On oscillatory and asymptotic behavior of fourth order non-linear neutral delay dynamic equations. (English) Zbl 1241.34096
Summary: Oscillatory and asymptotic properties of solutions of a class of nonlinear fourth order neutral dynamic equations of the form
\[
(r(t)(y(t)+p(t)y(\alpha(t)))^{\Delta^2})^{\Delta^2}+ q(t)G(y(\beta(t)))=0\tag{H}
\]
and
\[
(r(t)(y(t)+p(t)y(\alpha(t)))^{\Delta^2})^{\Delta^2}+ q(t)G(y(\beta(t)))=f(t)\tag{NH}
\]
for \(t\in[t_0,\infty]_\mathbb{T}\), \(t_0\geq 0\), where \(\mathbb{T}\) is a time scale such that \(\sup\mathbb{T}=\infty\), \(t_0\in\mathbb{T}\) are studied under the assumption \(\int^\infty_{t_0}\frac{t}{r(t)}\Delta t=\infty\) for various ranges of \(p(t)\). Sufficient conditions are obtained for the existence of bounded positive solutions of (NH) by using Schauder’s fixed point theorem.
MSC:
34N05 | Dynamic equations on time scales or measure chains |
34K11 | Oscillation theory of functional-differential equations |
34K40 | Neutral functional-differential equations |
34K25 | Asymptotic theory of functional-differential equations |
47N20 | Applications of operator theory to differential and integral equations |
Keywords:
oscillation; neutral dynamic equations; existence of positive solutions; asymptotic behaviour; time scaleReferences:
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[3] | Thandapani, E.; Arockiasamy, I. M., Oscillatory and asymptotic behaviour of fourth order non-linear neutral delay difference equations, Indian J. Pure Appl. Math., 32, 109-123 (2001) · Zbl 1004.39005 |
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