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This paper is concerned with boundary value problems for a fourth-order nonlinear difference equation. Sufficient condition are obtained for the existence of at least three solutions, via variational methods and critical point theory. One example is included to illustrate the result. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the existence and multiplicity of nontrivial solutions with the φ p -Laplacian for the discrete 2 n -th order periodic boundary value issue. To support these conclusions, we have employed variational techniques and contemporary critical point theory. A few new findings are expanded upon and enhanced. We give an example to show how our key findings can be applied. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is the study of existence and multiplicity of solutions for a nonlinear discrete boundary value problems involving the p-laplacian. Design/methodology/approach – The approach is based on variational methods and critical point theory. Findings – Theorem 1.1. Theorem 1.2. Theorem 1.3. Theorem 1.4. Originality/value – The paper is original and the authors think the results are new.

Abstract In this paper, we consider the existence of multiple solutions for discrete boundary value problems involving the mean curvature operator by means of Clark’s Theorem, where the nonlinear terms do not need any asymptotic and superlinear conditions at 0 or at infinity. Further, the existence of a positive solution has been considered by the strong comparison principle. As an application, some examples are given to illustrate the obtained results.

Purpose -- The purpose of this paper is the study of existence and multiplicity of solutions for a nonlinear discrete boundary value problems involving the p-laplacian. Design/methodology/approach -- The approach is based on variational methods and critical point theory. Findings -- Theorem 1.1. Theorem 1.2. Theorem 1.3. Theorem 1.4. Originality/value -- The paper is original and the authors think the results are new. [ABSTRACT FROM AUTHOR]

Let \(n\in\mathbb{N}^{*}\), and \(N\geq n\) be an integer. We study the spectrum of discrete linear \(2n\)-th order eigenvalue problems \[\begin{cases}\sum_{k=0}^{n}(-1)^{k}\Delta^{2k}u(t-k) = \lambda u(t) ,\quad & t\in[1, N]_{\mathbb{Z}}, \\ \Delta^{i}u(-(n-1))=\Delta^{i}u(N-(n-1)),\quad & i\in[0, 2n-1]_{\mathbb{Z}},\end{cases}\] where \(\lambda\) is a parameter. As an application of this spectrum result, we show the existence of a solution of discrete nonlinear \(2n\)-th order problems by applying the variational methods and critical point theory.

In this paper, we consider the existence of multiple solutions for discrete boundary value problems involving the mean curvature operator by means of Clark's Theorem, where the nonlinear terms do not need any asymptotic and superlinear conditions at 0 or at infinity. Further, the existence of a positive solution has been considered by the strong comparison principle. As an application, some examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

Discrete function theory in higher-dimensional setting has been in active development since many years. However, available results focus on studying discrete setting for such canonical domains as half-space, while the case of bounded domains generally remained unconsidered. Therefore, this paper presents the extension of the higher-dimensional function theory to the case of arbitrary bounded domains in R n . On this way, discrete Stokes’ formula, discrete Borel–Pompeiu formula, as well as discrete Hardy spaces for general bounded domains are constructed. Finally, several discrete Hilbert problems are considered. [ABSTRACT FROM AUTHOR]

In this paper we study the existence and the nonexistence of solutions for the boundary value problems of a class of nonlinear second-order discrete equations depending on a parameter. Variational (the mountain pass technique) and non-variational methods are applied.

In the framework of variational methods, we use a two non-zero critical points theorem to obtain the existence of two positive solutions to Dirichlet boundary value problems for difference equations involving the discrete p -Laplacian operator. [ABSTRACT FROM AUTHOR]

In the present paper, we obtain the existence of nontrivial weak solutions to a class of discrete boundary value problems of p(k)-Kirchhoff-type which is a discrete variant of continuous p(x)-Kirchholi-type problem. We apply the variational principle along with the critical point theory. [ABSTRACT FROM AUTHOR]

In this paper we study the existence and the nonexistence of solutions for the boundary value problems of a class of nonlinear second-order discrete equations depending on a parameter. Variational (the mountain pass technique) and non-variational methods are applied. [ABSTRACT FROM AUTHOR]

