UDC 517.917

ON A PRIORI ESTIMATES FOR THE DERIVATIVES OF BOUNDED FUNCTIONS SATISFYING SECOND-ORDER DIFFERENTIAL INEQUALITIES

I. T. Kiguradze

In the study of boundary-value problems for the differential equation

\[ u''=f(t,u,u') \tag{1} \]

the following is essentially used in many works.

Nagumo’s Theorem*). Let \(a, b, r\), and \(\omega(t)\) be given numbers and a function, where \(-\infty<a<b<\infty\), \(r>0\), and the function \(\omega(t)\) is positive and continuous on the interval \([0,\infty)\), and

\[ \int_0^\infty \frac{t\,dt}{\omega(t)}=\infty . \]

Then there exists a number \(l\) such that, for any function \(u(t)\), absolutely continuous together with its derivative on the interval \([a,b]\) and satisfying on this interval the inequalities

\[ |u(t)|\le r \]

and

\[ |u''(t)|\le \omega(|u'(t)|), \tag{2} \]

we shall have \(|u'(t)|\le l\) for \(t\in[a,b]\).

The theorem just stated is applicable to the differential equation (1) only in the case when the function \(f(t,x,y)\) satisfies, in every finite part of the plane \(t,x\), an inequality of the form

\[ |f(t,x,y)|\le \omega(|y|). \]

This restriction is rather severe, since it excludes from consideration equations whose right-hand sides are unbounded functions of \(t\).

Thus there naturally arises the problem of an a priori estimate for the derivatives of bounded functions satisfying a more general differential inequality than (2). The present note is devoted to this question.

Below we shall everywhere assume that \(a,b,\alpha,\beta,\sigma_1(t),\sigma_2(t)\), and \(\omega(t,x,y)\) are given numbers and functions satisfying the following conditions: \(-\infty<a\le\alpha<\beta\le b<\infty\), \(\sigma_1(t)\) and \(\sigma_2(t)\) are continuous on the interval \([a,b]\), and \(\sigma_1(t)\le\sigma_2(t)\) for \(t\in[a,b]\), while \(\omega(t,x,y)\) is a nonnegative function satisfying the Carathéodory conditions in the domain: \(a\le t\le b\), \(0\le x\le\rho\), \(0\le y\le\rho\), for every \(\rho\in(0,\infty)\).

*) It should be noted that an analogous result for the case \(\omega(t)=1+t^2\) was first obtained by S. N. Bernstein (see [1], p. 192).

We shall also agree on the following notation used below:

1. By \(A'_\sigma(a,b)\) we denote the set of all functions \(u(t)\) that are absolutely continuous together with their first derivatives on the interval \([a,b]\) and satisfy the condition

\[ \sigma_1(t) \leq u(t) \leq \sigma_2(t) \quad \text{for } t \in [a,b]. \tag{3} \]

1. By \(d(t,\tau)\) we mean the function defined as follows:

\[ d(\tau,t)=d(t,\tau)=\max_{t_1,t_2\in(t,\tau)}|\sigma_2(t_2)-\sigma_1(t_1)|. \tag{4} \]

1. The norm of a function \(\psi(t)\in L^p(t_1,t_2)\) is denoted by \(N(\psi;p,t_1,t_2)\) or \(N(\psi;p,t_2,t_1)\). Thus,

\[ N(\psi;p,t_1,t_2)=N(\psi;p,t_2,t_1)= \begin{cases} \left|\displaystyle\int_{t_1}^{t_2}|\psi(t)|^p\,dt\right|^{1/p}, & \text{for } 1\leq p<\infty,\\[1.2em] \operatorname*{vrai\,max}\limits_{t\in(t_1,t_2)}|\psi(t)|, & \text{for } p=\infty. \end{cases} \]

1. By \(y(\varphi;y_0,t_0,t)\) and \(z(\varphi;z_0,t_0,t)\) we denote, respectively, the upper solutions of the following problems:

\[ \frac{dy}{dt}=\omega(t,\varphi(t),y), \tag{5} \]

\[ y(t_0)=y_0 \tag{5'} \]

and

\[ \frac{dz}{dt}=-\omega(t,\varphi(t),z),\quad z(t_0)=z_0. \]

Here the solutions are assumed to be defined on the entire interval \([a,b]\), since at those points of the interval \([a,b]\) where they are not defined in the usual sense, their values are taken to be \(+\infty\).

