A THEOREM ON A DIFFERENTIAL INEQUALITY FOR A PERIODIC BOUNDARY VALUE PROBLEM FOR AN ORDINARY DIFFERENTIAL EQUATION OF THE SECOND ORDER

A. Ya. Khokhryakov, B. M. Arkhipov

In recent years a number of results have appeared that are devoted to the theorem on differential inequalities for multipoint boundary value problems of Vallée-Poussin type \([1—7]\) for an ordinary differential equation, and also for a boundary value problem of Sturm–Liouville type \([8, 9, 11, 12]\). Each boundary condition in such problems, as is well known, is written as applied to only one of the boundary points. It is therefore natural to wish to try to extend the theorem on differential inequalities to problems of another kind—problems in which one or several boundary conditions would be connected with several boundary points. The simplest problem of this kind, in our view, is the periodic boundary value problem. The study of the latter for the simplest classes of differential equations of the third and fourth orders is carried out in \([4, 14—15]\). Recently V. A. Churikov \([13]\) obtained an interesting result on the behavior of the Green’s function for one class of boundary value problems for a linear differential equation of order \(n\).

In the present paper, as in \([14—15]\), the periodic boundary value problem for a differential equation of the second order is considered. Conditions for the existence and sign-constancy of the Green’s function of such a problem are investigated, and on this basis a formulation of the main theorem on a differential inequality is given. As an application of the theorems, a sufficient condition is presented for the existence of a unique periodic solution of a nonlinear differential equation, and an estimate of such a solution is also given.

In the present paper we shall use the following concepts and notation:

1. The maximal nonoscillation interval for fixed \(\alpha\) of the equation
   \[ L[y]\equiv y''+p(x)y'+q(x)y=0 \]
   will be denoted by \([\alpha,\varphi(\alpha)]\) \([15]\).

2. We shall call a function \(\Phi(x)\) sign-constant on some interval of variation of \(x\) if \(\Phi(x)\) preserves its sign on this interval.

3. By \(K(x,s)\) we shall denote the Cauchy function of the operation \(L[y]\). It is known that if \([\alpha,\beta]\) is a nonoscillation interval of the equation \(L[y]=0\), then \(K(x,s)>0\) in the triangle \(\alpha<s<x\le \beta\), and \(K(x,s)<0\) in the triangle \(\alpha\le x<s<\beta\).

4. \(u_0(x,s)\), \(u_1(x,s)\) are functions forming such a fundamental system of solutions of the equation \(L[y]=0\) that, for fixed \(s\),
   \[ u_i^{(k)}(s,s)=\left.\frac{\partial^k}{\partial x^k}u_i(s,s)\right|= \begin{cases} 0, & \text{for } k\ne i,\\ 1, & \text{for } k=i \end{cases} \quad (i,k=0,1), \]

\[ W(x,s)= \begin{vmatrix} u_0(x,s) & u_1(x,s)\\ u_0'(x,s) & u_1'(x,s) \end{vmatrix}, \qquad K(x,s)=\frac{1}{W(s,\alpha)} \begin{vmatrix} u_0(s,\alpha) & u_1(s,\alpha)\\ u_0(x,\alpha) & u_1(x,\alpha) \end{vmatrix} \]

are the Wronskian and the Cauchy function of the equation \(L[y]=0\).

1. We shall mark the corresponding functions for the adjoint equation with an asterisk. Then the following relations hold ([16], p. 213):

\[ u_0^*(x,s)=-\frac{1}{W(x,s)}\,u_1(x,s),\qquad u_1^*(x,s)=\frac{1}{W(x,s)}\,u_0(x,s), \]

\[ K(x,s)=u_0(x,\alpha)u_0^*(s,\alpha)+u_1(x,\alpha)u_1^*(s,\alpha)=K^*(s,x). \]

Hence, in particular, it is seen that \(K(x,s)\) and \(\dfrac{\partial}{\partial x}K(x,s)\), for fixed \(x\), will be solutions of the adjoint equation \(L^*[y]=0\).

1. Consider the boundary-value problem

\[ L[y]\equiv y''+p(x)y'+q(x)y=0, \tag{1} \]

\[ y(\alpha)-y(\beta)=0,\qquad y'(\alpha)-y'(\beta)=0, \tag{2} \]

where \(p(x), q(x)\) are continuous functions.

Below we consider the question of the existence and behavior of the Green’s function of problem (1), (2), under the condition that the coefficient \(q(x)\) has constant sign (\(\ne 0\)) on the interval \([\alpha,\beta]\) and that the equation \(L[y]=0\) is nonoscillatory [15].

The existence of the Green’s function of problem (1), (2) follows from

Lemma 1. Let \(q(x)\) satisfy the above-stated conditions on the interval \([\alpha,\beta]\). Then the boundary-value problem (1), (2) has the unique solution \(u(x)\equiv 0\).

