish the global structure of positive solutions of fourth-order periodic boundary-value problems u⁗(t) + Mu(t) = λf (t, u(t)), t ∈ [0, T], uk(0) = u(k)(T), k = 0, 1, 2, 3, with M ∈ (0, 4(2πnM4/T)4) and u(4)(t) − Mu(t) + λg(t, u(t)) = 0, t ∈ [0, T], uk(0) = u(k)(T), k = 0, 1, 2, 3, with M ∈ (0, (2πM4/T)4);here g, f ∈ C([0,T] × [0, ∞), [0, ∞)), M is constant, and λ> 0 is a real parameter. The main results are based on a global bifurcation theorem.