ON INTEGRAL INEQUALITIES FOR A SYSTEM OF VOLTERRA EQUATIONS WHOSE RIGHT-HAND SIDE IS A DISCONTINUOUS OPERATOR

LI MUN SU

In the study of questions in the qualitative theory of differential equations connected with estimates of solutions (existence, uniqueness, asymptotic behavior of a solution), the method of integral [1, 2] and differential inequalities is widely used. The applicability of differential inequalities to the study of the system of equations

\[ y_i' = f_i(t, y_1, \ldots, y_n), \]

where \(f_i(t, y_1, \ldots, y_n)\) is discontinuous with respect to \(y_1, \ldots, y_n\), was shown in [3, 4]. Since a system of differential equations reduces to a system of integral equations, the assertions given below make it possible to transfer the method of integral inequalities to equations with a discontinuous right-hand side.

The present publication is a continuation of our preceding article [8]. In the following work we shall give a number of applications of theorems on integral inequalities and discuss various definitions of a solution of an integral equation with a discontinuous operator.

We shall use the following definitions and notation:

1. If \(x=\{x^1,\ldots,x^n\}\), then \(\|x\|=\max_{1\le i\le n}|x^i|\).

2. If
   \[ H(x)=\{H^1(x^1,\ldots,x^n),\ldots,H^n(x^1,\ldots,x^n)\} \]
   is a measurable vector-function, bounded almost everywhere for \(\|x\|<c\), then

\[ M_x\{H(x)\}=\left\{\lim_{\delta\to 0}\operatorname{Vrai}\max_{z\in R(x,\delta)} H^i(z)\right\}, \]

\[ m_x\{H(x)\}=\left\{\lim_{\delta\to 0}\operatorname{Vrai}\min_{z\in R(x,\delta)} H^i(z)\right\}, \]

where \(R(x,\delta)\) is the ball of radius \(\delta\) with center at the point \(x\).

1. In the system of integral equations

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

the vector-function \(f(t)=\{f^1(t),\ldots,f^n(t)\}\) is continuous for \(t\in[a,b)\), and \(\|f(a)\|<c\).

The vector-function

\[ K(t,s,x)=\{K^1(t,s,x^1,\ldots,x^n),\ldots,K^n(t,s,x^1,\ldots,x^n)\} \]

is defined in \(G: a \leq s \leq t < b,\ \|x\| < c\) and satisfies the following conditions:

A) \(K(t,s,x)\) is measurable with respect to \(s\) for all \(t\) and almost all \(x\), and measurable with respect to \(x\) for all \(t\) and almost all \(s\).

B) Whatever positive number \(\gamma < c\) is taken, there exist functions \(\mu_\gamma(t,s)\) and \(\nu_\gamma(t,t_1,s)\) summable with respect to \(s\) on \([a,b)\) \((a \leq s \leq t \leq t_1 < b)\), such that

\[ \operatorname{Vrai}\max_{\|x\|\leq \gamma}\|K(t,s,x)\|\leq \mu_\gamma(t,s), \tag{2} \]

\[ \operatorname{Vrai}\max_{\|x\|\leq \gamma} \|M_x\{K(t,s,x)\}-m_x\{K(t_1,s,x)\}\| \leq \nu_\gamma(t,t_1,s), \]

\[ \operatorname{Vrai}\max_{\|x\|\leq \gamma} \|m_x\{K(t,s,x)\}-M_x\{K(t_1,s,x)\}\| \leq \nu_\gamma(t,t_1,s), \tag{3} \]

and for every \(\varepsilon>0\) one can find a \(\delta>0\) such that, when \(|t-t_1|<\delta\), the inequality

\[ \int_a^t \nu_\gamma(t,t_1,s)\,ds+\int_t^{t_1}\mu_\gamma(t_1,s)\,ds<\varepsilon \]

is satisfied.

We note that the functions \(\mu_\gamma(t,s)\) and \(\nu_\gamma(t,t_1,s)\) do not decrease as \(\gamma\) increases.

C) \(K(t,s,x)\) is nondecreasing in \(x\) for all \(t\) and almost all \(s\).

1. A vector function \(v_\tau(t)\), defined on \([a,\tau]\), \(\tau \leq b\), will be called a solution of (1) on \([a,\tau]\) (cf. [3, 4]) if it is continuous and, for \(t\in [a,\tau]\), the inequalities \(\|v_\tau(t)\|<c\),

\[ \int_a^t m_x\{K(t,s,v_\tau(s))\}\,ds+f(t)\leq v_\tau(t)\leq \]

\[ \leq \int_a^t M_x\{K(t,s,v_\tau(s))\}\,ds+f(t) \]

are satisfied.

1. If \(\tau<\xi,\ \tau,\xi\in(a,b)\) and \(v_\tau(t)=v_\xi(t)\) for \(t\in[a,\tau)\), then \(v_\xi(t)\) is called an extension of the solution \(v_\tau(t)\) to \([a,\xi]\), while \(v_\tau(t)\) is called a part of the solution \(v_\xi(t)\).

2. A solution \(v_\tau(t)\) is called nonextendable beyond \([a,\tau)\) if \(v_\tau(t)\) is not a part of any solution of system (1).

3. The inequality \(z\geq y\) \((z>y)\) between \(n\)-dimensional vectors \(z=\{z^i\}\), \(y=\{y^i\}\) means that \(z^i\geq y^i\) \((z^i>y^i)\), \(i=1,\ldots,n\). The symbol \(y\to x+0\) means that \(y^i\to x^i\), with \(y^i>x^i\) \((i=1,\ldots,n)\).