Proof. Suppose, for definiteness, that \(q(x)>0\). Assume that problem (1), (2) has a nontrivial solution \(u(x)\). Obviously, it cannot have zeros on the interval \([\alpha,\beta]\). Without loss of generality, one may suppose \(u(x)>0\). Since \(u(x)\) satisfies conditions (2), on \([\alpha,\beta]\) \(u'(x)\) (the derivative of the solution \(u(x)\)) will have at least one zero, say \(\xi\), where \(u(\xi)=\min u(x)\) for \(x\in[\alpha,\beta]\).

Denote \(u(\xi)=c\). Clearly, \(c>0\), and \(L[c]=q(x)c>0\). Then the solution \(u(x)\) will satisfy the conditions \(y(\xi)=c,\ y'(\xi)=0\). Hence, by the theorem on a differential inequality for the Cauchy problem [17], we shall have \(u(x)<c\) for \(x\in[\alpha,\beta]\). But the inequality \(u(x)<c\) contradicts the definition of the number \(c\). The lemma is proved for \(q(x)>0\).

The case \(q(x)<0\) is treated analogously.

Corollary 1. Whatever continuous function \(f(x)\) is taken, the nonhomogeneous problem

\[ L[y]=f(x),\qquad y(\alpha)-y(\beta)=y'(\alpha)-y'(\beta)=0 \tag{I} \]

has the unique solution

\[ y(x)=\int_{\alpha}^{\beta}\Gamma(x,s)f(s)\,ds, \tag{3} \]

where \(\Gamma(x,s)\) is the Green’s function of problem (1), (2).

Remark 1. In the case of sign-changing \(q(x)\), generally speaking, uniqueness of the solution of problem (1), (2) is violated.

Indeed, it is easy to verify that the equation

\[ y''+\bigl(\lambda\sin x-\lambda^2\cos^2 x\bigr)y=0, \tag{4} \]

where \(\lambda\) is some number other than the trivial one, has the nontrivial solution \(u(x)=\exp(\lambda\sin x)\), satisfying the periodic boundary conditions \(y(0)-y(2\pi)=y'(0)-y'(2\pi)=0\). It is not hard to see that, for sufficiently small \(\lambda\), the interval \((0,2\pi)\) will be an interval of nonoscillation for equation (4).

Remark 2. The interval of existence of the Green’s function for the boundary-value problem (1), (2) \([\alpha,\beta]\) depends on the coefficients of the operation \(L[y]\). Thus, for example, the interval of existence of the Green’s function will be \((\alpha,\beta)\), where \(\beta<2\inf \rho(x)\), \(x\in(\alpha,\rho(\alpha))\), if \(p(x)\equiv0\), \(q(x)>0\) is a periodic function of period \(\omega=\beta-\alpha\). (Concerning \(\rho(\alpha)\), see [15]).

Indeed, suppose that the boundary-value problem (1), (2) under the indicated conditions has a nontrivial solution \(u(x)\). Then on some subinterval \([\xi,\xi+\omega]\) the function \(u(x)\) will have at least three zeros. The latter is impossible, since two adjacent zeros of the solution \(u(x)\) would lie in an interval of nonoscillation. Hence \(u(x)\equiv0\), and there exists the Green’s function of problem (1), (2).

Corollary 2. There exists a unique solution \(w(x)\) of the problem

\[ y''+p(x)y'+q(x)y=0,\qquad y(\alpha)-y(\beta)=0,\qquad y'(\alpha)-y'(\beta)=1, \tag{5} \]

where \(q(x)\) is a sign-constant function; moreover the solution \(w(x)>0\) \((<0)\), if \(q(x)\geq0\) \((\leq0)\).

Proof. Obviously, the solution \(w(x)\) of problem (5) exists and is unique. It is also obvious that \(w(x)\) cannot be sign-constant.

Let \(q(x)\geq0\). Suppose that \(w(x)<0\) for \(\alpha\leq x\leq\beta\). Then either \(w'(\alpha)\geq0\) or \(w'(\alpha)<0\) is possible:

a) Let \(w'(\alpha)\geq0\). The substitution \(w(x)=Y(x)+w(\alpha)\) transforms problem (5) into the problem

\[ Y''+p(x)Y'+q(x)Y=-w(\alpha)q(x), \tag{6} \]

\[ Y(\alpha)=Y(\beta)=0,\qquad Y'(\alpha)-Y'(\beta)=1, \]

whose solution can be written by Cauchy’s formula as

\[ Y(x)=w'(\alpha)K(x,\alpha)+\int_{\alpha}^{x}K(x,s)\{-w(\alpha)q(s)\}\,ds, \tag{7} \]

where \(K(x,s)\) is the Cauchy function of the operation \(L[y]\). Since \(K(x,s)>0\) for \(\alpha\leq s<x\leq\beta\), it follows, as is seen from (7), that \(Y(\beta)>0\). This last inequality contradicts the boundary condition \(Y(\beta)=0\) from (6).

b) Let now \(w'(\alpha)<0\). From the second boundary condition of problem (5) it follows that \(w'(\beta)<0\). Again, as in a), passing to problem (6), the solution of problem (6) can be written by the formula

\[ Y(x)=w'(\beta)K(x,\beta)+\int_{\beta}^{x}K(x,s)\{-w(\alpha)q(s)\}\,ds. \]

Since \(K(x,s)<0\) for \(\alpha\leq x<s<\beta\), it follows that \(Y(\alpha)>0\). Again we arrive at a contradiction with the boundary condition of problem (6).

