Oscillations in deviating difference equations using an iterative technique

The paper deals with the oscillation of the first-order linear difference equation with deviating argument and nonnegative coefficients. New sufficient oscillation conditions, involving limsup, are given, which essentially improve all known results, based on an iterative technique. We illustrate the results and the improvement over other known oscillation criteria by examples, numerically solved in Matlab.


Introduction
Consider the difference equation with a variable retarded argument of the form x(n)+p(n)x τ (n) = , n ∈ N  , ( E ) and the (dual) difference equation with a variable advanced argument of the form ∇x(n)-q(n)x σ (n) = , n ∈ N, E where N  and N are the sets of nonnegative integers and positive integers, respectively. Equations (E) and (E ) are studied under the following assumptions: everywhere (p(n)) n≥ and (q(n)) n≥ are sequences of nonnegative real numbers, (τ (n)) n≥ is a sequence of integers such that τ (n) ≤ n -, ∀n ∈ N  and lim n→∞ τ (n) = ∞ (.) and (σ (n)) n≥ is a sequence of integers such that σ (n) ≥ n + , ∀n ∈ N.
By a solution of (E), we mean a sequence of real numbers (x(n)) n≥-w which satisfies (E) for all n ≥ . It is clear that, for each choice of real numbers c -w , c -w+ , . . . , c - , c  , there exists a unique solution (x(n)) n≥-w of (E) which satisfies the initial conditions x(-w) = c -w , x(-w + ) = c -w+ , . . . , x(-) = c - , x() = c  . When the initial data is given, we can obtain a unique solution to (E) by using the method of steps.
By a solution of (E ), we mean a sequence of real numbers (x(n)) n≥ which satisfies (E ) for all n ≥ .
A solution (x(n)) n≥-w (or (x(n)) n≥ ) of (E) (or (E )) 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. An equation is oscillatory if all its solutions oscillate.
In arising is whether we can state oscillation criteria considering the argument τ (n) (or σ (n)) to be not necessarily monotone. In the present paper, we achieve this goal by establishing criteria which, up to our knowledge, essentially improve all other known results in the literature.
Throughout this paper, we are going to use the following notations: Clearly, the sequences h(n) and ρ(n) are nondecreasing with τ (n) ≤ h(n) ≤ n - for all n ≥  and σ (n) ≥ ρ(n) ≥ n +  for all n ≥ , respectively.

Chronological review for retarded difference equations
with P  (n) = p(n), then all solutions of (E) are oscillatory.

Chronological review for advanced difference equations
with q  (n) = q(n), then all solutions of (E ) oscillate.

Main results
We study further (E) and (E ), and derive new sufficient oscillation conditions, involving lim sup, which essentially improve all the previous results.