No. 1. Let us establish some properties of \(M_x\{H(x)\}\) and \(m_x\{H(x)\}\).

Lemma 1. If \(H(x)\) is nondecreasing and \(x_2>x_1\), then \(m_x\{H(x_2)\}\geq M_x\{H(x_1)\}\).

Proof. Consider \(R(x_1,\delta)\) and \(R(x_2,\delta)\), where

\[ 0<\delta<\min_i\left[\frac{x_2^i-x_1^i}{2}\right]. \]

Obviously, \(z>y\) if \(z\in R(x_2,\delta)\) and \(y\in R(x_1,\delta)\). Since \(H(x)\) is nondecreasing, \(H(z)\geq H(y)\). Hence, by the definition of \(M_x\{H(x)\}\) and \(m_x\{H(x)\}\), we have:

\[ m_x\{H(x_2)\}\geq M_x\{H(x_1)\}, \]

if \(x_2>x_1\).

Remark. The strict inequality \(x_2 > x_1\) is essential, as the following example shows:

\[ H(x)=\{5\operatorname{sgn} x^1+2\operatorname{sgn} x^2,\;4\operatorname{sgn} x^1+\operatorname{sgn} x^2\}, \]

where

\[ m_x\{H(x_2)\}<M_x\{H(x_1)\} \]

for \(x_2=\{0,3\}\) and \(x_1=\{0,2\}\).

Lemma 2. If \(H(x)\) is nondecreasing, then \(M_x\{H(x)\}\) and \(m_x\{H(x)\}\) are nondecreasing.

Proof. Since \(H(x)\) is nondecreasing, we have

\[ M_x\{H(x)\}=\lim_{y\to x+0} H(y). \]

From this equality and by virtue of the monotonicity of \(H(x)\), we obtain

\[ M_x\{H(x_2)\}\geq M_x\{H(x_1)\}, \]

if \(x_2\geq x_1\). For \(m_x\{H(x)\}\) the proof is analogous.

Lemma 3. \(M_x\{H(x)\}\) is upper semicontinuous and \(m_x\{H(x)\}\) is lower semicontinuous.

Proof. Let

\[ \overline{\lim_{x\to x_0}}\, M_x\{H(x)\}=a. \]

Obviously,

\[ M_x\{H(x_0)\}\leq a+\varepsilon. \]

On the other hand, in every neighborhood of the point \(x_0\) there is a point \(y\) such that

\[ \operatorname{Vrai\,max}_{z\in R(y,\delta)} H(z)>a-\varepsilon, \]

i.e., in every neighborhood of it there is a set \(E\) of positive measure on which \(H(x)>a-\varepsilon\). Hence such a set exists in every neighborhood of the point \(x_0\). Thus,

\[ M_x\{H(x_0)\}\geq a-\varepsilon. \]

Whence,

\[ [M_x\{H(x_0)\}]=a, \]

which proves the upper semicontinuity of \(M_x\{H(x)\}\). The proof of the lower semicontinuity of \(m_x\{H(x)\}\) is analogous.

Lemma 4. If \(H(t,x)\) is finite, nondecreasing, upper (lower) semicontinuous with respect to \(x\) for almost all \(t\), and \(\{x_i(t)\}\) is a monotonically decreasing (increasing) sequence of functions \(x_i(t)\) continuous on \([a,b_1]\), converging to \(x(t)\), then

\[ \lim_{i\to\infty} H(t,x_i(t))=H(t,x(t)) \]

for almost all \(t\in[a,b_1]\).

Proof. Denote by \(E\) the set of points \(t\) at which \(H(t,x)\) is semicontinuous in \(x\). Obviously, \(mE=b_1-a\).

By virtue of the semicontinuity and finiteness of \(H(t,x)\), for \(x\in E\) we have:

\[ H(t,x(t))=\overline{\lim_{i\to\infty}} H(t,x_i(t)). \]

But since \(H(t,x)\) is nondecreasing in \(x\) and \(\{x_i(t)\}\) is a monotonically decreasing sequence, then

\[ \lim_{i\to\infty}\overline{H}(t,x_i(t))=\lim_{i\to\infty}H(t,x_i(t)) \]

on the set \(E\). Thus,

\[ \lim_{i\to\infty}H(t,x_i(t))=H(t,x(t)) \]

almost everywhere in \([a,b_1]\).

Lemma 5. If \(H(t,x)\) is semicontinuous in \(x\) for almost all \(t\), measurable in \(t\) for all \(x\) in \(G_1: a\le t\le b_1<b,\ |x|\le \gamma<c\), and \(x(t)\) is any function continuous on \([a,b_1]\), \(|x(t)|\le \gamma\), then \(H(t,x(t))\) is measurable on \([a,b_1]\).

Proof. Suppose \(H(t,x)\) is upper semicontinuous in \(x\) for \(t\in E\subset [a,b_1]\). Obviously, \(mE=b_1-a\). Then for any \(t\in E\) there exists a decreasing sequence of functions \(\{H_i(t,x)\}\), continuous in \(x\), such that

\[ \lim_{i\to\infty}H_i(t,x)=H(t,x). \]

\(H_i(t,x(t))\) is measurable on the set \(E\) (see, for example, [10]). Hence \(H(t,x(t))\) is measurable on \(E\). But since \(b_1-a-mE=0\), \(H(t,x(t))\) is measurable on \([a,b_1]\). The proof of the measurability of \(H(t,x(t))\) in the case of lower semicontinuity is analogous.

The lemma is proved.