In the case \(q(x)\leq0\) the proof is analogous.

The corollary is established.

Lemma 2. Let the coefficient \(q(x) \geqslant 0\) in the operator \(L[y]\). Then the local minimum (maximum) of any nontrivial solution of the equation \(L[y]=0\) is less (greater) than zero.

Proof. Let a nontrivial solution \(y(x)\) attain a local minimum at the point \(\xi\), and let \(y(\xi)>0\). Using the substitution \(y(x)=Y(x)+y(\xi)\), we write the equation \(L[y]=0\) in the following form:

\[ Y''+p(x)Y'+q(x)Y=-y(\xi)q(x). \]

The solution \(y(x)\) passes into \(Y(x)\), for which \(Y(\xi)=0,\ Y'(\xi)=0\), and therefore, with the aid of the Cauchy function \(K(x,s)\) of the operation \(L[y]=0\), the solution \(Y(x)\) can be written by Cauchy’s formula

\[ Y(x)=\int_{\xi}^{x} K(x,s)\{-y(\xi)q(s)\}\,ds. \]

Hence \(Y(x)<0\) for \(x>\xi\), since \(K(x,s)\geqslant 0\) for \(\xi<s<x<\beta\), \(y(\xi)>0\). But this inequality contradicts the assumption that at the point \(\xi\) \(Y(x)\), and consequently also \(y(x)\), has a local minimum.

Analogously it is proved that the maximum is positive.

The lemma is proved.

Lemma 3. Let \((\alpha,\beta)\) be a nonoscillation interval of the equation

\[ y''+Q(x)y=0 \]

and let \(Q(x)\geqslant 0\) \((\not\equiv 0)\). Then the boundary value problem

\[ y''+Q(x)y=0, \tag{8} \]

\[ y(\alpha)-\lambda y(\beta)=0,\qquad y'(\alpha)-\lambda y'(\beta)=0, \tag{9} \]

where \(\lambda>0\), has the unique solution \(u_\lambda(x)\equiv 0\).

Proof. Suppose that there exists a nontrivial solution \(u_\lambda(x)\) of problem (8), (9).

Let \(\lambda\geqslant 1\). Then the conditions (9) will be satisfied only when \(y(\alpha)\cdot y(\beta)>0,\ y'(\alpha)\cdot y'(\beta)>0\). Since \(u_\lambda(x)\) is a solution of the homogeneous equation (8), without loss of generality one may assume that \(u_\lambda(x)>0\) for \(x\in[\alpha,\beta]\). Hence, from equation (8), we have

\[ u'_\lambda(\alpha)-u'_\lambda(\beta)=\int_{\alpha}^{\beta} Q(s)u_\lambda(s)\,ds>0. \tag{10} \]

At the same time, either \(u'_\lambda(\alpha)\geqslant 0\) or \(u'_\lambda(\alpha)<0\) is possible. If \(u'_\lambda(\alpha)\geqslant 0\), then on \([\alpha,\beta]\) there must exist a positive local minimum. The latter, by Lemma 2, is impossible when \(Q(x)\geqslant 0\). If, however, \(u'_\lambda(\alpha)<0\), then \(u'_\lambda(\alpha)-u'_\lambda(\beta)\leqslant 0\), since \(\lambda\geqslant 1\). But then the inequality \(u'_\lambda(\alpha)-u'_\lambda(\beta)\leqslant 0\) contradicts the inequality (10). Hence, it follows that \(u_\lambda(x)\equiv 0\) for \(\lambda\geqslant 1\).

The case \(\lambda<1\) is checked in a completely analogous way.

The lemma is proved.

Remark. Repeating the proof of Lemma 3, it is easy to establish the uniqueness of the solution of the problem \(y''+Q(x)y=0,\ y(\alpha)-y(\beta)=0,\ y'(\alpha)-\lambda y'(\beta)=0\), if \(Q(x)\geqslant 0\) and \(\lambda>0\).

1. We pass to the study of the behavior of the Green function of the periodic boundary value problem (1), (2).

Let \(\Gamma(x,s)\) be the Green function of problem (1), (2). The function \(\Gamma(x,s)\) in the square \(x,s\in[\alpha,\beta]\) satisfies the identity

\[ \int_{\alpha}^{\beta} \Gamma(x,s)q(s)\,ds=1, \tag{11} \]

where \(q(x)\) is the coefficient of the operation \(L[y]\). Indeed, the problem

\[ L[y]=q(x),\qquad y(\alpha)-y(\beta)=0,\qquad y'(\alpha)-y'(\beta)=0 \tag{12} \]

has the unique solution \(u(x)\equiv 1\). But then, by Green’s formula, we have

\[ u(x)=\int_{\alpha}^{\beta}\Gamma(x,s)q(s)\,ds . \]

Hence (11) follows.

