ON A DIFFERENTIAL INEQUALITY FOR A BOUNDARY VALUE PROBLEM

V. V. OSTROUMOV

Theorems on differential and integral inequalities have acquired great importance in analysis as a rich source of estimates. Questions concerning differential inequalities for boundary value problems are the subject of recent works of the Izhevsk seminar [1—5].

However, these questions, even for linear problems, are far from completely solved.

Below we consider the question of the existence and sign of the Green’s function for the problem

\[ L[y]\equiv y^{(n)}-\sum_{k=0}^{n-1} g_k y^{(k)}=f(t); \tag{1} \]

\[ y^{(j)}(\alpha_i)=A_i^j,\quad j=0,\ldots,m_i-1;\qquad \sum_{i=1}^{p} m_i=n;\qquad \alpha_1<\cdots<\alpha_p, \tag{2} \]

where the \(g_k(t)\) are \((n-1)\)-times continuously differentiable.

Following the terminology of [1—3], denote by \(r_{m_1,\ldots,m_p}(\alpha)=\sup \beta\), where \(\beta\) is such that, for \(\alpha_i\in[\alpha,\beta)\), problem (1)—(2) has a unique solution. We shall call problem (1)—(2) a problem of type \((m_1,\ldots,m_p)\). Let starred notation correspond to the operation

\[ L^*[y]\equiv y^{(n)}-\sum_{k=0}^{n-1}(-1)^{\,n-k}(g_k y)^{(k)}. \]

Lemma 1.

\[ r_{n-k,k}(t)=r^*_{k,n-k}(t). \]

Proof. Integrating by parts, we obtain

\[ \int_{\alpha_1}^{\alpha_2} z(L)[y]\,dt = \sum_{j=0}^{n-1} y^{(j)}V_{n-j-1}[z]\Big|_{\alpha_1}^{\alpha_2} + (-1)^n\int_{\alpha_1}^{\alpha_2} yL^*[z]\,dt, \]

where

\[ V_0[z]=z;\qquad V_j[y]=(-1)^j\left[y^{(j)}-\sum_{k=0}^{j-1}(-1)^k(g_{n-k-1}y)^{(k)}\right]; \quad j=1,\ldots,n-1. \]

It is known that, if \(y(t)\) is the unique solution of the problem

\[ L[y]=f(t), \]

\[ y^{(j)}(\alpha_1)=0,\quad j=0,\ldots,n-k-1, \]

\[ y^{(j)}(\alpha_2)=0,\quad j=0,\ldots,k-1, \]

then \(z(t)\) is the unique solution of the problem

\[ \begin{gathered} L^*[y]=f(t),\\ V_j[y(\alpha_1)]=0,\quad j=0,\ldots,k-1,\\ V_j[y(\alpha_2)]=0,\quad j=0,\ldots,n-k-1. \end{gathered} \tag{3} \]

It is not difficult to verify that the conditions (3) are equivalent to the conditions

\[ y^{(j)}(\alpha_1)=0,\quad j=0,\ldots,k-1, \]

\[ y^{(j)}(\alpha_2)=0,\quad j=0,\ldots,n-k-1. \]

Therefore \(r_{n-k,k}(\alpha)\leq r^*_{k,n-k}(\alpha)\). Similarly we obtain \(r_{n-k,k}(\alpha)\geq r^*_{k,n-k}(\alpha)\).

Lemma 2.

\[ r_{m_1,\ldots,m_p}(\alpha)\geq \max\left\{ \min_{k=1,\ldots,n-m_1}\bigl[r_{n-k,k}(\alpha)\bigr], \min_{k=1,\ldots,n-m_p}\bigl[r_{k,n-k}(\alpha)\bigr] \right\}. \]

Proof. Denote
\[ \min_{k=1,\ldots,n-m_1}\bigl[r_{n-k,k}(\alpha)\bigr]=\omega. \]

Suppose that \(r_{m_1,\ldots,m_p}(\alpha)<\omega\). Then, according to the definition of \(r_{m_1,\ldots,m_p}(\alpha)\), there exist \(\alpha_i\in[\alpha,\omega)\) such that the homogeneous problem \((m_1,\ldots,m_p)\)

\[ L[y]=0, \]

\[ y^{(j)}(\alpha_i)=0,\quad j=0,\ldots,m_i-1, \]

will have a nontrivial solution \(y(t)\not\equiv0\).