Lemma 6. Let \(K(t,s,x)\) satisfy conditions A) and B) in \(G_1\). If \(\{x_i(t)\}\) is a family of vector functions continuous on \([a,b_1]\) and \(\|x_i(t)\|\le \gamma<c\), then

\[ \left\{\int_a^t M_x\{K(t,s,x_i(s))\}\,ds\right\} \quad\text{and}\quad \left\{\int_a^t m_x\{K(t,s,x_i(s))\}\,ds\right\} \]

are equicontinuous on \([a,b_1]\).

Proof. Let \(t\in [a,b_1]\) and \(\|x_i(t)\|\le \gamma\). Denote

\[ \Theta_i(t)=\int_a^t M_x\{K(t,s,x_i(s))\}\,ds. \]

By Lemma 3 and Lemma 5, such a function \(\Theta_i(t)\) exists. From the inequality

\[ \|\Theta_i(t_1)-\Theta_i(t_2)\|\le \int_a^{t_1}\|M_x\{K(t_2,s,x_i(s))\}- \]

\[ - M_x\{K(t_1,s,x_i(s))\}\|\,ds+ \int_{t_1}^{t_2}\|M_x\{K(t_2,s,x_i(s))\}\|\,ds\le \]

\[ \le \int_a^{t_1}\psi_{\gamma+\rho}(t_1,t_2,s)\,ds+ \int_{t_1}^{t_2}\psi_{\gamma+\rho}(t_2,s)\,ds, \]

where \(a\le t_1\le t_2\le b_1\) and \(\rho<c-\gamma\), it follows that \(\{\Theta_i(t)\}\) is equicontinuous on \([a,b_1]\). The proof of the equicontinuity of

\[ \left\{\int_a^t m_x\{K(t,s,x_i(s))\}\,ds\right\} \]

on \([a,b_1]\) is analogous.

No. 2. Let us consider the assertion on the integral inequality.

Theorem 1. If a continuous vector-function \(z(t)\) \((\|z(t)\|<c)\) is such that

\[ z(t)>\int_a^t m_x\{K(t,s,z(s))\}\,ds+f(t) \]

\[ \left(z(t)<\int_a^t M_x\{K(t,s,z(s))\}\,ds+f(t)\right) \]

for \(t\in[a,b)\), then \(z(t)>u(t)\) \((z(t)<u(t))\) in \([a,b)\), where \(u(t)\) is any solution of system (1) defined in \([a,b)\).

Proof. Since \(z(a)>f(a)\) and \(u(a)=f(a)\), we have \(z(a)>u(a)\). By virtue of the continuity of \(z(t)\) and \(u(t)\), the inequality \(z(t)>u(t)\) holds in a neighborhood of the point \(a\). Suppose that the theorem is false, i.e., there is a \(t_1<b\) such that \(z(t_1)=u(t_1)\) and \(z(t)>u(t)\) for \(t\in[a,t_1)\),

\[ \int_a^{t_1} m_x\{K(t_1,s,z(s))\}\,ds+f(t_1)<z(t_1), \]

\[ \int_a^{t_1} m_x\{K(t_1,s,u(s))\}\,ds+f(t_1)\le u(t_1)\le \]

\[ \le \int_a^{t_1} M_x\{K(t_1,s,u(s))\}\,ds+f(t_1). \]

Subtracting the right-hand side of the second inequality from the first inequality, we obtain

\[ z(t_1)-u(t_1)> \int_a^{t_1}\bigl[m_x\{K(t_1,s,z(s))\}-M_x\{K(t_1,s,u(s))\}\bigr]\,ds. \]

By Lemma 1,

\[ m_x\{K(t_1,s,z(s))\}-M_x\{K(t_1,s,u(s))\}\ge 0 \]

for almost all \(s\in[a,t_1]\). Thus, \(z(t_1)>u(t_2)\). The contradiction obtained proves the lemma.

Remark. The condition \(z(a)>f(a)\) is essential, as the following example shows:

\[ x(t)=\int_0^t \operatorname{sgn}x(s)\,ds-\frac{t}{2},\quad \text{where }\operatorname{sgn}x= \begin{cases} 1 & \text{for } x>0,\\ 0 & \text{for } x=0,\\ -1 & \text{for } x<0. \end{cases} \]

Indeed, \(u=\dfrac{t}{2}\) is a solution of the equation. If \(z(t)=-t\), then in \((0,\infty)\) we have

\[ z(t)>\int_0^t m_x\{\operatorname{sgn}z(s)\}\,ds-\frac{t}{2}, \]

but

\[ z(t)<\frac{t}{2}. \]

No. 3. Following [1], we shall call \(u(t)\) an upper (lower) solution on \([a,b)\) of system (1) if:

a) \(u(t)\) is a solution of system (1) defined in \([a,b)\);

b) \(u(t)\ge v_\tau(t)\) \((u(t)\le v_\tau(t))\) on \([a,\tau)\subset[a,b)\) for every solution \(v_\tau(t)\).

Theorem 2. Suppose that in \([a,b)\) there exists a pair of continuous vector-functions

\[ z_i(t)\quad (\|z_i(t)\|\leq c,\ t\in [a,b),\ i=1,2), \]

satisfying the integral inequalities

\[ z_1(t)>\int_a^t m_x\{K(t,s,z_1(s))\}\,ds+f(t), \tag{4} \]

\[ z_2(t)<\int_a^t M_x\{K(t,s,z_2(s))\}\,ds+f(t), \tag{5} \]

then on \([a,b)\) there exist an upper solution \(u_1(t)\) and a lower solution \(u_2(t)\) of system (1), and

\[ z_1(t)>u_1(t)\geq u_2(t)>z_2(t)\quad \text{in }[a,b). \]

Proof. Without loss of generality, put \(f(t)\equiv 0\). Denote

\[ \varphi_1(t)=z_1(t)-\int_a^t m_x\{K(t,s,z_1(s))\}\,ds>0 \tag{6} \]

and

\[ \varphi_2(t)=\int_a^t M_x\{K(t,s,z_2(s))\}\,ds-z_2(t)>0. \tag{7} \]

The continuity of \(\varphi_1(t)\) and \(\varphi_2(t)\) on \([a,b)\) follows from Lemma 6. From (7) and the inequality

\[ z_1(t)>\int_a^t m_x\{K(t,s,z_1(s))\}\,ds-\varphi_2(t) \]

by Theorem 1 we have \(z_1(t)>z_2(t)\) in \([a,b)\).

