V. A. PLISS

ON THE BOUNDEDNESS OF SOLUTIONS OF SOME NONLINEAR DIFFERENTIAL EQUATIONS OF THE THIRD ORDER

(Presented by Academician V. I. Smirnov on 6 III 1961)

In the paper [1] Ezeilo considered the equation

[
\dddot{x}+a\ddot{x}+b\dot{x}+f(x)=p(t)
\tag{1}
]

and proved the following assertion. Let (p(t)) be continuous, and let (f(x)) be continuously differentiable for all (t) and (x), respectively. Suppose the following conditions are satisfied:

1) (a>0,\ b>0).

2) (f(x)\operatorname{sign}x>0) for (|x|\geq 1).

3) (\lim_{|x|\to\infty}|f(x)|=\infty).

4) There exists a constant (c) ((0<c<ab)) such that (f'(x)\leq c) for (|x|\geq 1).

5) There exists a constant (A>0) such that, for all (t),

[
|p(t)|<A,\quad \left|\int^t p(t)\,dt\right|<A.
]

Then one can specify a constant (D>0) such that, for any solution of equation (1) with initial data at (t=t_0,\ x=x_0,\ \dot{x}=\dot{x}_0,\ \ddot{x}=\ddot{x}_0), one has

[
|x|<D,\quad |\dot{x}|<D,\quad |\ddot{x}|T(t_0,x_0,\dot{x}_0,\ddot{x}_0).
\tag{2}
]

In the present note we formulate several theorems on the boundedness of solutions of nonlinear equations of the third order. Theorem 1 is a direct generalization of Ezeilo’s theorem.

(1^\circ). Consider the differential equation

[
\dddot{x}+a\ddot{x}+b\dot{x}+f(x)=P_1(x,\dot{x},\ddot{x},t).
\tag{3}
]

Put (y=ax+\dot{x},\ z=bx+a\dot{x}+\ddot{x}); then equation (3) is replaced by the system

[
\dot{x}=y-ax,\quad
\dot{y}=z-bx,\quad
\dot{z}=-f(x)+P(x,y,z,t).
\tag{4}
]

Theorem 1. Let the functions (f(x)) and (P_1(x,\dot{x},\ddot{x},t)) be continuous and satisfy the uniqueness condition for solutions of equation (3) for all (x,\dot{x},\ddot{x},t). Suppose the following conditions are satisfied:

1) (a>0,\ b>0).

2) (0<\dfrac{f(x)}{x}<ab) for (|x|\geq 1).

3) (\lim_{|x|\to\infty}|f(x)-abx|=\infty).

4) (\lim_{|x|\to\infty}|f(x)|=\infty).

5) There exists a constant (A>0) such that for all (x,\dot x,\ddot x,t) one has
(|P_1(x,\dot x,\ddot x,t)|<A).

Then one can specify a positive constant (D) such that, for any solution of equation (3),

[
|x|<D,\qquad |\dot x|<D,\qquad |\ddot x|<D
\quad \text{for } t\geq T(t_0,x_0,\dot x_0,\ddot x_0),
\tag{5}
]

where (t_0,x_0,\dot x_0,\ddot x_0) are the initial data of the chosen solution.

The proof of the theorem is based on considering the function

[
v_1=\frac12(a^2x-ay+z)^2+\frac12(z-bx)^2+\frac b2 y^2
+a\int_0^x |f(x)-abx|\,dx
\tag{6}
]

and its total time derivative, taken by virtue of the differential equations of system (4):

[
\dot v_1=-a(a^2x-ay+z)^2+[abx-f(x)][2(a^2x-ay+z)-bx]+
]
[
+[(a^2-b)x-ay+2z]\,P(x,y,z,t).
\tag{7}
]

It can be shown that along every solution, after a sufficiently long interval of time, the function (v_1) becomes smaller than some constant quantity independent of the choice of the solution. From this it is already not difficult to derive the assertion of the theorem.

(2^\circ). Consider the equation

[
\dddot z+a\ddot z+\varphi(\dot z)+bz=G_1(z,\dot z,\ddot z,t).
]

We shall assume that (b>0). The scale of the quantity (t) can be changed so that (b=1). Therefore we shall study the equation

[
\dddot z+a\ddot z+\varphi(\dot z)+z=G_1(z,\dot z,\ddot z,t).
\tag{8}
]

Put (z=-x,\ y=\dot x+ax,\ \varphi(x)=-\psi(-x)); then we obtain the system

[
\dot x=y-ax,\qquad \dot y=z-\psi(x)+G(x,y,z,t),\qquad \dot z=-x.
\tag{9}
]

Theorem 2. Let (\varphi(z)) and (G_1(z,\dot z,\ddot z,t)) be continuous and satisfy the uniqueness condition for solutions of equation (8) for all (z,\dot z,\ddot z,t). Suppose, in addition, that the following conditions are fulfilled:

1) (a>0).

2) (\dfrac{\varphi(x)}{x}>\dfrac1a) for (|x|\geq 1).

3) (|a\varphi(x)-x|\to\infty) as (|x|\to\infty).

4) There exists a constant (A) such that for all (z,\dot z,\ddot z,t) one has
[
|G_1(z,\dot z,\ddot z,t)|<A.
]

Then there exists (D>0) such that, for any solution of equation (8),

[
|z|<D,\qquad |\dot z|<D,\qquad \ddot z<D
\quad \text{for } t\geq T(z_1,\dot z_0,\ddot z_0,t_0).
\tag{10}
]

To prove this theorem one should consider the function (see work ((^2)))

[
v_2=\frac12 y^2+\frac a2 z^2-zx+\int_0^x \psi(x)\,dx
\tag{11}
]

and its time derivative, taken by virtue of system (9):

[
\dot v_2=x[x-a\psi(x)]+yG(x,y,z,t).
\tag{12}
]

3°. Finally, let us consider the equation

[
\dddot{\xi}+g(\ddot{\xi})+b\dot{\xi}+a\xi
=
Q_1(\xi,\dot{\xi},\ddot{\xi},t).
]

Assuming (b>0), we can change the scale of the time variable (t) in such a way that in our equation (b=1). Therefore, let us consider the equation

[
\dddot{\xi}+g(\ddot{\xi})+\dot{\xi}+a\xi
=
Q_1(\xi,\dot{\xi},\ddot{\xi},t).
\tag{13}
]

Put (x=\ddot{\xi}), (y=-(\dot{\xi}+a\xi)), (z=-a\dot{\xi}); then we obtain the system

[
\dot{x}=y-g(x)+Q(x,y,z,t),\qquad
\dot{y}=z-x,\qquad
\dot{z}=-ax.
\tag{14}
]

Theorem 3. Suppose that the function (g(x)) is continuously differentiable for all (x), and that (Q_1(\xi,\dot{\xi},\ddot{\xi},t)) is continuous and satisfies the uniqueness condition for solutions of equation (13) for all (\xi,\dot{\xi},\ddot{\xi},t). Suppose, moreover, that the following conditions are satisfied:

1) (a>0).
2) (g'(x)>a+\varepsilon) for (|x|\geqslant 1), where (\varepsilon>0) is a constant.
3) There exists a constant (A>0) such that
[
|Q_1(\xi,\dot{\xi},\ddot{\xi},t)|<A
]
for all (\xi,\dot{\xi},\ddot{\xi},t).

Then one can indicate a constant (D>0) such that, for any solution of equation (13),

[
|\xi|