Ultimate Bounded-ness of Periodic Solutions of A Class of Lienard Type p- Laplacian Equation with Multiple Deviating Arguments

Abstract

In this article, we investigate quite formally the existence of solutions followed by investigating the
ultimate bounded-ness of the following Lienard type p-Laplacian equation with multiple deviating
arguments :
(p(x0))0 + f1(x)(p(x0))2 + f2(x)(p(x0))
+(t)
Xn
i=1
g(t; x(t .. i(t))) = e(t)
9>=
>;
(L)
Zwhere > 0; f1; f2; e 2 C(RI ;RI ); (t) > 0; i(t) > 0 are two T-periodic functions with T
0
e(t)dt = 0; T > 0, g(t; x)is continuous and g(t; :) = g(t + T; :) and for p > 1, p : IR !
IR; ((p(x0)))0 =
d
dt
fj
dx
dt
jp..2 dx
dt
g is a one-dimensional p-Laplacian. The study employs a combina-
tion of the Manasevich-Mawhin continuation theorem in settling the question of existence of periodic
solutions and the Lyapunov second method in providing a framework for obtaining bounded-ness of
such solutions. An example illustrates our results.