ON A BOUNDARY VALUE PROBLEM FOR A SYSTEM OF TWO DIFFERENTIAL EQUATIONS

G. N. MIL’SHTEIN

1. Consider the system of two first-order differential equations:

\[ \frac{dX}{dt}=P(t,X,Y), \qquad \frac{dY}{dt}=Q(t,X,Y). \tag{1.1} \]

It is required to find a solution \(\{X(t),Y(t)\}\) of system (1.1) satisfying the boundary conditions

\[ X(a)=X_0,\qquad X(b)=X_1. \tag{1.2} \]

An extensive literature is devoted to the two-point boundary value problem (1.1)—(1.2) (see, for example, [1—4]). Especially many works concern the boundary value problem for the second-order equation

\[ X''=Q(t,X,X'), \tag{1.3} \]

which reduces to system (1.1) with \(P(t,X,Y)\equiv Y\).

We shall assume that the right-hand sides of equations (1.1), \(P(t,X,Y)\) and \(Q(t,X,Y)\), are defined and continuous together with their first partial derivatives with respect to the variables \(X\) and \(Y\) in the strip \(S\),

\[ S:\ a\leq t\leq b,\quad -\infty<X<+\infty,\quad -\infty<Y<+\infty. \]

It is also assumed that the partial derivatives are bounded and satisfy in the strip \(S\) the following inequalities:

\[ A_1\leq \frac{\partial P}{\partial X}\leq A_2,\qquad B_1\leq \frac{\partial P}{\partial Y}\leq B_2, \]

\[ C_1\leq \frac{\partial Q}{\partial X}\leq C_2,\qquad D_1\leq \frac{\partial Q}{\partial Y}\leq D_2, \tag{1.4} \]

where the constants \(B_1\) and \(B_2\) are either both positive or both negative, while the remaining constants are arbitrary. In what follows, for definiteness, we shall take the constants \(B_1\) and \(B_2\) to be positive. The condition of existence and boundedness of the derivatives in the strip \(S\) ensures the existence, uniqueness, and extendability to the interval \([a,b]\) of solutions of system (1.1) satisfying arbitrary initial data [5,6]. This condition also ensures the existence of derivatives with respect to the initial data for solutions of system (1.1). We shall consider the solution \(\{X(c,t),Y(c,t)\}\) of system (1.1) satisfying the initial conditions

\[ X(a)=X_0,\qquad Y(a)=c \tag{1.5} \]

for arbitrary but fixed \(X_0\) and with varying parameter \(c\).

It is clear that, in order for the boundary-value problem (1.1)—(1.2) to have a unique solution for all \(X_1(-\infty<X_1<+\infty)\), it is necessary and sufficient that the function \(X(c,b)\), as a function of the parameter \(c\), be strictly monotone and that it assume all values in the infinite interval \((-\infty,+\infty)\). We formulate the following obvious proposition as a lemma.

Lemma 1. In order that the solution of the boundary-value problem (1.1)—(1.2), under the assumptions adopted, exist and be unique, it is sufficient that one of the following two inequalities be satisfied:

\[ X'_c(c,b)\geqslant \mu>0, \tag{1.6} \]

\[ X'_c(c,b)\leqslant \nu<0. \tag{1.7} \]

The derivatives of the solution with respect to the initial value \(c\), \(X'_c(c,t)\), \(Y'_c(c,t)\), satisfy the system of differential equations

\[ \frac{dX'_c}{dt} = \frac{\partial P}{\partial X}(t,X(c,t),Y(c,t))\cdot X'_c + \frac{\partial P}{\partial Y}(t,X(c,t),Y(c,t))\cdot Y'_c, \]

\[ \frac{dY'_c}{dt} = \frac{\partial Q}{\partial X}(t,X(c,t),Y(c,t))\cdot X'_c + \frac{\partial Q}{\partial Y}(t,X(c,t),Y(c,t))\cdot Y'_c \tag{1.8} \]

and the initial data

\[ X'_c(c,a)=0,\qquad Y'_c(c,a)=1. \tag{1.9} \]

1. Alongside system (1.8), consider the system of differential equations with control \(u=\{u_{11},u_{12},u_{21},u_{22}\}\)

\[ \frac{dx}{dt}=u_{11}x+u_{12}y,\qquad \frac{dy}{dt}=u_{21}x+u_{22}y. \tag{2.1} \]