To prove the existence of an upper solution, consider the sequence \(\{x_i(t)\}\), setting

\[ x_1(t)=\int_a^t m_x\{K(t,s,z_1(s))\}\,ds+\frac{\varphi_1(t)}{2}, \]

\[ x_{i+1}(t)=\int_a^t M_x\{K(t,s,x_i(s))\}\,ds+\frac{\varphi_1(t)}{i+1}. \tag{8} \]

Obviously, \(z_1(t)>x_1(t)\) in \([a,b)\). From (7) and the inequality

\[ x_1(t)>\int_a^t m_x\{K(t,s,z_1(s))\}\,ds-\varphi_2(t)\geq \]

\[ \geq \int_a^t m_x\{K(t,s,x_1(s))\}\,ds-\varphi_2(t) \]

by Theorem 1 we have \(x_1(t)>z_2(t)\) in \([a,b)\). Thus, \(z_1(t)>x_1(t)>z_2(t)\) in \([a,b)\). The continuity of the vector-function \(x_1(t)\) on \([a,b)\) follows from Lemma 6. By Lemma 1 we have:

\[ m_x\{K(t,s,z_1)\}-M_x\{K(t,s,x_1)\}\geq 0. \]

Therefore

\[ x_1(t)-x_2(t)=\int_a^t \bigl[m_x\{K(t,s,z_1(s))\}-M_x\{K(t,s,x_1(s))\}\bigr]\,ds\geq 0. \]

Since

\[ x_2(t)-z_2(t)=\int_a^t [M_x\{K(t,s,x_1(s))\}- \]

\[ -M_x\{K(t,s,z_2(s))\}]\,ds+\frac{\varphi_1(t)}{2}+\varphi_2(t), \]

then, by Lemma 2, we have \(x_2(t)>z_2(t)\) in \([a,b)\). Suppose that \(x_i(t)\) and \(x_{i+1}(t)\) are continuous and

\[ z_1(t)>x_i(t)\geq x_{i+1}(t)>z_2(t), \]

then, by Lemma 6, \(x_{i+2}(t)\) is defined and continuous in \([a,b)\), and the relation

\[ z_1(t)>x_{i+1}(t)>x_{i+2}(t)>z_2(t),\qquad t\in [a,b). \]

holds. Indeed,

\[ x_{i+1}(t)-x_{i+2}(t)=\int_a^t [M_x\{K(t,s,x_i(s))\}- \]

\[ -M_x\{K(t,s,x_{i+1}(s))\}]\,ds+\frac{\varphi_1(t)}{(i+1)(i+2)}>0. \]

From (7), (8), and Theorem 1 we obtain that \(x_{i+2}(t)>z_2(t)\). Thus the sequence of vector-functions \(\{x_i(t)\}\), continuous in \([a,b)\), decreases monotonically and is bounded below. Denote by \(x(t)\) the limit of this sequence. Let \([a,b_1]\subset [a,b)\) and \(\|z_i(t)\|\leq \gamma<c,\ i=1,2,\ t\in [a,b_1]\). Then from the relation \(z_1(t)>x_i(t)>z_2(t)\) we have \(\|x_i(t)\|\leq \gamma\) for \(t\in [a,b_1]\), i.e. the sequence \(\{x_i(t)\}\) is uniformly bounded. From Lemma 6 follows the equicontinuity of the sequence \(\{x_i(t)\}\) on \([a,b_1]\). Thus \(x(t)=\lim_{i\to\infty}x_i(t)\) is continuous in \([a,b_1]\subset [a,b)\) and, consequently, by the arbitrariness of \(b_1\), is continuous in \([a,b)\). By Lemma 4 and Lebesgue’s theorem on passage to the limit under the integral sign, we have

\[ \int_a^t m_x\{K(t,s,x(s))\}\,ds\leq x(t)=\int_a^t M_x\{K(t,s,x(s))\}\,ds. \]

By Theorem 1, \(x_i(t)>u(t)\), where \(u(t)\) is any solution of system (1). Consequently, \(x(t)\geq u(t)\) in \([a,b)\).

The proof of the existence of a lower solution is analogous.

Remark 1. For sufficiently small \(\varepsilon>0\), on \([a,a+\varepsilon)\) there exist \(z_1(t)\) and \(z_2(t)\) satisfying the conditions of Theorem 2. Consequently, on \([a,a+\varepsilon)\) there exist upper and lower solutions of system (1).