On the sides of the square \(x,s\in[\alpha,\beta]\), \(\Gamma(x,s)\) does not vanish. Otherwise there would be a nontrivial solution having at least two zeros on the interval \([\alpha,\beta]\).

Theorem 1. Let \(q(x)\) be a continuous function of constant sign for \(x\in[\alpha,\beta]\), and let \([\alpha,\beta]\) be an interval of nonoscillation of the equation \(L[y]=0\), i.e. \(\beta<\rho(\alpha)\). Then the Green’s function \(\Gamma(x,s)\) of problem (1), (2) has constant sign in the square \(x,s\in[\alpha,\beta]\), and \(q(x)\Gamma(x,s)>0\) for \(x,s\in[\alpha,\beta]\).

Proof. 1) Let \(q(x)\le 0\), and let \(f(x)\) be a continuous function. Consider the solution \(y(x)\) of problem (1), which can be written in terms of the Cauchy function \(K(x,s)\) as follows:

\[ y(x)=\int_{\alpha}^{x}K(x,s)f(s)\,ds+c_1K(x,\alpha)+c_2\omega(x) =\int_{\alpha}^{\beta}\Gamma(x,s)f(s)\,ds, \tag{13} \]

where \(K(x,s)\) is the Cauchy function of the operation \(L[y]\), and \(\omega(x)\) is the solution of boundary-value problem (5). Choosing the coefficients \(c_1\) and \(c_2\) in such a way that the solution (13) of the equation \(L[y]=0\) actually satisfies the boundary conditions (2), we obtain the following expression for the Green’s function:

\[ \Gamma(x,s)=\widetilde K(x,s)-\frac{K(\beta,s)}{K(\beta,\alpha)}K(x,\alpha)+ \]

\[ +\left[K'(\beta,s)+\frac{K(\beta,s)}{K(\beta,\alpha)}\left(1-K'(\beta,\alpha)\right)\right]\omega(x), \tag{14} \]

where

\[ \widetilde K(x,s)= \begin{cases} 0, & \text{for } \alpha\le x\le s\le \beta,\\ K(x,s), & \text{for } \alpha\le s\le x\le \beta. \end{cases} \]

Since \(q(x)\le 0\), \(\Gamma(x,s)\) has no zeros on the sides of the square \(x,s\in[\alpha,\beta]\), and

\[ \int_{\alpha}^{\beta}\Gamma(\alpha,s)q(s)\,ds = \int_{\alpha}^{\beta}\Gamma(\beta,s)q(s)\,ds =1, \]

we have \(\Gamma(\alpha,s)=\Gamma(\beta,s)<0\). The assertion of the theorem will be established for \(q(x)\le 0\) if we show the absence of zeros of the function \(\Gamma(x,s)\) and its negativity in the triangle \(\alpha\le x\le s\le \beta\).

In formula (14) we note that

\[ -\frac{K(\beta,s)}{K(\beta,\alpha)}K(x,\alpha)<0 \quad \text{for } \alpha\le x\le s\le \beta, \]

and

\[ v(x)=\frac{\partial}{\partial x}K(\beta,s) +\frac{K(\beta,s)}{K(\beta,\alpha)} \left[1-\frac{\partial}{\partial x}K(\beta,\alpha)\right]\ge 0, \]

since \(v(x)\) is a solution of the adjoint equation \(L^*[y]\), and \(v(\alpha)=v(\beta)=1\); therefore, in accordance with Corollary 2 of Lemma 1, we have \(\omega(x)<0\). Consequently, \(\Gamma(x,s)<0\) for \(\alpha\le x\le s\le \beta\). Hence \(\Gamma(x,s)<0\) for \(q(x)\le 0\) and \(x,s\in[\alpha,\beta]\).

1. Let \(q(x) \geqslant 0\). Without loss of generality, we may assume \(\alpha\beta > 0\), since otherwise, by a corresponding change of the variable \(x\), we arrive at the condition \(\alpha\beta > 0\).

Introduce the function \(g(x)\), defined for \(-\infty < x < +\infty\) and satisfying the following conditions: whatever \(x=x_0\), \(y=y_0\), \(y'=y_0'\) may be, there exists a unique solution of the Cauchy problem:
\[ y''+p(x)y'+g(x)y=0,\qquad x=x_0,\qquad y(x_0)=y_0,\qquad y'(x_0)=y_0'; \]
\[ g(x)=q(x)\quad \text{for } x\in[\alpha,\beta], \]
\[ g(-x)=-q(x)\quad \text{for } x\in[\alpha,\beta]. \]

The boundary-value problem
\[ y''+p(x)y'+g(x)y=f(x),\qquad y(\alpha)-y(\beta)=0,\qquad y'(\alpha)-y'(\beta)=0, \tag{15} \]
where \(f(x)\) is a continuous function \((f(-x)=f(x),\ f(x)\geqslant 0)\), like problem (1), (2), has a unique solution and, consequently, has a Green’s function \(G(x,s)\). In this case the solution \(Y(x)\) of problem (15) can be written as
\[ Y(x)=\int_{\alpha}^{\beta}G(x,s)f(s)\,ds. \tag{16} \]