We impose on the control the same restrictions as the restrictions (1.4) on the derivatives of the functions \(P\) and \(Q\), i.e.,

\[ A_1\leqslant u_{11}\leqslant A_2,\qquad 0<B_1\leqslant u_{12}\leqslant B_2, \]

\[ C_1\leqslant u_{21}\leqslant C_2,\qquad D_1\leqslant u_{22}\leqslant D_2. \tag{2.2} \]

As the class of admissible controls we choose the class of measurable controls defined on an arbitrary time interval \([t_0,t_1]\) and satisfying the constraints (2.2) at each instant of time. By a solution of system (2.1) under some admissible control \(u(t)\) we mean a solution in the extended sense [7]. Let \(\{x(t),y(t)\}\) be a solution of system (2.1) under some control, satisfying at the initial time \(t=0\) the conditions

\[ x(0)=0,\qquad y(0)=1. \tag{2.3} \]

Denote by \(A\) the greatest of the constants

\[ |A_i|,\ B_2,\ |C_i|,\ |D_i|\quad (i=1,2). \]

Lemma 2. For any admissible control the function \(x(t)\) is positive on the interval

\[ \left(0,\ \frac{1}{2A}\ln\frac{2B_1+A}{B_1+A}\right]. \]

Proof. Write equations (2.1) in integral form

\[ x=\int_0^t (u_{11}x+u_{12}y)\,dt,\qquad y=1+\int_0^t (u_{21}x+u_{22}y)\,dt. \tag{2.4} \]

From the obvious inequality

\[ |x|+|y|\leq 1+2A\int_0^t (|x|+|y|)\,dt, \]

using a well-known lemma ([6], p. 19), we obtain two inequalities:

\[ |x|\leq \exp 2At,\qquad |y|\leq \exp 2At. \tag{2.5} \]

Now from (2.4)

\[ |x|\leq \exp 2At-1,\qquad |y-1|\leq \exp 2At-1 \tag{2.6} \]

and, consequently,

\[ y\geq 2-\exp 2At. \tag{2.7} \]

Let us estimate the derivative of the function \(x(t)\), equal to \(u_{11}x+u_{12}y\),

\[ u_{11}x+u_{12}y\geq B_1(2-\exp 2At)-A(\exp 2At-1). \tag{2.8} \]

The final proof of the lemma follows from the fact that the function on the right-hand side of inequality (2.8) is positive on the interval

\[ \left(0,\ \frac{1}{2A}\ln\frac{2B_1+A}{B_1+A}\right). \]

We formulate a lemma whose proof is easily obtained by the traditional method of reducing a system of differential equations to a system of integral equations and using certain elementary facts from functional analysis.

Lemma 3. Suppose there is a sequence of admissible controls \(u^n(t)\) \((n=1,2,\ldots)\), defined on one and the same time interval \([0,t_1]\), and the corresponding sequence of solutions \(\{x^n(t),y^n(t)\}\) of system (2.1), satisfying the same initial data. Then there exist subsequences \(x^{n_k}(t)\) and \(y^{n_k}(t)\) that converge uniformly on the interval \([0,t_1]\), respectively, to functions \(x(t)\) and \(y(t)\) such that \(\{x(t),y(t)\}\) is a solution of the control system (2.1) for some admissible control \(u(t)\).

1. For the control system (2.1), consider the time-optimal problem of transferring the point from position (2.3) to the axis \(x=0\) in positive time. It follows from Lemma 2 that this positive time is strictly greater than

\[ \frac{1}{2A}\ln\frac{2B_1+A}{B_1+A}. \]

Now the use of Lemma 3 makes it possible, without particular difficulties, to prove the following theorem.

Theorem 1. If there exists an admissible control \(u(t)\) that transfers the point \((0,1)\) of the phase plane to the axis \(x=0\) in finite time greater than zero, then there also exists an optimal control transferring the point \((0,1)\) to the axis \(x=0\) in a time \(T>0\).

The remainder of this paragraph is devoted to the proof of the following theorem.

Theorem 2. In order that there exist a control transferring, by means of the control system, the point \((0,1)\) to the axis \(x=0\) in the least-

for any positive time, it is necessary and sufficient that each integral in the sum of integrals

\[ J=\int_{-\infty}^{0}\frac{dz}{B_2 z^2+(A_1-D_2)z-C_1} +\int_{0}^{+\infty}\frac{dz}{B_2 z^2+(A_2-D_1)z-C_1} \tag{3.1} \]

converge. In the case of convergence of both integrals, the optimal transition time \(T\) coincides with the value of the sum of the integrals (3.1).