1. By \(\rho_a(t)\), \(\rho_b(t)\), and \(\rho_{\alpha\beta}(t)\) we denote functions satisfying the inequalities

\[ \rho_a(t)\geq y(\varphi;0,a,t) \quad \text{for } t\in[a,b], \tag{6} \]

\[ \rho_b(t)\geq z(\varphi;0,b,t) \quad \text{for } t\in[a,b] \]

and

\[ \rho_{\alpha\beta}(t)\geq \begin{cases} z(\varphi;y_{\alpha\beta},\beta,t), & \text{for } t\in[a,\beta],\\ y(\varphi;y_{\alpha\beta},\alpha,t), & \text{for } t\in[\alpha,b], \end{cases} \tag{7} \]

where

\[ y_{\alpha\beta}=\min_{t_1,t_2\in(\alpha,\beta)} \frac{|\sigma_2(t_2)-\sigma_1(t_1)|}{|t_2-t_1|}, \tag{8} \]

for any continuous and nonnegative function \(\varphi(t)\) on the interval \([a,b]\) satisfying the condition

\[ \int_{t_1}^{t_2}\varphi(\tau)\,d\tau\leq d(t_1,t_2) \quad \text{for any } t_1,t_2\in[a,b]. \tag{9} \]

Theorem 1. If \(u(t)\in A'_\sigma(a,b)\),

\[ u'(a)=0 \tag{10} \]

and

\[ u''(t)\operatorname{sign}u'(t)\leq \omega(t,|u'(t)|,|u''(t)|) \quad \text{for } t\in[a,b], \tag{11} \]

then

\[ |u'(t)| \leqslant \rho_a(t) \quad \text{for } t \in [a,b]. \tag{12} \]

Proof. Suppose the contrary, i.e., for some \(t_0 \in (a,b)\) we have

\[ |u'(t_0)| > \rho_a(t_0). \tag{13} \]

Since \(\rho_a(t)\) is a nonnegative function, by (10) and (13) there exist numbers \(t_1\) and \(t_2\), \(a \leqslant t_1 < t_0 \leqslant t_2 \leqslant b\), such that

\[ |u'(t_1)| = 0 \tag{14} \]

and

\[ u'(t) \ne 0 \quad \text{for } t \in (t_1,t_2), \tag{15} \]

with either \(t_2=b\), or \(u'(t_2)=0\). Put

\[ \varphi(t)= \begin{cases} 0, & t \notin [t_1,t_2],\\ |u'(t)|, & t \in [t_1,t_2], \end{cases} \tag{16} \]

from (11) we obtain

\[ \frac{d|u'(t)|}{dt} \leqslant \omega\bigl(t,\varphi(t), |u'(t)|\bigr) \quad \text{for } t \in [t_1,t_2]. \tag{17} \]

Hence, by (14), it follows\(^*\)

\[ |u'(t)| \leqslant y(\varphi;0,t_1,t) \quad \text{for } t \in [t_1,t_2]. \]

But in view of the nonnegativity of \(\omega(t,\varphi(t),y)\) we have

\[ y(\varphi;0,t_1,t) \leqslant y(\varphi;0,a,t) \quad \text{for } t \in [t_1,b]. \]

Consequently,

\[ |u'(t)| \leqslant y(\varphi;0,a,t) \quad \text{for } t \in [t_1,t_2]. \]

By virtue of (3), (4), and (15), it is clear from (16) that the function \(\varphi(t)\) satisfies condition (9). Therefore, according to condition (6), from the last inequality we conclude that

\[ |u'(t)| \leqslant \rho_a(t) \quad \text{for } t \in [t_1,t_2]. \]

Thus, \(|u'(t_0)| \leqslant \rho_a(t_0)\), which contradicts inequality (13). The contradiction obtained proves the theorem.

For \(\omega(t,x,y) \equiv \omega(t,y)\), Theorem 1 immediately gives the following

Corollary 1. If \(u(t) \in A_\sigma'(a,b)\), \(u'(a)=0\), and

\[ u''(t)\operatorname{sign}u'(t) \leqslant \omega\bigl(t,|u'(t)|\bigr) \quad \text{for } t \in [a,b], \tag{18} \]

then

\[ |u'(t)| \leqslant y_0(t) \quad \text{for } t \in [a,b], \tag{19} \]

where \(y_0(t)\) is the upper solution of the problem

\[ \frac{dy}{dt}=\omega(t,y), \qquad y(a)=0. \tag{20} \]

In particular, if \(\omega(t,y)=\psi_1(t)y+\psi_2(t)\), where \(\psi_k(t)\geqslant 0\) for \(t \in [a,b]\) and \(\psi_k(t)\in L(a,b)\) \((k=1,2)\), then

\[ |u'(t)| \leqslant \int_a^t \exp\left[\int_\tau^t \psi_1(s)\,ds\right]\psi_2(\tau)\,d\tau \quad \text{for } t \in [a,b]. \]

\(^*\) See [4], comparison lemma.