Let us rewrite problem (15), using the substitutions \(x=-x,\ y=y\); we obtain
\[ y''-p(-x)y'-q(x)y=0,\qquad y(-\alpha)-y(-\beta)=0, \]
\[ -y'(\alpha)+y'(-\beta)=0. \tag{17} \]

In this case formula (16) may be rewritten as
\[ Y(x)=\int_{\alpha}^{\beta}G(-x,s)f(s)\,ds. \tag{18} \]

Obviously, (18) will be a solution of problem (17). On the other hand, the solution of this problem can be written through the Green’s function \(G_1(x,s)\) of problem (17):
\[ Y(-x)=\int_{-\alpha}^{-\beta}G_1(-x,s)f(s)\,ds, \tag{19} \]
where the sign of the function \(G_1(-x,s)\), according to item 1), is negative in the square \(-x,s\in[-\beta,-\alpha]\). Hence it follows that \(Y(x)>0\) for any continuous nonnegative function \(f(x)\).

In formula (18), returning to the former variable \(x\), we note that \(G(x,s)>0\) for \(x,s\in[\alpha,\beta]\).

We shall show that \(G(x,s)\equiv \Gamma(x,s)\) for \(x,s\in[\alpha,\beta]\), thereby proving the validity of Theorem 1 for \(q(x)\geqslant 0\). Let \(y(x)\) be the solution of problem (1), \(v(x)\) the solution of problem (15), and \(f(x)\) a continuous function. Consider
\[ \eta(x)=y(x)-v(x)=\int_{\alpha}^{\beta}\{\Gamma(x,s)-G(x,s)\}f(s)\,ds. \tag{20} \]

On the other hand, \(\eta(x)\) satisfies the problem
\[ L[\eta]=0,\qquad \eta(\alpha)-\eta(\beta)=0,\qquad \eta'(\alpha)-\eta'(\beta)=0. \]

and therefore, according to Lemma 1, \(\eta(x)\equiv 0\) for \(x\in [\alpha,\beta]\). Hence, from formula (20), in view of the arbitrariness of \(f(x)\), it follows that the identity \(\Gamma(x,s)\equiv G(x,s)\) holds for \(x,s\in[\alpha,\beta]\).

Theorem 1 is proved.

Corollary 1. If \(p(x)\), \(q(x)\), \(f(x)\) are continuous periodic functions of period \(\omega<\rho(\alpha)-\alpha\), and \(q(x)\) is of one sign, then the equation \(L[y]=f(x)\) has a unique periodic solution of period \(\omega\).

Remark 1. The condition \(\beta<\rho(\alpha)\) in Theorem 1 is essential. Moreover, the inequality \(\beta<\rho(\alpha)\) is a necessary condition for preservation of the sign of the Green’s function \(\Gamma(x,s)\) of problem (1), (2).

Indeed, by the definition of \(\rho(\alpha)\), some solution \(u(x)\) of the equation \(L[y]\) will satisfy the conditions \(u(\alpha)=u(\rho(\alpha))\), \(u'(\alpha)=\gamma>0\), \(u'(\rho(\alpha))=\gamma_1<0\). According to the theorem on continuous dependence of a solution on the initial data, for any \(\varepsilon>0\) one can indicate a number \(\delta>0\) such that the solution \(v(x)\) of the equation \(L[y]=0\) with initial conditions \(v(\alpha)<0\), \(|v(\alpha)|<\delta\), \(|v'(\alpha)-\gamma|<\delta\), at some point \(\beta\) of the interval \((\rho(\alpha),\rho(\alpha)+\varepsilon)\), will satisfy the equality \(v(\alpha)=v(\beta)\). In addition, by means of \(\delta\) one can guarantee the inequalities \(v'(\beta)<0\), \(K(\beta,\alpha)\ne 0\).

Taking now in formula (14), in place of \(w(x)\), the function \(v(x)\) just considered, we note that the function

\[ \Gamma(x,s)=\widetilde K(x,s)-\frac{K(\beta,s)}{K(\beta,\alpha)}K(x,\alpha)+ \]

\[ +\left[K'(\beta,s)+\frac{K(\beta,s)}{K(\beta,\alpha)}\bigl(1-K'(\beta,s)\bigr)\right]\frac{v(x)}{v'(\alpha)-v'(\beta)} \]

will be the Green’s function of problem (1), (2). Hence it is clear that the behavior of

\[ \Gamma(x,\alpha)=\frac{v(x)}{v'(\alpha)-v'(\beta)} \]

on \([\alpha,\beta]\) will be determined by the behavior of the function \(v(x)\). Obviously, \(v(x)\) will be sign-changing if the numbers \(\gamma\) and \(\delta\) are chosen in the corresponding manner. Consequently, \(\Gamma(x,s)\) changes sign in the square \([\alpha,\beta]\), where \(\beta>\rho(\alpha)\).

