On the Problem of a Differential Inequality

N. V. Azbelev, Z. B. Tsalyuk

Many facts in the theory of differential equations are consequences of more general statements about integral equations, and in considering a number of questions concerning a differential equation the latter is first reduced to an integral equation. On the basis of statements about an integral inequality, we give below a solution of N. N. Luzin’s problem [1] on a differential inequality for equations of order higher than the first [2, 3, 4].

1. Consider the system of Volterra equations

\[ x(t)=\int_0^t K[t,s,x(s)]\,ds+\psi(t), \tag{1} \]

where the vector-function \(K[t,s,x]=\{K_i(t,s,x_1,\ldots,x_n)\}\), \(i=1,\ldots,n\), is defined for \(0\le s\le t<T\), \(\|x\|<c\), and the vector-function \(\psi(t)\) is continuous on \([0,T)\) and \(\|\psi\|<c\). Below, continuous solutions of system (1) are considered. The inequality \(z>y\) (\(z\ge y\)) between \(n\)-dimensional vectors \(z=\{z_i\}\), \(y=\{y_i\}\) is understood as the inequalities \(z_i>y_i\) (\(z_i\ge y_i\)), \(i=1,\ldots,n\).

In [5] a number of statements about system (1) were proved under the assumption that \(K[t,s,x]\) satisfies the Carathéodory conditions. We formulate the results of [5] used below in the form of the following theorem.

Theorem 1. Let \(K[t,s,x]\) be nondecreasing in \(x\). Then

a) System (1) has an upper \(\overline u\) and a lower \(\underline u\) solution, i.e. such solutions that \(\overline u\ge u\ge \underline u\) for any solution \(u\); the upper and lower solutions can be extended to the boundary of the domain of definition of \(K\).

b) If \(\overline u\) (\(\underline u\)) is defined on \([0,T)\), then on this interval the integral inequality

\[ \varphi(t)=z(t)-\int_0^t K[t,s,z(s)]\,ds+\psi(t)\ge 0 \quad (\varphi(t)\le 0) \]

implies the inequality \(z\ge \underline u\) \((z\le \overline u)\). Moreover, if \(\varphi>0\) \((\varphi<0)\) for \(t\in[0,T)\), then \(z>\underline u\) \((z<\overline u)\) on \((0,T)\).

The condition of the theorem on the monotonicity of \(K\) is essential, but not necessary. Indeed, let, for example, \(u(t)\) be a solution of the equation

\[ x(t)=\int_0^t \sin(t-s)x(s)\,ds+\psi(t), \quad t\in[0,\infty). \tag{2} \]

If

\[ \varphi(t)=z(t)-\int_0^t \sin(t-s)z(s)\,ds+\psi(t), \]

then

\[ z-u=\varphi+\int_0^t (t-s)\varphi(s)\,ds\ge 0 \]

for \(\varphi \geq 0\). Thus, for (2) the assertion of Theorem 1 is valid, although here \(K=x\sin(t-s)\) does not satisfy the condition of monotonicity in \(x\). It is easy to see that the assertion of Theorem 1 is valid for any equation of the form

\[ x(t)=\int_{0}^{t}\{\sin(t-s)x(s)+Q(t,s,x(s))\}\,ds+\psi(t), \]

where \(Q\) is nondecreasing in \(X\). Moreover, the last remark is a special case of a general comparison theorem (cf. [3, 6]), which can be proved on the basis of the results of [4].

Theorem 1 bis. Let \(K[t,s,x]=R[t,s,x]+Q[t,s,x]\), where \(Q\) is nondecreasing in \(x\), and \(R\) is such that for the system

\[ x(t)=\int_{0}^{t}R[t,s,x(s)]\,ds+\psi(t) \tag{3} \]

the assertion of Theorem 1 is valid. Then the assertion of Theorem 1 is also valid for (1).

In addition to the condition of monotonicity of \(R\) in \(x\), we unfortunately do not possess sufficiently general effective criteria guaranteeing the validity of the assertion of Theorem 1 for the system (3). Therefore below we shall rely only on Theorem 1.