Corollary 2. If \(u(t)\in A_{\delta}'(a,b)\), \(u'(a)=0\), and

\[ u''(t)\operatorname{sign} u'(t)\leq \psi(t)\omega(|u'(t)|)\quad \text{for } t\in [a,b], \tag{21} \]

where \(\psi(t)\geq 0\) for \(t\in [a,b]\) and \(\psi(t)\in L(a,b)\), while the function \(\omega(t)\) is nonnegative and continuous on the interval \([0,\infty)\) and for some \(l\in(0,\infty)\) satisfies the condition

\[ \int_0^l \frac{dt}{\omega(t)}=\infty, \tag{22} \]

then

\[ |u'(t)|\leq \min \left(l,\omega_l\int_a^t \psi(\tau)d\tau\right)\quad \text{for } t\in [a,b], \tag{23} \]

where \(\omega_l=\max\limits_{t\in[0,l]}\omega(t)\).

Proof. According to (22), there exists a positive number \(\varepsilon\) such that

\[ \int_0^l \frac{dt}{\varepsilon+\omega(t)}>\int_a^b \psi(t)dt. \tag{24} \]

Putting \(\omega(t,y)=\psi(t)[\varepsilon+\omega(y)]\), for the solution \(y_0(t)\) of problem (20) we shall have

\[ \int_0^{y_0(t)}\frac{d\tau}{\varepsilon+\omega(\tau)} = \int_a^t \psi(\tau)d\tau. \]

Hence, according to (19) and (24), it follows that

\[ |u'(t)|\leq l\quad \text{for } t\in [a,b], \]

and, taking this into account, estimate (23) is obtained directly from condition (21).

Corollary 3. If \(u(t)\in A_{\delta}'(a,b)\), \(u'(a)=0\), and

\[ u''(t)\operatorname{sign} u'(t) \leq \psi_1(t)|u'(t)|^{1+1/q}+\psi_2(t) \quad \text{for } t\in [a,b], \tag{25} \]

where \(q>1\), \(\psi_k(t)\geq 0\) for \(t\in [a,b]\) \((k=1,2)\),

\[ \psi_1(t)\in L^p(a,b),\qquad \frac{1}{p}+\frac{1}{q}=1,\qquad \psi_2(t)\in L(a,b), \]

then for any \(t_0\in(a,b)\) we shall have

\[ |u'(t)|\leq \exp\left[l+d^{1/q}(t_0,t)N(\psi_1;p,t_0,t)\right] \int_a^t \psi_2(\tau)d\tau \quad \text{for } t\in [a,b], \tag{26} \]

where

\[ l=d^{1/q}(a,t_0)N(\psi_1;p,a,t_0). \tag{27} \]

Proof. From (25) it is clear that, for the function

\[ \omega(t,x,y)=\psi_1(t)x^{1/q}y+\psi_2(t), \]

inequality (11) is satisfied. On the other hand, equation (5) has the form

\[ \frac{dy}{dt}=\psi_1(t)\varphi^{1/q}(t)y+\psi_2(t). \]

Therefore, for the solution \(y(\varphi;0,a,t)\) of problem (5), \((5')\) with \(y_0=0\), we have

\[ y(\varphi;0,a,t) = \int_a^t \exp\left[\int_\tau^t \psi_1(s)\varphi^{1/q}(s)\,ds\right] \psi_2(\tau)d\tau \leq \]

\[ \leqslant \exp \left[\int_a^t \psi_1(\tau)\varphi^{1/q}(\tau)\,d\tau\right]\int_a^t \psi_2(\tau)\,d\tau \quad \text{for } t \in [a,b]. \]

But, according to Hölder’s inequality and condition (9), we have

\[ \int_a^t \psi_1(\tau)\varphi^{1/q}(\tau)\,d\tau = \int_a^{t_0} \psi_1(\tau)\varphi^{1/q}(\tau)\,d\tau + \int_{t_0}^t \psi_1(\tau)\varphi^{1/q}(\tau)\,d\tau \leqslant \]

\[ \leqslant l+d^{1/q}(t_0,t)N(\psi_1;p,t_0,t) \quad \text{for } t \in [a,b], \]

where \(l\) is the number defined by equality (27). Consequently,

\[ y(\varphi;0,a,t) \leqslant \exp\{l+d^{1/q}(t_0,t)N(\psi_1;p,t_0,t)\}\int_a^t \psi_2(\tau)\,d\tau = \]

\[ =\rho_a(t) \quad \text{for } t \in [a,b]. \]

Hence, according to inequality (12), the validity of estimate (26) follows immediately.

