Advances in Difference Equations

List of Papers (Total 3,023)

One parameter family of linear difference equations and the stability problem for the numerical solution of ODEs

The study of the stability properties of numerical methods leads to considering linear difference equations depending on a complex parameter q. Essentially, the associated characteristic polynomial must have constant type for q ∈ ℂ-. Usually such request is proved with the help of computers. In this paper, by using the fact that the associated polynomials are solutions of a...

On stability zones for discrete-time periodic linear Hamiltonian systems

The main purpose of the paper is to give discrete-time counterpart for some strong (robust) stability results concerning periodic linear Hamiltonian systems. In the continuous-time version, these results go back to Liapunov and Žukovskii; their deep generalizations are due to Kreĭn, Gel'fand, and Jakubovič and obtaining the discrete version is not an easy task since not all...

Periodic solutions for a coupled pair of delay difference equations

Based on the fixed-point index theory for a Banach space, positive periodic solutions are found for a system of delay difference equations. By using such results, the existence of nontrivial periodic solutions for delay difference equations with positive and negative terms is also considered.

Thermodynamic modeling, energy equipartition, and nonconservation of entropy for discrete-time dynamical systems

We develop thermodynamic models for discrete-time large-scale dynamical systems. Specifically, using compartmental dynamical system theory, we develop energy flow models possessing energy conservation, energy equipartition, temperature equipartition, and entropy nonconservation principles for discrete-time, large-scale dynamical systems. Furthermore, we introduce a new and dual...

On a (2,2)-rational recursive sequence

We investigate the asymptotic behavior of the recursive difference equation yn+1 = (α+βyn)/(1+yn-1) when the parameters α < 0 and β ∈ ℝ. In particular, we establish the boundedness and the global stability of solutions for different ranges of the parameters α and β. We also give a summary of results and open questions on the more general recursive sequences yn+1 = (a + byn)/(A...

Periodic solutions of a discrete-time diffusive system governed by backward difference equations

A discrete-time delayed diffusion model governed by backward difference equations is investigated. By using the coincidence degree and the related continuation theorem as well as some priori estimates, easily verifiable sufficient criteria are established for the existence of positive periodic solutions.

Stability of periodic solutions of first-order difference equations lying between lower and upper solutions

We prove that if there exists α ≤ β, a pair of lower and upper solutions of the first-order discrete periodic problem Δu(n) = f(n,u(n));n ∈ IN ≡ {0,...,N-1},u(0) = u(N), with f a continuous N-periodic function in its first variable and such that x + f(n,x) is strictly increasing in x, for every n ∈ IN, then, this problem has at least one solution such that its N-periodic...

On the oscillation of certain third-order difference equations

We establish some new criteria for the oscillation of third-order difference equations of the form , where Δ is the forward difference operator defined by Δx(n) = x(n+1) - x(n).

On the algebraic difference equations un+2un = ψ(un+1) in , related to a family of elliptic quartics in the plane

We continue the study of algebraic difference equations of the type un+2un = ψ(un+1), which started in a previous paper. Here we study the case where the algebraic curves related to the equations are quartics Q(K) of the plane. We prove, as in "on some algebraic difference equations un+2un = ψ(un+1) in , related to families of conics or cubics: generalization of the Lyness...

Positive periodic solutions of functional discrete systems and population models

We apply a cone-theoretic fixed point theorem to study the existence of positive periodic solutions of the nonlinear system of functional difference equations x(n+1) = A(n)x(n) + f(n,xn).

First- and second-order dynamic equations with impulse

We present existence results for discontinuous first- and continuous second-order dynamic equations on a time scale subject to fixed-time impulses and nonlinear boundary conditions.

Hyers-Ulam stability of the linear recurrence with constant coefficients

Let X be a Banach space over the field ℝ or ℂ, a1,...,ap ∈ ℂ, and (bn)n≥0) a sequence in X. We investigate the Hyers-Ulam stability of the linear recurrence xn+p = a1xn+p-1 + ⋯ + ap-1xn+1 + apxn + bn, n ≥ 0, where x0,x1,...,xp-1 ∈ X.

Construction of upper and lower solutions for singular discrete initial and boundary value problems via inequality theory

We present new existence results for singular discrete initial and boundary value problems. In particular our nonlinearity may be singular in its dependent variable and is allowed to change sign.

