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UDC 517.911
ON THE QUESTION OF THE EXISTENCE OF BOUNDED SOLUTIONS OF A SECOND-ORDER DIFFERENTIAL EQUATION
E. G. KHALIKOV
Consider the equation
\[ N[y] \equiv \ddot y - f(t,y,\dot y)=0, \]
\[ f(t,y,\dot y)= \begin{cases} F(t,x,\dot y), & y<x,\\ F(t,y,\dot y), & x\le y\le z,\\ F(t,z,\dot y), & y>z, \end{cases} \]
where \(x(t), z(t)\) are certain solutions of equation (1), satisfying the inequality \(x(t)<z(t)\) for \(-\infty<t<\infty\). The function \(F(t,y,\dot y)\) is continuous in the domain \(G(x\le y\le z,\ -\infty<t,\dot y<\infty)\) and satisfies the Lipschitz condition with respect to \(y,\dot y\) in any bounded domain \(P\subset G\).
Under the assumption that the boundary-value problem
\[ N[y]=0, \tag{1} \]
\[ y(t_0)=c_0,\quad y(t_0+h)=c_1, \tag{2} \]
\[ c_0\in[x(t_0),z(t_0)],\quad c_1\in[x(t_0+h),z(t_0+h)] \]
has no more than one solution for fixed \(c_0,c_1,t_0,h\in(-\infty,\infty)\), the following assertion is true:
Theorem 1. Through every point \((t_0,p_0)\), where \(t_0\in(-\infty,\infty)\), \(p_0\in(x(t_0),z(t_0))\), there passes at least one solution \(y(t)\) of equation (1) satisfying the inequality
\[ x(t)<y(t)<z(t),\quad -\infty<t<\infty. \tag{3} \]
Proof. By virtue of the Lipschitz condition [1] there exists a unique solution \(y(t)\equiv y(t,p_0,p_1)\) of equation (1), satisfying the conditions
\[ y(t_0)=p_0,\quad \dot y(t)=p_1 \tag{4} \]
and depending continuously on \(p_0,p_1\). Consequently, for arbitrary \(\varepsilon>0\) and fixed \(T\in(-\infty,\infty)\) there exists \(\Delta>0\) such that, when \(\max_j |p_j-r_j|\le\Delta\) \((j=0,1)\),
\[ |y(T,p_0,p_1)-y(T,r_0,r_1)|<\varepsilon. \tag{5} \]
We shall say that a solution \(y(t)\) of equation (1), satisfying the conditions \(y(t_0)=p_0,\ x(t_0)<p_0<z(t_0)\), for \(t\ge t_0\) \((t\le t_0)\) belongs
1) to class \(A\) \((A')\), if \(y(\tau)=z(\tau)\) \((y(\tau')=z(\tau'))\) for some \(\tau>t_0\) \((\tau'<t_0)\);
2) to class \(C\) \((C')\), if \(y(\theta)=x(\theta)\) \((y(\theta')=x(\theta'))\) for some \(\theta>t_0\) \((\theta'<t_0)\);
3) to class \(B\) \((B')\), if \(x(t)<y(t)<z(t)\), \(t_0\le t<\infty\) \((-\infty<t\le t_0)\).
As a consequence of the uniqueness of the solution of problems (1), (2) and (1), (4), we have \(y(t)>z(t)\) for \(t>\tau\) for \(y(t)\in A\), and \(y(t)<x(t)\) for \(t>\theta\) for \(y(t)\in C\).
We divide the proof into two parts. First we shall prove the nonemptiness of the classes \(B\) and \(B'\), and then the existence of a solution belonging to both these classes, i.e. the assertion of the theorem itself.
The classes \(A\) and \(C\) are nonempty. To see this, it is sufficient to show that the problem (1), (2) has a solution for some \(h\).
We write problem (1), (2) in the form of the system
\[ Y(t)=\int_{t_0}^{t_0+h} H(t,s;h) f(s,Y(s))\,ds+\varphi(t), \tag{6} \]
where
\[ Y(t)=\{y(t),\dot y(t)\},\qquad \varphi(t)=\left\{c_0+\frac{c_1-c_0}{h}(t-t_0),\frac{c_1-c_0}{h}\right\}, \]
\[ H(t,s;h)= \left\{ \begin{array}{cc} G(t,s;h), & 0\\[4pt] 0, & \dfrac{\partial}{\partial t}G(t,s;h) \end{array} \right\}, \]
\(G(t,s;h)\) is the Green’s function of the problem
\[ \ddot y=0,\qquad y(t_0)=c_0,\qquad y(t_0+h)=c_1. \]
Applying the Schauder principle to system (6), we obtain that, for sufficiently small \(|h|\), problem (1), (2) is solvable.