Let us note that, since for the integrands in the integrals of the sum (3.1) it is easy to indicate an antiderivative for known values of \(A_1, A_2, B_2, C_1, D_1, D_2\), in the case of convergence of the integrals the optimal time is calculated without difficulty. It is also not difficult to indicate cases of divergence of at least one of the integrals in (3.1). This will occur if at least one of the following three conditions is satisfied:

a)

\[ C_1 \geq 0; \tag{3.2} \]

b) \(C_1<0\), but the inequalities hold:

\[ (D_1-A_2)^2+4B_2C_1 \geq 0,\quad D_1-A_2 \geq 0; \tag{3.3} \]

c) \(C_1<0\), but the inequalities hold:

\[ (D_2-A_1)^2+4B_2C_1 \geq 0,\quad D_2-A_1 \leq 0. \tag{3.4} \]

We now pass to the proof of the theorem. Assuming that an optimal control exists, let us find it. We write the necessary conditions in the form of Pontryagin’s maximum principle [8], which are satisfied by the optimal control and the optimal trajectory of the problem under consideration. Introduce a function \(H\), depending on the variables \(x, y, u_{ij}\) \((i=1,2;\ j=1,2)\), \(\varphi, \psi, \psi_0\):

\[ H=\varphi(u_{11}x+u_{12}y)+\psi(u_{21}x+u_{22}y)+\psi_0. \tag{3.5} \]

With the help of this function we compose a system of differential equations for the auxiliary unknowns \(\varphi\) and \(\psi\):

\[ \frac{d\varphi}{dt}=-\frac{\partial H}{\partial x}=-u_{11}\varphi-u_{21}\psi,\quad \frac{d\psi}{dt}=-\frac{\partial H}{\partial y}=-u_{12}\varphi-u_{22}\psi. \tag{3.6} \]

If \(u(t)=\{u_{11}(t), u_{12}(t), u_{21}(t), u_{22}(t)\}\) is an optimal control, and \(\{x(t),y(t)\}\) is an optimal trajectory of the problem under consideration, then, according to Pontryagin’s maximum principle, there exists a solution (3.6) \(\{\varphi(t),\psi(t)\}\) corresponding to the control \(u(t)\), supplemented by a nonpositive constant \(\psi_0\), such that, first, the vector \(\{\varphi(t),\psi(t),\psi_0\}\) is nonzero on the interval \([0,T]\), and, second, for any \(0\leq t\leq T\) the maximum condition is satisfied

\[ \varphi(t)(u_{11}(t)x(t)+u_{12}y(t))+\psi(t)(u_{21}(t)x(t)+ \]

\[ +u_{22}y(t))+\psi_0 =\max_{u_{ij}}\,[\varphi(t)(u_{11}x(t)+u_{12}y(t))+ \]

\[ +\psi(t)(u_{21}x(t)+u_{22}y(t))+\psi_0]=0. \tag{3.7} \]

Moreover, at the end of the trajectory \(\{x(t),y(t)\}\) \(x(T)=0\), and the transversality condition is satisfied, which in the present case takes the form \(\psi(T)=0\), so that for \(t=T\) we have

\[ x(T)=0,\quad \psi(t)=0. \tag{3.8} \]

Thus, in order to find the optimal solution, one must solve the systems of equations (2.1) and (3.6) under the boundary conditions (2.3) and (3.8) and under condition (3.7).

Let us first show that $\psi_0 \ne 0$. Indeed, suppose that $\psi_0 = 0$. Then, by virtue of (3.8), from (3.7) at $t = T$ we obtain

\[ \varphi(T) u_{12}(T) y(T) = 0 . \tag{3.9} \]

But $y(T) \ne 0$ (since otherwise the solution of the system (2.1) $\{x(t), y(t)\}$ would be trivial), and $u_{12}(T) > 0$. Therefore it follows from (3.9) that $\varphi(T) = 0$. In that case the solution of the homogeneous system of equations (3.6) $\{\varphi(T), \psi(T)\}$ is trivial, and the vector $\{\varphi(t), \psi(t), \psi_0\}$ is zero on the interval $[0,T]$, which is impossible. Thus, $\psi_0 \ne 0$. We may set $\psi_0 = -1$. Without rewriting equality (3.7), we shall assume that $\psi_0$ in this equality is equal to $-1$.

Let us note the fact that the linear systems of homogeneous equations (2.1) and (3.6) are adjoint for any control. Hence it follows that, for solutions of these systems, the equality

\[ x(t)\cdot \varphi(t) + y(t)\cdot \psi(t) \equiv C, \tag{3.10} \]

holds, where $C$ is some constant.