Remark 2. Theorem 1 makes it possible to establish a very simple estimate for the solution \(y(x)\) of problem (1). Let \(q(x)\), \(f(x)\) be continuous functions of one sign, \(q(x)\le f(x)\) \((q(x)\ge f(x))\), and let \([\alpha,\beta]\) be an interval of nonoscillation of the equation \(L[y]=0\). Then

\[ 1<y(x)\quad (0<y(x)<1)\quad \text{for } x\in[\alpha,\beta]. \tag{21} \]

Indeed, from the formulas

\[ y(x)=\int_{\alpha}^{\beta}\Gamma(x,s)f(s)\,ds>0,\qquad \int_{\alpha}^{\beta}\Gamma(x,s)q(s)\,ds=1 \]

we obtain

\[ y(x)-1=\int_{\alpha}^{\beta}\Gamma(x,s)\{f(s)-q(s)\}\,ds. \]

The indicated estimates (21) follow from this.

Remark 3. An assertion of Mikušinski’s theorem type [18] holds. Denote by \(z(x)\) and \(u(x)\), respectively, the solutions of the problems

\[ y''+p_i(x)y'+q_i(x)y=f(x),\qquad y(\alpha)-y(\beta)=0, \tag{22} \]

\[ y'(\alpha)-y'(\beta)=0,\qquad i=1,2, \]

where \(q_1(x), q_2(x)\) are functions of constant sign and \(q_1(x)\cdot q_2(x)>0\)* for \(x\in[\alpha,\beta]\), \(\beta<\min\{\rho_1(\alpha),\rho_2(\alpha)\}\). If, moreover, there exists a number \(k>0\) such that \(q_1(x)\le kf(x)\le q_2(x)\), then \(z(x)>u(x)\) \((<)\) for \(x\in[\alpha,\beta]\) and \(q_1(x)>0\) \((q_2(x)<0)\).

Indeed, if \(q_1(x)>0\) and \(\Gamma_1(x,s)\) is the Green’s function of problem ((22) \(i=1\)), then

\[ z(x)=\int_\alpha^\beta \Gamma_1(x,s)f(s)\,ds. \]

On the other hand, taking into account the property of the Green’s function of the periodic boundary-value problem, we have

\[ kz(x)-1=\int_\alpha^\beta \Gamma_1(x,s)\{kf(s)-q_1(s)\}\,ds>0. \]

Hence \(z(x)>\dfrac{1}{k}\) for \(x\in[\alpha,\beta]\). Similarly we verify the inequality \(u(x)<\dfrac{1}{k}\) for \(x\in[\alpha,\beta]\). Comparing the last two inequalities, we arrive at the inequality \(z(x)>u(x)\) for \(x\in[\alpha,\beta]\) and \(q_1(x)>0\). If \(q_2(x)<0\), then, repeating only the above arguments, we obtain the inequality \(z(x)<u(x)\).

It is not difficult to see that Remark 3 will remain valid for periodic boundary-value problems written for differential equations of higher orders, provided only that the Green’s functions of such problems exist and have constant sign.

Theorem 1 also makes it possible to prove a theorem on a differential inequality for the periodic boundary-value problem (1):

\[ L[y]=f(x),\qquad y(\alpha)-y(\beta)=0,\qquad y'(\alpha)-y'(\beta)=0. \]

Theorem 2. Let the twice continuously differentiable function \(z(x)\) satisfy the boundary conditions (2) and the differential inequality
\[ L[z(x)]-f(x)=\varphi(x)>0\;(\le 0) \]
for \(x\in[\alpha,\beta]\), and let \(\beta<\rho(\alpha)\). Then \(z(x)>y(x)\) \((\le)\) when \(q(x)>0\), and \(z(x)\le y(x)\) \((\ge)\) when \(q(x)<0\), if \(y(x)\) is the solution of problem (1).

Proof. Let \(q(x)>0\) and let \(L[z]-f(x)>0\). Denote by \(\eta(x)=z(x)-y(x)\). Obviously, \(\eta(x)\) will satisfy the conditions

\[ L[\eta]=\varphi(x)>0,\qquad \eta(\alpha)-\eta(\beta)=0,\qquad \eta'(\alpha)-\eta'(\beta)=0, \]

and therefore

\[ \eta(x)=\int_\alpha^\beta \Gamma(x,s)\varphi(s)\,ds, \]

since \(\Gamma(x,s)>0\) is the Green’s function of problem (1), (2). Consequently, \(\eta(x)=z(x)-y(x)>0\), i.e. \(z(x)>y(x)\). If \(\varphi(x)\equiv0\), then, obviously, \(z(x)=y(x)\). The remaining cases are considered similarly. The theorem is proved.

