Oscillations in Difference Equations with Deviating Arguments and Variable Coefficients

and Applied Analysis 3 Set z i (n) = x (τ i (n)) x (n) , ∀n ≥ n 3 , 1 ≤ i ≤ m, φ i = lim inf n→∞ z i (n) , 1 ≤ i ≤ m. (18) It is obvious that z i (n) > 1, φ i ≥ 1 for i = 1, 2, . . . , m. (19) Now we will show that φ i < ∞ for i = 1, 2, . . . , m. Indeed, assume that φ i = ∞ for some i, i = 1, 2, . . . , m. For this i, by (ER), we have Δx (n) + p i (n) x (τ i (n)) ≤ 0, ∀n ≥ n 3 . (20) At this point, we will establish the following claim. Claim 1 (cf. [8]). For each n ≥ n 3 , there exists an integer n∗ i ≥ n for each i = 1, 2, . . . , m such that τ i (n ∗ i ) ≤ n − 1, and

A solution (()) ≥− (or (()) ≥0 ) of (E R ) (or (E A )) is called oscillatory, if the terms () of the sequence are neither eventually positive nor eventually negative.Otherwise, the solution is said to be nonoscillatory.
Proof.Assume, for the sake of contradiction, that (()) ≥− is a nonoscillatory solution of (E R ).Then 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 for all large .Let  1 ≥ − be an integer such that () > 0 for all  ≥  1 .Then, there exists  2 ≥  1 such that In view of this, (E R ) becomes which means that the sequence (()) is eventually decreasing.
Next choose a natural number  3 >  2 such that It is obvious that Now we will show that   < ∞ for  = 1, 2, . . ., .Indeed, assume that   = ∞ for some ,  = 1, 2, . . ., .For this , by (E R ), we have At this point, we will establish the following claim.
Dividing both sides of (E R ) by (), for  ≥  3 , we obtain or Summing up (38) from   () to  − 1 for  = 1, 2, . . ., , we find or Combining ( 39) and (41), we obtain or ()   () ,  = 1, 2, . . ., . (43) Taking limit inferiors on both sides of the above inequalities (43), we obtain and by adding we find Set Clearly Since for the function  has a maximum at the critical point since the quadratic form Since ( 1 ,  2 , . . .,   ) ≥ 0, the maximum of  at the critical point should be nonnegative.Thus, that is, max Hence or which contradicts (14).The proof of the theorem is complete.
Proof.Assume, for the sake of contradiction, that (()) ≥− is a nonoscillatory solution of (E R ).Then 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 for all large .Let  1 ≥ − be an integer such that () > 0 for all  ≥  1 .Then, there exists  2 ≥  1 such that In view of this, (E R ) becomes which means that the sequence (()) is eventually decreasing.
Taking into account the fact that   < ∞ for  = 1, 2, . . .,  (see proof of Theorem 3), by using (44) and the fact that we obtain Adding these inequalities we have ()) or which contradicts (56).The proof of the theorem is complete.

Advanced Difference Equations.
Similar oscillation theorems for the (dual) advanced difference equation (E A ) can be derived easily.The proofs of these theorems are omitted, since they follow a similar procedure as in Section 2.1.

Corollary 7.
Assume that Then all solutions of (E) oscillate.
Remark 8.A research question that arises is whether Theorems 3-6 are valid, even in the case where the coefficients () oscillate (see [15,16]).Then our results would be comparable to those in [15,16].This is a question that we currently study and expect to have some results soon.

Examples
The following two examples illustrate that the conditions for oscillations (65) and (66) are independent.They are chosen in such a way that only one of them is satisfied.
It is easy to see that That is, condition (66) of Corollary 7 is satisfied and therefore all solutions of (74) oscillate.However, That is, condition (65) of Corollary 7 is not satisfied.Observe that Thus and therefore none of the conditions ( 9), (12), and (13) are satisfied.
At this point, we give an example with general retarded arguments illustrating the main result of Theorem 3. Similarly, one can construct examples to illustrate Theorems 4-6.