Remark 2. From the proof of the theorem it is easy to see that the upper solution \(u_1(t)\) and the lower solution \(u_2(t)\) satisfy, respectively, the equalities

\[ u_1(t)=\int_a^t M_x\{K(t,s,u_1(s))\}\,ds+f(t) \]

and

\[ u_2(t)=\int_a^t m_x\{K(t,s,u_2(s))\}\,ds+f(t). \]

Remark 3. As is known, for a first-order differential equation, upper and lower solutions exist without предполо-

of monotonicity of the right-hand side (see, for example, [3, 4]). At the Izhevsk seminar, in connection with the consideration of uniqueness theorems, the question was raised whether equation (1) for \(n=1\) possesses an analogous property. The following example shows that the monotonicity condition on \(K(t,s,x)\) with respect to \(x\) is essential,

\[ x(t)=\int_0^t 2\sin t\,\sqrt[3]{\frac{s x(s)}{\sin s}}\,ds . \tag{9} \]

It is easy to show that \(u_1(t)=t^2\sin t\) and \(u_2(t)=-t^2\sin t\) are solutions of (9); \(\sin t\,\sqrt[3]{\dfrac{s\cdot x}{\sin s}}\) is not decreasing in \(x\) for \(t\in[0,\pi)\), and \(u_1(t)\) and \(u_2(t)\) are, on \([0,\pi)\), respectively upper and lower solutions of (9). But for \(t\in[0,2\pi)\) the integrand in (9) is not monotone, and (9) no longer has either an upper or a lower solution on \([0,2\pi)\).

No. 4. Let us consider the following refinement of the assertion on the integral inequality given above.

Theorem 3. Let \(u(t)\) be a lower (upper) solution of system (1) on \([a,b)\). If a vector function \(z(t)\) continuous on \([a,b)\) \((\|z(t)\|<c,\ t\in[a,b))\) satisfies the integral inequality

\[ \int_a^t m_x\{K(t,s,z(s))\}\,ds+f(t)\leq z(t) \]

\[ \left( z(t)\leq \int_a^t M_x\{K(t,s,z(s))\}\,ds+f(t) \right), \]

then \(z(t)\geq u(t)\) \((z(t)\leq u(t))\) on \([a,b)\).

Proof. Let \(f(t)\equiv 0\) and \(\|z(t)\|\leq \gamma<c\) on \([a,b_1]\), \(b_1<b\). Consider the sequence of solutions of the systems

\[ x_n(t)=\int_a^t m_x\{K(t,s,x_n(s))\}\,ds-\frac{1}{n}. \tag{10} \]

By virtue of the remark to Theorem 2, there exists an \(\varepsilon>0\) such that on \([a,a+\varepsilon]\) there exists a solution of the system

\[ x_1(t)=\int_a^t m_x\{K(t,s,x_1(s))\}\,ds-1. \]

From the inequalities

\[ z(t)>\int_a^t m_x\{K(t,s,z(s))\}\,ds-\frac{1}{n}, \]

\[ x_1(t)<\int_a^t M_x\{K(t,s,x_1(s))\}\,ds-\frac{1}{n}, \]

by virtue of Theorem 2 there exist upper and lower solutions of system (10) on \([a,a+\varepsilon]\) for \(n>1\). We shall show that the sequence of solutions of (10) increases monotonically and is bounded above. Indeed,

\[ \int_a^t m_x\{K(t,s,x_{n+1}(s))\}\,ds-\frac{1}{n}<x_{n+1}(t) \]

and

\[ \int_a^t m_x\{K(t, s, x_n(s))\}\,ds-\frac1n=x_n(t) \]

By Theorem 1, \(x_{n+1}(t)>x_n(t)\). Let us prove that \(\{x_n(t)\}\) is bounded above by the vector function \(z(t)\). Indeed, from the inequality

\[ \int_a^t m_x\{K(t, s, z(s))\}\,ds-\frac1n<z(t) \]

and by Theorem 1, \(x_n(t)<z(t)\). Since \(\{x_n(t)\}\) increases monotonically and is bounded above, there exists a limit \(x(t)=\lim_{n\to\infty}x_n(t)\). From the relations \(x_1(t)<x_n(t)<z(t)\) and \(\|x_1(t)\|\leq \gamma<c\) on \([a,a+\varepsilon]\), it follows that \(\|x_n(t)\|\leq \gamma\) for \(t\in[a,a+\varepsilon]\). The uniform continuity of \(\{x_n(t)\}\) on \([a,a+\varepsilon]\) follows from Lemma 6. By Lemma 4 and Lebesgue’s theorem on passage to the limit under the integral sign, we have

\[ \int_a^t m_x\{K(t, s, x(s))\}\,ds=x(t)\leq \int_a^t M_x\{K(t, s, x(s))\}\,ds. \]

Since \(x_n(t)<z(t)\), it follows that \(x(t)\leq z(t)\) on \([a,a+\varepsilon]\). Hence \(u(t)\leq z(t)\) on \([a,a+\varepsilon]\).

Suppose that the theorem is false. Then there exists a maximal \(\tau\) such that \(u(t)\leq z(t)\) on \([a,\tau)\). Consider the system of integral equations

\[ x(t)=\int_\tau^t K(t, s, x(s))\,ds+\varphi(t), \tag{11} \]

where

\[ \varphi(t)=\int_a^\tau m_x\{K(t, s, u(s))\}\,ds. \]

