Discrete Boundary Value Problems Research Articles

Second order discrete boundary value problem with the $(p_1(k); p_2(k))$-Laplacian

In this paper we investigate existence and non-existence of solutions for a Dirichlet boundary value problem involving the $(p_1(k), p_2(k))$-Laplacian operator when variational methods are applied to obtain the results.

Three solutions for a discrete fourth-order boundary value problem with four parameters

This paper presents several sufficient conditions for the existence of at least three classical solutions of a boundary value problem for a fourth-order difference equation. Fourth-order boundary value problems act as models for the bending or deforming of elastic beams. In different fields of research, such as computer science, mechanical engineering, control systems, artificial or biological neural networks, economics and many others,the mathematical modelling of important questions leads naturally to the consideration of nonlinear difference equations. Our technical approach is based on variational methods. An example is included in the paper.Numerical computations of the example confirm our theoretical results.

Multiplicity of Solutions for Discrete 2n-TH Order Periodic Boundary Value Problem with φp-Laplacian

The purpose of this paper is to investigate the existence and multiplicity of nontrivial solutions with the φp-Laplacian for the discrete 2n-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.

Existence of Multiple Weak Solutions to a Discrete Fractional Boundary Value Problem

The existence of at least three weak solutions to a discrete fractional boundary value problem containing a p-Laplacian operator and subject to perturbations is proved using variational methods. Some applications of the main results are presented. The results obtained generalize some recent results on both discrete fractional boundary value problems and p-Laplacian boundary value problems. Examples illustrating the results are given.

Positive Solutions to the Discrete Boundary Value Problem of the Kirchhoff Type

The paper aims to study a discrete boundary value problem of the Kirchhoff type based on the critical point theory and the strong maximum principle. Compared to the existing literature, the existence and multiplicity of positive solutions to the problem are considered according to the behavior of the nonlinear term f in some points between the zero and positive infinity, which is a new attempt. Under different assumptions of the nonlinear term f, we obtain the determined open intervals of the parameter λ, such that the problem has at least three positive solutions or at least two positive solutions in different intervals. In the end, two concrete examples are used to illustrate our main conclusions.

Multiplicity of solutions for discrete 2n-th order periodic boundary value problem

Multiplicity of solutions for discrete 2n-th order periodic boundary value problem

On a general discrete boundary value problem for an elliptic pseudo-differential equation in a quadrant

On a general discrete boundary value problem for an elliptic pseudo-differential equation in a quadrant

On Discrete Boundary-Value Problems and Their Approximation Properties

Discrete analogs of pseudo-differential operators and equations in discrete Sobolev–Slobodetsky spaces are considered. Using suitable discrete boundary conditions, we prove the unique solvability of the discrete boundary-value problem.

LYAPUNOV TYPE INEQUALITY FOR DISCRETE FRACTIONAL BOUNDARY VALUE PROBLEM

In this paper we consider the discrete fractional boundary value problem. The Green’s function and it’s properties are used to find maximum value of function. With the help of maximum value of the function Lyapunov type inequality is obtain for this problem.

Variational Approaches to a Discrete Elliptic Problem with a Weight

In this article, we are concerned with the existence of at least one, two and three distinct solutions for discrete boundary value problems driven by the Laplacian. The proof of the main result depends on variational methods. By using a consequence of the local minimum theorem due Bonanno we investigate the existence of at least one solution and two solutions for the problem with the weight. Furthermore, by using two critical point theorems, one due Averna and Bonanno, and another due Bonanno we explore the existence of two and three solutions for the problem. We also provide two examples in order to illustrate the main results.

A variational framework for second-order backward discrete boundary value problems

A variational framework for second-order backward discrete boundary value problems

On a discrete boundary value problem in a quarter-plane

On a discrete boundary value problem in a quarter-plane

Infinitely many large energy solutions to a class of nonlocal discrete elliptic boundary value problems