Let \(\{u_i\}\), \(i=0,\ldots,n-1\), be a fundamental system of the equation \(L[y]=0\) such that
\[ u_i^{(j)}(\alpha_1)=\delta_{i,n-j-1},\quad j=0,\ldots,n-1. \]
Then
\[ y(t)=\sum_{i=0}^{n-1}c_i u_i,\quad \text{where } c_i=y^{(n-i-1)}(\alpha_1),\quad i=0,\ldots,n-1. \]
Since \(y^{(j)}(\alpha_1)=0\), \(j=0,\ldots,m_i-1\), it follows that
\[ y(t)=\sum_{i=0}^{n-m_1-1}c_i u_i. \]

It is not difficult to verify that a necessary and sufficient condition for the uniqueness of the solution of the problem \((n-k,k)\) is the nonvanishing of the determinant

\[ D_{n-k,k}(\alpha_1,\alpha_2)= \left| \begin{array}{cccc} u_0(\alpha_1) & \cdots & u_{n-1}(\alpha_1)\\ \cdot & \cdots & \cdot\\ \cdot & \cdots & \cdot\\ \cdot & \cdots & \cdot\\ u_0^{(n-k-1)}(\alpha_1) & \cdots & u_{n-1}^{(n-k-1)}(\alpha_1)\\ u_0(\alpha_2) & \cdots & u_{n-1}(\alpha_2)\\ \cdot & \cdots & \cdot\\ \cdot & \cdots & \cdot\\ \cdot & \cdots & \cdot\\ u_0^{(k-1)}(\alpha_2) & \cdots & u_{n-1}^{(k-1)}(\alpha_2) \end{array} \right|. \]

or, taking into account the definition of \(\{u_i\}\),

\[ D_{n-k,k}=(-1)^k \begin{vmatrix} u_0(a_2) & \cdots & u_{k-1}(a_2)\\ \cdot & \cdots & \cdot\\ \cdot & \cdots & \cdot\\ u_0^{(k-1)}(a_2) & \cdots & u_{k-1}^{(k-1)}(a_2) \end{vmatrix} =(-1)^k W_{k-1}(a_2), \]

where \(a_2 \in (a_1, r_{n-k,k}(a_1))\).

Hence, and from the condition of the theorem, we have: \(W_{k-1}(t)\), \(k=1,\ldots,n-m_1\), do not vanish for \(t\in(a,\omega)\). Therefore, by Mammen’s criterion [6, 7],

\[ y(t)=\sum_{i=0}^{n-m_1-1} c_i u_i \]

cannot have in \([a,\omega)\) \(n-m_1\) zeros, counting a multiple zero as many times as its multiplicity. The contradiction obtained proves the inequality

\[ r_{m_1,\ldots,m_p}(a)\ge \min_{k=1,\ldots,n-m_1}\,[r_{n-k,k}(a)]. \]

The inequality

\[ r_{m_1,\ldots,m_p}(a)\ge \min_{k=1,\ldots,n-m_p}\,[r_{k,n-k}(a)] \]

is proved analogously.

Let us introduce the definition of the interval of nonoscillation \([a,r(a))\) of the system of functions \(v_0,\ldots,v_{k-1}\). Let \(r(a)=\sup \beta\), where \(\beta\) is such that, for \(t\in[a,\beta)\), any nontrivial combination of the system \(v_0,\ldots,v_{k-1}\) has no more than \(k-1\) zeros, counting a multiple zero as many times as its multiplicity. It can be shown that, for a fundamental system of the equation of \(n\)-th order \(L[y]=0\),

\[ r(a)=\min_{k=1,\ldots,n-1}\,[r_{n-k,k}(a)]. \]

An estimate of \(r_{n-k,k}(a)\) in terms of the coefficients of the operation may be obtained on the basis of Theorem 5 of [8].

Consider the Green’s function \(G(t,s)\) of problem (1)—(2), putting \(A_i^j=0\). For brevity of notation set

\[ \min_{k=1,\ldots,n-m_1+1}[r_{n-k,k}(a_1)]=\omega, \]

\[ \min_{k=1,\ldots,m_1}[r_{k,n-k}(a_1)]=\gamma,\quad \max[\omega,\gamma]=\psi,\quad \min[r_{m_1,\ldots,m_p}(a_1),\psi]=\varphi \]

and prove the following generalization of [1—3, 5].

Theorem 1. If \(a_p<\varphi\), then for \(s\in(a_1,a_p)\), \(t\in(a_i,a_{i+1})\),

\[ \operatorname{sgn}G(t,s)=(-1)^{\,n-\sum_{k=1}^{i}m_k}. \]

Proof. The solution of the problem under consideration has the form

\[ y(t)=\int_{a_1}^{a_p} G(t,s)f(s)\,ds. \]

If \(y(t)\) preserves its sign on \([a,b]\) for any \(f>0\), then \(G(t,s)\) preserves its sign for \(s\in(a_1,a_p)\), \(t\in(a,b)\).

