IJPAM: Volume 63, No. 3 (2010)

NONOSCILLATION OF A SECOND ORDER SUBLINEAR
DIFFERENTIAL EQUATION WITH ``MAXIMA"

D. Kolev$^1$, N. Markova$^2$, S. Nenov$^3$
$^{1,3}$Department of Mathematics
University of Chemical Technology and Metallurgy
8, Kliment Ohridski Blvd., Sofia, 1756, BULGARIA
$^1$e-mail: [email protected], [email protected]
$^1$url: http://math.uctm.edu/ mkolev
$^3$e-mail: [email protected]
$^2$Department of Mathematics, Physics and Chemistry
Technical University
Sliven, 8800, BULGARIA
e-mail: [email protected]


Abstract.In this paper we consider a second order functional differential equation

\begin{displaymath}(r(t)x')' +
q(t)f\left(\max\limits_{s\in [\sigma (t),\tau (t)]}x(s)\right)=b(t),
\end{displaymath}

containing a function $f$ of sublinear rate and depending on the maximum of the unknown $x(t)$ and defined on some interval taken before coming the present time $t$. We call this equation with ``maxima". Criteria for existence of nonoscillating solutions are established under requirement that $r$, $r'$, $q$, $b$ should be continuous on some sets, and $f$ has sublinear rate. These differential equations could be seen in lots of mathematical models in theoretical physics, optimal control, chemistry, mechanics of materials, biology, ecology, etc.

Received: May 20, 2010

AMS Subject Classification: 34K15

Key Words and Phrases: differential equations with ``maxima'', second order ODEs, sublinear differential equation, nonoscillation, oscillation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 63
Issue: 3