Advances in Difference Equations

List of Papers (Total 2,995)

Existence and uniqueness of solutions of higher-order antiperiodic dynamic equations

We prove existence and uniqueness results in the presence of coupled lower and upper solutions for the general nth problem in time scales with linear dependence on the ith Δ-derivatives for i = 1,2,…,n, together with antiperiodic boundary value conditions. Here the nonlinear right-hand side of the equation is defined by a function f(t,x) which is rd-continuous in t and continuous...

Existence and global stability of positive periodic solutions of a discrete predator-prey system with delays

We study the existence and global stability of positive periodic solutions of a periodic discrete predator-prey system with delay and Holling type III functional response. By using the continuation theorem of coincidence degree theory and the method of Lyapunov functional, some sufficient conditions are obtained.

Stability for delayed generalized 2D discrete logistic systems

This paper is concerned with delayed generalized 2D discrete logistic systems of the form , where σ and τ are positive integers, is a real function, which contains the logistic map as a special case, and m and n are nonnegative integers, where and . Some sufficient conditions for this system to be stable and exponentially stable are derived.

On the growth rate of generalized Fibonacci numbers

Let α(t) be the limiting ratio of the generalized Fibonacci numbers produced by summing along lines of slope t through the natural arrayal of Pascal's triangle. We prove that is an even function.

Trichotomy, stability, and oscillation of a fuzzy difference equation

We study the trichotomy character, the stability, and the oscillatory behavior of the positive solutions of a fuzzy difference equation.

On the solvability of initial-value problems for nonlinear implicit difference equations

Our aim is twofold. First, we propose a natural definition of index for linear nonautonomous implicit difference equations, which is similar to that of linear differential-algebraic equations. Then we extend this index notion to a class of nonlinear implicit difference equations and prove some existence theorems for their initial-value problems.

Maximum principles for a family of nonlocal boundary value problems

We study a family of three-point nonlocal boundary value problems (BVPs) for an nth-order linear forward difference equation. In particular, we obtain a maximum principle and determine sign properties of a corresponding Green function. Of interest, we show that the methods used for two-point disconjugacy or right-disfocality results apply to this family of three-point BVPs.

Global asymptotic stability of solutions of cubic stochastic difference equations

Global almost sure asymptotic stability of solutions of some nonlinear stochastic difference equations with cubic-type main part in their drift and diffusive part driven by square-integrable martingale differences is proven under appropriate conditions in ℝ1. As an application of this result, the asymptotic stability of stochastic numerical methods, such as partially drift...

On monotonicity of solutions of discrete-time nonnegative and compartmental dynamical systems

Nonnegative and compartmental dynamical system models are widespread in biological, physiological, and pharmacological sciences. Since the state variables of these systems are typically masses or concentrations of a physical process, it is of interest to determine necessary and sufficient conditions under which the system states possess monotonic solutions. In this paper, we...

A functional-analytic method for the study of difference equations

We will give the generalization of a recently developed functional-analytic method for studying linear and nonlinear, ordinary and partial, difference equations in the and spaces, p∈ℕ, . The method will be illustrated by use of two examples concerning a nonlinear ordinary difference equation known as the Putnam equation, and a linear partial difference equation of three variables...

Modelling AIDS epidemic and treatment with difference equations

We propose two models for the description of the dynamics of an AIDS epidemic and of the effect of a combined-drugs AIDS treatment based on difference equations. We show that our interacting population model, despite its extreme simplicity, describes adequately the evolution of an AIDS epidemic. A cellular-automaton analogue of the discrete system of equations is presented as...

Rothe time-discretization method applied to a quasilinear wave equation subject to integral conditions

This paper presents a well-posedness result for an initial-boundary value problem with only integral conditions over the spatial domain for a one-dimensional quasilinear wave equation. The solution and some of its properties are obtained by means of a suitable application of the Rothe time-discretization method.

On the existence and uniqueness of solutions to boundary value problems on time scales

This work formulates existence, uniqueness, and uniqueness-implies-existence theorems for solutions to two-point vector boundary value problems on time scales. The methods used include maximum principles, a priori bounds on solutions, and the nonlinear alternative of Leray-Schauder.

ʗm-smoothness of invariant fiber bundles for dynamic equations on measure chains

We present a new self-contained and rigorous proof of the smoothness of invariant fiber bundles for dynamic equations on measure chains or time scales. Here, an invariant fiber bundle is the generalization of an invariant manifold to the nonautonomous case. Our main result generalizes the “Hadamard-Perron theorem” to the time-dependent, infinite-dimensional, noninvertible, and...

Smooth and discrete systems—algebraic, analytic, and geometrical representations

What is a differential equation? Certain objects may have different, sometimes equivalent representations. By using algebraic and geometrical methods as well as discrete relations, different representations of objects mainly given as analytic relations, differential equations can be considered. Some representations may be suitable when given data are not sufficiently smooth, or...

Lyapunov functionals construction for stochastic difference second-kind Volterra equations with continuous time

The general method of Lyapunov functionals construction which was developed during the last decade for stability investigation of stochastic differential equations with aftereffect and stochastic difference equations is considered. It is shown that after some modification of the basic Lyapunov-type theorem, this method can be successfully used also for stochastic difference...

Finite difference schemes with monotone operators

Several existence theorems are given for some second-order difference equations associated with maximal monotone operators in Hilbert spaces. Boundary conditions of monotone type are attached. The main tool used here is the theory of maximal monotone operators.

On a class of fourth-order nonlinear difference equations

We consider a class of fourth-order nonlinear difference equations. The classification of nonoscillatory solutions is given. Next, we divide the set of solutions of these equations into two types: F+- and F−-solutions. Relations between these types of solutions and their nonoscillatory behavior are obtained. Necessary and sufficient conditions are obtained for the difference...

Vector dissipativity theory for discrete-time large-scale nonlinear dynamical systems

In analyzing large-scale systems, it is often desirable to treat the overall system as a collection of interconnected subsystems. Solution properties of the large-scale system are then deduced from the solution properties of the individual subsystems and the nature of the system interconnections. In this paper, we develop an analysis framework for discrete-time large-scale...

An asymptotic result for some delay difference equations with continuous variable

We consider a nonhomogeneous linear delay difference equation with continuous variable and establish an asymptotic result for the solutions. Our result is obtained by the use of a positive root with an appropriate property of the so called characteristic equation of the corresponding homogeneous linear (autonomous) delay difference equation. More precisely, we show that, for any...