1. We pass to the consideration of the nonlinear problem

\[ N[y]\equiv y''+f(x,y)=0, \tag{23} \]

\[ y(\alpha)-y(\beta)=0,\qquad y'(\alpha)-y'(\beta)=0, \tag{24} \]

\[ \text{* The case } q_1(x)\cdot q_2(x)<0 \text{ is of no interest, since from Theorem 1 it is seen that } z(x)\cdot u(x)<0 \text{ for sign-constant } f(x). \]

assuming that \(f(x,y)\) is defined and continuous in some convex domain \(G\) of the \((x,y)\)-plane and satisfies in this domain conditions \(L_1\) and \(L_2\), i.e., Azelev’s condition [17]. Since \(f(x,y)\) satisfies Azelev’s condition in \(G\), there exist continuous functions \(q_1(x)\) and \(q_2(x)\) such that

\[ q_1(x)(y_1-y_2)\leq f(x,y_1)-f(x,y_2)\leq q_2(x)(y_1-y_2) \]

for all \(y_1,y_2\in G,\quad y_1\geq y_2\).

Let \([\alpha,\rho(\alpha))\) be the maximal interval of nonoscillation of the equation
\[ y''+q_i(x)y=0,\quad i=1,2. \]
Put \(\rho(\alpha)=\min[\rho_1(\alpha),\rho_2(\alpha)]\).

There is the following assertion on a differential inequality, which generalizes Theorem 2 to the case of a nonlinear boundary-value problem.

Theorem 3. Let \(f(x,y)\) satisfy in the domain \(G\) Azelev’s conditions \(L_1,L_2\), and let \(q_1(x), q_2(x)\) be sign-constant functions of one sign and \(\beta<\rho(\alpha)\). Suppose, further, that there exists a pair of twice continuously differentiable functions \(z_1(x)\) and \(z_2(x)\), satisfying the boundary conditions (2) (whose graphs lie in the domain \(G\)), such that \(N[z_1]\geq 0\) and \(N[z_2]\leq 0\). Then the boundary-value problem (23)—(24) has a unique solution \(y(x)\in G\), which on \([\alpha,\beta]\) satisfies the inequalities \(z_1(x)\leq y(x)\leq z_2(x)\), if \(q_2(x)\leq 0\), and \(z_1(x)\geq y(x)\geq z_2(x)\), if \(q_1(x)\geq 0\).

We omit the proof of Theorem 3, since it is analogous to the proof of Theorem 5 of [1] (see also [14]).

Corollary. Let all the conditions of Theorem 3 be fulfilled, and let \(f(x,y)\) be periodic with respect to \(x\) with period \(0<\omega<\rho(\alpha)-\alpha\), i.e., \(f(x+\omega,y)=f(x,y)\). Then the differential equation

\[ y''+f(x,y)=0 \]

has a unique periodic solution \(y(x)\) of period \(\omega\), which will satisfy the inequalities

\[ z_1(x)\leq y(x)\leq z_2(x)\quad \text{for } q_2(x)\leq 0 \]

and

\[ z_1(x)\geq y(x)\geq z_2(x)\quad \text{for } q_1(x)\geq 0. \]

Remark to Theorem 3. The requirement imposed on the functions \(q_1(x)\) and \(q_2(x)\) in Theorem 3 is essential. As the example considered below shows, in the case when \(q_1(x)\) and \(q_2(x)\) are sign-constant and \(q_1(x)q_2(x)\leq 0\), Theorem 3 guarantees only the existence of a solution of problem (23)—(24), but not the uniqueness of the solution.

As an example, consider the nonlinear equation [19]

\[ N[x]\equiv \ddot{x}+\frac{l\cos t}{1+l\cos t}\cdot x+\mu\sin\frac{x}{1+l\cos t}-4l\sin t=0. \tag{25} \]

We shall show that, under certain restrictions on \(\mu\) and \(l\), this equation has a \(2\pi\)-periodic solution \(x(t)\) satisfying the inequalities

\[ -\frac{3}{2}\pi(1+l\cos t)\leq x(t)\leq \frac{3}{2}\pi(1+l\cos t). \tag{26} \]

Our assertion will evidently be established if we show the existence of a solution of the boundary-value problem

\[ N[x]\equiv \ddot{x}+f(t,x),\quad x(\alpha)-x(\alpha+2\pi)=0,\quad \dot{x}(\alpha)-\dot{x}(\alpha+2\pi)=0, \tag{27} \]

where

\[ f(t,x)=\frac{l\cos t}{1+l\cos t}\cdot x+\mu\sin\frac{x}{1+l\cos t}-4l\sin t, \]

and that this solution satisfies inequalities (26).

The nonlinear function \(f(t,x)\) satisfies Azbelev’s conditions in the domain \(G:-\infty<t<+\infty,\ -A\leq x\leq A\), where \(A\) is a certain positive number. Indeed, whatever the two functions \(v>u\) whose graphs lie in the domain \(G\), the inequalities

\[ -\frac{\mu-l\cos t}{1+l\cos t}\,(v-u)\leq f(t,v)-f(t,u)\leq \frac{\mu+l\cos t}{1+l\cos t}\,(v-u), \tag{28} \]

will hold if \(\mu>4l\). Let us note in passing that the coefficients

\[ q_1(t)=-\frac{\mu-l\cos t}{1+l\cos t},\qquad q_2(t)=\frac{\mu+l\cos t}{1+l\cos t} \]

of the conditions \(L_1\) and \(L_2\) are sign-constant, but of opposite signs. Since \(\mu>4l\), the interval \([\alpha,\alpha+2\pi]\), obviously, is an interval of nonoscillation of the equation \(\ddot x+q_1(t)x=0\).