We shall say that the number \(\alpha=\dot y(t_0)\) belongs to the set \(S\) (\(R\)) if, for fixed \(p_0\), the solution \(y(t,p_0,\alpha)\equiv y(t,\alpha)\in A\) \((y(t,\alpha)\in C)\).
Suppose that the class \(B\) is empty, i.e. \(B=0\). Then every number \(\alpha=\dot y(t_0)\) will belong to one and only one of the sets \(S\) or \(R\). As a consequence of the uniqueness of the solution of problem (1), (2), we have \(\alpha_S>\alpha_R\) for all \(\alpha_S\in S\) and \(\alpha_R\in R\). According to Dedekind’s theorem [2], there exists a number \(\Gamma\) such that either \(\Gamma=\min \alpha_S\in S\), or \(\Gamma=\max \alpha_R\in R\). Let \(\Gamma\in S\). Then \(y(t,\Gamma)\in A\), and for a fixed \(T\) \((\tau<T<\infty)\) and any \(\Delta>0\) we have the inequality
\[ y(T,\Gamma)-y(T,\Gamma-\Delta)>z(T)-y(T,\Gamma-\Delta)>\rho, \tag{7} \]
where \(\rho=y(T,\Gamma)-z(T)>0\). We see that (7) contradicts (5). The assumption \(\Gamma\in R\) leads to an analogous contradiction. Hence \(y(t,\Gamma)\in B\), i.e. \(B\ne0\). Analogously we obtain \(B'\ne0\). From Dedekind’s theorem it follows that there exist
\[ \alpha_1=\min\alpha_M\;(\beta_1=\min\beta_{M'}),\qquad \alpha_2=\max\alpha_M\;(\beta_2=\max\beta_{M'}), \]
which belong to the set \(M\) \((M')\), such that \(y(t,\alpha_M)\in B\) \((y(t,\beta_{M'})\in B')\), for all \(\alpha_M\in M\) \((\beta_{M'}\in M')\).
Suppose that the theorem is not true, i.e. no solution \(y(t)\) of equation (1) can belong to the classes \(B\) and \(B'\) simultaneously. Then either \(\beta_1>\alpha_2\), or \(\alpha_1>\beta_2\). Otherwise, the intervals \([\alpha_1,\alpha_2]\), \([\beta_1,\beta_2]\) have no common point.
Let \(\beta_1>\alpha_2\). By the conditions of the theorem there exists a solution \(w(t)\) of equation (1) satisfying the conditions \(w(t_0)=p_0\), \(\beta_1>w(t_0)>\alpha_2\). Since \(\dot w(t_0)>\alpha_2\), we have \(w(t)\in A\) for \(t>t_0\). From the inequality \(w(t_0)<\beta_1\) it follows that \(w(t)\in A'\) for \(t\le t_0\). Therefore there exist \(\tau_0<t_0\) and \(\tau_1>t_0\) such that \(w(\tau_0)=z(\tau_0)\), \(w(\tau_1)=z(\tau_1)\), \(\tau_1-\tau_0=h<\infty\), i.e. the problem
\[ N[z]=0,\qquad z(\tau_0)=c_0,\qquad z(\tau_0+h)=c_1 \]
has a second solution \(w(t)\ne z(t)\), which contradicts the condition of the theorem. Hence \(\beta_1\le\alpha_2\). Analogously we obtain \(\alpha_1\le\beta_2\). Thus the system of inequalities
\[ \alpha_1\le\gamma\le\alpha_2,\qquad \beta_1\le\gamma\le\beta_2 \]
has a solution.
For any solution \(y(t)\) of equation (1) satisfying the conditions \(y(t_0)=p_0\), \(\dot y(t_0)=\gamma\), we shall have (3)
\[ x(t)<y(t)<z(t),\qquad -\infty<t<\infty. \]
Remark 1. From the proof it is clear that, replacing in the condition of the theorem the inequality \(x(t)<z(t)\) by \(x(t)\le z(t)\), instead of (3) we obtain
\[ \inf\{x(t),z(t)\}<y(t)<\sup\{x(t),y(t)\},\qquad -\infty<t<\infty, \]
except, possibly, at one point, where the latter inequalities are replaced by equalities.
Remark 2. It follows from the proof that the problem
\[ \ddot y-f(t,y,\dot y)=0,\qquad y(t_0)=p_0,\qquad x(t)<y(t)<z(t), \]
\[ t_0\le t<\infty\quad (-\infty<t\le t_0) \]
has upper and lower solutions for fixed \(t_0\in(-\infty,\infty)\), \(p_0\in(x(t_0),z(t_0))\).