1. Here we shall consider the assertion, following from Theorem 1, on the existence of upper and lower solutions and on a differential inequality for the nonlinear equation

\[ N[y]=y^{(n)}-f(t,y,\ldots,y^{(r)})=0, \tag{4} \]

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

As is known, for \(n=1\) and under very general assumptions on \(f\), equation (4) has an upper \(\bar u\) and a lower \(\underline u\) solution, and the following assertion (Chaplygin’s theorem) on a differential inequality is valid:

if \(v(0)\geq 0\) and \(v'\geq f(t,v)\), then \(v\geq \underline u\); but if \(v'>f(t,v)\) for \(t\in[0,T)\), then \(v>\underline u\) in \((0,T)\).

The estimates following from this theorem, and its generalizations to systems of first-order equations [3, 6, 7], have formed the basis of many investigations in the qualitative theory of equations and approximate methods (see, for example, [3, 8]).

For \(n>1\), the differential inequality \(N[v]\geq 0\), \(v^{(k)}(0)=0\), \(k=0,\ldots,n-1\), does not guarantee the estimate \(v\geq y\) for a solution \(y\) of equation (4), even in the linear case. Indeed, if \(N[y]=y^{(n)}-\sum_{k=0}^{n-1}g_k(t)y^{(k)}-\psi(t)\), then, by Cauchy’s formula,

\[ v-y=\int_{0}^{t} C(t,s)N[v(s)]\,ds, \]

where \(C(t,s)\) is the Cauchy function [4, 10] of the equation \(N[y]=0\). Therefore, in \((0,T)\) the estimate \(v\geq y\) follows from the differential inequality \(N[v]\geq 0\) only under the condition that \(C(t,s)\geq 0\) in the triangle \(0\leq s\leq t<T\). Thus there arose the problems [1]: a) to single out a sufficiently broad class of equations for which there exists, independent of \(v\), an interval (the so-called “interval of applicability” of Chaplygin’s theorem) in which the differential inequality \(N[v]\geq 0\) guarantees the inequal-

...the property \(v \gg y\); b) to give a lower estimate for the length of such an interval. Below a solution of these problems is proposed.

In the absence of qualifications we assume:

1. The function \(f\) is defined in the domain \(G: 0 \le t < T,\ a_k < y^{(k)} < b_k,\ 0 > a_k=\operatorname{const},\ 0 < b_k=\operatorname{const},\ k=0,\ldots,r\), and satisfies in this domain the Carathéodory condition [5, 7].

2. The operation \(N[y]\) satisfies condition \(L_1\) [2, 4], i.e.
   \[ N[y]=L[y]-M(t,y,\ldots,y^{(r)}), \tag{5} \]
   where
   \[ L[y]=y^{(n)}-\sum_{k=0}^{r} g_k(t)y^{(k)} \]
   is an operation with coefficients \(g_k\) summable on \([0,T)\), while \(M(t,y,\ldots,y^{(r)})\) is defined in \(G\) and is nondecreasing with respect to \(y^{(k)}\) \((k=0,\ldots,r)\).

3. Let \(m\) be some number, \(r \le m \le n-1\). The Cauchy function \(C(t,s)\) of the operation \(L[y]\) and its derivatives \(C^{(i)}(t,s)\) with respect to the first argument are nonnegative for \(0 \le s \le t < T,\ i=0,\ldots,m\).

Put \(y^{(i)}=x_i\) \((i=0,\ldots,m)\). Since from (4), (5) and Cauchy’s formula we have
\[ y(t)=\int_0^t C(t,s)M(s,y(s),\ldots,y^{(r)}(s))\,ds, \]
equation (4) is equivalent to the system
\[ x_i(t)=\int_0^t C^{(i)}(t,s)M(s,x_0(s),\ldots,x_m(s))\,ds,\quad i=0,\ldots,m. \tag{6} \]

This system satisfies the conditions of Theorem 1. Therefore (4) has, on some interval \([0,\tau)\), an “\(m\)-upper” \(\overline y\) and an “\(m\)-lower” \(\underline y\) solution [9], i.e. solutions such that
\[ \overline y^{(i)} \gg y^{(i)} \gg \underline y^{(i)},\quad i=0,\ldots,m \]
for any solution \(y\).