Corollary 4. If \(u(t)\in A'_{\delta}(a,b)\), \(u'(a)=0\), and condition (21) is satisfied, where \(\psi(t)\geqslant 0\) for \(t\in [a,b]\) and \(\psi(t)\in L^p(a,b)\), while the function \(\omega(t)\) is continuous and positive on the interval \([0,\infty)\), and

\[ \int_0^\infty \frac{t^{1/q}}{\omega(t)}\,dt=\infty, \quad \text{where } \frac{1}{p}+\frac{1}{q}=1, \]

then, for any \(t_0\in(a,b)\), we have

\[ |u'(t)|\leqslant M[\omega;q,0,l+d^{1/q}(t_0,t)N(\varphi;p,t_0,t)] \quad \text{for } t\in[a,b], \tag{28} \]

where \(M(\omega;q,x,z)\) is the function defined from the identity

\[ \int_x^{M(\omega;q,x,z)} \frac{t^{1/q}\,dt}{\omega(t)} = z \tag{29} \]

and \(l\) is the number defined by equality (27).

Proof. Put

\[ \omega(t,x,y)=\psi(t)(\varepsilon+x)^{1/q}\frac{\omega(y)}{(\varepsilon+y)^{1/q}}, \]

where \(\varepsilon\) is an arbitrarily small positive number. Then from (21) we conclude that condition (11) is satisfied.

Since equation (5) in the present case has the form

\[ \frac{dy}{dt} = \psi(t)\bigl(\varepsilon+\varphi(t)\bigr)^{1/q} \frac{\omega(y)}{(\varepsilon+y)^{1/q}}, \]

it follows that

\[ \int_0^{y(\varphi;0,a,t)} \frac{(\varepsilon+\tau)^{1/q}}{\omega(\tau)}\,d\tau = \int_a^t \psi(\tau)\bigl(\varepsilon+\varphi(\tau)\bigr)^{1/q}\,d\tau . \]

But, taking into account that

\[ \int_a^t \psi(\tau)(\varepsilon+\varphi(\tau))^{1/q}\,d\tau \leqslant \]

\[ \le [\varepsilon t_0 + d(a,t_0)]^{1/q} N(\psi;p,a,t_0) + [\varepsilon t + \]
\[ +d(t_0,t)]^{1/q} N(\psi;p,t_0,t) \quad \text{for } t \in [a,b], \]

according to (29), from the last equality we find

\[ y(\varphi;0,a,t) \le M\left[\omega;q,0,\left(1+\frac{\varepsilon t_0}{d(a,t_0)}\right)^{1/q} l + \right. \]
\[ \left. +(\varepsilon t+d(t_0,t))^{1/q}N(\psi;p,t_0,t)\right] = \rho_a(t) \quad \text{for } t \in [a,b]. \]

Hence, by virtue of the arbitrariness of \(\varepsilon\) and inequality (12), the validity of estimate (28) follows.

In a completely analogous way to Theorem 1 one proves also

Theorem 1′. If \(u(t) \in A'_{\sigma}(a,b)\), \(u'(b)=0\), and

\[ u''(t)\operatorname{sign} u'(t) \ge -\omega(t,|u'(t)|,|u'(t)|) \quad \text{for } t \in [a,b], \]

then \(|u'(t)| \le \rho_b(t)\) for \(t \in [a,b]\).

From Theorem 1′ there follow corollaries analogous to the corollaries of Theorem 1, which we do not give here.

Theorem 2. If \(u(t) \in A'_{\sigma}(a,b)\),

\[ u(a)=u(b)=0 \tag{30} \]

and

\[ u''(t)\operatorname{sign} u(t) \ge -\omega(t,|u'(t)|,|u'(t)|) \quad \text{for } t \in [a,b], \tag{31} \]

then

\[ |u'(t)| \le \max[\rho_a(t),\rho_b(t)] \quad \text{for } t \in [a,b]. \tag{32} \]

Proof. Suppose the contrary, that for some \(t_0 \in [a,b]\) we have

\[ |u'(t_0)| > \max[\rho_a(t_0),\rho_b(t_0)],\quad u(t_0)\ne 0. \tag{33} \]

Let \((t_1,t_2)\) be the maximal interval containing \(t_0\) on which the condition

\[ u'(t)u(t)\ne 0 \]

is satisfied. According to (30) it is clear that

\[ u'(t_1)u(t_1)=u'(t_2)u(t_2)=0. \tag{34} \]

