Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations
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...
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2010-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2010/598068 |
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| author | L. K. Kikina I. P. Stavroulakis |
| author_facet | L. K. Kikina I. P. Stavroulakis |
| author_sort | L. K. Kikina |
| collection | DOAJ |
| description | 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<liminft→∞{Q(t)P(g(t))}≤1/4 and limsupt→∞{Q(t)P(g(t))}<1, for the second-order functional equation, are presented. |
| format | Article |
| id | doaj-art-b412fff662aa4338975b376e93840484 |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-b412fff662aa4338975b376e938404842025-02-03T05:46:50ZengWileyInternational Journal of Differential Equations1687-96431687-96512010-01-01201010.1155/2010/598068598068Oscillation Criteria for Second-Order Delay, Difference, and Functional EquationsL. K. Kikina0I. P. Stavroulakis1Department of Mathematics, University of Gjirokastra, 6002 Gjirokastra, AlbaniaDepartment of Mathematics, University of Ioannina, 451 10 Ioannina, GreeceConsider 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<liminft→∞{Q(t)P(g(t))}≤1/4 and limsupt→∞{Q(t)P(g(t))}<1, for the second-order functional equation, are presented.http://dx.doi.org/10.1155/2010/598068 |
| spellingShingle | L. K. Kikina I. P. Stavroulakis Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations International Journal of Differential Equations |
| title | Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations |
| title_full | Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations |
| title_fullStr | Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations |
| title_full_unstemmed | Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations |
| title_short | Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations |
| title_sort | oscillation criteria for second order delay difference and functional equations |
| url | http://dx.doi.org/10.1155/2010/598068 |
| work_keys_str_mv | AT lkkikina oscillationcriteriaforsecondorderdelaydifferenceandfunctionalequations AT ipstavroulakis oscillationcriteriaforsecondorderdelaydifferenceandfunctionalequations |