Discrete Boundary Value Problems Research Articles

Boundary value problems for a second-order difference equation involving the mean curvature operator

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.

Discrete Boundary Value Problems as Approximate Constructions

Discrete Boundary Value Problems as Approximate Constructions

Existence of homoclinic solutions for difference equations on integers via variational method

In this paper, we are concerned with the existence of at least three distinct solutions for discrete boundary value problems. The proof of the main result is based on variational methods. We also provide two examples in order to illustrate the main results.

A discrete boundary value problem with point interaction

This paper is concerned with a boundary value problem (BVP) for discrete Sturm-Liouville equation with point interaction and boundary conditions depending on a hyperbolic eigenvalue parameter. This paper presents some spectral and scattering properties of this BVP in terms of Jost solution, scattering solutions, scattering function, continuous and discrete spectrum. In addition, the resolvent operator of the BVP is obtained to get the properties of eigenvalues. Furthermore, an example is considered as a special case of the main problem to demonstrate the effectiveness of our results.

On the Ulam-Hyers-Rassias stability of two structures of discrete fractional three-point boundary value problems: Existence theory

<abstract><p>We prove existence and uniqueness of solutions to discrete fractional equations that involve Riemann-Liouville and Caputo fractional derivatives with three-point boundary conditions. The results are obtained by conducting an analysis via the Banach principle and the Brouwer fixed point criterion. Moreover, we prove stability, including Hyers-Ulam and Hyers-Ulam-Rassias type results. Finally, some numerical models are provided to illustrate and validate the theoretical results.</p></abstract>

Green's function for a discrete fractional boundary value problem

Green's function for a discrete fractional boundary value problem

Nontrivial solutions for nonlinear discrete boundary value problems of the fourth order

We study the existence of multiple nontrivial solutions for nonlinear fourth order discrete boundary value problems. We first establish criteria for the existence of at least two nontrivial solutions of the problems and obtain conditions to guarantee that the two solutions are sign-changing. Under some appropriate assumptions, we further prove that the problems have at least three nontrivial solutions, which are positive, negative, and sign-changing, respectively. We include two examples to illustrate the applicability of our results. Our theorems are proved by employing variational approaches, combined with the classic mountain pass lemma and a result from the theory of invariant sets of descending flow.

Existence and uniqueness of solutions to discrete,third-order three-point boundary value problems

The purpose of this article is to move towards a more complete understanding of the qualitative properties of solutions to discrete boundary value problems. In particular, we introduce and develop sufficient conditions under which the existence of a unique solution for a third-order difference equation subject to three-point boundary conditions is guaranteed. Our contributions are realized in the following ways. First, we construct the corresponding Green’s function for the problem and formulate some new bounds on its summation. Second, we apply these properties to the boundary value problem by drawing on Banach’s fixed point theorem in conjunction with interesting metrics and appropriate inequalities. We discuss several examples to illustrate the nature of our advancements.

Discrete Fractional Boundary Value Problems and Inequalities

Abstract In this paper, a general nonlinear discrete fractional boundary value problem is considered, of order between one and two. The main result is an existence theorem, proving the existence of at least one solution to the boundary value problem, subject to validity of a certain key inequality that allows unrestricted growth in the problem. The proof of this existence theorem is accomplished by using Brouwer's fixed point theorem as well as two other main results of this paper, namely, first, a result showing that the solutions of the boundary value problem are exactly the solutions to a certain equivalent integral representation, and, second, the establishment of solutions satisfying certain a priori bounds provided the key inequality holds. In order to establish the latter result, several novel discrete fractional inequalities are developed, each of them interesting in itself and of potential future use in different contexts. We illustrate the usefulness of our existence results by presenting two examples.

The Solvability of the Discrete Boundary Value Problem on the Half-Line.

This paper provides conditions for the existence of a solution to the second-order nonlinear boundary value problem on the half-line of the form with where , . To achieve our goal, we use Schauder’s fixed-point theorem and the perturbation technique for a Fredholm operator of index 0. Moreover, we construct the necessary condition for the existence of a solution to the considered problem.

On the Multiplicity of Solutions for the Discrete Boundary Problem Involving the Singular ϕ -Laplacian

In this paper, we consider the multiplicity of solutions for a discrete boundary value problem involving the singular ϕ -Laplacian. In order to apply the critical point theory, we extend the domain of the singular operator to the whole real numbers. Instead, we consider an auxiliary problem associated with the original one. We show that, if the nonlinear term oscillates suitably at the origin, there exists a sequence of pairwise distinct nontrivial solutions with the norms tend to zero. By our strong maximum principle, we show that all these solutions are positive under some assumptions. Moreover, the solutions of the auxiliary problem are solutions of the original one if the solutions are appropriately small. Lastly, we give an example to illustrate our main results.

On Discrete Solutions for Elliptic Pseudo-Differential Equations

We consider discrete analogue for simplest boundary value problem for elliptic pseudo-differential equation in a half-space with Dirichlet boundary condition in Sobolev–Slobodetskii spaces. Based on the theory of discrete boundary value problems for elliptic pseudo-differential equations we give a comparison between discrete and continuous solutions for certain model boundary value problem.

