Qualitative behavior of solutions of difference equations with several oscillating coefficients

Sufficient conditions which guarantee the convergence of the nonoscillatory solutions or oscillation of all solutions of a difference equation with several deviating arguments and oscillating coefficients are presented. Corresponding difference equations of both retarded and advanced type are studied. Examples illustrating the results are also given.


Introduction
In this paper, we study the convergence and oscillation of all solutions of the retarded difference equation of the form {σ i (n)} n∈N , 1 ≤ i ≤ m, are sequences of integers such that σ i (n) ≥ n + 1, n ∈ N, 1 ≤ i ≤ m, (1.2) denotes the forward difference operator x(n) = x(n + 1) − x(n) and ∇ denotes the backward difference operator ∇x(n) = x(n) − x(n − 1).
In the last few decades, the asymptotic and oscillatory behavior of all solutions of difference equations has been extensively studied when the coefficients p i (n) are nonnegative. However, for the general case when p i (n) are allowed to oscillate, it is difficult to study the oscillation of (E R ) and (E A ), since the differences x(n) and ∇x(n) of any nonoscillatory solution of (E R ) or (E A ) are always oscillatory. Therefore, the results on convergence and oscillation of difference and differential equations with oscillating coefficients are relatively scarce. Thus, a small number of papers are dealing with this case. See, for example, [2][3][4][6][7][8][10][11][12][13][14][15][16][17][18][19] and the references cited therein. For the general theory of difference equations, the reader is referred to the monographs [1,5,9].
By a solution of the retarded difference equation (E R ), we mean a sequence of real numbers {x(n)} n≥−w which satisfies (E R ) for all n ∈ N 0 . Here, It is clear that, for each choice of real numbers c −w , c −w+1 , . . ., c −1 , c 0 , there exists a unique solution {x(n)} n≥−w of (E R ) which satisfies the initial conditions By a solution of the advanced difference equation (E A ), we mean a sequence of real numbers {x(n)} n∈N 0 which satisfies (E A ) for all n ∈ N.
A solution {x(n)} n≥−w (or {x(n)} n∈N 0 ) of the difference equation (E R ) (or (E A )) is called oscillatory if the terms x(n) of the sequence are neither eventually positive nor eventually negative. Otherwise, the solution is said to be nonoscillatory.
Strong interest in (E R ) with several variable retarded arguments is motivated by the fact that it represents a discrete analogue of the differential equation with several variable retarded arguments (see [4] and the references cited therein) where for every i ∈ {1, . . . , m}, p i is an oscillating continuous real-valued function in the interval [0, ∞), and τ i is a continuous real-valued function on [0, ∞) such that while (E A ) represents a discrete analogue of the advanced differential equation (see [4] and the references cited therein) where, for every i ∈ {1, . . . , m}, p i is an oscillating continuous real-valued function in the interval [1, ∞), and σ i is a continuous real-valued function on [1, ∞) such that For m = 1, (E R ) and (E A ) take the forms x(n) + p(n)x(τ (n)) = 0, n ∈ N 0 and ∇x(n) − p(n)x(σ (n)) = 0, n ∈ N, respectively, and they represent the discrete analogues of the differential equations (see [4] and the references cited therein) and respectively, where τ (t) ≤ t, σ (t) ≥ t, and the coefficient p is a continuous function which is allowed to oscillate. In the case of m = 1 and τ 1 (n) = n − k, σ 1 (n) = n + k, (E R ) and (E A ) take the forms respectively. These equations represent the discrete analogues of the differential equations (see [8,13] and the references cited therein) and respectively, where τ and σ are positive constants and the coefficient p is a continuous function which is allowed to oscillate. In 1982, Ladas et al. [8] studied the differential equations (1.6) and (1.7) with constant arguments and established the following theorems.
then all solutions of (1.6) oscillate.
then all solutions of (1.7) oscillate.
In 1984, Fukagai and Kusano [4] extended the above results to the differential equations (1.3) and (1.4) as follows.
In 1992, Qian et al. [11] studied the differential equation (1.5) with constant retarded argument and established the following theorem.
In this paper, we study the difference equations (E R ) and (E A ) with several variable (retarded or advanced) arguments and oscillating coefficients and establish sufficient conditions for the convergence and oscillation of all solutions of these equations. We also provide examples to illustrate the results derived in this paper. Throughout, we use the notation where α ∈ R, and note that clearly (1.8)

