Bounded Solutions to Nonhomogeneous Linear Second-Order Difference Equations

Stevo Stević

doi:10.3390/sym9100227

Abstract

By using some solvability methods and the contraction mapping principle are investigated bounded, as well as periodic solutions to some classes of nonhomogeneous linear second-order difference equations on domains N 0 , Z ∖ N 2 and Z . The case when the coefficients of the equation are constant and the zeros of the characteristic polynomial associated to the corresponding homogeneous equation do not belong to the unit circle is described in detail.

Highlights

Let Z denote the set of all integers, Nk := {n ∈ Z : n ≥ k}, k ∈ Z, and N = N1
Motivated by the recent studies of the solvability, quite recently in [30], we have studied, among other problems, the existence of bounded solutions to the difference equation x n +2 − q n x n = f n, n ∈ N0, (3)
Formula (4), as well as another method, we have shown in [30], among other results, that the equation in the case qn = q, n ∈ N0, has a unique bounded solution in the case when |q| > 1, and used the obtained formula for the bounded solution as a motivation for introducing an operator which along with the contraction mapping principle ([4]) helps in showing the existence of a unique bounded solution to Equation (3) under some conditions posed on the sequencen∈N0

Summary

Introduction

Let us mention that beside showing the solvability of difference equations and systems by finding closed-form formulas for their solutions, in the cases when it is not possible to find them, one can try to find some of their invariants which can be useful in studying of the long-term behavior of their solutions ([21,22]). (a) We solve the equation by the method of variation of constants ([8,20]) As it is well-known, the corresponding homogeneous equation, in this case, has the general solution in the following form: xn = cλ1n + dλ2n , n ∈ N0 , where λ1,2 are the zeros of the characteristic polynomial.

Tλ x

Discussion