From the inequality

\[ \|\varphi(t_1)-\varphi(t_2)\|\leq \int_a^\tau \|m_x\{K(t_1, s, u(s))\}-m_x\{K(t_2, s, u(s))\}\|\,ds\leq \]

\[ \leq \int_a^\tau \operatorname{Vrai\,max}_{\|y\|\leq\gamma+\rho} \|m_x\{K(t_1, s, y)\}-m_x\{K(t_2, s, y)\}\|\,ds\leq \]

\[ \leq \int_a^\tau \nu_{\gamma+\rho}(t_1,t_2,s)\,ds \leq \int_a^{t_1}\nu_{\gamma+\rho}(t_1,t_2,s)\,ds, \]

where \(\tau\leq t_1\leq t_2<b\), it follows that \(\varphi(t)\) is continuous on \([\tau,b)\). Obviously, \(\|\varphi(\tau)\|<c\). Since

\[ \int_a^t m_x\{K(t, s, z(s))\}\,ds= \]

\[ =\int_\tau^t m_x\{K(t, s, z(s))\}\,ds+\int_a^\tau m_x\{K(t, s, z(s))\}\,ds\geq \]

\[ \geq \int_\tau^t m_x\{K(t, s, z(s))\}\,ds+\int_a^\tau m_x\{K(t, s, u(s))\}\,ds, \]

then

\[ z(t) \geqslant \int_{\tau}^{t} m_x\{K(t,s,z(s))\}\,ds+\varphi(t). \]

By virtue of the first part of the proof, there exists a positive \(h\) such that on \([\tau,\tau+h)\) the inequality \(z(t)>u_1(t)\) holds, where \(u_1(t)\) is a lower solution of (11). Consider the vector-function

\[ \omega(t)= \begin{cases} u(t), & \text{for } t\in [a,\tau),\\ u_1(t), & \text{for } t\in [\tau,\tau+h). \end{cases} \]

By virtue of the remark to Theorem 2 we have

\[ u(t)=\int_a^t m_x\{K(t,s,u(s))\}\,ds \]

and

\[ u_1(t)=\int_{\tau}^{t} m_x\{K(t,s,u_1(s))\}\,ds+ \int_a^{\tau} m_x\{K(t,s,u(s))\}\,ds. \]

Consequently, \(\omega(t)\) is continuous on \([a,\tau+h)\) and is a solution of (1), and moreover \(z(t)\geqslant \omega(t)\) for \(t\in [a,\tau+h)\). In addition, \(\omega(t)\geqslant u(t)\) on \([a,\tau+h)\), since \(u(t)\) is a lower solution of (1). Hence \(z(t)\geqslant u(t)\), if \(t\in [a,\tau+h)\). This contradicts the definition of the point \(\tau\). For the upper solution the arguments are analogous.

The theorem is proved.

Corollary. Let \(K_1(t,s,x)\) satisfy conditions A), B), and C), and let \(f_1(t)\) be continuous on \([a,b)\) and \(\|f_1(a)\|\leq c\). If

\[ K_1(t,s,x)\leqslant M_x\{K(t,s,x)\} \]

almost everywhere in \(G\) and \(f_1(t)\leqslant f(t)\) on \([a,b)\), then \(u_1(t)\leqslant u(t)\), where \(u(t)\) is the upper solution of (1), and \(u_1(t)\) is the upper solution of the system

\[ x(t)=\int_a^t K_1(t,s,x(s))\,ds+f_1(t), \tag{12} \]

Proof. Since

\[ K_1(t,s,x)\leqslant M_x\{K(t,s,x)\}, \]

then

\[ M_x\{K_1(t,s,x)\}\leqslant M_x\{K(t,s,x)\}. \]

Therefore

\[ \xi(t)\leqslant \int_a^t M_x\{K(t,s,\xi(s))\}\,ds+f(t), \]

where \(\xi(t)\) is any solution of (12). By virtue of Theorem 3, \(\xi(t)\leqslant u(t)\). Hence \(u_1(t)\leqslant u(t)\).

Remark. If at least one of the inequalities

\[ K_1(t,s,x)<M_x\{K(t,s,x)\} \]

or \(f_1(t)<f(t)\) is satisfied, then \(\xi(t)<v(t)\), where \(\xi(t)\) is any solution of (12), and \(v(t)\) is the lower solution of (1). This remark follows from Theorem 2.

No. 5. The following assertion on the prolongability of an upper (lower) solution holds.

Theorem 4. There exists a nonextendable solution \(v_d(t)\) such that \(v_d(t)\) is an upper solution of system (1).

Proof. The set \(D\) of those points \(\tau \in (a,b)\) such that on \([a,\tau)\) system (1) has an upper solution is nonempty by the remark to Theorem 2. Let \(\{\tau_n\}\) be some monotonically increasing sequence of points \(\tau_n \in D\), for which the point \(d=\sup D\) is a limit point. Let \(v_n(t)\) be an upper solution on \([a,\tau_n)\). Obviously, \(v_{n+1}(t)=v_n(t)\) for \(t\in [a,\tau_n)\). Define \(v_d(t)\) as follows: if \(t\in [a,\tau_n)\), then \(v_d(t)=v_n(t)\). Then \(v_d(t)\) is an upper solution on \([a,d)\). We shall show that \(v_d(t)\) is nonextendable. Suppose the contrary, i.e. \(d<b\), but \(v_d(t)\) is extendable.

Consider the system of equations

\[ x(t)=\int_d^t K(t,s,x(s))\,ds+\varphi(t), \tag{13} \]

where

\[ \varphi(t)=\int_a^d M_x\{K(t,s,v_d(s))\}\,ds. \]

It is easy to show that \(\varphi(t)\) is continuous on \([a,b)\). Since \(v_d(t)\) is an upper solution of (1), by the remark to Theorem 2 we have

\[ v_d(t)=\int_a^t M_x\{K(t,s,v_d(s))\}\,ds. \]

Obviously,

\[ \lim_{t\to d-0} v_d(t)=\varphi(d) \quad \text{and} \quad \|\varphi(d)\|<c. \]

By the remark to Theorem 2 there exists a positive \(h\) such that system (13) has on \([d,d+h)\) an upper solution \(\omega(t)\). Just as in Theorem 3, one can prove that

\[ u(t)= \begin{cases} v_d(t), & \text{for } t\in [a,d),\\ \omega(t), & \text{for } t\in [d,d+h), \end{cases} \]

is a solution of system (1) on \([a,d+h)\). Let \(y_\tau(t)\) \((\tau\in(a,d+h))\) be some solution of system (1), and let \(\eta\) be the least upper bound of such numbers \(\vartheta\) that \(y_\tau(t)\le u(t)\) for \(t\in [a,\vartheta)\). Since \(d=\sup D\), \(u(t)\) is not an upper solution of (1) on \([a,d+h)\). Therefore there exists a solution \(y_\tau(t)\) of system (1) for which \(\eta<\tau\) \((\eta\ge d)\).

