caa a22 4500 606200983 CHVBK 20210128100946.0 cr unu---uuuuu 210128e20150901xx s 000 0 eng 10.1007/s00025-014-0425-z doi (NATIONALLICENCE)springer-10.1007/s00025-014-0425-z An Oscillation Criterion for First Order Difference Equations [Elektronische Daten] [Özkan Öcalan, Sermin Öztürk] This paper is concerned with the oscillatory behavior of first order difference equation with general argument $$\Delta x(n) + p(n)x\left( \tau (n)\right) = 0,\quad n = 0,1,\ldots \qquad\qquad (\star)$$ Δ x ( n ) + p ( n ) x τ ( n ) = 0 , n = 0 , 1 , ... ( ⋆ ) where $${{(p(n))_{n\geq 0}}}$$ ( p ( n ) ) n ≥ 0 is a sequence of nonnegative real numbers and $${{(\tau (n))_{n\geq 0}}}$$ ( τ ( n ) ) n ≥ 0 is a sequence of integers. Let the numbers k and L be defined by $$k = \liminf\limits_{n \rightarrow \infty} \sum \limits_{j=\tau (n)}^{n-1}p(j)$$ k = lim inf n → ∞ ∑ j = τ ( n ) n - 1 p ( j ) and $$L=\limsup\limits_{n\rightarrow \infty} \sum\limits_{j=\tau(n)}^{n}p(j).$$ L = lim sup n → ∞ ∑ j = τ ( n ) n p ( j ) . It is proved that, when L<1 and $${{0 < k \leq \frac{1}{e},}}$$ 0 < k ≤ 1 e , all solutions of Equation ( $${{\star}}$$ ⋆ ) oscillate if the condition $$L > 2k+\frac{2}{{\rm \lambda} _{1}}-1$$ L > 2 k + 2 λ 1 - 1 where $${{{\rm \lambda} _{1}\in \lbrack 1,e]}}$$ λ 1 ∈ [ 1 , e ] is the unique root of the equation $${{{\rm \lambda} =e^{k\lambda },}}$$ λ = e k λ , is satisfied. Springer Basel, 2014 Delay difference equation nationallicence oscillation nationallicence Öcalan Özkan Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, ANS Campus, 03200, Afyon, Turkey aut Öztürk Sermin Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, ANS Campus, 03200, Afyon, Turkey aut Results in Mathematics Springer Basel 68/1-2(2015-09-01), 105-116 1422-6383 68:1-2<105 2015 68 25 https://doi.org/10.1007/s00025-014-0425-z text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00025-014-0425-z text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Öcalan Özkan Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, ANS Campus, 03200, Afyon, Turkey aut NATIONALLICENCE 700 1- Öztürk Sermin Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, ANS Campus, 03200, Afyon, Turkey aut NATIONALLICENCE 773 0- Results in Mathematics Springer Basel 68/1-2(2015-09-01), 105-116 1422-6383 68:1-2<105 2015 68 25