oscillation criteria for second-order delay, difference, and functional equations
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2010
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Abstract
Consider the second-order linear delay differential equation x′′(t)+p(t)x(τ(t))=0, t≥t0, where p∈C([t0,∞),ℝ+), τ∈C([t0,∞),ℝ), τ(t) is nondecreasing, τ(t)≤t for t≥t0 and limt→∞τ(t)=∞, the (discrete analogue) second-order difference equation Δ2x(n)+p(n)x(τ(n))=0, where Δx(n)=x(n+1)−x(n), Δ2=Δ∘Δ, p:ℕ→ℝ+, τ:ℕ→ℕ, τ(n)≤n−1, and limn→∞τ(n)=+∞, and the second-order functional equation x(g(t))=P(t)x(t)+Q(t)x(g2(t)), t≥t0, where the functions P, Q∈C([t0,∞),ℝ+), g∈C([t0,∞),ℝ), g(t)≢t for t≥t0, limt→∞g(t)=∞, and g2 denotes the 2th iterate of the function g, that is, g0(t)=t, g2(t)=g(g(t)), t≥t0. The most interesting oscillation criteria for the second-order linear delay differential equation, the second-order difference equation and the second-order functional equation, especially in the case where liminft→∞∫τ(t)tτ(s)p(s)ds≤1/e and limsupt→∞∫τ(t)tτ(s)p(s)ds<1 for the second-order linear delay differential equation, and 0
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| Reference Key |
kikina2010internationaloscillation
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| Authors | ;L. K. Kikina;I. P. Stavroulakis |
| Journal | Biological reviews of the Cambridge Philosophical Society |
| Year | 2010 |
| DOI |
10.1155/2010/598068
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