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

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

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),

\[ |\xi|<D,\qquad |\dot{\xi}|<D,\qquad |\ddot{\xi}|<D \quad \text{for } t\geqslant T(\xi_0,\dot{\xi}_0,\ddot{\xi}_0,t_0). \tag{15} \]

For the proof one considers the function (see (3))

\[ v_3= \frac12(z-x)^2+\frac12 y^2+axy-yg(x)+\frac12 g^2(x) - a\int_0^x g(x)\,dx. \tag{16} \]

The derivative of this function by virtue of system (14) is equal to

\[ \dot v_3 = -[g'(x)-a][y-g(x)]-(z-x)Q(x,y,z,t)- \]
\[ \qquad -[g'(x)-a][y-g(x)]\,Q(x,y,z,t). \tag{17} \]

4°. From the form of the functions \(v_1, v_2, v_3\) and from the well-known Brouwer theorem on the existence of fixed points, one can derive the assertion of the following theorems.

Theorem 4. Let the conditions of Theorem 1 be satisfied. Let the function \(P_1(x,\dot{x},\ddot{x},t)\) have period \(\omega\) in \(t\), i.e.
\[ P_1(x,\dot{x},\ddot{x},t+\omega)=P_1(x,\dot{x},\ddot{x},t). \]
Then equation (3) has at least one \(\omega\)-periodic solution.

Theorem 5. Let the conditions of Theorem 2 be satisfied and, in addition, let the function \(G_1\) have period \(\omega\) in \(t\). Then equation (8) has at least one \(\omega\)-periodic solution.

Theorem 6. Let the conditions of Theorem 3 be satisfied and let the function \(Q_1\) have period \(\omega\) in \(t\). Then equation (13) has at least one \(\omega\)-periodic solution.

Leningrad State University
named after A. A. Zhdanov

Received
8 II 1961

CITED LITERATURE

1. J. O. Ezeilo, Proc. Lond. Math. Soc., 9, No. 33, 74 (1959).
2. E. A. Barbashin, Prikl. matem. i mekh., 16, issue 5 (1952).
3. V. A. Pliss, Some Problems in the Theory of Stability of Motion in the Large, L., 1958.