Since in our case $x(T)=0$, $\psi(T)=0$, we have $C=0$, and equality (3.10) becomes

\[ x(t)\cdot \varphi(t) + y(t)\cdot \psi(t) \equiv 0 . \tag{3.11} \]

We shall show that on the interval $(0,T)$ the function $\psi(t)$ does not vanish and, consequently, does not change sign. Indeed, suppose that for some $0<\tau<T$ we have $\psi(\tau)=0$. If also $\varphi(\tau)=0$, then the vector $\{\varphi(t), \psi(t)\}$ would be zero and equality (3.7) would not be satisfied. Therefore $\varphi(\tau)\ne 0$. From (3.11) it follows that $x(\tau)=0$. But the function $x(t)$, by Lemma 2 and by the very meaning of the problem, is positive on the interval $(0,T)$. The contradiction obtained proves that $\psi(t)$ is indeed different from zero on $(0,T)$.

Let us determine the sign of the function $\psi(t)$. Since $x(T)=0$ and $x(t)$ is positive for $0<t<T$, while $y(T)\ne 0$, it is easy to see from the first equation of the system (2.1) that $y(T)<0$. Equality (3.7) at the moment $T$ takes the form

\[ \varphi(T) u_{12}(T) y(T) = 1, \tag{3.12} \]

whence it follows that $\varphi(T)<0$.

Now from the system of equations (3.6) and a result analogous to Lemma 2, it is easy to see that the function $\psi(t)$ for moments $t$ close to $T$ and $t<T$ is negative. Consequently, $\psi(t)<0$ for $0<t<T$. Since $\psi(t)\cdot x(t)$ is negative for $0<t<T$, the maximization condition (3.7) allows us to find the coordinate $u_{21}$ of the optimal vector-control:

\[ u_{21}(t) \equiv C_1 . \tag{3.13} \]

Since the functions $x(t)$ and $\psi(t)$ take different signs on $(0,T)$, it follows from (3.11) that the functions $\varphi(t)$ and $y(t)$ have the same signs, and, moreover, they vanish simultaneously. In what follows we shall show that the vanishing of the functions $\varphi(t)$ and $y(t)$ can occur only once. Therefore, except for a single instant of time, $\varphi(t)y(t)>0$, and the maximization condition (3.7) for the coordinate $u_{12}$ of the optimal vector-control gives the value $B_2$

\[ u_{12}(t) \equiv B_2 . \tag{3.14} \]

We shall prove that if \(C_1 \geqslant 0\), then the optimal problem under consideration has no solution. Indeed, suppose that the problem under consideration has an optimal solution. The above arguments show in this case that \(x(t)>0\) for \(0<t<T\) and \(y(T)<0\). But from the second equation of system (2.1), taking (2.3) into account, there follows the equality

\[ y(t)=e^{\int_0^t u_{22}(\xi)\,d\xi} \left(1+\int_0^t e^{-\int_0^\tau u_{22}(\xi)\,d\xi}\,u_{21}(\tau)x(\tau)\,d\tau\right), \tag{3.15} \]

which shows that \(y(t)>0\) on the interval \([0,T]\), and we have arrived at a contradiction. Thus, in the case when \(C_1 \geqslant 0\), \(x(t)>0\) for all \(t>0\) under any admissible control defined on \([0,+\infty)\). In the subsequent arguments it is assumed that \(C_1<0\), and the assumption is still retained that the optimal problem under consideration has a solution.

Since \(y(T)<0\), while \(y(0)=1\), there is a smallest time \(t=\vartheta\) such that \(y(\vartheta)=0\). In this case, from the second equation of system (2.1), for \(\vartheta \leqslant t \leqslant T\), there follows the equality

\[ y(t)=C_1\int_\vartheta^t e^{\int_\tau^t u_{22}(\xi)\,d\xi}\,x(\tau)\,d\tau, \tag{3.16} \]

which proves that the function \(y(t)\) is negative on the interval \((\vartheta,T]\). Thus, the function \(y(t)\) indeed vanishes only once on the interval \([0,T]\), and equality (3.14) is proved.

