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BRIEF COMMUNICATIONS
A CERTAIN THEOREM AND A PROBLEM FROM THE QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS
A. V. DRAGILEV
The present note arose in connection with the author’s reading of Chapter 2 of Figueiredo’s book [1]. Section 2 contains a generalization of Theorem 2.1 from the above-mentioned book. In Section 3 the reader’s attention is drawn to an assertion more general than Theorem 2.2 of [1], the question of proving or disproving which remains open. Section 1 is auxiliary.
- In the following theorem the results of Theorems 3 and 4 of § 2 of the author’s paper [2] are combined as applied to the equation, more special than that considered in [2],
\[ \frac{d^2u}{dt^2}+F\left(\frac{du}{dt}\right)+u=0. \tag{1} \]
In this form the result is essentially close to Theorem 1 of the paper by A. F. Filippov [3].
Theorem 1. The system
\[ \frac{du}{dt}=v,\qquad \frac{dv}{dt}=-F(v)-u, \tag{2} \]
corresponding to equation (1), belongs to the class \(D^*\), if
1) \(F(v)\) is piecewise continuous and ensures uniqueness of solutions of system (2);
2) \(\dfrac{F(v)}{v}\geq a>-2\) for \(|v|>C\), where \(C\) is some positive number;
3) there exists \(v_1>0\) such that
\[ \text{a) }\quad F(v)\geq F(-v)\quad \text{for } v\geq v_1, \]
\[ \text{b) }\quad \int_0^{v_1} [F(v)-F(-v)]\,v\,dv \geq \alpha>0. \]
- Theorem 2. The system
\[ \frac{dx}{dt}=y,\qquad \frac{dy}{dt}=-f(x,y)y-g(x), \tag{3} \]
corresponding to the equation
\[ \frac{d^2x}{dt^2}+f\left(x,\frac{dx}{dt}\right)\frac{dx}{dt}+g(x)=0, \tag{4} \]
belongs to the class \(D\), if the following conditions are fulfilled:
1) \(f(x,y)\) and \(g(x)\) are piecewise continuous and ensure uniqueness of solutions of system (3). The points of discontinuity of \(f(x,y)\) may be placed on several straight lines \(x=\mathrm{const}\);
2) \(\operatorname{sgn} g(x)=\operatorname{sgn} x\);
3) \(g(x)\leq a^2x\) \((a>0)\) for \(x\geq x_0>0\) and
\[ \int_0^{+\infty} g(\xi)\,d\xi=+\infty; \]
\[ \text{* Definition of a system of class } D \text{ see in [4].} \]
4) \(g(x)\leqslant \beta^{2}x\) \((\beta>0)\) for \(x\leqslant -x_{0}\);
5) \(f(x,y)\geqslant \omega\) \((\omega>0)\) for \(x>x_{0}\);
6) \(f(x,y)\geqslant -\omega_{1}\) \((\omega_{1}>0)\)* for \(x<-x_{0}\);
7) \(f(x,y)\geqslant -M\) for \(|x|\leqslant x_{0}\);
8) \(\omega_{1}<2\beta\);
9) \[ \frac{\omega}{\alpha}>\frac{\omega_{1}}{\beta}. \]
The proof below is given for the particular case of system (3), namely for the system
\[ \frac{dx}{dt}=y,\qquad \frac{dy}{dt}=-\varphi(x)y-g(x), \tag{5} \]
in which \(\varphi(x)=-\omega_{1}\) for \(x<-x_{0}\), \(\varphi(x)=-M\) for \(|x|\leqslant x_{0}\), \(\varphi(x)=\omega\) for \(x>x_{0}\). The passage to the general case is carried out with the aid of the comparison theorem (Theorem 1, § 1 of [2]).
Proof. The substitution
\[ y=-u-\Phi(x),\quad \left(\Phi(x)=\int_{0}^{x}\varphi(\xi)\,d\xi\right) \tag{6} \]
transforms system (5) into the system
\[ \frac{dx}{dt}=-u-\Phi(x),\qquad \frac{du}{dt}=g(x). \tag{7} \]
The substitution
\[ v=\sqrt{\,2\int_{0}^{x}g(\xi)\,d\xi\,}\,\operatorname{sgn}x, \tag{8} \]
accompanied by the corresponding transformation of \(t\), reduces the study of system (7), and consequently also of (5), to the study of the system
\[ \frac{dv}{dt}=-u-\Phi[x(v)],\qquad \frac{du}{dt}=v. \tag{9} \]
Theorem 1 can be applied to system (9), putting \(F(v)=\Phi[x(v)]\). Using (8), (9), and also conditions 1)—7) of Theorem 2, it is easy to obtain that, for arbitrarily small \(\varepsilon>0\), there exists \(v_{0}>0\) such that for \(v\geqslant v_{0}\) the inequalities
\[ F(v)\geqslant \frac{\omega-\varepsilon}{\alpha+\varepsilon}\,v,\qquad F(-v)\leqslant \frac{\omega_{1}+\varepsilon}{\beta-\varepsilon}\,v, \]
hold; these, together with conditions 8) and 9) of Theorem 2, ensure the fulfillment of conditions 2) and 3) of Theorem 1. The uniqueness of the solutions of system (9) follows from the mutual one-to-one character of the substitutions (6) and (8).
- Problem. Prove or disprove the following assertion. The system
\[ \frac{dx}{dt}=y,\qquad \frac{dy}{dt}=-f(x,y)\cdot y-g(x)+e(t), \]
corresponding to the equation
\[ \frac{d^{2}x}{dt^{2}}+f\left(x,\frac{dx}{dt}\right)\frac{dx}{dt}+g(x)=e(t), \]
belongs to the class \(D\), if
* If \(\omega_{1}\leqslant 0\), Theorem 2 is true under much weaker restrictions on \(g(x)\) and, in essence, is not new. Compare, for example, [5], p. 153.
1) \(f(x,y)\) and \(g(x)\) satisfy all the conditions of Theorem 2, with condition 3) of that theorem replaced by the stronger condition \(3')\) \(\gamma^2 x \leq g(x) \leq \alpha^2 x\) \((0<\gamma<\alpha)\) for \(x \geq x_0\);
2) \(e(t)\) is defined for all \(t \geq 0\), piecewise continuous, and bounded.
References
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Figueiredo R. P. Contribution to the theory of certain non-linear differential equation. Lisboa, 1960.
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Dragilev A. V. Mat. sb., 63 (105), No. 2, 1964, p. 320.
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Filippov A. F. Mat. sb., 30 (72), No. 1, 1952, pp. 171–180.
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Pliss V. A. Nonlocal Problems in the Theory of Oscillations. Publishing House “Nauka,” Moscow–Leningrad, 1964.
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Nemytskii V. V., Stepanov V. V. Qualitative Theory of Differential Equations. Gostekhizdat, 1949.
Received by UMN on April 9, 1964.
Tartu State University