If

\[ u'(t)u(t)<0 \quad \text{for } t \in (t_1,t_2), \tag{35} \]

then \(|u(t_1)|>|u(t_2)|\), and therefore from (34) we obtain

\[ u'(t_1)=0. \tag{36} \]

On the other hand, by virtue of (31) and (35), it is clear that condition (17) is satisfied for the function \(\varphi(t)\), defined as follows:

\[ \varphi(t)= \begin{cases} 0, & \text{for } t \in [t_1,\overline{t}_2],\\ |u'(t)|, & \text{for } t \in [t_1,\overline{t}_2], \end{cases} \]

where \(\overline{t}_2\) is the exact upper bound of those \(t \in (t_2,b]\) for which \(u'(t)\ne 0\).

But from (17) and (36) it follows that

\[ |u'(t)| \le y(\varphi;0,t_1,t) \le y(\varphi;0,a,t) \quad \text{for } t \in [t_1,t_2]. \]

Since \(\varphi(t)\) satisfies condition (9), from this we obtain

\[ |u'(t)| \le \rho_a(t) \quad \text{for } t \in [t_1,t_2]. \]

Consequently, \(|u'(t_0)| \le \rho_a(t_0)\), which contradicts condition (33). The contradiction obtained proves that condition (35) cannot occur.

Quite analogously, the condition

\[ u'(t)u(t)>0 \quad \text{for } t\in(t_1,t_2). \]

also cannot hold.

Thus, the assumption that inequality (33) is satisfied is false. The theorem is proved.

Corollary 1. If \(u(t)\in A'_\sigma(a,b)\), \(u(a)=u(b)=0\), and

\[ u''(t)\operatorname{sign}u(t)\geq -\omega(t,|u'(t)|) \quad \text{for } t\in[a,b], \]

then

\[ |u'(t)|\leq \max(y_0(t),z_0(t)) \quad \text{for } t\in[a,b], \]

where \(y_0(t)\) and \(z_0(t)\) are respectively the upper solutions of the problems

\[ \frac{dy}{dt}=\omega(t,y), \qquad y(a)=0 \]

and

\[ -\frac{dz}{dt}=-\omega(t,z), \qquad z(b)=0. \]

In particular, if \(\omega(t,y)=\psi_1(t)y+\psi_2(t)\), where \(\psi_k(t)\geq 0\) for \(t\in[a,b]\) and \(\psi_k(t)\in L[a,b]\) \((k=1,2)\), then

\[ |u'(t)|\leq \exp\left[\int_a^b \psi_1(\tau)\,d\tau\right] \int_a^b \psi_2(\tau)\,d\tau \quad \text{for } t\in[a,b]. \]

Corollary 2. If \(u(t)\in A'_\sigma(a,b)\), \(u(a)=u(b)=0\), and

\[ u''(t)\operatorname{sign}u(t)\geq -\psi(t)\omega(|u'(t)|) \quad \text{for } t\in[a,b], \tag{37} \]

where the functions \(\psi(t)\) and \(\omega(t)\) satisfy the hypotheses of Corollary 2 of Theorem 1, then

\[ |u'(t)|\leq l \quad \text{for } t\in[a,b]. \]

Corollary 3. If \(u(t)\in A'_\sigma(a,b)\), \(u(a)=u(b)=0\), and

\[ u''(t)\operatorname{sign}u(t) \geq -\psi_1(t)|u'(t)|^{1+1/q}-\psi_2(t) \quad \text{for } t\in[a,b], \]

where the functions \(\psi_1(t)\) and \(\psi_2(t)\) satisfy the hypotheses of Corollary 3 of Theorem 1, then

\[ |u'(t)|\leq \exp\left[d^{1/q}(a,b)N(\psi_1;p,a,b)\right] \int_a^b \psi_2(\tau)\,d\tau \quad \text{for } t\in[a,b]. \]

Corollary 4. If \(u(t)\in A'_\sigma(a,b)\), \(u(a)=u(b)=0\), and inequality (37) is satisfied, where the functions \(\psi(t)\) and \(\omega(t)\) satisfy the hypotheses of Corollary 4 of Theorem 1, then

\[ |u'(t)|\leq M\left[\omega;q,0,d^{1/q}(a,b)N(\psi;p,a,b)\right] \quad \text{for } t\in[a,b]. \]