Let some function \(v(t)\), with absolutely continuous \(v^{(n-1)}\) on \([0,T)\), satisfy the conditions
\[ a_k<v^{(k)}<b_k,\quad k=0,\ldots,r,\quad t\in(0,T), \tag{7} \]
\[ N[v]=\Theta(t)\gg 0\;(\Theta\le 0)\quad \text{almost everywhere on }[0,T). \tag{8} \]

Denote \(v^{(i)}=z_i,\ i=0,\ldots,m\), and assume that \(v^{(n-1)}(0)\ge 0,\ v^{(k)}(0)=0,\ k=0,\ldots,n-2\). From (5), (8), and Cauchy’s formula we have
\[ z_i(t)=\int_0^t C^{(i)}(t,s)M(s,z_0(s),\ldots,z_m(s))\,ds+ \]
\[ +\varphi_i(t),\quad i=0,\ldots,m, \tag{9} \]
where
\[ \varphi_i(t)=\int_0^t C^{(i)}(t,s)\Theta(s)\,ds+v^{(n-1)}(0)C^{(i)}(t,0). \]

Consequently, the vector-function \(z(t)=\{z_i\}=\{v,\ldots,v^{(m)}\}\), when substituted in system (6), gives a sign-definite residual \(\varphi=\{\varphi_i\}\ge 0\). Hence, and from assertion b) of Theorem 1, we obtain the inequalities
\[ v^{(i)}\gg \overline y^{(i)},\quad i=0,\ldots,m. \]

Thus we have proved

Theorem 2. Equation (4) has, on some interval \([0,\tau)\), an \(m\)-upper and an \(m\)-lower solution.

If an \(m\)-lower (\(m\)-upper) solution \(y\) is defined on \([0,T)\), the function \(v(t)\) satisfies (7), (8) and the initial conditions \(v^{(k)}(0)=0,\ k=0,\ldots,n-2,\ v^{(n-1)}(0)\geq 0\) \((v^{(n-1)}(0)\leq 0)\), then \(v^{(i)}\geq y^{(i)}\) \((v^{(i)}\leq y^{(i)})\), \(i=0,\ldots,m,\ t\in(0,T)\).

Remark 1. From (6) and the results of [5] it follows that an \(m\)-upper (\(m\)-lower) solution \(y\) can be continued to the boundary of the domain \(G\). Thus, if \(y\) is not defined on the whole interval \([0,T)\), then there exist \(\tau\in(0,T)\), \(k\leq r\), and a sequence \(t_j\to\tau\), such that \(y\) is defined on \([0,\tau)\) and either \(y^{(k)}(t_j)\to \tau\), or \(y^{(k)}(t_j)\to b_k\).

Remark 2. The answer to the question of the essential nature of condition \(L_1\) is analogous to the remark on the essential nature of the monotonicity condition \(K\) in Theorem 1. Namely, in condition \(L_1\) the linear operation \(L[y]\) may be replaced by a nonlinear \(R[y]\), requiring that Chaplygin’s theorem be valid for the equation \(R[y]=0\).

Remark 3. If the comparison function \(v\) satisfies the initial inequalities \(v^{(k)}(0)\geq 0\) \((v^{(k)}(0)\leq 0)\), \(k=0,\ldots,n-1\), then the vector function \(z=\{v,\ldots,v^{(m)}\}\), when substituted in (6), gives the residual \(\varphi=\{\varphi_i\}\), where

\[ \varphi_i(t)=\int_0^t C^{(i)}(t,s)\Theta(s)\,ds+w^{(i)}(t), \]

\[ L[w]=0,\qquad w^{(k)}(0)=v^{(k)}(0),\quad k=0,\ldots,n-1. \]

Therefore the assertion on the differential inequality in Theorem 2 can be supplemented as follows (cf. [4]):

if \(v\) satisfies the initial inequalities \(v^{(k)}(0)\geq 0,\ k=0,\ldots,n-1\), and \(v^{(i)}(t)\geq 0,\ t\in(0,T),\ i=0,\ldots,m\), then \(v^{(i)}\geq y^{(i)}\), \(i=0,\ldots,m\), on the interval \((0,T)\).

Putting \(\Theta\equiv 0\), we obtain from this an assertion on the monotone dependence of the solutions of equation (4) on the initial conditions [9].