Remark 3. Criteria that guarantee uniqueness of the solution of problem (1), (2) are closely connected with the question of the applicability of theorems on differential inequalities. Such criteria are available in [3, 4, 5] and in a number of other works of the Izhevsk seminar.
Consider the particular case: \(x(t)=a=\mathrm{const}\), \(z(t)=b=\mathrm{const}\). Suppose that equation (1) satisfies the conditions of Theorem 1. Suppose further that
a) \(f(t,p,0)\ne 0,\quad a<p<b;\quad t\in(-\infty,\infty);\)
b) \(f(t,y,-\mu)\ne 0\) \((f(t,y,\mu)\ne 0),\quad a<y<b,\quad \mu>0;\)
c) \(\displaystyle \lim_{t\to\pm\infty} f(t,y,0)\ne 0,\quad a<y<b;\)
d) for any fixed \(t\in(-\infty,\infty)\) the equation \(f(t,r_n,0)=0\) can have only a finite number of roots \(r_n,\ a<r_n<b\).
Under these assumptions the following assertion on a transition solution holds (cf. [5]).
Theorem 2. There exists a solution \(y(t)\) of equation (1) such that
\[ \lim_{t\to-\infty} y(t)=a,\qquad \lim_{t\to\infty} y(t)=b \]
\[ \left( \lim_{t\to-\infty} y(t)=b,\qquad \lim_{t\to\infty} y(t)=a \right). \]
Proof. By condition a) and Theorem 1 there exists a solution \(y(t)\) of equation (1) such that \(a<y<b,\ y(t)\ne \mathrm{const},\ -\infty<t<\infty\). Condition b) means that there is not a single point \(t=\xi\) where \(y'(\xi)<0\) \((y'(\xi)>0),\ y'(\xi)=0\) for any solution \(y(t)\) of equation (1). Therefore the solution \(y(t)\) does not decrease (does not increase) for \(t\) greater than some fixed \(T\in(-\infty,\infty)\). Consequently, there exists \(\lim_{t\to\infty} y(t)=c\le b\).
It can be shown that condition d) is sufficient for the existence of the limits
\[
\lim_{t\to\pm\infty} \ddot y(t)=\lim_{t\to\pm\infty}\dot y(t)=0.
\]
Hence, by virtue of equation (1), we have
\[ \lim_{t\to\infty} \ddot y(t)=\lim_{t\to\infty} f(t,y,\dot y)=\lim_{t\to\infty} f(t,c,0)=0. \]
According to c), we obtain \(\displaystyle \lim_{t\to\infty} y(t)=b\) \(\left(\displaystyle \lim_{t\to\infty} y(t)=a\right)\). Similarly we obtain \(\displaystyle \lim_{t\to-\infty} y(t)=a\) \(\left(\displaystyle \lim_{t\to-\infty} y(t)=b\right)\).
Let us now clarify the question of existence of a solution of the problem
\[ \ddot y-m(t)\dot y+F(t,y)=0, \tag{8} \]
\[ \lim_{t\to-\infty} y(t)=0,\qquad \lim_{t\to\infty} y(t)=1, \tag{9} \]
which arises in combustion theory [5, 6]. Here \(F(t,0)\equiv F(t,1)\equiv 0,\ F(t,y)>0,\ m(t)>0,\ (y\in(0,1),\ t\in(-\infty,\infty))\). The functions \(m(t)\) and \(F(t,y)\) are continuously differentiable with respect to \(y\) and \(t\) in the domain \((0\le y\le 1,\ -\infty<t<\infty)\). From [3, 4] it follows that the inequality
\[ \frac{1}{4}\left[m^2(t)+\frac{1}{t^2}\right]-\frac{1}{2}\frac{dm}{dt}\ge \frac{\partial F}{\partial y} \tag{10} \]
guarantees uniqueness of the solution of the boundary-value problem (1), (2). Verification of the remaining conditions of Theorem 2 presents no difficulty. Consequently, when (10) is satisfied, problem (8), (9) has a solution (cf. [5]).
In conclusion, the author expresses deep gratitude to the participants of the Izhevsk seminar for a number of valuable remarks on this paper.
References
-
Matveev N. M. Differential equations. Leningrad University Press, 1963.
-
Fikhtengol'ts G. M. Course of differential and integral calculus, 1, Fizmatgiz, 1962.
-
Azbelev N. V., Tsalyuk Z. B. Differential Equations, 1, No. 4, 431—438, 1965.
-
Plaks S. A. DAN SSSR, 148, No. 6, 1963.
-
Klokov Yu. A. Boundary-value problems with a condition at infinity for equations of mathematical physics. Monograph. Riga, 1963.
-
Gel'fand I. M. UMN, 14, issue 2, 87—159, 1959.
Received by the editors
December 1, 1965
Izhevsk Mechanical Institute