Theorem 3. If \(u(t)\in A'_\sigma(a,b)\),

\[ u''(t)\operatorname{sign}u'(t) \geq -\omega(t,|u'(t)|,|u'(t)|) \quad \text{for } t\in[a,\alpha], \tag{38_1} \]

\[ |u''(t)|\leq \omega(t,|u'(t)|,|u'(t)|) \quad \text{for } t\in[\alpha,\beta], \tag{38_0} \]

and

\[ u''(t)\operatorname{sign}u'(t) \leq \omega(t,|u'(t)|,|u'(t)|) \quad \text{for } t\in[\beta,b], \tag{38_2} \]

then

\[ |u'(t)|\leq \rho_{\alpha\beta}(t) \quad \text{for } t\in[a,b]. \tag{39} \]

Proof. We shall first show that for some \(t_0 \in [\alpha,\beta]\) we have
\[ |u'(t_0)| \le y_{\alpha\beta}. \tag{40} \]
Indeed, otherwise, in view of (8), for some \(s_1,s_2 \in [\alpha,\beta]\) we would have
\[ |u'(t)| > \frac{|\sigma_2(s_2)-\sigma_1(s_1)|}{|s_2-s_1|} \quad \text{for } t \in [\alpha,\beta]. \]
Hence we would obtain
\[ |u(s_2)-u(s_1)| > |\sigma_2(s_2)-\sigma_1(s_1)|, \]
which contradicts condition (3).

Suppose now that
\[ |u'(t^*)| > \rho_{\alpha\beta}(t^*) \quad \text{for some } t^* \in [t_0,b]. \tag{41} \]
Since \(\rho_{\alpha\beta}(t) \ge y_{\alpha\beta}\) for \(t \in [a,b]\), it follows from inequalities (40) and (41) that there exist numbers \(t_1\) and \(t_2\), \(t_0 \le t_1 < t^* < t_2 \le b\), such that
\[ |u'(t_1)| \le y_{\alpha\beta} \tag{42} \]
and
\[ u'(t) \ne 0 \quad \text{for } t \in (t_1,t_2). \]

Let \((\underline t,\overline t)\) be the maximal interval containing \((t_1,t_2)\) in which \(u'(t)\ne 0\). It is clear that the function \(\varphi(t)\), defined as follows:
\[ \varphi(t)= \begin{cases} 0, & \text{for } t \in [t,\overline t],\\ |u'(t)|, & \text{for } t \in [\underline t,t], \end{cases} \]
satisfies condition (9). On the other hand, according to \((38_0)\) and \((38_2)\), it is clear that condition (17) is satisfied. But by virtue of (7) and (42), from (17) we obtain
\[ |u'(t)| \le y(\varphi;y_{\alpha\beta},t_1,t) \le y(\varphi;y_{\alpha\beta},\alpha,t) \le \rho_{\alpha\beta}(t) \quad \text{for } t \in [t_1,t_2]. \]
Consequently, \(|u'(t^*)|\le \rho_{\alpha\beta}(t^*)\), which contradicts condition (41). The contradiction obtained proves that
\[ |u'(t)| \le \rho_{\alpha\beta}(t) \quad \text{for } t \in [t_0,b]. \]
In exactly the same way one proves that \(|u'(t)|\le \rho_{\alpha\beta}(t)\) for \(t \in [a,t_0]\). The theorem is proved.

Corollary 1. If \(u(t)\in A'_\sigma(a,b)\),
\[ u''(t)\operatorname{sign}u'(t) \ge -\omega(t,|u'(t)|) \quad \text{for } t \in [a,\alpha], \]
\[ |u''(t)| \le \omega(t,|u'(t)|) \quad \text{for } t \in [\alpha,\beta] \]
and
\[ u''(t)\operatorname{sign}u'(t) \le \omega(t,|u'(t)|) \quad \text{for } t \in [\beta,b], \]
then
\[ |u'(t)| \le \begin{cases} z_\beta(t), & \text{for } t \in [a,\beta],\\ \max [y_\alpha(t),z_\beta(t)], & \text{for } t \in [\alpha,\beta],\\ y_\alpha(t), & \text{for } t \in [\beta,b], \end{cases} \]
where \(y_\alpha(t)\) and \(z_\beta(t)\) are the upper solutions of the following problems:
\[ \frac{dy}{dt}=\psi(t,y), \qquad y(\alpha)=y_{\alpha\beta} \]
and
\[ -\frac{dz}{dt}=-\psi(t,y), \qquad z(\beta)=y_{\alpha\beta}. \]

