Optimal oscillation criteria for first order difference equations with delay argument

Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument

“…Using the upper bound of the ratio ( ( ))/ ( ) for possible nonoscillatory solutions ( ) of (1) , presented in [7][8][9][10], the above result was essentially improved in 2014 by Stavroulakis [30].…”

“…Remark 21 (see [8]). Note that condition (43) is not a limitation since, by (C 1 ), if is a nondecreasing function, all solutions of (1) oscillate.…”

“…Remark 22 (see [8]). Condition (C 2 ) is optimal for (1) under the assumption that lim →+∞ ( − ( )) = ∞, since in this case the set of natural numbers increases infinitely in the interval [ ( ), − 1] for → ∞.…”

“…In [8] an example is also presented to show that the condition (C 2 ) is optimal, in the sense that it cannot be replaced by the nonstrong inequality.…”

On the oscillation of the solutions to delay and difference equations

“…In the case of the equation (1) with a general delay argument τ (n), from C h a t z a r a k i s, K o p l a t a d z e and S t a v r o u l a k i s [1], it follows the following This result generalizes the oscillation criterion (C 2 ) . Also C h a t z a r a k i s, K o p l a t a d z e and S t a v r o u l a k i s [2] extended the oscillation criterion (C 3 ) to the general case of the equation (1) . More precisely, the following theorem has been established in [2].…”