.. Retarded difference equations
The following simple result is stated to explain why we can consider only the case where λ  >  is the smaller root of the transcendental equation λ = e αλ with  < α ≤ /e. Theorem  Assume that there exists a subsequence θ (n), n ∈ N of positive integers such that Then all solutions of (E) are oscillatory.
Proof Assume, for the sake of contradiction, that (x(n)) n≥-w is a nonoscillatory solution of (E). Then it is either eventually positive or eventually negative. As (-x(n)) n≥-w is also a solution of (E), we may restrict ourselves only to the case where x(n) >  for all large n. Let n  ≥ -w be an integer such that x(n) >  for all n ≥ n  . Then there exists n  ≥ n  such that x(τ (n)) > , ∀n ≥ n  . In view of this, equation (E) becomes which means that the sequence (x(n)) is eventually nonincreasing. Taking into account the fact that (.) holds, equation (E) gives where θ (n) → ∞ as n → ∞, which contradicts the assumption that x(n) >  for all n ≥ n  .
The proofs of our main results are essentially based on the following lemmas. The first lemma is taken from []. For the sake of completeness, we cite its proof here.
The proof of the lemma is complete.
is an eventually positive solution of (E). Then where λ  is the smaller root of the transcendental equation λ = e αλ .
Proof Assume that (x(n)) n≥-w is an eventually positive solution of (E). Then there exists n  ≥ -w such that x(n), x(τ (n)) >  for all n ≥ n  . In view of this, equation (E) becomes which means that (x(n)) is an eventually nonincreasing sequence of positive numbers. Taking into account that  < α ≤ /e, it is clear that there exists ε ∈ (, α) such that We will show that where λ  (ε) is the smaller root of the equation Assume, for the sake of contradiction, that (.) is incorrect. Then there exists ε  >  such that On the other hand, for any δ >  there exists n(δ) such that Summing up last inequality from h(n) to n -, we get Combining (.) and (.), we have i.e., Therefore, Combining the last inequality with (.), we obtain The proof of the lemma is complete.
with P  (n) = λ  p(n) and λ  is the smaller root of the transcendental equation λ = e αλ , then all solutions of (E) are oscillatory.
Proof Assume that (x(n)) n≥-w is an eventually positive solution of (E). Then there exists n  ≥ -w such that x(n), x(τ (n)) >  for all n ≥ n  . In view of this, equation (E) becomes which means that (x(n)) is an eventually nonincreasing sequence of positive numbers. Taking this into account along with the fact that τ (n) ≤ h(n), (E) implies Observe that (.) implies that for each >  there exists a n( ) such that Combining the inequalities (.) and (.) we obtain Applying the discrete Grönwall inequality, we obtain , for all n ≥ n( ). (.) Dividing (E) by x(n) and summing up from k to n -, we take Combining (.) and (.), we obtain In view of (.), (.) gives Summing up (E) from τ (n) to n -, we have , (.) so, combining (.) and (.), we find Multiplying the last inequality by p(n), we get which, in view of (E), becomes i.e., .
Repeating the above argument leads to a new estimate, .
Continuing by induction, for sufficiently large n we get Summing up (E) from h(n) to n, we have Combining (.) and (.), we have, for all sufficiently large n, The inequality is valid if we omit x(n + ) >  in the left-hand side: Since may be taken arbitrarily small, this inequality contradicts (.). The proof of the theorem is complete. Proof Assume, for the sake of contradiction, that (x(n)) n≥-w is an eventually positive solution of (E). Then, as in the proof of Theorem , for sufficiently large n, (.) is satisfied, i.e., That is, By Lemma , inequality (.) holds. So the last inequality leads to Since may be taken arbitrarily small, this inequality contradicts (.). The proof of the theorem is complete.
Remark  It is clear that the left-hand sides of both conditions (.) and (.) are identical, also the right-hand side of condition (.) reduces to (.) in the case that α = . So it seems that Theorem  is the same as Theorem  when α = . However, one may notice that condition  < α ≤ /e is required in Theorem  but not in Theorem .
where P (n) is defined by (.), then all solutions of (E) are oscillatory.
Proof Assume, for the sake of contradiction, that (x(n)) n≥-w is an eventually solution of (E). Then, as in the proof of Theorem , for sufficiently large n, (.) is satisfied. Therefore Summing up (E) from h(n) to n, we have which, in view of (.), gives Thus, for all sufficiently large n, Letting n → ∞, we take which, in view of (.), gives Since may be taken arbitrarily small, this inequality contradicts (.). The proof of the theorem is complete.
Remark  If P (n) ≥  then (.) guarantees that all solutions of (E) are oscillatory. In fact, (.) gives which means that x(n+) ≤ . This contradicts x(n) >  for all n ≥ n  . Thus, in Theorems ,  and  we consider only the case P (n) < . Another conclusion that can be drawn from the above, is that if at some point through the iterative process, we get a value of , for which P (n) ≥ , then the process terminates, since in any case, all solutions of (E) will be oscillatory. The value of , that is, the number of iterations, obviously depends on the coefficient p(n) and the form of the non-monotone argument τ (n).