Further, for the functions \(z_1(t)=-\dfrac{3}{2}\pi(1+l\cos t)\), \(z_2(t)=\dfrac{3}{2}\pi(1+l\cos t)\), the following conditions will hold:

\[ N[z_1]=\mu-4l\sin t\geq 0,\qquad N[z_2]=-\mu-4l\sin t\leq 0, \]

\[ z_i(\alpha)-z_i(\alpha+2\pi)=0,\qquad z_i'(\alpha)-z_i'(\alpha+2\pi)=0,\qquad i=1,2. \tag{29} \]

Repeating now the method of proof of the existence of a solution in Theorem 5 of [1], taking into account the conditions listed above, (28) and (29), we become convinced of the existence of a solution \(x(t)\) of problem (27), and at the same time establish inequalities (26).

We shall show that there exist at least two solutions of problem (27) satisfying inequalities (26). Indeed, noting the inequalities

\[ N\left[\frac{\pi}{2}(1+l\cos t)\right]=\mu-4l\sin t\geq 0,\qquad N\left[\frac{3}{2}\pi(1+l\cos t)\right]\leq 0, \]

on the one hand, and the inequalities

\[ N\left[-\frac{3}{2}\pi(1+l\cos t)\right]\geq 0,\qquad N\left[-\frac{\pi}{2}(1+l\cos t)\right]\leq 0, \]

on the other, we are convinced of the existence of such solutions \(x_1(t)\) and \(x_2(t)\) of problem (27) that

\[ -\frac{3}{2}\pi(1+l\cos t)\leq x_1(t)\leq -\frac{\pi}{2}(1+l\cos t), \]

\[ \frac{\pi}{2}(1+l\cos t)\leq x_2(t)\leq \frac{3}{2}\pi(1+l\cos t). \]

Hence it follows that there exist at least two \(2\pi\)-periodic solutions of equation (25) for which inequalities (26) hold.

It is not difficult to note that equation (25) has an infinite set of \(2\pi\)-periodic solutions.

References

1. Azbelev N. V., Khokhryakov A. Ya. and Tsalyuk Z. B. Theorems on differential inequalities for boundary-value problems. Matem. sb., 59(101), 1962.

2. Levin A. Yu. DAN SSSR, 148, No. 3, 1963.

3. Teptin A. L. Proceedings of the Izhevsk Mathematical Seminar, issue 1, 1963, pp. 42–45.

4. Aliev R. G. A theorem on a differential inequality for ordinary differential equations of the fourth order with multipoint boundary conditions. Reports of the Second Siberian Conference on Mathematics and Mechanics. Tomsk, 1962.

5. Chichkin E. S. Izv. vuzov, Matem., No. 2, 1962.

6. Aliev R. G., Inozemtsev I. N. On the distribution of zeros of solutions of a linear differential equation of the fifth order. Proceedings of the Izhevsk Mathematical Seminar, issue 1, 1963.

7. Ostroumov V. V. An existence and uniqueness theorem for a nonlinear boundary-value problem with Wallis–Poussin conditions. Proceedings of the Izhevsk Mathematical Seminar, issue 1, 1963.

8. Pak S. A. DAN SSSR, 143, No. 6, 1963.

9. Pak S. A. On the question of a differential inequality for the Sturm–Liouville problem. Reports of the Third Siberian Conference on Mathematics and Mechanics. Tomsk, 1964.

10. Ostroumov V. V. Some questions on the distribution of zeros and the sign of Green’s function. Reports of the Third Siberian Conference on Mathematics and Mechanics. Tomsk, 1964.

11. Anishchenko R. I. Siberian Mathematical Journal, 5, No. 3, 1964.

12. Teptin A. L. and Yuberev N. N. Siberian Mathematical Journal, 5, No. 5, 1964.

13. Churikov V. A. On the existence and sign of Green’s function for the Sturm–Liouville problem. Reports of the Third Siberian Conference on Mathematics and Mechanics. Tomsk, 1960.

14. Khokhryakov A. Ya. On the periodic boundary-value problem for a differential equation of the third order. Matem. sb., (105):4, 1964.

15. Azbelev N. V. and Tsalyuk Z. B. On the distribution of zeros of solutions of a linear differential equation of the third order. Matem. sb., 51(93), 1960.

16. Stepanov V. V. A Course of Differential Equations. State Publishing House of Physical and Mathematical Literature, Moscow, 1958.

17. Azbelev N. V. and Tsalyuk Z. B. On Chaplygin’s problem. Ukrainian Mathematical Journal, 10, No. 1, 1958.

18. Mikusinski J. Sur l’équation \(x^{(n)} + A(t)x = 0\), Ann. polon. math., 1, No. 2, 1955.

19. Kill’ I. D. On a periodic solution of one nonlinear equation. PMM, 27, issue 6, 1963.

Received December 12, 1964

Mogilev Machine-Building Institute,
Mogilev Pedagogical Institute