Retarded equations
Convergence of all nonoscillatory solutions of (E R ) is described by the following result.
then every nonoscillatory solution of (E R ) tends to a finite limit, and this limit is zero provided Proof Assume that the solution {x(n)} n≥−w of (E R ) is nonoscillatory. Then it is either eventually positive or eventually negative. As {−x(n)} n≥−w is also a solution of (E R ), we may restrict ourselves to the case where x(n) > 0 for all large n. Let n 1 ≥ −w be an integer such that x(n) > 0 for all n ≥ n 1 . Then there exists n 2 ≥ n 1 such that In view of (1.8), (E R ) shows that and thus for all n ≥ n 3 . Summing up (2.6) from n 3 to ϕ(n) − 1 and taking into account (2.3) and (2.4), for sufficiently large n, we obtain i.e., which means that {x(ϕ(n))} is bounded. This contradicts (2.4). Therefore {x(n)} is bounded. Hence there exists L > 0 such that 0 < x(n) ≤ L for all n ∈ N 0 . Now we show that lim n→∞ x(n) exists. Indeed, summing up (2.5) from n 3 to n ≥ n 3 , we obtain m i=1 n j=n 3 Therefore, by (2.5), we see that exists, that is, exists and is finite. If, additionally, (2.2) holds, then lim n→∞ x(n) = 0. Indeed, assume, for the sake of contradiction, that lim n→∞ x(n) > 0. Hence Summing up (2.5) from n 3 to n ≥ n 3 , we obtain which, in view of (2.2), means that lim n→∞ x(n) = −∞, a contradiction. Therefore lim n→∞ x(n) = 0.
The following result is an immediate corollary of Theorem 2.1.
Proof Assume, for the sake of contradiction, that {x(n)} n≥−w is an eventually positive solution of (E R ). Then, in view of (2.7) and (2.9), it is clear that there exists j 0 ∈ N such that In view of (2.10) and (2.11), (E R ) gives This guarantees that the sequence x is decreasing on m i=1 [τ (τ (n i ( j 0 ))), n i ( j 0 )] ∩ N. Now, summing up (E R ) from τ (n( j 0 )) to n( j 0 ) and taking into account (2.12) and that τ is increasing and x is decreasing, we obtain which is a contradiction.
A slight modification in the proof of Theorem 2.4 leads to the following result about retarded difference inequalities.

Theorem 2.5 Assume that all conditions of Theorem 2.4 hold. Then there are
(i) no eventually positive solutions of the difference inequality (ii) no eventually negative solutions of the difference inequality

Advanced equations
Convergence of all nonoscillatory solutions of (E A ) is described by the following result.

Theorem 3.1 If there exists n
then every nonoscillatory solution of (E A ) tends to a finite limit, and this limit is zero provided Proof The proof is an easy modification of the proof of Theorem 2.1 and hence is omitted.
The following result is an immediate corollary of Theorem 3.1.

Corollary 3.3 If (3.1) holds, then every unbounded solution of (E A ) oscillates.
Now we present a new sufficient condition for the oscillation of all solutions of (E A ), under the assumption that σ i is increasing for all i ∈ {1, . . . , m}.  Proof Assume, for the sake of contradiction, that x is an eventually positive solution of (E A ). Then, in view of (3.3) and (3.5), it is clear that there exists j 0 ∈ N such that In view of (3.6) and (3.7), (E A ) gives Now, summing up (E A ) from n( j 0 ) to σ (n( j 0 )) and taking into account (3.8) and that both σ and x are increasing, we obtain which is a contradiction.
A slight modification in the proof of Theorem 3.4 leads to the following result about advanced difference inequalities. (ii) no eventually negative solutions of the difference inequality

Examples
We give four examples illustrating the four main results of this paper. Observe that

Example 4.1 Consider the retarded difference equation
is negative for odd n and positive for even n, while is positive for odd n and negative for even n.
which means that (2.1) of Theorem 2.1 is satisfied. Hence, every nonoscillatory solution of (4.1) tends to a (finite) limit. In fact, α is one such solution since it satisfies (4.1) for all n ≥ 3 and lim n→∞ α(n) = 3/4.
In view of (2.8), it is obvious that τ (n) = n − 2. Observe that for Observe that that is, (2.9) of Theorem 2.4 is satisfied, and therefore all solutions of (4.2) oscillate. Observe that is negative for odd n and positive for even n, while p 2 (n) = (−1) n+1 6 · 2 n + 11 + − 1 2 n 2 11 · 4 n − 2 , n ∈ N is positive for odd n and negative for even n. Hence which means that (3.1) of Theorem 3.1 is satisfied. Hence, every nonoscillatory solution of (4.3) tends to a (finite) limit. In fact, α is one such solution since it satisfies (4.3) for all n ∈ N and lim n→∞ α(n) = 11/6. In view of (3.4), it is obvious that σ (n) = n + 1. Observe that for  Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.