.. Advanced difference equations
Similar oscillation theorems for the (dual) advanced difference equation (E ) can be derived easily. The proofs of these theorems are omitted, since they are quite similar to the proofs for a retarded equation.
The following simple result is stated to explain why we can consider only the case where λ  >  is the smaller root of the transcendental equation λ = e βλ with  < β ≤ /e. Theorem  Assume that there exists a subsequence θ (n), n ∈ N of positive integers such that Then all solutions of (E ) are oscillatory.

Theorem  Assume that (.) and (.) hold, and ρ(n) is defined by (.). If for some
with Q  (n) = λ  q(n) and λ  is the smaller root of the transcendental equation λ = e βλ , then all solutions of (E ) are oscillatory.
Theorem  Assume that (.) and (.) hold, ρ(n) is defined by (.) and where Q(n) is defined by (.), then all solutions of (E ) are oscillatory.
Remark  It is clear that the left-hand sides of both conditions (.) and (.) are identical, also the right hand side of condition (.) reduces to (.) in the case that β = . So it seems that Theorem  is the same as Theorem  when β = . However, one may notice that condition  < β ≤ /e is required in Theorem  but not in Theorem . Remark  Similar comments to those in Remark  can be made for Theorems ,  and , concerning equation (E ).

.. Difference inequalities
A slight modification in the proofs of Theorems - and - leads to the following results about deviating difference inequalities.
has no eventually positive solutions; has no eventually negative solutions.

Discussion
In the present paper we are concerned with the oscillation of a linear delay or advanced difference equation with non-monotone argument. New sufficient conditions have been established for the oscillation of all solutions of (E) and (E ). These conditions include (.), (.), (.), (.), (.) and (.) of Theorems , , , ,  and , respectively, and are based on an iterative method.
The main advantage of these conditions is that they improve all the oscillation conditions in the literature. Conditions (.) and (.) improve the non-iterative conditions that are listed in the introduction, namely conditions (.), (.) and (.), respectively. This conclusion becomes evident immediately by inspecting the left-hand side of (.), (.) and the left-hand side of (.), (.) and (.).
The improvement of (.) and (.) as to the other iterative conditions, namely (.) (for r > ), (.) (for > ), (.) (for > ) and (.) (for r > ), (.) (for > ), is that they require far fewer iterations to establish oscillation, than the other conditions. This advantage can easily be verified computationally, by running the Matlab programs and comparing the number of iterations required by each condition to establish oscillation (see Section ).
Similar observations and comments can be made for conditions (.) and (.). It is to be pointed out that conditions (.) and (.) are of a type different from all the known oscillation conditions. Nevertheless, in Example , it is shown that (.) implies oscillation, while other known ones fail.

Examples and comments
In this section, examples illustrate cases when the results of the present paper imply oscillation while previously known results fail. The examples not only illustrate the significance of main results, but also serve to indicate the high degree of improvement, compared to the previous oscillation criteria in the literature. All the calculations were made in Matlab.

Figure 1 The graphs of τ (n) and h(n).
Example  (Taken and adapted from []) Consider the retarded difference equation with (see Figure  attains its maximum at n = μ + , μ ∈ N  , for every ∈ N. Specifically, . By using an algorithm on Matlab software, we obtain  It is easy to see that Clearly, q(n) =  , = . < /λ  ., i.e., (.) is satisfied. Observe that the function F : N  → R + defined as attains its maximum at n = μ + , μ ∈ N  , for every ∈ N. Specifically,

Conclusions
In this paper, new sufficient oscillation conditions for all solutions of (E) and (E ) have been established. These conditions have been derived using an iterative technique. As a result, the conditions in this paper significantly improve on the previously reported conditions that are reviewed in the introduction. The results are illustrated by two examples, showing that our conditions achieve a significant improvement over the known conditions. That improvement gets even greater by appropriately selecting the coefficients p(n) and q(n) and the non-monotone arguments τ (n) and σ (n). The conditions in this paper involve lim sup. Thus, an apparent research objective for future work can be establishing similar iterative techniques, for oscillation conditions, involving lim inf.