Discrete Neumann Boundary Value Problem Research Articles

S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems

Abstract By using the bifurcation method, we study the existence of an S-shaped connected component in the set of positive solutions for discrete second-order Neumann boundary value problem. By figuring the shape of unbounded connected component of positive solutions, we show that the Neumann boundary value problem has three positive solutions suggesting suitable conditions on the weight function and nonlinearity.

Existence of multiple solutions to second-order discrete Neumann boundary value problems

By using the invariant set of descending flow and variational method, we establish the existence of multiple solutions to a class of second-order discrete Neumann boundary value problems. The solutions include sign-changing solutions, positive solutions, and negative solutions. An example is given to illustrate our results.

Discrete Neumann boundary value problem for a nonlinear equation with singular \u03d5-Laplacian

Let Isubsetmathbb{R} be an open interval with 0in I, and let gin C^{1}(I, (0,+infty)). Let Ninmathbb{N} be an integer with Ngeq4, [2, N-1]_{mathbb{Z}}:={2, 3,ldots,N-1}. We are concerned with the existence of solutions for the discrete Neumann problem \t\t\t{∇(kn−1△vk1−(△vk)2)=nkn−1[−g′(ψ−1(vk))1−(△vk)2+g(ψ−1(vk))H(ψ−1(vk),k)],k∈[2,N−1]Z,Δv1=0=ΔvN−1\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} \\nabla(k^{n-1}\\frac{\\triangle v_{k}}{\\sqrt{1-(\\triangle v_{k})^{2}}} )=nk^{n-1}[-\\frac{ g'(\\psi^{-1}(v_{k}))}{\\sqrt{1-(\\triangle v_{k})^{2}}}+g(\\psi^{-1}(v_{k}))H(\\psi^{-1}(v_{k}),k)],\\quad k\\in[2, N-1]_{\\mathbb{Z}},\\\\ \\Delta v_{1}=0=\\Delta v_{N-1} \\end{cases} $$\\end{document} which is a discrete analogue of the Neumann problem about the rotationally symmetric spacelike graphs with a prescribed mean curvature function in some Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes, where psi(s):=int_{0}^{s}frac{dt}{g(t)}, psi ^{-1} is the inverse function of ψ, and H:mathbb{R}times[2, N-1]_{mathbb{Z}}tomathbb{R} is continuous with respect to the first variable. The proofs of the main results are based upon the Brouwer degree theory.

Positive solutions of discrete Neumann boundary value problems with sign-changing nonlinearities

Our concern is the existence of positive solutions of the discrete Neumann boundary value problem $$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} -\Delta^{2} u(t-1)=f(t, u(t)), \quad t\in[1,T]_{\mathbb{Z}}, \Delta u(0)=\Delta u(T)=0, \end{array}\displaystyle \right . \end{aligned}$$ where $f: [1,T]_{\mathbb{Z}}\times\mathbb{R}^{+}\to\mathbb{R}$ is a sign-changing function. By using the Guo-Krasnosel’skiĭ fixed point theorem, the existence and multiplicity of positive solutions are established. The nonlinear term $f(t,z)$ may be unbounded below or nonpositive for all $(t,z)\in[1,T]_{\mathbb{Z}}\times\mathbb{R}^{+}$ .

Nonexistence and Existence Results for a Fourth-Order p-Laplacian Discrete Neumann Boundary Value Problem

In this paper, a fourth-order nonlinear p-Laplacian difference equation is considered. Using the critical point theory, we establish various sets of sufficient conditions of the nonexistence and existence of solutions for Neumann boundary value problem and give some new results. The existing results are generalized and significantly complemented.

Existence and multiplicity of solutions of second-order discrete Neumann problem with singular ϕ-Laplacian operator

In this paper, we obtain the existence and multiplicity of solutions for discrete Neumann boundary value problem with singular ϕ-Laplacian operator ∇( Δ u k 1 − κ ( Δ u k ) 2 )+ r k u k +f(k, u k ,Δ u k )=0, 2≤k≤N−1, Δ u 1 =0=Δ u N − 1 by using upper and lower solutions method and Brouwer degree theory, where κ>0 is a constant, r=( r 2 ,…, r N − 1 )∈ R N − 2 , and f is a continuous function. We also give some examples to illustrate the main results.MSC:34B10, 34B18.

Existence and nonexistence results for a fourth-order discrete neumann boundary value problem

In this paper, a fourth-order nonlinear difference equation is considered. By making use of the critical point theory, we establish various sets of sufficient conditions for the existence and nonexistence of solutions for Neumann boundary value problem and give some new results. Results obtained generalize and complement the existing ones.

The existence of solutions for a second-order discrete Neumann problem with a p-Laplacian

In this paper, we investigate the solutions of second-order discrete Neumann boundary value problem with a p-Laplacian. By using critical point theory the existence results are obtained.

Solvability of discrete Neumann boundary value problems

In this article we gain solvability to a nonlinear, second-order difference equation with discrete Neumann boundary conditions. Our methods involve new inequalities on the right-hand side of the difference equation and Schaefer's Theorem in the finite-dimensional space setting.