This paper is concerned with the oscillatory and nonoscillatory
behavior of solutions of fourth order quasilinear differential
equations of the form
By a solution of (1) we mean a real-valued function
such that
and
and
satisfies (1) at every point of
, where
and
may depend on
. Such a solution
of (1) is called nonoscillatory if
is eventually
positive or eventually negative. A solution
of (1) is
called oscillatory if it has an infinite sequence of zeros clustering
at
. Equation (1) itself is called oscillatory if
all of its solutions are oscillatory.
The main objective is to investigate the oscillatory and nonoscillatory
behavior of solutions of (1). We first study the structure of
the set of nonoscillatory solutions of (1). It is observed that a
solution which is asymptotic to a positive constant as
is ``minimal'' in the set of all eventually positive solutions
of (1), and a solution
which is asymptotic to a
positive constant multiple of the function
In the case , equation (1) is
Now, consider the second order quasilinear differential equation
(a) Equation (6) has a solution satisfying (7)
if and only if
(b) Equation (6) has a solution satisfying (8)
if and only if
(c) Let . Equation (6) has a nonoscillatory
solution
if and only if (9) is satisfied.
(d) Let . Equation (6) has a nonoscillatory
solution
if and only if (10) is satisfied.
The results in the present paper for the fourth order equation (1) provide parallel results to the second order equation (6).