1. For a number of applications, for example in the study of the question of uniqueness of a solution, it is important to have an upper estimate of the upper and lower solution. Such estimates do not follow from Theorem 2. A corresponding refinement of Theorem 2 may be obtained on the basis of the following supplement to the theorem on an integral inequality.

We shall say that the vector function \(K[t,s,x]=\{K_i(t,s,x_1,\ldots,x_n)\}\) satisfies condition \(M(j,t_0)\) if

\[ \int_0^{t_0} K_j(t_0,s,u_1(s),\ldots,u_n(s))\,ds < \]

\[ < \int_0^{t_0} K_j(t_0,s,z_1(s),\ldots,z_n(s))\,ds \]

for any continuous \(z\) and \(u\) such that \(\|z\|<c,\ \|u\|<c,\ z(s)>u(s)\) for \(s\in(0,t_0)\).

Lemma 1. Let \(z(t)\) and \(u(t)\) be continuous vector functions on \([0,T)\),

\[ \|z\|<c,\quad \|u\|<c,\quad \varphi(t)=z(t)-\int_0^t K[t,s,z(s)]\,ds\geq \psi(t)=u(t)-\int_0^t K[t,s, \]

\[ u(s)]\,ds,\quad t\in(0,T), \]

and \(z>u\) in \((0,\varepsilon)\) for some \(\varepsilon>0\). Suppose, further, that \(K[t,s,x]\) is nondecreasing in \(x\) and satisfies condition \(M(j,t_0)\) for those \(j\) and \(t_0\) for which \(\varphi_j(t_0)=\psi_j(t_0)\). Then \(z>u\) on the whole interval \((0,T)\).

Proof. Suppose the contrary and denote by \(h\) the least upper bound of those \(\tau\) for which \(z>u\) for \(t\in(0,\tau)\). Obviously, \(\varepsilon\leq h<T\), \(z>u\) for \(t\in(0,h)\), and \(z(h)\geq u(h)\), and there exists such a \(j\) that \(z_j(h)=u_j(h)\). But from the condition of the lemma there follows the inequality \(z_j(h)=\)

\[ = \int_0^h K_j\bigl(h,s,z_1(s),\ldots,z_n(s)\bigr)\,ds+\varphi_j(h)> \int_0^h K_j\bigl(h,s,u_1(s),\ldots,u_n(s)\bigr)\,ds+ \]
\[ +\psi_j(h)=u_j(h), \]
contradicting the definition of \(h\).

Theorem 3. Let an \(m\)-upper (\(m\)-lower) solution \(y\) of equation (4) be defined on \([0,T)\), and let the comparison function \(v\) satisfy conditions (7), (8), with
\[ v^{(k)}(0)=0,\quad k=0,\ldots,n-2,\qquad v^{(n-1)}(0)>0\quad \bigl(v^{(n-1)}(0)<0\bigr). \]
If the residual \(\Theta=N[v]\) is not equivalent to zero in any right neighborhood of the point \(t=0\), then
\[ v^{(i)}>y^{(i)}\quad \bigl(v^{(i)}<y^{(i)}\bigr) \]
for \(t\in(0,T)\), \(i=0,\ldots,m\).

Proof. Put \(z=\{v,\ldots,v^{(m)}\}\) and \(x=\{y,\ldots,y^{(m)}\}\), and consider equalities (6) and (9). As is known (see, for example, [4, 9]), the set of zeros of the function \(C^{(i)}(t,s)\), for fixed \(t\), is nowhere dense. Therefore \(\varphi(t)>0\) in \((0,T)\). From the initial inequality
\[ v^{(n-1)}(0)>y^{(n-1)}(0) \]
it follows that \(v^{(i)}>y^{(i)}\), \(i=0,\ldots,m\), i.e. \(z>x\) on some interval \((0,\varepsilon)\). Applying now Lemma 1 to (6) and (9), we obtain the inequality \(z>x\) on the whole interval \((0,T)\).

Remark 1. If in (5) the function \(M(t,y,\ldots,y^{(r)})\) increases with respect to \(y^{(k)}\), \(k=0,\ldots,r\), then the assertion of Theorem 3 is valid without the assumption that the residual \(\Theta\) is not equivalent to zero, since in this case the vector-function
\[ K[t,s,x]=\{C(t,s)M(s,x_0,\ldots,x_m),\ldots,C^{(m)}(t,s)M(s,x_0,\ldots,x_m)\} \]
satisfies condition \(M(j,t_0)\) for all \(j=0,\ldots,m\), \(t\in(0,T)\).