We study the existence of solutions to nonlinear discrete boundary value problems with the discrete p-Laplacian, potential, and nonlinear source terms. Using variational methods, we demonstrate that there exist at least two positive solutions. The existence strongly depends on the smallest positive eigenvalue of Dirichlet eigenvalue problems and the growth conditions of the source terms. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we discuss the existence of a solution for a resonance problem involving the discrete -Laplacian. Using asymptotic behaviors of at infinity, a nontrivial solution is established. Moreover, we also find a necessary condition for asymptotic behavior of at zero to guarantee the existence of at least two nontrivial solutions. [Copyright &y& Elsevier]

An application is made to the second-order right focal discrete boundary value problem of a recent extension of the Leggett–Williams fixed point theorem, which requires neither of the functional boundaries to be invariant. A non-trivial example is also provided. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, we show the existence of a positive solution of nonlinear discrete boundary value problems for the discrete -Laplacian with potential term. Moreover, the multiplicity of solutions for the problems is also investigated. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the existence and the uniqueness of discrete boundary value problems involving the p-Laplacian with potential terms. To this end, we deal with the strong comparison principle and the Dirichlet principle for the p-Laplacian with potential terms on weighted graphs with boundary. Moreover, we provide a lower bound of the potential terms which guarantees the existence of solutions of the boundary value problem for the operator. Finally, we also discuss inverse conductivity and potential problems for the operator. [ABSTRACT FROM AUTHOR]

Abstract: We present here a systematic study of general boundary value problems on weighted networks that includes the variational formulation of such problems. In particular, we obtain the discrete version of the Dirichlet Principle and we apply it to the analysis of the inverse problem of identifying the conductivities of the network in a very general framework. Our approach is based on the development of an efficient vector calculus on weighted networks which mimetizes the calculus in the smooth case. The key tool is an adequate construction of the tangent space at each vertex. This allows us to consider discrete vector fields, inner products and general metrics. Then, we obtain discrete versions of derivative, gradient, divergence and Laplace–Beltrami operators, satisfying analogous properties to those verified by their continuous counterparts. On the other hand we develop the corresponding integral calculus that includes the discrete versions of the Integration by Parts technique and Green’s Identities. Finally, we apply our discrete vector calculus to analyze the consistency of difference schemes used to solve numerically a Robin boundary value problem in a square. [Copyright &y& Elsevier]

Abstract: Multiple solutions for a discrete boundary value problem involving the -Laplacian are established. Our approach is based on critical point theory. [Copyright &y& Elsevier]

Abstract: In this paper, we deal with a class of semipositone discrete boundary value problems via critical point theory developed by Chang, and obtain nonexistence and multiplicity results on sublinear nonlinearities and an existence result on superlinear nonlinearities, respectively. [Copyright &y& Elsevier]

Abstract: In this paper, we characterize the eigenvalues and show existence of positive solutions to discrete boundary value problem (here , and is a parameter) [Copyright &y& Elsevier]

We introduce a determinant that encompasses the classical Vandermonde determinant, the generalized Vandermonde determinant, and the recently introduced exponential Vandermonde determinant when the exponents are nonnegative integers. An explicit factorization of such a determinant will be established. This factorization enables us to develop a computationally tractable necessary and sufficient condition for the existence of a unique solution of a Hermite (l-point) discrete boundary value problem. [ABSTRACT FROM AUTHOR]

Difference equations which may arise as discrete approximations to two-point boundary value problems for systems of second-order, ordinary differential equations are investigated and conditions are formulated under which solutions to the discrete problem are unique. Some existence, uniqueness implies existence, and convergence theorems for solutions to the discrete problem are also presented. [Copyright &y& Elsevier]

We investigate difference equations which arise as discrete approximations to two-point boundary value problems for systems of second-order, ordinary differential equations. We formulate conditions under which all solutions to the discrete problem satisfy certain a priori bounds which are independent of the step-size. As a result, the nonexistence of spurious solutions are guaranteed. Some existence and convergence theorems for solutions to the discrete problem are also presented. [Copyright &y& Elsevier]