<abstract xmlns="http://www.w3.org/1999/xhtml"> Consider the first-order linear delay differential equation <italic>x</italic>ʹ(<italic>t</italic>) + <italic>p</italic>(<italic>t</italic>)<italic>x(τ</italic>(<italic>t</italic>)) = 0<italic>, t≥ </italic>t&lt;sub>0&lt;/sub><italic>, </italic>(1) where <italic>p, τ </italic>∈<italic> C</italic> ([<italic>t&lt;sub></italic>0&lt;/sub><italic>,∞, </italic>ℝ&lt;sup>+&lt;/sup>, <italic>τ</italic>(<italic>t</italic>) is nondecreasing, <italic>τ</italic>(<italic>t</italic>) <italic>&lt; t </italic>for <italic>t ≥ t&lt;sup></italic>0&lt;/sup> and lim&lt;sub><italic>t→∞&lt;/sub> τ</italic>(<italic>t</italic>) = <italic>∞</italic>, and the (discrete analogue) difference equation Δ<italic>x</italic>(<italic>n</italic>) + <italic>p</italic>(<italic>n</italic>)<italic>x</italic>(<italic>τ</italic>(<italic>n</italic>)) = 0<italic>, n</italic>= 0<italic>, </italic>1<italic>, </italic>2<italic>,…, </italic>(1)ʹ where Δ<italic>x</italic>(<italic>n</italic>) = <italic>x</italic>(<italic>n </italic>+ 1) <italic>− x</italic>(<italic>n</italic>), <italic>p</italic>(<italic>n</italic>) is a sequence of nonnegative real numbers and <italic>τ</italic>(<italic>n</italic>) is a nondecreasing sequence of integers such that <italic>τ</italic>(<italic>n</italic>) <italic>≤ n − </italic>1 for all <italic>n ≥ </italic>0 and lim&lt;sub><italic>n→∞&lt;/sub> τ</italic>(<italic>n</italic>) = <italic>∞</italic>. Optimal conditions for the oscillation of all solutions to the above equations are presented.</abstract>

Consider the first-order linear delay differential equation xʹ(t) + p(t)x(τ(t)) = 0, t≥ t0, (1) where p, τ ∈ C ([t0,∞, ℝ+, τ(t) is nondecreasing, τ(t) < t for t ≥ t0 and limt→∞ τ(t) = ∞, and the (discrete analogue) difference equation Δx(n) + p(n)x(τ(n)) = 0, n= 0, 1, 2,…, (1)ʹ where Δx(n) = x(n + 1) − x(n), p(n) is a sequence of nonnegative real numbers and τ(n) is a nondecreasing sequence of integers such that τ(n) ≤ n − 1 for all n ≥ 0 and limn→∞ τ(n) = ∞. Optimal conditions for the oscillation of all solutions to the above equations are presented.

On the oscillation of the solutions to delay and difference equations

Département de Mathématiques et Informatique Faculté des Sciences Université Hassan II Ain Chock Casablanca MAROC

Department of Mathematics University of Ioannina 451-10 Ioannina GREECE

Tatra Mountains Mathematical Publications

{"article-title":"On the oscillation of the solutions to delay and difference equations"}

Consider the first-order linear delay differential equation xʹ(t) + p(t)x(τ(t)) = 0, t≥ t0, (1) where p, τ ∈ C ([t0,∞, ℝ+, τ(t) is nondecreasing, τ(t) 0 and limt→∞ τ(t) = ∞, and the (discrete analogue) difference equation Δx(n) + p(n)x(τ(n)) = 0, n= 0, 1, 2,…, (1)ʹ where Δx(n) = x(n + 1) − x(n), p(n) is a sequence of nonnegative real numbers and τ(n) is a nondecreasing sequence of integers such that τ(n) ≤ n − 1 for all n ≥ 0 and limn→∞ τ(n) = ∞. Optimal conditions for the oscillation of all solutions to the above equations are presented.

Consider the first-order linear delay differential equation xʹ(t) + p(t)x(τ(t)) = 0, t≥ t0, (1) where p, τ ∈ C ([t0,∞, ℝ+,...

On the oscillation of the solutions to delay and difference equations

Published Online: Nov 12, 2012

Page range: 173 - 187

DOI: https://doi.org/10.2478/v10127-009-0036-3

Keywords
delay equation, difference equation, oscillatory solution, nonoscillatory solution

This content is open access.

On the oscillation of the solutions to delay and difference equations

Published Online: Nov 12, 2012

Page range: 173 - 187

DOI: https://doi.org/10.2478/v10127-009-0036-3

Keywordsdelay equation, difference equation, oscillatory solution, nonoscillatory solution

This content is open access.

Keywords
delay equation, difference equation, oscillatory solution, nonoscillatory solution