From the maximization condition (3.7) it is now not difficult to see that the optimal control has one switching, and this switching occurs at the moment when \(y\) vanishes, i.e., at \(t=\vartheta\). The still unknown coordinates \(u_{11}(t)\) and \(u_{22}(t)\) of the optimal vector-control, found from condition (3.7), have the form

\[ u_{11}(t)= \begin{cases} A_2, & 0\leqslant t<\vartheta,\\ A_1, & \vartheta\leqslant t\leqslant T; \end{cases} \tag{3.17} \]

\[ u_{22}(t)= \begin{cases} D_1, & 0\leqslant t<\vartheta,\\ D_2, & \vartheta\leqslant t\leqslant T. \end{cases} \tag{3.18} \]

It follows from the very course of the arguments that if the optimal control exists, then it is uniquely determined by formulas (3.13), (3.14), (3.17), and (3.18). In order now to find the time of optimal regulation, one must find the solution of system (2.1) under the initial conditions (2.3) and under the optimal control. The time of optimal regulation will coincide with the moment when the function \(x(t)\) vanishes. Avoiding the lengthy computations that arise when seeking the time \(T\) in this way, we proceed differently. Introduce the ratio of the coordinates of the optimal solution \(\{x(t),y(t)\}\),

\[ z(t)=\frac{y(t)}{x(t)}, \]

defined on \((0,T)\). By direct differentiation it is easy to verify that the function \(z(t)\) satisfies the differential equation

\[ \frac{dz}{dt}=\varphi(z)=C_1+(D(z)-A(z))z-B_2z^2, \tag{3.19} \]

where

\[ A(z)= \begin{cases} A_1, & -\infty<z\leqslant 0,\\ A_2, & 0<z<+\infty; \end{cases} \tag{3.20} \]

\[ D(z)= \begin{cases} D_2, & -\infty<z\leqslant 0,\\ D_1, & 0<z<+\infty. \end{cases} \tag{3.21} \]

The function \(\varphi(z)\ne 0\), since under the contrary assumption it would not be difficult to arrive at a contradiction with the fact that \(z(t)\) on the interval \((0,T)\) changes from \(+\infty\) to \(-\infty\). The solution of equation (3.19) tending to \(+\infty\) as \(t\to+\infty\) satisfies the relation

\[ \int_{+\infty}^{z}\frac{dz}{\varphi(z)}=t. \tag{3.22} \]

Since as \(t\to T-0\), \(z(t)\to-\infty\), it follows from (3.22) that

\[ \int_{+\infty}^{-\infty} \frac{dz}{C_1+(D(z)-A(z))z-B_2z^2}=T. \tag{3.23} \]

The integral in (3.23) coincides with the sum of the integrals in (3.1). Thus, under the assumption of the existence of a solution of the optimal problem, the convergence of the sum of the integrals (3.1) and the coincidence of this sum with the time of optimal control have been proved.

Now suppose that each integral in the sum (3.1) converges or, what is the same, the integral (3.23) converges. In this case none of the conditions (3.2), (3.3), and (3.4) is satisfied. The function \(\varphi(z)\) is therefore everywhere negative and satisfies the inequality \(\varphi(z)<p\), where \(p\) is some negative number. Consider the solution \(\{x(t),y(t)\}\) of system (2.1) under the control

\[ u=\{u_{11},u_{12},u_{21},u_{22}\}=\{A_2,B_2,C_1,D_1\} \]

on the time interval \([0,\vartheta]\), where

\[ \vartheta= \int_{+\infty}^{0} \frac{dz}{C_1+(D_1-A_2)z-B_2z^2}, \]

satisfying the initial conditions (2.3). On this interval the function

\[ z(t)=\frac{y(t)}{x(t)} \]

decreases monotonically from \(+\infty\) to \(0\), and since the function \(x(t)\) is positive on \((0,\vartheta]\), we have \(y(\vartheta)=0\). Then, under the control \(u=\{A_1,B_2,C_1,D_2\}\), consider the solution of system (2.1) on the interval \([\vartheta,T]\), satisfying the conditions \((x(\vartheta),0)\) for \(t=\vartheta\). On this interval \(z(t)\) decreases monotonically from \(0\) to \(-\infty\), and, consequently, \(x(T)=0\). Thus there exists a control which transfers the point \((0,1)\) in a finite time \(T>0\) to the axis \(x=0\). Hence, by Theorem 1, an optimal control also exists. Theorem 2 is completely proved.

Let us single out a consequence of this theorem that is important for what follows.

Corollary of Theorem 2. If one of the conditions (3.2), (3.3), (3.4) is satisfied, or, what is the same, at least one of the integrals of the sum (3.1) diverges, then for any admissible control defined on any time interval \([0,H]\), the coordinate \(x(t)\) of the solution \(\{x(t),y(t)\}\) of system (2.1), satisfying the initial conditions (2.3), is positive on \((0,H]\).

