Order Discrete Boundary Value Problem 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.

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.

A Variational Framework for a Second Order Discrete Boundary Value Problem with Mixed Periodic Boundary Conditions

In this paper, a Banach space and an associated functional are presented to study the existence of solutions to a second order discrete boundary value problem with mixed periodic boundary conditions. The proofs make use of critical point theory. Some illustrative examples are provided.

Multiple and particular solutions of a second order discrete boundary value problem with mixed periodic boundary conditions

Multiple and particular solutions of a second order discrete boundary value problem with mixed periodic boundary conditions

Existence of solutions for a second order discrete boundary value problem with mixed periodic boundary conditions

In this paper, a second order discrete boundary value problem with a pair of mixed periodic boundary conditions is considered. Sufficient conditions on the existence and multiplicity of solutions are obtained by using variational methods. A particular Banach space and an associated functional are presented to overcome the asymmetry of the mixed periodic boundary conditions. Examples are also given to illustrate the applications of the main result.

On the dependence on parameters for second order discrete boundary value problems with the p(k)-Laplacian

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.

Investigation of matrix nullity for the second order discrete nonlocal boundary value problem

In this paper we investigate the relation between the matrix nullity of the second order discrete boundary value problem and nonlocal boundary conditions. The obtained classification and examples are also presented.

Continuous dependence on parameters for second order discrete BVP’s

Abstract Using Fan’s Min-Max Theorem we investigate existence of solutions and their dependence on parameters for some second order discrete boundary value problem. The approach is based on variational methods and solutions are obtained as saddle points to the relevant Euler action functional.

A note on a fourth order discrete boundary value problem

Using variational methods we investigate the existence of solutions and their dependence on parameters for certain fourth order difference equations.

Positive solutions for systems of nonlinear discrete boundary value problems

Existence of eigenvalues yielding positive solutions for systems of second order discrete boundary value problems (two-point, three-point and generalized three-point) are established. The results are obtained by the use of a Guo–Krasnoselskii fixed point theorem in cones.

On variational methods for periodic discrete problems

We prove existence results for a general 2n-th order discrete boundary value problem subjected to periodic conditions. In contrast to published work, we exploit the finite dimension of function spaces and some simple results from matrix theory to construct a nonlinear equation in which is eventually studied with the degree theory and simple variational methods. Existence and uniqueness theorems, which are obtained in this way, improve the existing results (even in the second-order case) by including non-continuous right-hand sides, higher order problems and non-invertible difference operators.

Existence of solutions of nonlinear fourth order discrete boundary value problem†

This article is concerned with the existence of solutions of boundary value problems for nonlinear fourth-order difference equations of the type , P n is nonzero and real value. The authors apply the Linking Theorem in the critical point theory and give new sufficient conditions for the existence of solutions of boundary value problems of the above equation.

Difference equations in abstract spaces

AbstractExistence results are presented for second order discrete boundary value problems in abstract spaces. Our analysis uses only Sadovskii's fixed point theorem.

Upper and lower solutions method for higher order discrete boundary value problems

Upper and lower solutions method for higher order discrete boundary value problems