Consider the system of equations

\[ x(t)=\int_\eta^t K(t,s,x(s))\,ds+\psi(t), \tag{14} \]

where

\[ \psi(t)=\int_a^\eta M_x\{K(t,s,u(s))\}\,ds. \]

Obviously, \(\psi(t)\) is continuous on \([\eta,b)\) and \(\|\psi(\eta)\|<c\). Since \(K(t,s,x)\) is monotone in \(x\), by the remark to Theorem 2 system (14) has an upper solution \(r(t)\) on \([\eta,\eta+\varepsilon)\). Since \(y_\tau(t)\) is a solution of system (1), we have

\[ y_\tau(t)\le \int_a^t M_x\{K(t,s,y_\tau(s))\}\,ds = \int_\eta^t M_x\{K(t,s,y_\tau(s))\}\,ds+ \]

\[ + \int_a^\eta M_x\{K(t,\ s,\ y_\tau(s))\}\,ds \leq \int_\eta^t M_x\{K(t,\ s,\ y_\tau(s))\}\,ds+ \]

\[ + \int_a^\eta M_x\{K(t,\ s,\ u(s))\}\,ds = \int_\eta^t M_x\{K(t,\ s,\ y_\tau(s))\}\,ds+\psi(t). \]

By virtue of Theorem 3, \(y_\tau(t)\leq r(t)\) for \(t\in[\eta,\eta+\varepsilon)\). Define the vector-function

\[ \rho(t)= \begin{cases} \omega(t), & \text{for } t\in[d,\eta),\\ r(t), & \text{for } t\in[\eta,\eta+\varepsilon). \end{cases} \]

The continuity of \(\rho(t)\) on \([d,\eta+\varepsilon)\) is obvious. Let us show that \(\rho(t)\) is a solution of system (13). Let \(t\in[\eta,\eta+\varepsilon)\); then, by virtue of the remark to Theorem 2, we have

\[ r(t)=\int_\eta^t M_x\{K(t,\ s,\ r(s))\}\,ds+\psi(t) \]

or

\[ r(t)= \int_\eta^t M_x\{K(t,\ s,\ r(s))\}\,ds+ \int_d^\eta M_x\{K(t,\ s,\omega(s))\}\,ds+ \]

\[ +\int_a^d M_x\{K(t,\ s,\upsilon_d(s))\} = \int_d^t M_x\{K(t,\ s,\rho(s))\}\,ds+\varphi(t). \]

Thus,

\[ \rho(t)=\int_d^t M_x\{K(t,\ s,\rho(s))\}\,ds+\varphi(t). \]

Since \(\omega(t)\) is an upper solution of (13), it follows that

\[ \omega(t)\geq \rho(t)\geq y_\tau(t)\quad \text{for } t\in[d,\eta+\varepsilon_1), \]

where

\[ \eta+\varepsilon_1=\min[d+h,\eta+\varepsilon]. \]

Thus, \(u(t)\geq y_\tau(t)\) on \([a,\eta+\varepsilon_1)\). This contradicts the definition of the point \(\eta\). The contradiction obtained proves the noncontinuability of the solution \(\upsilon_d(t)\). The arguments in the case of a lower solution are analogous.
The theorem is proved.

No. 6. Under the condition of continuity of \(K(t,s,x)\) with respect to \(x\), a number of comparison theorems are known (see, for example, [1, 2]), which play an important role in obtaining various estimates needed in the study of questions of uniqueness and asymptotic behavior of solutions.

On the basis of the results presented above, it is possible to transfer to an equation with discontinuity a number of the comparison theorems just mentioned.

In system (1) we shall assume that \(K(t,s,x)\) satisfies conditions A) and B). Along with system (1), consider the equation

\[ \xi(t)=\int_a^t \omega(t,\ s,\ \xi(s))\,ds+\|f(t)\|, \tag{15} \]

where \(\omega(t,s,\xi)\) satisfies conditions A), B), C) and \(\omega(t,s,\xi)=0\) for \(\xi\leq 0\). In addition, in Theorem 5 (bis) we shall assume that \(\omega(t,s,\xi)\) is defined in \(G_2:\ a\leq s\leq t<b,\ |\xi|<2c\).

Theorem 5. Suppose

\[ \|K(t,\ s,\ x)\|\leq M_x\{\omega(t,\ s,\ \|x\|)\} \]

almost everywhere in \(G\). Suppose further that all solutions \(x(t)\) of system (1) and the upper solution \(u(t)\) of equation (15) are defined in \([a,b)\); then in \([a,b)\) we have the estimate \(\|x(t)\|\leqslant u(t)\).