Let \(\{u_i\}\), \(i=0,\ldots,n-1\), be a fundamental system of the equation \(L[y]=0\), and let \(u_i^{(j)}(\alpha_1)=\delta_{i,n-j-1}\), \(j=0,\ldots,n-1\). Then

\[ y(t)=\int_{\alpha_1}^{t} K(t,s)f(s)\,ds+\sum_{i=0}^{\,n-m_1-1} c_i u_i, \]

where \(K(t,s)\) is the Cauchy function of the operator \(L[y]\).

By hypothesis, \(y(t)\) has \(n-m_1\) zeros at the points \(\alpha_2,\ldots,\alpha_p\), counting a multiple zero as many times as its multiplicity. Therefore, if \((\alpha_1,\alpha_p]\) belongs to an interval of nonoscillation of the system
\[ \int_{\alpha_1}^{t} K(t,s)f(s)\,ds,\quad u_0,\ldots,u_{n-m_1-1}, \]
then \(y(t)\) will not vanish for \(t\in(\alpha_i,\alpha_{i+1})\). A sufficient condition for nonoscillation of this system is the nonvanishing of the Wronskians [6, 7]:

\[ W_0(t)=\int_{\alpha_1}^{t} K(t,s)f(s)\,ds, \]

\[ W_i(t)=\left|\int_{\alpha_1}^{t} K(t,s)f(s)\,ds,\ u_0,\ldots,u_{i-1}\right|, \qquad i=1,\ldots,n-m_1. \]

By the definition of the Cauchy function, it does not vanish for \(t,s\in[a,r_{n-1,1}(\alpha))\).

Rewrite
\[ W_i(t)=\int_{\alpha_1}^{t}\left|K(t,s),u_0,\ldots,u_{i-1}\right|f(s)\,ds =\int_{\alpha_1}^{t}D_i(t,s)f(s)\,ds. \]

\(W_i(t)\) will not vanish for \(t\in[a,b]\) if \(D_i(t,s)\) preserves its sign for \(s\in(\alpha_1,t)\), \(t\in[a,b]\):

\[ D_i(t,s)=\left|\sum_{j=0}^{n-1} v_j(s)u_j(t),\ u_0,\ldots,u_{i-1}\right|= \]

\[ =\sum_{j=i}^{n-1} v_j(s)\left|u_j(t),\ u_0(t),\ldots,u_{i-1}(t)\right|, \]

where \(\{v_j\}\), \(j=0,\ldots,n-1\), is a fundamental system of the equation

\[ L^*[y]=0. \]

According to the definition of \(r_{m_1,\ldots,m_p}(t)\), \(d_i(s)=D_i(t_{\mathrm{cp}},s)\ne0\) for
\[ t_{\mathrm{cp}}\in\bigl(\alpha,r_{n-i-1,i+1}(\alpha)\bigr). \]
Counting shows that

\[ d_i^{(j)}(\alpha_1)=0,\qquad j=0,\ldots,i-1; \]

\[ d_i^{(j)}(\alpha_2)=0,\qquad j=0,\ldots,n-i+2. \]

Hence, according to the definition of \(r_{m_1,\ldots,m_p}(t)\), the following functions do not vanish:

\[ d_i(s),\qquad s\in\left(\alpha_1,r^*_{i,1,n-i-1}(\alpha_1)\right); \]

\[ D_i(t,s),\qquad \alpha<s<t<\min\left[r^*_{i,1,n-i-1}(\alpha_1),\ r_{n-i-1,i+1}(\alpha_1)\right]=\theta; \]

\[ W_i(t),\qquad t\in(\alpha_1,\theta). \]

Finally, \((\alpha_1,\alpha_p)\) will belong to an interval of nonoscillation of the system

\[ \int_{\alpha_1}^{t} K(t,s) f(s)\,ds,\quad u_0,\ldots,u_{n-m_1-1}, \]

if

\[ \alpha_p < \min_{i=1,\ldots,n-m_1} \left[ r_{i,\,1,\,n-i-1}^{*}(\alpha_1),\quad r_{n-i-1,\,i+1}(\alpha_1)\right] = \xi . \]

According to Lemmas 1 and 2,

\[ \xi > \max \left\{ \min_{k=1,\ldots,n-m_1+1} [r_{n-k,k}(\alpha_1)],\quad \min_{k=1,\ldots,m_1} [r_{k,n-k}(\alpha_1)] \right\} = \psi . \]

Thus \(G(t,s)\) preserves its sign for \(s\in(\alpha_1,\alpha_p)\), \(t\in(\alpha_i,\alpha_{i+1})\), if

\[ \alpha_p < \min [r_{m_1,\ldots,m_p}(\alpha_1),\psi] = \varphi . \]

This sign is determined by Chichkin’s formula [2]: for \(s\in(\alpha_1,\alpha_p)\), \(t\in(\alpha_i,\alpha_{i+1})\),

\[ \operatorname{sgn} G = (-1)^{\,n-\sum_{k=1}^{i} m_k}. \]

Otherwise, by virtue of its continuous dependence on the boundary conditions, the Green’s function would not preserve its sign in the indicated rectangles.