In particular, if \(\omega(t,y)=\psi_1(t)y+\psi_2(t)\), where \(\psi_k(t)\geq 0\) and \(\psi_k(t)\in L(a,b)\) \((k=1,2)\), then

\[ |u'(t)|\leq \begin{cases} \displaystyle \exp\left[\int_t^\beta \psi_1(\tau)\,d\tau\right] \left\{ y_{\alpha\beta} + \int_t^\beta \exp\left[-\int_\tau^\beta \psi_1(s)\,ds\right]\psi_2(\tau)\,d\tau \right\} & \text{for } t\in[a,\alpha], \\[2.2ex] \displaystyle \exp\left[\int_\alpha^\beta \psi_1(\tau)\,d\tau\right] \left\{ y_{\alpha\beta} + \int_\alpha^\beta \psi_2(\tau)\,d\tau \right\} & \text{on } [\alpha,\beta], \\[2.2ex] \displaystyle \exp\left[\int_\alpha^t \psi_1(\tau)\,d\tau\right] \left\{ y_{\alpha\beta} + \int_\alpha^t \exp\left[-\int_\alpha^\tau \psi_1(s)\,ds\right]\psi_2(\tau)\,d\tau \right\} & \text{for } t\in[\beta,b]. \end{cases} \]

Corollary 2. If \(u(t)\in A_\sigma'(a,b)\),

\[ u''(t)\operatorname{sign}u'(t)\geq -\psi(t)\omega(|u'(t)|)\quad \text{for } t\in[a,\alpha], \]

and

\[ |u''(t)|\leq \psi(t)\omega(|u'(t)|)\quad \text{for } t\in[\alpha,\beta] \]

and

\[ u''(t)\operatorname{sign}u'(t)\leq \psi(t)\omega(|u'(t)|)\quad \text{for } t\in[\beta,b], \]

where \(\psi(t)\geq 0\) for \(t\in[a,b]\) and \(\psi(t)\in L(a,b)\), while the function \(\omega(t)\) is nonnegative and continuous on the interval \([0,\infty)\) and for some \(l\in(y_{\alpha\beta},\infty)\) satisfies the condition

\[ \int_{t_0}^{l}\frac{d\tau}{\omega(\tau)}=\infty \quad \text{for every } t_0\in(0,l), \]

then

\[ |u'(t)|\leq l \quad \text{for } t\in[a,b]. \]

Corollary 3. If \(u(t)\in A_\sigma'(a,b)\),

\[ u''(t)\operatorname{sign}u'(t)\geq -\psi_1(t)|u'(t)|^{1+\frac{1}{q_1}}-\psi(t) \quad \text{for } t\in[a,\alpha], \]

and

\[ |u''(t)|\leq \psi_0(t)|u'(t)|^{1+\frac{1}{q_0}}+\psi(t) \quad \text{for } t\in[\alpha,\beta] \]

and

\[ u''(t)\operatorname{sign}u'(t)\leq \psi_2(t)|u'(t)|^{1+1/q_2}+\psi(t) \quad \text{for } t\in[\beta,b], \]

where \(\psi(t)\geq 0\), \(\psi_k(t)\geq 0\) \((k=0,1,2)\) for \(t\in[a,b]\), \(\psi(t)\in L(a,b)\),

\[ \psi_k(t)\in L^{p_k}(a,b),\qquad q_k\geq 1 \quad\text{and}\quad \frac{1}{p_k}+\frac{1}{q_k}=1 \quad (k=0,1,2), \]

then for arbitrary \(t_1\) and \(t_2\), \(t_1\in[a,\alpha]\) and \(t_2\in[\beta,b]\), we have

\[ |u'(t)|\leq \begin{cases} \displaystyle \exp\!\left[l_1+d^{1/q_1}(t_1,t)N(\psi_1;p_1,t_1,t)\right] \left( l_0+\int_t^\alpha \psi(\tau)\,d\tau \right) & \text{for } t\in[a,\alpha], \\[2.2ex] \displaystyle l_0 & \text{for } t\in[\alpha,\beta], \\[2.2ex] \displaystyle \exp\!\left[l_2+d^{1/q_2}(t_2,t)N(\psi_2;p_2,t_2,t)\right] \left( l_0+\int_\beta^t \psi(\tau)\,d\tau \right) & \text{for } t\in[\beta,b], \end{cases} \]

where

\[ l_0=\exp\!\left[d^{1/q_0}(\alpha,\beta)N(\psi_0;p_0,\alpha,\beta)\right] \left[ y_{\alpha\beta}+\int_\alpha^\beta \psi(\tau)\,d\tau \right], \]

\[ l_1=d^{1/q_1}(t_1,\alpha)N(\psi_1;p_1,\alpha,\beta),\qquad l_2=d^{1/q_2}(\beta,t_2)N(\psi_2;p_2,\alpha,\beta). \tag{43} \]

