Three Solutions for a Partial Discrete Dirichlet Problem Involving the Mean Curvature Operator

Partial difference equations have received more and more attention in recent years due to their extensive applications in diverse areas. In this paper, we consider a Dirichlet boundary value problem of the partial difference equation involving the mean curvature operator. By applying critical point theory, the existence of at least three solutions is obtained. Furthermore, under some appropriate assumptions on the nonlinearity, we respectively show that this problem admits at least two or three positive solutions by means of a strong maximum principle. Finally, we present two concrete examples and combine with images to illustrate our main results.

Chaotic Dynamics of Partial Difference Equations with Polynomial Maps

In this paper, chaotic dynamics of a class of partial difference equations are investigated. With the help of the coupled-expansion theory of general discrete dynamical systems, two chaotification schemes for partial difference equations with polynomial maps are established. These controlled equations are proved to be chaotic either in the sense of Li–Yorke or in the sense of both Li–Yorke and Devaney. One example is provided to illustrate the theoretical results with computer simulations for demonstration.

Multidimensional sampling theorems for multivariate discrete transforms

This paper is devoted to the establishment of two-dimensional sampling theorems for discrete transforms, whose kernels arise from second order partial difference equations. We define a discrete type partial difference operator and investigate its spectral properties. Green’s function is constructed and kernels that generate orthonormal basis of eigenvectors are defined. A discrete Kramer-type lemma is introduced and two sampling theorems of Lagrange interpolation type are proved. Several illustrative examples are depicted. The theory is extendible to higher order settings.

Forms of Solutions for Some Two-Dimensional Systems of Rational Partial Recursion Equations

In this paper, we offer the closed-form expressions of systems of second-order partial difference equations. We will utilize an alternative approach to verify the results by (odd-even) dual mathematical induction. We research and enforce the specific solutions of partial difference formulas and ordinary difference formulas as a straight effect.

Existence of chaos for partial difference equations via tangent and cotangent functions

This paper is concerned with the existence of chaos for a type of partial difference equations. We establish four chaotification schemes for partial difference equations with tangent and cotangent functions, in which the systems are shown to be chaotic in the sense of Li–Yorke or of both Li–Yorke and Devaney. For illustration, we provide three examples are provided.

Chaotification of First-Order Partial Difference Equations

This paper is concerned with chaotification of first-order partial difference equations. Two criteria of chaos for the difference equations with general controllers are established, and all the controlled systems are proved to be chaotic in the sense of Li–Yorke or of both Li–Yorke and Devaney by applying the coupled-expanding theory of general discrete dynamical systems. The controllers used in this paper can be easily constructed, facilitating the chaotification of first-order partial difference equations. For illustration, two illustrative examples are provided.

Multiple Solutions for Partial Discrete Dirichlet Problems Involving the p-Laplacian

Due to the applications in many fields, there is great interest in studying partial difference equations involving functions with two or more discrete variables. In this paper, we deal with the existence of infinitely many solutions for a partial discrete Dirichlet boundary value problem with the p-Laplacian by using critical point theory. Moreover, under appropriate assumptions on the nonlinear term, we determine open intervals of the parameter such that at least two positive solutions and an unbounded sequence of positive solutions are obtained by using the maximum principle. We also show two examples to illustrate our results.

A Transformation Method for Delta Partial Difference Equations on Discrete Time Scale

The aim of this study is to develop a transform method for discrete calculus. We define the double Laplace transforms in a discrete setting and discuss its existence and uniqueness with some of its important properties. The delta double Laplace transforms have been presented for integer and noninteger order partial differences. For illustration, the delta double Laplace transforms are applied to solve partial difference equation.