Approximation Properties of Discrete Boundary Value Problems for Elliptic Pseudo-Differential Equations

Discrete fourth-order boundary value problems with four parameters

Multiple nontrivial solutions for a class of higher dimensional discrete boundary value problems

The theorem of the existence and uniqueness of the solution of the boundary value problem for the equation in partial derivatives of the fourth order with variable coefficients containing the product of the mixed parabolic-hyperbolic operator and the differential operator of the oscillation string with discontinuous conditions of gluing in the pentagon to the plane is proved. By the method of reducing the order of equations, the solvability of the boundary value problem is reduced to the solution of the Tricomi problem for the mixed parabola-hyperbolic equation with variable coefficients and discontinuous gluing conditions. Solving this problem is reduced to the solution of Fredholm’s integral equation of the second order relative to the trace of the derivative function on y along the line of variation of the equation type. In the hyperbolic part of the domain, the representation of the solution of the problem for the hyperbolic equation with the smallest terms was obtained by using the Riemann function method. In the parabolic part of the domain, the solution of the first boundary value problem for the parabolic equation with the smallest terms is obtained by the method of successive approximations and the Green’s function. As a result, the solution of the problem is realized by the method of solving the Gursa problem and the first boundary value problem for the equation of string oscillation.

In this paper, we prove the existence of at least one non-trivial solution for a discrete nonlinear boundary value problem with $phi_c$-Laplacian. The approach is based on variational methods.

In this paper we investigate the existence of positive solution of the following nonlinear discrete third-order two-point boundary value problem. whereis continuous and there existssuch that . Our approach relies on the Krasnosel'skii fixed point theorem. An example is given to demonstrate the application of the theorem obtained.

In this paper, we consider the existence of solutions for the discrete mixed boundary value problems involvingp,q-Laplacian operator. By using critical points theory, we obtain the existence of at least two positive solutions for the boundary value problem under appropriate assumptions on the nonlinearity.

In this article, the existence of nontrivial solutions for the discrete elliptic boundary value problems is considered by using the extremum principle. Such system admits at least 2n nontrivial solutions when the nonlinear term is superlinear or sublinear. An explanation example is also given. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

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.

We propose two new methods of constructing the solutions of linear multi‐point discrete boundary value problems. These methods are applied to solve some continuous two‐point boundary value problems which are known to be numerically unstable.

Existence and uniqueness of positive solutions for fourth-order nonlinear singular continuous and discrete boundary value problems

This paper is concerned with the existence of positive solutions to a discrete fractional boundary value problem. By using the Krasnosel’skii and Schaefer fixed point theorems, the existence results are established. Additionally, examples are provided to illustrate the effectiveness of the main results. MSC:26A33, 39A10, 47H07.

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.

In this paper, we consider the existence of positive solutions for a discrete third-order boundary value problems, which has the sign-changing Green's function. The approach we use is the Guo-Krasnoselskii fixed point theorem in a cone.

Solutions of the discrete boundary value problem of physical geodesy are discussed. Dirac impulses, Wiener–Hopf predictions and "reflexive predictions" are compared. The dramatic computational gains with "reflexive filtering" are discussed.

Under appropriate algebraic conditions on the nonlinearity, using variational methods and critical point theory we discuss the existence of one, two and three solutions for nonlinear discrete Dirichlet boundary value problems driven by ϕc-Laplacian operator involving two parameters λ and µ, without imposing the symmetry or oscillating behavior at infinity on the the nonlinearity, which has applications in the dynamic model of combustible gases, the capillarity problem in hydrodynamics, and the flux-limited diffusion phenomenon. Some applications and examples illustrate the obtained results.