Oscillation criteria for linear difference equations with several variable delays

Abstract

We obtain new sufficient criteria for the oscillation of all solutions of linear delay difference equations with several (variable) finite delays. Our results relax numerous well-known limes inferior-type oscillation criteria from the literature by letting the limes inferior be replaced by the limes superior under some additional assumptions related to slow variation. On the other hand, our findings generalize an oscillation criterion recently given for the case of a constant, single delay.

A Survey on Sharp Oscillation Conditions for Delay Difference Equations

“…The concept of slowly varying is used in [38] to improve the above results and also it is assumed that holds for all (26) The following result in the next theorem improves Theorem 3.…”

“…Theorem 6 [38,Theorem 6] Suppose that condition (26) holds and that there exists a positive constant 𝑀 such that 0 ≤ 𝑝 (𝑛) ≤ 𝑀 holds for all 1 ≤ 𝑖 ≤ 𝑘. Assume further that there exists a sequence (𝜏 * (𝑛)) ∈ such that 𝜏 (𝑛) ≤ 𝜏 * (𝑛) ≤ 𝑛 − 1 holds for all 1 ≤ 𝑖 ≤ 𝑘 and 𝑛 ∈ 𝑁, and that the function…”

“…Theorem 7 [38,Theorem 7] Suppose that for all and there exist a positive constant 𝑀 and a positive integer 𝑙 * ≤ 𝑙 such that 0 ≤ 𝑝 (𝑛) ≤ 𝑀 hold for all 1 ≤ 𝑖 ≤ 𝑘 and the function 𝑝 (𝑗) * is slowly varying, and moreover, the inequality…”

“…Then all solutions of equation (1.2) are oscillatory.Similarly Theorem 5 is improved by the following result.Theorem 9[38, Theorem 10] Suppose that for all and there exists 𝑀 > 0 such that 0 ≤ 𝑝 (𝑛) ≤ 𝑀 hold for all 1 ≤ 𝑖 ≤ 𝑘. Furthermore, assume that the function every solution of equation (E) is oscillatory.The last two Theorems improve Theorem 1 because it is a special case with 𝑘 = 1 and 𝜏 * (𝑛) = 𝜏 (𝑛).The following Example compares Theorems 6 and 3.Example 10 [38, Example 3.1] Let us consider equation (1.…”

“…𝐴(𝑛) is not slowly varying and one could not apply Theorem 6 with this choice of 𝜏 * .Finally, an application of Theorem 9 is given.Example 11[38, Example 3.2] Consider the constant delay equation(27) with 𝑙 = 1 and 𝑙 = 2 and coefficient functions positive constants 𝑐 , 𝑐 and 𝑑 . The coefficients are evidently nonnegative and bounded, moreover, condition(26) is satisfied.…”