Power series techniques for a special Schrödinger operator and related difference equations

We address finding solutions y ∈ ʗ2 (ℝ+) of the special (linear) ordinary differential equation xy''(x) + (ax2 + b) y' (x) + (cx + d)y(x) = 0 for all x ∈ ℝ+, where a, b, c, d ∈ ℝ are constant parameters. This will be achieved in three special cases via separation and a power series method which is specified using difference equation techniques. Moreover, we will prove that our...

Nonoscillatory half-linear difference equations and recessive solutions

Recessive and dominant solutions for the nonoscillatory half-linear difference equation are investigated. By using a uniqueness result for the zero-convergent solutions satisfying a suitable final condition, we prove that recessive solutions are the "smallest solutions in a neighborhood of infinity," like in the linear case. Other asymptotic properties of recessive and dominant...

Multiple positive solutions of singular discrete p-Laplacian problems via variational methods

We obtain multiple positive solutions of singular discrete p-Laplacian problems using variational methods.

Periodic solutions of nonlinear second-order difference equations

We establish conditions for the existence of periodic solutions of nonlinear, second-order difference equations of the form y(t + 2) + by (t + 1) + cy(t) = f (y(t)), where f: ℝ → ℝ and β > 0 is continuous. In our main result we assume that f exhibits sublinear growth and that there is a constant uf (u) > 0 such that |u| ≥ β whenever c = 1. For such an equation we prove that if N...

Exponential stability of dynamic equations on time scales

We investigate the exponential stability of the zero solution to a system of dynamic equations on time scales. We do this by defining appropriate Lyapunov-type functions and then formulate certain inequalities on these functions. Several examples are given.

On the appearance of primes in linear recursive sequences

We present an application of difference equations to number theory by considering the set of linear second-order recursive relations, , U0 = 0, U1 = 1, and , , where R and Q are relatively prime integers and n ∈ {0,1,...}. These equations describe the set of extended Lucas sequences, or rather, the Lehmer sequences. We add that the rank of apparition of an odd prime p in a...

Behavior of the positive solutions of fuzzy max-difference equations

We extend some results obtained in 1998 and 1999 by studying the periodicity of the solutions of the fuzzy difference equations xn+1 = max{A/xn, A/xn-1,...,A/xn-k}, xn+1 = max{A0/xn, A1/xn-1}, where k is a positive integer, A, Ai, i = 0,1, are positive fuzzy numbers, and the initial values xi, i = -k, -k + 1,...,0 (resp., i = -1,0) of the first (resp., second) equation are...

Eigenstructure of nonselfadjoint complex discrete vector Sturm-Liouville problems

We present a study of complex discrete vector Sturm-Liouville problems, where coefficients of the difference equation are complex numbers and the strongly coupled boundary conditions are nonselfadjoint. Moreover, eigenstructure, orthogonality, and eigenfunctions expansion are studied. Finally, an example is given.

Periodic boundary value problems on time scales

We extend the results concerning periodic boundary value problems from the continuous calculus to time scales. First we use the Schauder fixed point theorem and the concept of lower and upper solutions to prove the existence of the solutions and then we investigate a monotone iterative method which could generate some of them. Since this method does not work on each time scale, a...

Global attractor of coupled difference equations and applications to Lotka-Volterra systems

This paper is concerned with a coupled system of nonlinear difference equations which is a discrete approximation of a class of nonlinear differential systems with time delays. The aim of the paper is to show the existence and uniqueness of a positive solution and to investigate the asymptotic behavior of the positive solution. Sufficient conditions are given to ensure that a...

Solvability conditions for some difference operators

Infinite-dimensional difference operators are studied. Under the assumption that the coefficients of the operator have limits at infinity, limiting operators and associated polynomials are introduced. Under some specific conditions on the polynomials, the operator is Fredholm and has the zero index. Solvability conditions are obtained and the exponential behavior of solutions of...

On the recursive sequence

Our aim in this paper is to investigate the boundedness, global asymptotic stability, and periodic character of solutions of the difference equation xn+1 = (γxn-1 + δxn-2)/(xn + xn-2), n = 0,1,..., where the parameters γ and δ and the initial conditions are positive real numbers.