Remark 2. The essential nature of the condition \(v^{(n-1)}(0)\ne0\) is shown by the example
\[ y'=t^{-1/3}y^{1/3},\qquad y(0)=0. \]
Here \(y=t\) satisfies the equation (and is an upper solution). To the comparison function
\[ v=-\frac18 t \]
there corresponds the residual
\[ \Theta=\frac38=\mathrm{const}>0. \]
Nevertheless \(v<y\) on \((0,\infty)\).

The meaning of the condition \(v^{(n-1)}(0)\ne0\) in the preceding theorem is that this condition guarantees, in some neighborhood, the inequalities
\[ v^{(i)}>y^{(i)},\qquad i=0,\ldots,m. \]
Such inequalities can be obtained in another way, for example by requiring continuity on \([0,\varepsilon)\) of the difference \(v^{(n)}-y^{(n)}\) and the inequality \(v^{(n)}(0)>y^{(n)}(0)\). In the last example the difference \(v^{(n)}-y^{(n)}\) is discontinuous and \(\Theta\) is a positive constant. Nevertheless,
\[ v^{(n)}(0)-y^{(n)}(0)=\lim_{t\to0}\bigl[f(t,z,\ldots,z^{(r)})-f(t,y,\ldots,y^{(r)})\bigr]+\Theta(0)<0. \]

The last inequality is impossible if \(\Theta(0)>0\) and \(f\) is continuous. The consideration given leads to the following assertion, where the restriction \(v^{(n-1)}(0)\ne0\) is removed, but \(f\) is assumed to be continuous.

Theorem 4. Let an \(m\)-upper (\(m\)-lower) solution \(y\) of problem (4) be defined on \([0,T)\). Let, furthermore, the function \(v(t)\), continuously differentiable \(n\) times on \([0,T)\), satisfy condition (7), the initial conditions
\[ v^{(k)}(0)=0,\qquad k=0,\ldots,n-1, \]
and the differential inequality
\[ N[v]=\Theta>0\quad (\Theta<0),\qquad t\in[0,T), \]
with \(\Theta(0)\ne0\). Then
\[ v^{(i)}>y^{(i)}\quad \bigl(v^{(i)}<y^{(i)}\bigr) \]
for \(t\in(0,T)\), \(i=0,\ldots,m\).

Proof. Since
\[ v^{(n)}(0)-y^{(n)}(0)=\Theta(0)>0, \]
it follows, by virtue of the initial conditions, that \(v^{(i)}>y^{(i)}\), \(i=0,\ldots,m\), on some interval \((0,\varepsilon)\). Put \(z=\{v^{(i)}\}\), \(x=\{y^{(i)}\}\). Then \(z>x\) on \((0,\varepsilon)\). Applying Lemma 1 to (6) and (9), we obtain the assertion of the theorem.

Remark. The condition \(\Theta(0)\ne0\) is essential even in the class of equations with a monotonically increasing right-hand side (i.e. in the class of equations satisfying condition \(L_1\), where the coefficients of the operation

\(L[y]\) are equal to zero). Indeed, it is not hard to see that \(y=t^2\) \((y=-t^2)\) is the upper (lower) solution of the problem \(y' = 2\operatorname{sgn} y\sqrt{|y|},\ y(0)=0\). Put \(v=-t^4\) \((v=t^4)\). Then \(\Theta>0\) \((\Theta<0)\) for \(t\in\left(0,\dfrac12\right)\). Nevertheless \(v<y\) \((v>y)\) for \(t\in\left(0,\dfrac12\right)\).

In conclusion, let us note that Theorem 1 extends to the case when the vector-function \(K[t,s,x]\) is discontinuous with respect to \(x\). Therefore the above Theorems 2 and 3 also hold in the case when the function \(f\) has discontinuities with respect to \(y^{(k)}\), as was reported by Li Mun Su and A. B. Samarov at the Izhevsk seminar.

