Global stability and asymptotics of some classes of rational difference equations

In this note we prove that the positive solutions of some classes of rational difference equations are globally asymptotically stable. Using a Berg's result, we also find asymptotics of some solutions of these equations.     

Global stability of some symmetric difference equations

Suppose r∈(0,1],m∈N and 1?k₁<k₂<?<k_2m+1, and let S_2m+1={1,2,…,2m+1}. We show that every positive solution to the difference equation

     

Exponential type functional equation and its Hyers-Ulam stability

We give the solution of the functional equation f(x+y)+λf(x)f(y)=Φ(x,y) under some conditions. Also we show its Hyers-Ulam stability.     

Global stability for nonlinear difference equations with variable coefficients

Consider the following nonlinear difference equation with variable coefficients:

     

Hyers-Ulam stability of linear differential equations of first order, III

Let X be a complex Banach space and let I=(a,b) be an open interval. In this paper, we will prove the generalized Hyers-Ulam stability of the differential equation ty^′(t)+αy(t)+βtrx₀=0 for the class of continuously differentiable functions , where α, β and r are complex constants and x₀ is an element of X. By applying this result, we also prove the Hyers-Ulam stability of the Euler differential equation of second order.     

Global stability of some classes of higher-order nonlinear difference equations

We give elegant and short proofs of some recent results on global stability of some classes of higher-order nonlinear difference equations.     

A fixed point approach to stability of functional equations

We prove a simple fixed point theorem for some (not necessarily linear) operators and derive from it several quite general results on the stability of a very wide class of functional equations in single variable.     

The stability of d'Alembert and Jensen type functional equations

The aim of this paper is to study the stability problem of the d'Alembert type and Jensen type functional equations:

f(x+y)+f(x+σy)=2g(x)f(y),     

New global stability conditions for a class of difference equations

In this paper we consider some classes of difference equations, including the well-known Clark model, and study the stability of their solutions. In order to do that we introduce a property, namely semicontractivity, and study relations between ‘semi-contractive’ functions and sufficient conditions for the solution of the difference equation to be globally asymptotically stable. Moreover, we establish new sufficient conditions for the solution to be globally asymptotically stable, and we improve the ‘3/2 criteria’ type stability conditions.      

On the stability of a quartic functional equation

In this paper, we prove the generalized Hyers-Ulam stability for the following quartic functional equation

f(2x+y)+f(2x−y)=4f(x+y)+4f(x−y)+24f(x)−6f(y).     

Stability of nonautonomous difference equations with several delays

We consider the possibility to construct efficient stability criteria for solutions to difference equations with variable coefficients. We prove that one can associate a difference equation with a certain functional differential equation, whose solution has the same asymptotic behavior. We adduce examples, demonstrating the essential character of conditions of the obtained theorems and the exactness of the constant 3/2 which defines the boundary of the stability domain.     

On the stability of the linear differential equation of higher order with constant coefficients

We obtain a result on stability of the linear differential equation of higher order with constant coefficients in Aoki-Rassias sense. As a consequence we obtain the Hyers-Ulam stability of the above mentioned equation. A connection with dynamical sytems perturbation is established.     

Asymptotical stability of partial difference equations with variable coefficients

The dynamical systems considered have scalar state, are multivariate, linear, time-discrete, and time-variable and are described by an initial value problem for a class of evolutionary partial difference equations. The time set is the nonnegative part of the integer lattice in several dimensions. Parts of the asymptotical stability set in the parameter space spanned by the time-variable coefficients are explicitly found. To assess the quality of the sufficient stability criteria, a comparison with the exact stability set is made in an example.     

On the Hyers-Ulam stability of the linear differential equation

We obtain some results on generalized Hyers-Ulam stability of the linear differential equation in a Banach space. As a consequence we improve some known estimates of the difference between the perturbed and the exact solutions.     

Global asymptotic stability of a family of difference equations

We consider the family of difference equations of the form

     

Exponential stability of a partial difference equation with nonlinear perturbation

1.IntroductionPartialdifferenceequationshaveappearedinmanybranchesofmathematics.Indeed,LagrangeandLaplacehavediscussedsuchequationsinrelationtoprobability[1],Couralltetal.[z]haveconsideredtheminrelationtodifferentialequationsofmathematicalphysics.Inrecenty6ars,signalandimageprocessingtheory(seee.g.[3])alsomakesuseofthetheoryofpartialdchrenceequations.OscillationtheoryfortheseeqllationshasbeeninvestigatedbyanUmberofauthorsrecelltly[4].However,asystematicinvestigationofthestabilitytheoryofpart…     

On explicit conditions for the asymptotic stability of linear higher order difference equations

We derive some explicit sufficient conditions for the asymptotic stability of the zero solution in a general linear higher order difference equation, and compare our estimations with other related results in the literature. Our main result also applies to some nonlinear perturbations satisfying a kind of sublinearity condition.     

On the Hyers-Ulam-Rassias stability of functional equations in n-variables

In this paper we investigate a generalization of the Hyers-Ulam-Rassias stability for a functional equation of the form f(φ(X))=?(X)f(X)+ψ(X) and the stability in the sense of Ger for the functional equation of the form f(φ(X))=?(X)f(X), where X lie in n-variables. As a consequence, we obtain a stability result in the sense of Hyers-Ulam-Rassias, Gǎvruta, and Ger for some well-known equations such as the gamma, beta, and G-function type's equations.     

Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations

In this short paper the core of the direct method for proving stability of functional equations is described in a clear way and in a quite general form.