Nonoscillation and oscillation of solutions of a class of third order difference equations

In this paper, sufficient conditions are obtained for nonoscillation/oscillation of all solutions of a class of nonlinear third order difference equations of the form

     

On the global periodicity of discrete dynamical systems and application to rational difference equations

A new necessary condition for global periodicity of discrete dynamical systems and of difference equations is obtained here. This condition will be applied to contribute to solving the problem of global periodicity for second order rational difference equations.     

Homoclinic solutions for a class of fourth‐order difference equations

In this paper, a class of fourth‐order nonlinear difference equation is considered. By making use of the critical point theory, we establish various sets of sufficient conditions for the existence of homoclinic solutions and give some new results. One of our results generalizes and improves the results in the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence of periodic and subharmonic solutions for second order functional difference equations

In this paper, by using the critical point theory, the existence of periodic and subharmonic solutions to a class of second order functional difference equations is obtained. The main approach used in our paper is variational technique and the Saddle Point Theorem. The problem is to solve the existence of periodic and subharmonic solutions of second order forward and backward difference equations.     

Positive homoclinic solutions of n‐th order difference equations with sign‐changing Green's function

In this paper we, consider an n‐th order nonlinear difference equation with parameter dependence. An exhaustive study of the related Green's function is done. The exact expression of the function is given. The range of parameter for which either it has constant sign or it changes sign is obtained. Some existence results for the nonlinear problem are deduced by using the classical Krasnosel'skii's fixed point theorem on cones and fixed point index theory.     

On the dynamics of a higher order rational difference equations

The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the difference equation

$y_{n+1}=a{y}_n+b{y}_{n?t}+c{y}_{n?l}+\frac{d{y}_{n?k}+e{y}_{n?s}}{\alpha y_{n?k}+\beta y_{n?s}},n=0,1,\dots ,$

with positive parameters and non-negative initial conditions. Numerical examples to the difference equation are given to explain our results.     

Max‐type system of difference equations with positive two‐periodic sequences

This paper deals with the form and the periodicity of the solutions of the max‐type system of difference equations where , and are positive two‐periodic sequences and initial values x₀, x _{? 1}, y₀, y _{? 1} ∈ (0, + ∞ ). Copyright © 2014 John Wiley & Sons, Ltd.     

On the periodicity of solutions of max‐type difference equation

This paper is devoted to study the periodic nature of the solution of the following max‐type difference equation: where the initial conditions x_?2,x_?1,x₀ are arbitrary positive real numbers and is a periodic sequence of period two. Copyright © 2014 John Wiley & Sons, Ltd.     

Stability analysis of a system of second‐order difference equations

In this paper, we study the boundedness character and persistence, existence and uniqueness of the positive equilibrium, local and global behavior, and the rate of convergence of positive solutions of the following system of rational difference equations where the parameters α_i,β_i,a_i,b_i for i∈{1,2} and the initial conditions x_?1,x₀,y_?1,y₀ are positive real numbers. Some numerical example are given to verify our theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

Grid approximation of singularly perturbed parabolic equations in the presence of weak and strong transient layers induced by a discontinuous right-hand side

The initial value problem on a line for singularly perturbed parabolic equations with convective terms is investigated. The first-and the second-order space derivatives are multiplied by the parameters ?₁ and ?₂, respectively, which may take arbitrarily small values. The right-hand side of the equations has a discontinuity of the first kind on the set $\bar \gamma $ = [x = 0] × [0, T]. Depending on the relation between the parameters, the appearing transient layers can be parabolic or regular, and the “intensity” of the layer (the maximum of the singular component) on the left and on the right of $\bar \gamma $ can be substantially different. If the parameter ?₂ at the convective term is finite, the transient layer is weak. For the initial value problems under consideration, the condensing grid method is used to construct finite difference schemes whose solutions converge (in the discrete maximum norm) to the exact solution uniformly with respect to ?₁ and ?₂ (when ?₂ is finite and, therefore, the transient layers are weak, no condensing grids are required).     

Positive solutions for a system of difference equations with coupled multi-point boundary conditions

We investigate the existence and nonexistence of positive solutions for a system of nonlinear second-order difference equations with parameters subject to coupled multi-point boundary conditions.     

Estimate of solutions for differential and difference functional equations with applications to difference methods

The theorems on the estimate of solutions for nonlinear second-order partial differential functional equations mainly of parabolic type with Dirichlet’s condition and for the suitable explicit finite difference functional schemes are proved. The proofs are based on the comparison technique. The convergent difference method given is considered without an assumption of the global generalized Perron condition on the functional variable but with local one in some sense only. It is a consequence of our estimate theorems. The functional dependence is of the Volterra type.     

Alternating boundary layer type solutions of some singularly perturbed periodic parabolic equations with Dirichlet and Robin boundary conditions

In this paper, we continue the analysis of alternating boundary layer type solutions to certain singularly perturbed parabolic equations for which the degenerate equations (obtained by setting small parameter multiplying derivatives equal to zero) are algebraic equations that have three roots. Here, we consider spatially one-dimensional equations. We address special cases where the following are true: (a) boundary conditions are of the Dirichlet type with different values of unknown functions specified at different endpoints of the interval of interest; (b) boundary conditions are of the Robin type with an appropriate power of a small parameter multiplying the derivative in the conditions. We emphasize a number of new features of alternating boundary layer type solutions that appear in these cases. One of the important applications of such equations is related to modeling certain types of bioswitches. Special choices of Dirichlet and Robin type boundary conditions can be used to tune up such bioswitches. This article was submitted by the authors in English.     

正负系数中立型差分方程的正解的存在性

In this paper the sufficient conditions for the existence of positive solutions of the neutral difference equations with positive and negative coefficients are established. The results improve some known conclusions in the literature