1. The nonlinear theorems on the differential inequality given above are comparison theorems: for the equation \(N[y]=0\) the theorem is valid on the given interval \([0,T)\), if on this interval the assertion on the differential inequality is valid for the linear equation \(L[y]=0\), corresponding to the condition \(L_1\).

From the definition of condition \(L_1\) it follows that if \(N[y]\) satisfies this condition with the operation \(L[y]=y^{(n)}-\displaystyle\sum_{k=0}^{r} g_k(t)y^{(k)}\) and \(g_k\ge p_k=\mathrm{const}\), then condition \(L_1\) is also fulfilled with the operation \(L_1[y]=y^{(n)}-\displaystyle\sum_{k=0}^{r} p_k y^{(k)}\).

The equation \(L_1[y]=0\) with constant coefficients is integrable, and therefore the question of the size of the interval of applicability of Chaplygin’s theorem for \(L_1[y]=0\) has an exact solution.

The interval of applicability \([0,T)\) of Chaplygin’s theorem on the differential inequality for the linear equation \(L[y]=0\) is determined by the size of the triangle \(0\le s\le t<T\) in which the corresponding derivative of the Cauchy function \(C(t,s)\) is nonnegative. This follows directly from the Cauchy formula given above. From the definition of \(C(t,s)\) it follows that \(T>0\). A necessary and sufficient condition for the nonnegativity of the Cauchy function and its derivatives is given in [4, 10]. In addition, [10] indicates a method for constructing a sequence \(\{K_i(t,s)\}\to C(t,s)\), \(K_{i-1}\le K_i\), distinguished by an extraordinarily rapid rate of convergence (equal to the rate of convergence of the sequence \(1/(2^i-1)!\)). The monotonicity property of the sequence mentioned makes it possible to obtain an estimate for the size of the triangle \(0\le s\le t<T\) in which \(C(t,s)\ge 0\). Analogous estimates can be obtained on the basis of the results of [5]. Using one of these estimates, we propose below effective criteria for preservation of sign of the Cauchy function and its derivatives.

Let \(C(t,s)\) be the Cauchy function of the operation \(L[y]=y^{(n)}-\displaystyle\sum_{k=0}^{r} g_k y^{(k)}\). By \(V(t,s)\) we denote the Cauchy function of the auxiliary operation \(L_1[y]=y^{(n)}-\displaystyle\sum_{k=0}^{r} p_k y^{(k)}\). Write the equation \(L[y]=0\) in the form

\[ L_1[y]=\sum_{k=0}^{r}(g_k-p_k)y^{(k)}. \tag{10} \]

Set \(P_k(t)=\sup\{0,(p_k(t)-g_k(t))\}\) and consider the equation

\[ L_1[y]=\sum_{k=0}^{r} P_k y^{(k)}. \tag{11} \]

Denote by \(W(t,s)\) the Cauchy function of equation (11). Then, by Theorem 2 (see also the corollary to Theorem 4 of [4]), we have \(C^{(i)}(t,s)\geq 0\) \((i=0,\ldots,m)\) in the triangle \(0\leq s\leq t<T\), if \(W^{(m)}(t,s)\geq 0\) in this triangle.

From (11) and Cauchy’s formula we obtain a system of integral equations with respect to \(W^{(i)}(t,s)\), \(i=0,\ldots,m\):

\[ W^{(i)}(t,s)=\int_s^t V^{(i)}(t,\tau)\sum_{k=0}^{r} P_k(\tau)W^{(k)}(\tau,s)\,d\tau+V^{(i)}(t,s). \]

Integrating the last system, we have

\[ W^{(i)}(t,s)=\int_s^t \sum_{k=0}^{r} H_{ik}(t,\tau)W^{(k)}(\tau,s)\,d\tau+\Psi_i(t,s),\quad i=0,\ldots,m, \tag{12} \]

where

\[ \Psi_i(t,s)=V^{(i)}(t,s)+\int_s^t V^{(i)}(t,s)\sum_{k=0}^{r} P_k(\tau)V^{(k)}(\tau,s)\,d\tau \]

and \(H_{ik}(t,\tau)\) are certain nonnegative functions. For each fixed \(s\), the vector-function \(X(t,s)=\{C(t,s),\ldots,C^{(m)}(t,s)\}\) is a solution of the system (12) satisfying the conditions of Theorem 1. Therefore \(X\geq 0\) in the triangle \(0\leq s\leq t<T\), if in this triangle \(\Psi_i(t,s)\geq 0\), \(i=0,\ldots,m\). Thus Lemma 2 is proved.

