Interval Oscillation Criteria for Second-Order Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals
By using a generalized arithmetic-geometric mean inequality on time scales, we study the forced oscillation of second-order dynamic equations with nonlinearities given by Riemann-Stieltjes integrals of the...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2011-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/719628 |
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| Summary: | By using a
generalized arithmetic-geometric mean inequality
on time scales, we study the forced oscillation
of second-order dynamic equations with
nonlinearities given by Riemann-Stieltjes
integrals of the form [p(t)ϕα(xΔ(t))]Δ+q(t)ϕα(x(τ(t)))+∫aσ(b)r(t,s)ϕγ(s)(x(g(t,s)))Δξ(s)=e(t), where t∈[t0,∞)T=[t0,∞) ⋂ T, T is a time scale which is unbounded from above; ϕ*(u)=|u|*sgn u; γ:[a,b]T1→ℝ is a strictly increasing right-dense continuous function; p,q,e:[t0,∞)T→ℝ, r:[t0,∞)T×[a,b]T1→ℝ, τ:[t0,∞)T→[t0,∞)T, and g:[t0,∞)T×[a,b]T1→[t0,∞)T are right-dense continuous functions; ξ:[a,b]T1→ℝ is strictly increasing. Some interval oscillation criteria are established in both the cases of delayed and advanced arguments. As a special case, the work in this paper unifies and improves many existing results in the literature for equations with a finite number of nonlinear terms. |
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| ISSN: | 1085-3375 1687-0409 |