Corollary 4. If \(u(t)\in A'_\sigma(a,b)\),

\[ u''(t)\operatorname{sign}u'(t)\ge -\psi_1(t)\omega_1(|u'(t)|)\quad \text{for } t\in [a,\alpha], \]

\[ |u''(t)|\le \psi_0(t)\omega_0(|u'(t)|)\quad \text{for } t\in [\alpha,\beta] \]

and

\[ u''(t)\operatorname{sign}u'(t)\le \psi_2(t)\omega_2(|u'(t)|)\quad \text{for } t\in [\beta,b], \]

where \(\psi_k(t)\ge 0\) for \(t\in [a,b]\) and \(\psi_k(t)\in L^{p_k}(a,b)\) \((k=0,1,2)\), while the functions \(\omega_k(t)\) are positive and continuous on the interval \([0,\infty)\) and

\[ \int_0^\infty \frac{\tau^{1/q_k}}{\omega_k(\tau)}\,d\tau=\infty,\qquad \frac{1}{p_k}+\frac{1}{q_k}=1\quad (k=0,1,2), \]

then for arbitrary \(t_1\) and \(t_2\), \(t_1\in [a,\alpha]\) and \(t_2\in [\beta,b]\), we have

\[ |u'(t)|\le \begin{cases} M[\omega_1;q_1,l_0,l_1+d^{1/q_1}(t_1,t)N(\psi_1;p_1,t_1,t)] & \text{for } t\in [a,\alpha],\\ l_0 & \text{for } t\in [\alpha,\beta],\\ M[\omega_2;q_2,l_0,l_2+d^{1/q_2}(t_2,t)N(\psi_2;p_2,t_2,t)] & \text{for } t\in [\beta,b], \end{cases} \tag{44} \]

where \(l_0=M[\omega_0;q_0,y_{\alpha\beta},d^{1/q_0}(\alpha,\beta)N(\psi_0;p_0,\alpha,\beta)]\), and \(l_1\) and \(l_2\) are determined by equalities (43).

For \(\psi_k(t)\equiv 1,\ \omega_k(t)=\omega(t),\ q_k=1\ (k=0,1,2),\ \alpha=a\) and \(\beta=b\), the assertion above yields Nagumo’s theorem.

For \(\omega_k(t)\equiv 1,\ \psi_k(t)=u''(t),\ p_k=p\ (k=0,1,2),\ \alpha=a\) and \(\beta=b\), from (44) we find

\[ |u'(t)|\le \left\{ \left[\frac{|u(b)-u(a)|}{b-a}\right]^{\frac{q+1}{q}} +\frac{q+1}{q}\,[2d(a,b)]^{1/q}N(u'';p,a,b) \right\}^{\frac{q}{q+1}} \]

\[ \text{for } t\in [a,b]. \tag{45} \]

Since one may take as \(\sigma_1(t)\) and \(\sigma_2(t)\) the constants \(N(u;\infty,a,b)\) and \(-N(u;\infty,a,b)\), respectively, from inequality (45) we obtain

\[ N^{\frac{q+1}{q}}(u';\infty,a,b)\le \left[\frac{2N(u;\infty,a,b)}{b-a}\right]^{\frac{q+1}{q}} + \frac{q+1}{q}\,[2N(u;\infty,a,b)]^{1/q}N(u'';p,a,b). \]

Assuming that \(u(t)\in L^\infty(a,\infty)\) and \(u''(t)\in L^p(a,\infty)\), passage to the limit in the last inequality, as \(b\to\infty\), gives

\[ N^{\frac{q+1}{q}}(u';\infty,a,\infty) \le \frac{q+1}{q}\,[2N(u;\infty,a,\infty)]^{1/q}N(u'';p,a,\infty). \]

The inequality obtained, when \(q=1\), coincides with the well-known Landau–Hadamard inequality [2].

References

1. Bernstein S. N. Collected Works, vol. III, Publishing House of the Academy of Sciences of the USSR, 1960.
2. Nagumo M. Proc. Phys.-Math. Soc. Japan, 19, ser. 3, 861–866, 1937.
3. Hardy G. H., Littlewood J. E., Polya G. Inequalities, Moscow, IL, 1948.
4. Kiguradze I. T. Communications of the Academy of Sciences of the Georgian SSR, 39, No. 3, 513–518, 1965.

Received by the editors
November 14, 1966

Tbilisi State University