Asymptotic analysis for linear difference equations

We are concerned with asymptotic analysis for linear difference equations in a locally convex space. First we introduce the profile operator, which plays a central role in analyzing the asymptotic behaviors of the solutions. Then factorial asymptotic expansions for the solutions are given quite explicitly. Finally we obtain Gevrey estimates for the solutions. In a forthcoming paper we will develop the theory of cohomology groups for recurrence relations. The main results in this paper lay analytic foundations of such an algebraic theory, while they are of intrinsic interest in the theory of finite differences.

     

On the Gevrey order of formal solutions of nonlinear difference equations

We prove a Maillet type theorem for formal solutions of nonlinear difference systems, relating the Gevrey order of the formal solutions to the lowest level of an associated, linear difference operator.     

Periodic solutions of non-linear discrete Volterra equations with finite memory

In this paper we discuss the existence of periodic solutions of discrete (and discretized) non-linear Volterra equations with finite memory. The literature contains a number of results on periodic solutions of non-linear Volterra integral equations with finite memory, of a type that arises in biomathematics. The “summation” equations studied here can arise as discrete models in their own right but are (as we demonstrate) of a type that arise from the discretization of such integral equations. Our main results are in two parts: (i) results for discrete equations and (ii) consequences for quadrature methods applied to integral equations. The first set of results are obtained using a variety of fixed-point theorems. The second set of results address the preservation of properties of integral equations on discretizing them. The effect of weak singularities is addressed in a final section. The detail that is presented, which is supplemented using appendices, reflects the differing prerequisites of functional analysis and numerical analysis that contribute to the outcomes.     

Existence of complex ℓ2 solutions of linear delay systems of difference equations

We give sufficient conditions for the existence of complex ?₂ solutions of a non-homogeneous system of linear difference equations and of two general classes of delay systems of linear difference equations. In some cases, bounds of the established solutions are also given. As a consequence of the space ?₂ where we work, information can be obtained about the asymptotic behavior of the established solutions and, the asymptotic stability of the zero equilibrium point of the systems under consideration. The method we use is a functional-analytic one.     

Regular and extremal solutions for difference equations with generalized phi-Laplacian

Non-oscillatory solutions for second-order difference equations with generalized phi-Laplacian are studied. Solutions are classified according to the asymptotic behaviour as regular or extremal solutions. Their existence and possible coexistence are investigated as well. In particular, the existence of infinitely many extremal solutions for equations with the discrete mean curvature operator is proved by means of an iterative method. This paper is completed by examples and some open problems.     

On the computation of parameter derivatives of solutions of linear difference equations

A method is given to compute the parameter derivatives of recessive solutions of second-order inhomogeneous linear difference equations. The case of difference equations in which all solutions have the same rate of growth is also discussed.     

Homoclinic solutions in periodic difference equations with mixed nonlinearities

In this paper, by using critical point theory in combination with periodic approximations, we obtain some new sufficient conditions on the nonexistence and existence of homoclinic solutions for a class of periodic difference equations. Unlike the existing literatures that always assume that the nonlinear terms are only either superlinear or asymptotically linear at , but superlinear at 0, our nonlinear term can mix superlinear nonlinearities with asymptotically linear ones at both and 0. To the best of our knowledge, this is the first time to consider the homoclinic solutions of this class of difference equations with mixed nonlinearities. Our results are necessary in some sense, and extend and improve some existing ones even for some special cases. Copyright © 2015 John Wiley & Sons, Ltd.     

Global asymptotic behavior of positive solutions for exponential form difference equations with three parameters

In this paper, we study a class of second order difference equations with three paremeters. With positive initial values, the asymptotic behavior of positive solutions are investigated.     

Existence of periodic solutions for neutral functional differential equations with nonlinear difference operator

In this paper, the authors consider the problem of existence of periodic solutions for a second order neutral functional differential system with nonlinear difference D-operator. For such a system, since the possible periodic solutions may not be differentiable, our method is based on topological degree theory of condensing field, not based on Leray Schauder topological degree theory associated to completely continuous field.     

Positive periodic solutions of higher-dimensional functional difference equations with a parameter

By using Krasnoselskii's fixed point theorem and upper and lower solutions method, we find some sets of positive values λ determining that there exist positive T-periodic solutions to the higher-dimensional functional difference equations of the form where A(n)=diag[a₁(n),a₂(n),…,a_m(n)], h(n)=diag[h₁(n),h₂(n),…,h_m(n)], a_j,h_j :Z→R⁺, τ :Z→Z are T -periodic, j=1,2,…,m, T1, λ>0, x :Z→R^m, f :R₊^m→R₊^m, where R₊^m={(x₁,…,x_m)^TR^m, x_j0, j=1,2,…,m}, R⁺={xR, x>0}.     

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.     

On entire solutions of some type of nonlinear difference equations

In this article, the existence of finite order entire solutions of nonlinear difference equations f~n+ P_d(z, f) = p_1 e~(α1 z)+ p_2 e~(α2 z) are studied, where n ≥ 2 is an integer, Pd(z, f) is a difference polynomial in f of degree d(≤ n-2), p_1, p_2 are small meromorphic functions of ez, and α_1, α_2 are nonzero constants. Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order. As its applications, we also find some type of nonlinear difference equations having no finite order entire solutions.     

Solutions to the system of operator equations AXB=C=BXA

In this paper, we present some necessary and sufficient conditions for the existence of solutions, hermitian solutions and positive solutions to the system of operator equations AXB=C=BXA in the setting of bounded linear operators on a Hilbert space. Moreover, we obtain the general forms of solutions, hermitian solutions and positive solutions to the system above.     

Existence of multiple solutions for a second-order difference equation with a parameter

Applying the critical point theorem, we establish existence and multiple solutions for a second-order difference boundary value problem and show the explicit intervals of λ such that the equation has at least 2N distinct solutions.     

Existence of solutions for a nonlinear system with a parameter

In this paper, the existence of solutions for a system of nonlinear equations is considered. n² nonzero real solutions are obtained by using the critical point theory. Additionally, the Dirichlet boundary value problems of even order difference equations and partial difference equations are investigated.