Lemma 2. Let \(C(t,s)\) and \(V(t,s)\) be the Cauchy functions of the equations \(L[y]=0\) and \(L_1[y]=0\), respectively. Then \(C^{(i)}(t,s)\geq 0\), \(i=0,\ldots,m\), in the triangle \(0\leq s\leq t<T\), if in this triangle \(V^{(i)}(t,s)\geq 0\),

\[ V^{(i)}(t,s)+\int_s^t V^{(i)}(t,\tau)\sum_{k=0}^{r} P_k(\tau)V^{(k)}(\tau,s)\,d\tau\geq 0,\quad i=0,\ldots,m. \]

We shall give corollaries following from Lemma 2.

Put \(p_k=0\). Then \(V(t,s)=\dfrac{(t-s)^{n-1}}{(n-1)!}\), \(V^{(i)}(t,s)\geq 0\) for arbitrary

\(t\geq s\geq 0\). We take into account the inequality
\[ (t-\tau)^\alpha(\tau-s)^\beta \leq \frac{\alpha^\alpha\beta^\beta}{(\alpha+\beta)^{\alpha+\beta}}\, (t-s)^{\alpha+\beta},\quad s\leq \tau\leq t, \]
where \(\alpha\) and \(\beta\) are integers (we assume \(0^0=1\)), and denote \(g_k^-(t)=\sup\{0,-g_k(t)\}\). Then

\[ V^{(i)}(t,s)-\int_s^t V^{(i)}(t,\tau)\sum_{k=0}^{r} g_k^-(\tau)V^{(k)}(\tau,s)\,d\tau\geq \]

\[ \geq \frac{(t-s)^{\,n-i-1}}{(n-i-1)!} \left\{ 1-\sum_{k=0}^{r} \frac{(n-i-1)^{\,n-i-1}(n-k-1)^{\,n-k-1}(t-s)^{\,n-k-1}} {(n-k-1)!(2n-2-i-k)^{\,2n-2-i-k}} \times \right. \]

\[ \left. \times\int_s^t g_k^-(\tau)\,d\tau \right\}\geq 0 \]

for \(0 \leq s \leq t < T,\ i=0,\ldots,m,\ r \leq m \leq n-1\), if

\[ 1 \geq \sum_{k=0}^{r} \frac{(n-m-1)^{\,n-m-1}(n-k-1)^{\,n-k-1}T^{\,n-k-1}} {(n-k-1)!(2n-2-m-k)^{\,2n-2-m-k}} \int_{0}^{T} g_k(\tau)\,d\tau . \tag{13} \]

Thus it has been proved

Corollary 1. If the coefficients \(g_k\) of the equation
\[ y^{(n)}=\sum_{k=0}^{r} g_k(t)y^{(k)} \]
satisfy inequality (13), then \(C^{(i)}(t,s)\geq 0\) for \(0\leq s\leq t<T,\ i=0,\ldots,m,\ r\leq m\leq n-1\).

In an analogous way we obtain

Corollary 2. If
\[ 1 \geq \sum_{k=0}^{r} \frac{(n-m-1)!T^{\,n-k}}{(2n-m-k-1)!} \sup_{t\in[0,T)} |g_k(t)|, \]
then \(C^{(i)}(t,s)\geq 0\) for \(0\leq s\leq t<T,\ i=0,\ldots,m,\ r\leq m\leq n-1\).

We note that Theorem 3 of paper [11] follows from Corollary 1 as a special case when \(m=r=0\).

Criteria for preservation of the sign of \(C^{(i)}(t,s)\) close to the above corollaries of Lemma 2 were reported at the Izhevsk seminar by Yu. V. Komlenko, V. V. Ostroumov, and S. A. Pak. These criteria were obtained from estimates proposed in papers [12, 13].

References

1. Luzin N. N. Uspekhi matem. nauk, 6, issue 6, 1951, pp. 3—27.
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Received by the editors
January 4, 1965

Izhevsk Mechanical
Institute