Oscillatory and asymptotic behavior of $\frac{dx}{dt}+Q(t)G(x(t-\sigma(t)))=f(t)$
by Pitambar Das   P. K. Panda  

Vol. 6 No. 1 (2011) P.97~P.113

ABSTRACT

Consider the equation $$\frac{dx}{dt}+Q(t)G(x(t-\sigma(t)))=f(t)\qquad{(*)}$$ where $\textit{f},\sigma,\textit{Q}\in C ([0,\infty),[0,\infty)),G\in C(R,R),G(-x)=-G(x),xG(x)> o $ for $ x \neq 0,$ $\textit{G}$ is non-decreasing, $t >\sigma(t)$, $\sigma(t)$ is decreasing and $ t-\sigma(t)\rightarrow\infty$ as $t\rightarrow\infty$. When$ f(t)\equiv 0$, a sufficient condition in terms of the constants \begin{eqnarray*} k&=&\liminf_{t\rightarrow\infty}{\large\int_{t-\sigma(t)}^{t}{Q(s)ds}}\{\hbox{and}} \qquad L &=&\limsup_{t\rightarrow{\infty}}{\large\int_{t-\sigma(t)}^{t}{Q(s)ds}} \end{eqnarray*} is established for all solutions of $(*)$ to be oscillatory.The present results improve the earlier results of the literature by both weakening the conditions and considering a general non linear and non-homogeneous differential equation.

KEYWORDS
Nonlinear, non homogeneous, delay differential equation, oscillation, asymptotic behaviour

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 34K11,34C10

MILESTONES

Received: 2009-11-24
Revised : 2010-10-05
Accepted: 2010-10-05

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