On the basis of the results presented above, we prove the following assertion on a differential inequality of Chaplygin-theorem type, which generalizes and refines a number of results of [1—3, 5].

Theorem 2. Let \(\alpha_p<\varphi\). Suppose, further, that there exists a pair of functions \(z(t)\) and \(v(t)\) such that:

a) \(z\) and \(v\) have absolutely continuous derivatives of order \(n-1\) on \([\alpha_1,\alpha_p]\);

b) \(z\) and \(v\) satisfy the boundary conditions (2), with inequalities allowed:

\[ z^{(m_1-1)}(\alpha_1) \geq A_1^{m_1-1} \geq v^{(m_1-1)}(\alpha_1) \quad \left( z^{(m_1-1)}(\alpha_1) \leq A_1^{m_1-1} \leq v^{(m_1-1)}(\alpha_1) \right), \]

if \(n-m_1\) is an even (odd) number, and

\[ z^{(m_p-1)}(\alpha_p) \geq A_p^{m_p-1} \geq v^{(m_p-1)}(\alpha_p) \quad \left( z^{(m_p-1)}(\alpha_p) \leq A_p^{m_p-1} \leq v^{(m_p-1)}(\alpha_p) \right), \]

if \(m_p\) is an even (odd) number;

c) the differential inequalities are satisfied

\[ L[z]-f \geq 0 \quad \text{and} \quad L[v]-f \leq 0 . \]

Then, for the solution \(u\) of problem (1), (2), for \(t\in[\alpha_1,\alpha_p]\), the estimate

\[ \operatorname{sgn}(z-u)=\operatorname{sgn}(u-v)=\operatorname{sgn}G \]

holds.

Proof. It is easy to verify that

\[ z-u=w+\int_{\alpha_1}^{\alpha_p} G(t,s)\varphi(s)\,ds, \]

where \(w\) is the solution of the problem

\[ L[y]=0,\quad y^{(j)}(\alpha_i)=z^{(j)}(\alpha_i)-A_i^j \]

and \(\varphi(t)=L[z]-f(t)\).

Let us emphasize that, by the condition:

1) \(w\) has zeros at the points \(\alpha_1, \alpha_2, \ldots, \alpha_{p-1}, \alpha_p\), whose multiplicities are not less than, respectively, \(m_1 - 1, m_2, \ldots, m_{p-1}, m_p - 1\), a total of \(n - 2\) zeros;

2) in some neighborhood to the right of \(\alpha_1\) and to the left of \(\alpha_p\), \(\operatorname{sgn} w = \operatorname{sgn} G\).

Suppose that the theorem is false: there exists \(t \in [\alpha_1, \alpha_p]\) for which
\(\operatorname{sgn}(z - u) = -\operatorname{sgn} G\) or \(\operatorname{sgn} w = -\operatorname{sgn} G\).

By the continuity of \(w\), this implies the presence of at least two zeros of the function \(w\) (counting a multiple zero as many times as its multiplicity) in addition to those specified above. Thus, \(w \not\equiv 0\) has at least \(n\) zeros for \(t \in [\alpha_1,\alpha_p]\), i.e. \(r_{m_1-1,\ldots,m_{p-1}}(\alpha_1) < \varphi\), which contradicts Lemma 2 of the present paper.

The relation \(\operatorname{sgn}(u - v) = \operatorname{sgn} G\) is proved analogously.

References

1. N. V. Azbelev, A. Ya. Khokhryakov, Z. B. Tsalyuk. Matem. sb., 59 (101), supplement, 1962, pp. 125–144.

2. E. S. Chichkin. Izv. vuzov, matem., No. 2 (27), 1962, pp. 170–179.

3. R. G. Aliev. Theorems on a differential inequality for an ordinary differential equation of fourth order with multipoint boundary conditions. Reports of the Second Siberian Conference on Mathematics and Mechanics. Tomsk, 1962.

4. A. L. Teptin. Theorems on difference and differential inequalities for nonlinear boundary-value problems with Vallée-Poussin conditions. Trudy Izhevskogo seminara, issue 1, 1963.

5. R. G. Aliev, I. N. Inozemtseva. On the distribution of zeros of solutions of a linear differential equation of fifth order. Trudy Izhevskogo seminara, issue 1, 1963.

6. G. Mammana. Math. Zeitschr., 33, 1931, pp. 186–231.

7. I. S. Berezin and N. P. Zhidkov. Computational Methods, 1. Fizmatgiz, 1962, pp. 81–84.

8. N. V. Azbelev and Z. B. Tsalyuk. Dokl. Akad. Nauk SSSR, 156, No. 2, 1964.

Received by the editors
November 25, 1964

Izhevsk Mechanical
Institute