This paper is concerned with nonlocal discrete Kirchhoff type equation \begin{document}$ \begin{equation*} \left\{ \begin{aligned} &-[a+b(\sum\limits_{j = 1}^{n}\sum\limits_{i = 1}^{m+1}{|\Delta_1x(i-1, j)|}^2+\sum\limits_{i = 1}^{m}\sum\limits_{j = 1}^{n+1}{|\Delta_2x(i, j-1)|}^2)]\\ &\cdot \left(\Delta^2_1x(i-1, j)+\Delta^2_2x(i, j-1)\right) = f((i, j), x(i, j)), \; \forall (i, j)\in [1, m]\times [1, n], \\ &x(i, 0)\! = \!x(i, n\!+\!1)\! = \!0 \;\;\; i\!\in\! [0, m\!+\!1], \quad x(0, j)\! = \!x(m\!+\!1, j)\! = \!0 \;\;\; j\!\in\! [0, n\!+\!1], \end{aligned} \right. \end{equation*} $\end{document} via variational techniques and the (variant) fountain theorems. We obtain a series of results on the existence of infinitely many large energy solutions. Examples and numerical simulations are also provided to illustrate applications of our results.

Lyapunov and Hartman-type inequalities for higher-order discrete fractional boundary value problems

Lyapunov and Hartman-type inequalities for higher-order discrete fractional boundary value problems

Positive solutions of nabla fractional boundary value problem

In this article, we consider the following two-point discrete fractional boundary value problem with constant coefficient associated with Dirichlet boundary conditions. \begin{equation*} \begin{cases} -\big{(}\nabla^{\nu}_{\rho(a)}u\big{)}(t) + \lambda u(t) = f(t, u(t)), \quad t \in \mathbb{N}^{b}_{a + 2}, \\ u(a) = u(b) = 0, \end{cases} \end{equation*} where $1 < \nu < 2$, $a,b \in \mathbb{R}$ with $b-a\in\mathbb{N}_{3}$, $\mathbb{N}^b_{a+2} = \{a+2,a+3,\hdots,b\}$, $|\lambda| < 1$, $\nabla^{\nu}_{\rho(a)}u$ denotes the $\nu^{\text{th}}$-order Riemann--Liouville nabla difference of $u$ based at $\rho(a)=a-1$, and $f : \mathbb{N}^{b}_{a + 2} \times \mathbb{R} \rightarrow \mathbb{R}^{+}$. We make use of Guo--Krasnosels'ki\v{\i} and Leggett--Williams fixed-point theorems on suitable cones and under appropriate conditions on the non-linear part of the difference equation. We establish sufficient requirements for at least one, at least two, and at least three positive solutions of the considered boundary value problem. We also provide an example to demonstrate the applicability of established results.

Existence of a priori bounded solutions for discrete two-point boundary value problems

In this paper, a careful analysis on the existence of a priori bounded solutions for difference equations is provided. In particular, we study a second order nonlinear difference boundary value problem and its relationship with the continuous one and we obtain a version of these existence results with a priori bounds of the solution u and its derivative u′,u″ and we give an analytic expression of the solution u as the limit of a suitable sequence of functions.

Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator

Abstract In this article, by using critical point theory, we prove the existence of multiple T T -periodic solutions for difference equations with the mean curvature operator: − Δ ( ϕ c ( Δ u ( t − 1 ) ) ) + q ( t ) u ( t ) = λ f ( t , u ( t ) ) , t ∈ Z , -\Delta ({\phi }_{c}\left(\Delta u\left(t-1)))+q\left(t)u\left(t)=\lambda f\left(t,u\left(t)),\hspace{1em}t\in {\mathbb{Z}}, where Z {\mathbb{Z}} is the set of integers. As a T T -periodic problem, it does not require the nonlinear term is unbounded or bounded, and thus, our results are supplements to some well-known periodic problems. Finally, we give one example to illustrate our main results.

Infinitely Many Solutions for the Discrete Boundary Value Problems of the Kirchhoff Type

In this paper, we study the existence and multiplicity of solutions for the discrete Dirichlet boundary value problem of the Kirchhoff type, which has a symmetric structure. By using the critical point theory, we establish the existence of infinitely many solutions under appropriate assumptions on the nonlinear term. Moreover, we obtain the existence of infinitely many positive solutions via the strong maximum principle. Finally, we take two examples to verify our results.

Existence of infinitely many solutions of nonlinear fourth-order discrete boundary value problems

The fourth-order discrete Dirichlet boundary value problem is also a discrete elastic beam problem. In this paper, the existence of infinitely many solutions to this problem is investigated through the critical point theory. By an important inequality we established and the oscillatory behavior of f either near the origin or at infinity, we obtain the existence of infinitely many solutions, which either converge to zero or unbounded. In the end, two examples are presented to illustrate our results.

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.