In the early 1960s, Littlewood [

1] asked whether or not the solutions of the Duffing-type equation

are bounded for all time, that is, whether there are resonances that might cause the amplitude of the oscillations to increase without bound.

The first positive result of boundedness of solutions in the *superlinear case* (i.e.,
as
) was due to Morris [2]. By means of KAM theorem, Morris proved that every solution of the differential equation (1.1) is bounded if
, where
is piecewise continuous and periodic. This result relies on the fact that the nonlinearity
can guarantee the twist condition of KAM theorem. Later, several authors (see [3–5]) improved Morris's result and obtained similar result for a large class of superlinear function
.

When

satisfies

that is, the differential equation (1.1) is*semilinear*, similar results also hold, but the proof is more difficult since there may be resonant case. We refer to [6–8] and the references therein.

In [

8] Liu considered the following equation:

where
as
and
is a
-periodic function. After introducing new variables, the differential equation (1.3) can be changed into a Hamiltonian system. Under some suitable assumptions on
and
, by using a variant of Moser's small twist theorem [9] to the Pioncaré map, the author obtained the existence of quasi-periodic solutions and the boundedness of all solutions.

Then the result is generalized to a class of

-Laplacian differential equation.Yang [

10] considered the following nonlinear differential equation

where
is bounded,
is periodic. The idea is also to change the original problem to Hamiltonian system and then use a twist theorem of area-preserving mapping to the Pioncaré map.

The above differential equation essentially possess Hamiltonian structure. It is well known that the Hamiltonian structure and reversible structure have many similar property. Especially, they have similar KAM theorem.

Recently, Liu [

6] studied the following equation:

where

is a positive constant and

is

-periodic in

. Under some assumption of

and

, the differential equation (

1.5) has a reversible structure. Suppose that

satisfies

where

and

. Moreover,

where

is a constant. Note that here and below we always use

to indicate some constants. Assume that there exists

such that

Then, the following conclusions hold true.

(i)There exist
and a closed set
having positive measure such that for any
, there exists a quasi-periodic solution for (1.5) with the basic frequency
.

(ii)Every solution of (1.5) is bounded.

Motivated by the papers [

6,

8,

10], we consider the following

-Laplacian equation:

where
,
, and
are constants. We want to generalize the result in [6] to a class of
-Laplacian-type differential equations of the form (1.9). The main idea is similar to that in [6]. We will assume that the functions
and
have some parities such that the differential system (1.9) still has a reversible structure. After some transformations, we change the systems (1.9) to a form of small perturbation of integrable reversible system. Then a KAM Theorem for reversible mapping can be applied to the Poincaré mapping of this nearly integrable reversible system and some desired result can be obtained.

Our main result is the following theorem.

Theorem 1.1.

Suppose that

and

are of class

in their arguments and

-periodic with respect to t such that

Moreover, suppose that there exists

such that

for all
, for all
,
. Then every solution of (1.9) is bounded.

Remark 1.2.

Our main nonlinearity
in (1.9) corresponds to
in (1.5). Although it is more special than
, it makes no essential difference of proof and can simplify our proof greatly. It is easy to see from the proof that this main nonlinearity is used to guarantee the small twist condition.