Oscillations of Difference Equations with Several Oscillating Coefficients

and Applied Analysis 3 Theorem 5 (see [21, Theorem 2.1]). Assume (1) and that the sequences τ i are increasing for all i ∈ {1, . . . , m}. Suppose also that for each i ∈ {1, . . . , m} there exists a sequence {n i (j)} j∈N such that lim j→∞ n i (j) = ∞, p k (n) ≥ 0 ∀n ∈ m ⋂ i=1 { { { ⋃ j∈N [τ i (τ i (n i (j))) , n i (j)] ∩ N } } } ̸ = 0, 1 ≤ k ≤ m, (18) lim sup n→∞ m ∑

Strong interest in (E R ) is motivated by the fact that it represents a discrete analogue of the differential equation (see [1] and the references cited therein) where, for every  ∈ {1, . . ., },   is an oscillating continuous real-valued function in the interval [0, ∞), and   is a continuous real-valued function on [0, ∞) such that   () ≤ ,  ≥ 0, lim while, (E A ) represents a discrete analogue of the advanced differential equation (see [1] and the references cited therein) By a solution of (E R ), we mean a sequence of real numbers {()} ≥− which satisfies (E R ) for all  ∈ N 0 .Here, It is clear that, for each choice of real numbers  − ,  −+1 , . . .,  −1 ,  0 , there exists a unique solution {()} ≥− of (E R ) which satisfies the initial conditions (−) =  − , (− + 1) =  −+1 , . . ., (−1) =  −1 , and (0) =  0 .By a solution of the advanced difference equation (E A ), we mean a sequence of real numbers {()} ∈N 0 which satisfies ] is called oscillatory, if the terms () of the sequence are neither eventually positive nor eventually negative.Otherwise, the solution is said to be nonoscillatory.
If there is a constant  such that then all solutions of (5) oscillate.
For (E R ) and (E A ) with oscillating coefficients, recently, Bohner et al. [21,23] established the following theorems.

Retarded Equations
In this section, we present new sufficient conditions for the oscillation of all solutions of (E R ) when the conditions (14) and (20) are not satisfied, under the assumption that the sequences   are increasing for all  ∈ {1, . . ., }.To that end, the following lemma provides a useful tool.Lemma 7. Assume that (1) holds, the sequences   are increasing for all  ∈ {1, . . ., } and (()) ≥− is a nonoscillatory solution of (E R ).Suppose also that for each  ∈ {1, . . ., } there exists a sequence {  ()} ∈N , such that lim  → ∞   () = ∞, and (12) where  is defined by (13).Set where Proof.Since the solution {()} ≥− of (E R ) is nonoscillatory, it is either eventually positive or eventually negative.As {−()} ≥− is also a solution of (E R ), we may restrict ourselves only to the case where () > 0 eventually.By (12), it is obvious that there exists  0 ∈ N such that Also, by (24) we have where  is an arbitrary real number with 0 <  < .

Examples
The significance of the results is illustrated in the following examples.