Remark 1. It is not difficult to show that the time of the optimal transfer \(T(x,y)\) from an arbitrary point \((x,y)\), different from the origin, to the axis \(x=0\) (the Bellman function of the optimal problem under consideration) is given by formula (3.23) if \(x=0,\ y\ne0\). For the remaining values of \(x\) and \(y\), the function \(T(x,y)\) has the form

\[ T(x,y)=\int_{-\infty}^{\frac{y}{x}} \frac{dz}{B_2 z^2+\bigl(A(z)-D(z)\bigr)z-C_1}. \tag{3.24} \]

1. We prove the following lemma.

Lemma 4. Let \(T\) be the minimal positive time of transfer from the position \((0,1)\) to the axis \(x=0\) by means of the control system (2.1) (in the case of absence of a minimal time \(T\) it is here assumed that \(T=+\infty\)), and let \(0<h<T\). Then there exists a number \(\lambda>0\) such that, for any admissible control defined on the interval \([0,h]\), the coordinate \(x(t)\) of the corresponding solution satisfying the conditions (2.3) satisfies, at the instant \(h\), the inequality

\[ x(h)>\lambda . \tag{4.1} \]

Proof. Suppose the lemma is false. Then there exists a sequence \(u^n(t)\) of admissible controls, defined on the interval \([0,h]\), such that for the corresponding sequence of solutions \(\{x^n(t),y^n(t)\}\) of system (2.1), satisfying the initial conditions (2.3), the sequence \(x^n(h)\to0\) as \(n\to\infty\). Lemma 3 now makes it possible to conclude that there exists a control \(u(t)\) transferring the point \((0,1)\) to the axis \(x=0\) in time \(h\). The contradiction obtained proves the lemma.

We note that, by virtue of the autonomy of the control system (2.1), all the preceding results for the optimal problem can be obtained if, as the initial instant of time, one chooses \(t=a\). We formulate the main theorem of this paper, which follows easily from Theorem 2, Lemma 4, Lemma 1, and from the fact that system (1.8), for each \(c\), may be regarded, taking into account the restrictions (1.4), as a control system of the form (2.1).

Theorem 3. Let the right-hand sides of system (1.1), together with their first partial derivatives with respect to the variables \(X\) and \(Y\), be defined, continuous, and satisfy in the strip \(S\) the inequalities (1.4). Then, for arbitrary \(X_0\) and \(X_1\), the solution of the boundary-value problem (1.1)—(1.2) exists and is unique if either at least one of the integrals of the sum (3.1) diverges, or, equivalently, one of the relations (3.2), (3.3), (3.4) is fulfilled, or else each of the integrals of the sum (3.1) converges, but the value of the sum (3.1) exceeds the difference \(b-a\), i.e. \(b-a<T\).

Remark 2. It can be shown that the results obtained for the boundary-value problem (1.1)—(1.2), under the assumptions made, cannot be improved.

1. As an example of what has been set forth, consider the linear equation of second order

\[ x''+p(t)x'+q(t)x=0 \tag{5.1} \]

with continuous coefficients \(p(t)\) and \(q(t)\). The Vallée-Poussin theorem, as is known, asserts that for the distances \(h\) between two consecutive zeros of a solution \(x(t)\) of equation (5.1) there exists a positive lower bound, which is indicated in [9]. In contrast to previous results (see, for example, [10, 11]), the following theorem

gives an exact lower positive bound for the distances between two consecutive zeros of solutions of equations of the form (5.1). Here, too, in Theorem 4 the exact bound is established for the class of equations of the form (5.1) whose coefficients satisfy conditions (5.2).

Theorem 4. If the coefficients of equation (5.1) on a sufficiently large interval satisfy the conditions

\[ |p(t)| \leq M,\quad |q(t)| \leq N, \tag{5.2} \]

where \(N \ne 0\) (in the case \(N=0\), all solutions of equation (5.1) are nonoscillatory), then the exact lower bound \(T\) for the distances between two consecutive zeros of solutions of equations of the form (5.1) is given by the formula

\[ T = \begin{cases} \dfrac{4}{M}, & \text{if } M^2-4N=0,\\[1.2em] \dfrac{4}{\sqrt{4N-M^2}} \left( \dfrac{\pi}{2}-\operatorname{arctg}\dfrac{M}{\sqrt{4N-M^2}} \right), & M^2-4N<0,\\[1.2em] \dfrac{2}{\sqrt{M^2-4N}} \ln\dfrac{M+\sqrt{M^2-4N}}{M-\sqrt{M^2-4N}}, & M^2-4N>0, \end{cases} \tag{5.3} \]

and for \(M\ne 0\) the exact bound is not attained when, in equation (5.1), the coefficients \(p(t)\) and \(q(t)\) are continuous.