Infinitely Many Solutions for Discrete Boundary Value Problems with the p , q -Laplacian Operator

In this paper, we consider the existence and multiplicity of solutions for a discrete Dirichlet boundary value problem involving the p , q -Laplacian. By using the critical point theory, we obtain the existence of infinitely many solutions under some suitable assumptions on the nonlinear term. Also, by our strong maximum principle, we can obtain the existence of infinitely many positive solutions.

Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function

Abstract In this paper, we discuss the existence of positive solutions to a discrete third-order three-point boundary value problem. Here, the weight function a ( t ) a\left(t) and the Green function G ( t , s ) G\left(t,s) both change their sign. Despite this, we also obtain several existence results of positive solutions by using the Guo-Krasnoselskii’s fixed-point theorem in a cone.

Multiplicity of solutions for the discrete boundary value problem involving the p-Laplacian

PurposeThe 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/approachThe approach is based on variational methods and critical point theory.FindingsTheorem 1.1. Theorem 1.2. Theorem 1.3. Theorem 1.4.Originality/valueThe paper is original and the authors think the results are new.

On an anisotropic discrete boundary value problem of Kirchhoff type

ABSTRACT The authors prove the existence of at least one nontrivial solution to an anisotropic discrete boundary value problem of -Kirchhoff type and its associated parametric version. They provide an example to demonstrate the applicability of the main results. Their analysis mainly relies on variational methods and Ricceri's variational principle.

Operators and Equations: Discrete and Continuous

We consider discrete pseudo-differential equations with elliptic symbols and the corresponding discrete boundary-value problems in special canonical domains of multidimensional spaces. The solvability of such equations and boundary-value problems in discrete analogs of Sobolev–Slobodetsky spaces is examined.

On the existence of multiple solutions for a partial discrete Dirichlet boundary value problem with mean curvature operator

Abstract Apartial discrete Dirichlet boundary value problem involving mean curvature operator is concerned in this paper. Under proper assumptions on the nonlinear term, we obtain some feasible conditions on the existence of multiple solutions by the method of critical point theory. We further separately determine open intervals of the parameter to attain at least two positive solutions and an unbounded sequence of positive solutions with the help of the maximum principle.

Small Solutions of the Perturbed Nonlinear Partial Discrete Dirichlet Boundary Value Problems with (p,q)-Laplacian Operator

In this paper, we consider a perturbed partial discrete Dirichlet problem with the (p,q)-Laplacian operator. Using critical point theory, we study the existence of infinitely many small solutions of boundary value problems. Without imposing the symmetry at the origin on the nonlinear term f, we obtain the sufficient conditions for the existence of infinitely many small solutions. As far as we know, this is the study of perturbed partial discrete boundary value problems. Finally, the results are exemplified by an example.

Global bifurcation and constant sign solutions of discrete boundary value problem involving p-Laplacian

We study the unilateral global bifurcation result for the one-dimensional discrete p-Laplacian problem {−Δ[φp(Δu(t−1))]=λa(t)φp(u(t))+g(t,u(t),λ),t∈[1,T+1]Z,Δu(0)=u(T+2)=0,\\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}$$ \\textstyle\\begin{cases} -\\Delta [\\varphi _{p}(\\Delta u(t-1))]=\\lambda a(t)\\varphi _{p}(u(t))+g(t,u(t), \\lambda ),\\quad t\\in [1,T+1]_{Z}, \\\\ \\Delta u(0)=u(T+2)=0, \\end{cases} $$\\end{document} where Delta u(t)=u(t+1)-u(t) is a forward difference operator, varphi _{p}(s)=|s|^{p-2}s (1< p<+infty ) is a one-dimensional p-Laplacian operator. λ is a positive real parameter, a: [1,T+1]_{Z}to [0,+infty ) and a(t_{0})>0 for some t_{0}in [1,T+1]_{Z}, g :[1,T+1]_{Z}times mathbb{R}^{2}to mathbb{R} satisfies the Carathéodory condition in the first two variables. We show that (lambda _{1},0) is a bifurcation point of the above problem, and there are two distinct unbounded continua mathscr{C}^{+} and mathscr{C}^{-}, consisting of the bifurcation branch mathscr{C} from (lambda _{1},0), where lambda _{1} is the principal eigenvalue of the eigenvalue problem corresponding to the above problem. Let T>1 be an integer, Z denote the integer set for m, nin Z with m< n, [m, n]_{Z}:={m, m+1,ldots , n}.As the applications of the above result, we prove more details about the existence of constant sign solutions for the following problem: {−Δ[φp(Δu(t−1))]=λa(t)f(u(t)),t∈[1,T+1]Z,Δu(0)=u(T+2)=0,\\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}$$ \\textstyle\\begin{cases} -\\Delta [\\varphi _{p}(\\Delta u(t-1))]=\\lambda a(t)f(u(t)),\\quad t\\in [1,T+1]_{Z}, \\\\ \\Delta u(0)=u(T+2)=0, \\end{cases} $$\\end{document} where fin C(mathbb{R},mathbb{R}) with sf(s)>0 for sneq 0.