Proof. We write equation (5.1) in the form of a system

\[ \frac{dx}{dt}=y,\quad \frac{dy}{dt}=-q(t)x-p(t)y. \tag{5.4} \]

Let \(\{x(t),y(t)\}\) be such a nontrivial solution of system (5.6) that

\[ x(a)=0,\quad x(b)=0. \tag{5.5} \]

Since, in addition to the indicated nontrivial solution, the trivial solution of system (5.4) also satisfies the same conditions (5.5), the uniqueness of solutions of the boundary-value problem (5.4)—(5.5) is violated. Hence, by virtue of Theorem 3, it follows that the length of the interval \([a,b]\) is greater than or equal to the sum of the integrals (3.1). In the present case \(A_1=A_2=0\), \(B_2=1\), \(C_1=-N\), \(D_1=-M\), \(D_2=M\), and the sum of the integrals (3.1) coincides with the convergent integral

\[ T=2\int_0^\infty \frac{dz}{z^2+Mz+N}, \tag{5.6} \]

whose value in the various cases is given by formula (5.3). Thus it has been shown that the distance \(T\), given by formula (5.3), is indeed a lower bound for the distances between two consecutive zeros of solutions of equation (5.1). This distance \(T\) between two consecutive zeros of solutions of equation (5.1) will be attained if and only if the coefficients \(p(t)\) and \(q(t)\) have the form

\[ q(t)=N,\quad p(t)= \begin{cases} M, & a\leq t<a+\dfrac{T}{2},\\[0.8em] -M, & a+\dfrac{T}{2}\leq t\leq T. \end{cases} \tag{5.7} \]

This follows easily from the results of the second item, if system (5.4) is regarded as a system of differential equations with controls \(p\) and \(q\). Thus, in the case \(M \ne 0\), the bound \(T\) is indeed not attained for continuous coefficients \(p(t)\) and \(q(t)\). To prove that \(T\) is the exact lower bound, consider the sequence of discontinuous coefficients

\[ q_n(t)\equiv N,\quad p_n(t)= \begin{cases} M, & a\leq t\leq a+\dfrac{T}{2},\\[6pt] M-2Mn\left(t-a-\dfrac{T}{2}\right), & a+\dfrac{T}{2}\leq t\leq a+\dfrac{T}{2}+\dfrac{1}{n},\\[6pt] -M, & a+\dfrac{T}{2}+\dfrac{1}{n}\leq t<\infty . \end{cases} \tag{5.8} \]

If \(a\) and \(b_n\) are two consecutive zeros of the solution of equation (5.1) with coefficients (5.8), then it is not difficult to show that, as \(n\to\infty\), \(b_n-a\to b-a=T\). Theorem 4 is proved.

1. We indicate a method of successive approximations for finding the solution of the boundary value problem for the second-order equation (1.3) under the endpoint conditions (1.2). Equation (1.3) is reduced to the system of equations

\[ \frac{dX}{dt}=Y,\quad \frac{dY}{dt}=Q(t,X,Y). \tag{6.1} \]

We shall assume that, for the system of equations (6.1), all the conditions of Theorem 3 are satisfied. Hence the solution of the boundary value problem (6.1)—(1.2) exists and is unique for any \(X_1\). If, as in the first item, we consider the function \(X(c,b)\), then the solution of the boundary value problem (6.1)—(1.2) reduces to finding such a value of \(c\) for which the equality \(X(c,b)=X_1\) holds, i.e., the problem reduces to solving the equation

\[ J(c)=X(c,b)-X_1=0. \tag{6.2} \]

With a view to applying Newton’s method to the solution of equation (6.2), in addition to the condition \(J'(c)=X'_c(c,b)>0\), which is satisfied here, we need also the existence and constancy of sign of the second derivative of the function \(J(c)\).

Theorem 5. Suppose that for system (6.1) the conditions of Theorem 3 are satisfied and that the function \(Q(t,X,Y)\) has continuous partial derivatives of the second order with respect to the variables \(X\) and \(Y\) in the strip \(S\). If the function \(Q(t,X,Y)\) is convex (concave) with respect to the variables \(X,Y\), then

\[ J''(c)=X''_{c^2}(c,b)\geq 0\quad \bigl(J''(c)=X''_{c^2}(c,b)\leq 0\bigr). \]

Proof. The solution \(\{X(c,t),Y(c,t)\}\) of system (6.1) has second-order derivatives with respect to the parameter \(c\), which satisfy the following system of differential equations

\[ \frac{dX''_{c^2}}{dt}=Y''_{c^2}, \]

\[ \frac{dY''_{c^2}}{dt} = Q'_X X''_{c^2} + Q'_Y Y''_{c^2} + Q''_{X^2} X_c'^2 + 2Q''_{XY}X'_cY'_c + Q''_{Y^2}Y_c'^2, \tag{6.3} \]

where the initial data have the form

\[ X_{c^2}^{\prime\prime}(a)=0,\qquad Y_{c^2}^{\prime\prime}(a)=0. \tag{6.4} \]

Let, for each \(c\), \(\Phi(c,t)\) be the fundamental matrix of solutions of the homogeneous system of equations corresponding to system (6.3), normalized at \(t=a\), i.e. \(\Phi(c,a)=E\). Introduce the notation

\[ f(c,t)=Q_{X^2}^{\prime\prime}\cdot X_c^{\prime 2} +2Q_{XY}^{\prime\prime}X_c'Y_c' +Q_{Y^2}^{\prime\prime}\cdot Y_c^{\prime 2}. \]

Then, from Cauchy’s formula,

\[ X_{c^2}^{\prime\prime}(c,b)=\int_a^b \varphi_{12}(c,\tau,b)\,f(c,\tau)\,d\tau . \tag{6.5} \]

Here \(\varphi_{12}(c,\tau,b)\) is the corresponding element of the matrix \(\Phi(b)\cdot\Phi^{-1}(\tau)\). Consider the second column \((\varphi_{12},\varphi_{22})\) of the matrix \(\Phi(t)\cdot\Phi^{-1}(\tau)\). This column is a solution of the homogeneous system of equations corresponding to system (6.3), satisfying, for \(t=\tau\), the conditions

\[ \varphi_{12}(c,\tau,\tau)=0,\qquad \varphi_{22}(c,\tau,\tau)=1. \tag{6.6} \]

It follows from this that the function \(\varphi_{12}(c,\tau,t)\) is positive for \(0<t-\tau\le b-a\). Indeed, the function \(\varphi_{12}(c,\tau,t)\) is the first coordinate of the solution of the aforementioned homogeneous system of equations, issuing at \(t=\tau\) from the point \((0,1)\), while the quantity \(b-a\), as follows from the conditions of the theorem, is less than \(T\). In particular, the function \(\varphi_{12}(c,\tau,b)\) is positive for \(a\le\tau\le b\). Since, by virtue of the convexity (concavity) of the function \(Q(t,X,Y)\), the function \(f(c,\tau)\) is nonnegative (nonpositive) for \(a\le\tau\le b\), it follows from (6.5) that \(X_{c^2}^{\prime\prime}(c,b)\ge0\) \((X_{c^2}^{\prime\prime}(c,b)\le0)\). The theorem is proved.

If the conditions of Theorem 5 are satisfied, then, for the approximate search for the roots of equation (6.2), one may apply Newton’s method, which in the present case reduces to the following. Assign an arbitrary value to the parameter \(c_0\), and find the solution of the system of four equations composed of equations (6.1) and the equations

\[ \frac{dX_c'}{dt}=V_c',\qquad \frac{dY_c'}{dt}=Q_X'(t,X,Y)X_c' + Q_Y'(t,X,Y)Y_c', \tag{6.7} \]

satisfying the initial data

\[ X(a)=X_0,\qquad Y(a)=c_0,\qquad X_c'(a)=0;\qquad Y_c'(a)=1. \tag{6.8} \]

Having found the solution, we obtain the values

\[ J(c_0)=X(c_0,b)-X_1,\qquad J'(c_0)=X_c'(c_0,b). \]

Knowing the values \(J(c_0)\) and \(J'(c_0)\), in the usual way we find the next approximation \(c_1\) to the desired root and then proceed analogously. The obtained sequence \(c_n\) always converges, under the conditions of Theorem 5, to the root of equation (6.2).

References

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Received by the editors
April 22, 1965

Ural State University
named after A. M. Gorky