Proof. Since \(x(t)\) is a solution of (1), we have

\[ \int_a^t m_x\{K(t,s,x(s))\}\,ds+f(t)\leqslant x(t)\leqslant \]

\[ \leqslant \int_a^t M_x\{K(t,s,x(s))\}\,ds+f(t). \]

From the inequality

\[ \|K(t,s,x)\|\leqslant M_x\{\omega(t,s,\|x\|)\} \]

we obtain that

\[ M_x\{K^i(t,s,x)\}\leqslant M_x\{\omega(t,s,\|x\|)\} \]

and

\[ m_x\{K^i(t,s,x)\}\geqslant -\,M_x\{\omega(t,s,\|x\|)\}. \]

Then

\[ \int_a^t -\,M_x\{\omega(t,s,\|x(s)\|)\}\,ds-\|f(t)\|\leqslant x^i(t)\leqslant \]

\[ \leqslant \int_a^t M_x\{\omega(t,s,\|x(s)\|)\}\,ds+\|f(t)\| \]

or

\[ \|x(t)\|\leqslant \int_a^t M_x\{\omega(t,s,\|x(s)\|)\}+\|f(t)\|. \]

By virtue of Theorem 3, \(\|x(t)\|\leqslant u(t)\) in \([a,b)\).

The theorem is proved.

Theorem 5 (bis). Suppose

\[ \|K(t,s,x_1)-K(t,s,x_2)\|\leqslant M_x\{\omega(t,s,\|x_1-x_2\|)\} \tag{16} \]

almost everywhere in \(G\). Then, for any solutions \(x_1(t)\) and \(x_2(t)\) of system (1), the estimate

\[ \|x_1(t)-x_2(t)\|\leqslant u(t), \]

holds, where \(u(t)\) is an upper solution of equation (15).

Proof. Let \(f(t)\equiv 0\). Since \(x_1(t)\) and \(x_2(t)\) are solutions of (1), we have

\[ \int_a^t m_x\{K(t,s,x_1(s))\}\,ds\leqslant x_1(t)\leqslant \int_a^t M_x\{K(t,s,x_1(s))\}\,ds \]

and

\[ \int_a^t m_x\{K(t,s,x_2(s))\}\,ds\leqslant x_2(t)\leqslant \int_a^t M_x\{K(t,s,x_2(s))\}\,ds. \]

Then

\[ \int_a^t [m_x\{K(t,s,x_1(s))\}-M_x\{K(t,s,x_2(s))\}]\,ds\leqslant x_1(t)-x_2(t)\leqslant \]

\[ \leqslant \int_a^t [M_x\{K(t,s,x_1(s))\}-m_x\{K(t,s,x_2(s))\}]\,ds. \]

The inequality holds

\[ M_x\{K^i(t, s, x_1)\}-m_x\{K^i(t, s, x_2)\}\leq M_x\{\omega(t, s,\|x_1-x_2\|)\}. \]

Indeed, by condition (16) we have

\[ K^i(t, s, x_1)-K^i(t, s, x_2)\leq M_x\{\omega(t, s,\|x_1-x_2\|)\}. \]

On the other hand, according to the definition of Vrai max and Vrai min, for any \(\varepsilon>0\) there exist such \(y\in R(x_1,\delta)\) and \(z\in R(x_2,\delta)\) that

\[ \underset{x\in R(x_1,\delta)}{\operatorname{Vrai\,max}}\,K^i(t, s, x) - \underset{x\in R(x_2,\delta)}{\operatorname{Vrai\,min}}\,K^i(t, s, x) \leq \]

\[ \leq K^i(t, s, y)-K^i(t, s, z)+2\varepsilon \leq M_x\{\omega(t, s,\|y-z\|)\}+2\varepsilon \]

and the sets of the indicated \(y\) and \(z\) have positive measure. Then, as \(\delta\to0\), \(\varepsilon\to0\), \(y\to x_1\) and \(z\to x_2\). Thus,

\[ M_x\{K^i(t, s, x_1)\}-m_x\{K^i(t, s, x_2)\}\leq M_x\{\omega(t, s,\|x_1-x_2\|)\}. \]

Similarly we prove that

\[ m_x\{K^i(t, s, x_1)\}-M_x\{K^i(t, s, x_2)\}\geq -M_x\{\omega(t, s,\|x_1-x_2\|)\}. \]

Hence

\[ \int_a^t -M_x\{\omega(t, s,\|x_1(s)-x_2(s)\|)\}\,ds \leq x_1^i(t)-x_2^i(t)\leq \]

\[ \leq \int_a^t M_x\{\omega(t, s,\|x_1(s)-x_2(s)\|)\}\,ds \]

or

\[ \|x_1(t)-x_2(t)\|\leq \int_a^t M_x\{\omega(t, s,\|x_1(s)-x_2(s)\|)\}\,ds. \]

By virtue of Theorem 3 we have \(\|x_1(t)-x_2(t)\|\leq u(t)\).

The theorem is proved.

Remark. In condition (16) the non-strict inequality may be replaced by a strict one. Then, by virtue of Theorem 1, \(\|x_1(t)-x_2(t)\|<v(t)\), where \(v(t)\) is the lower solution of equation (15).

In conclusion we note the following:

1. The last theorems make it possible to carry over some results of [2] to equation (1).

2. From Theorem 2 follows the assertion on the differential inequality of [7].

3. Consider the system of differential equations

\[ \begin{aligned} x_1'&=f_1(t,x_1,\ldots,x_n),\\ &\cdots\cdots\cdots\\ x_n'&=f_n(t,x_1,\ldots,x_n), \end{aligned} \tag{17} \]

where \(f_i(t,x_1,\ldots,x_i,\ldots,x_n)\) satisfies A. Z. Azbelev’s condition \(L_1\) with respect to \(x_i\), i.e.

\[ f_i(t,x_1,\ldots,x_i,\ldots,x_n)=g_i(t)x_i+N(t,x_1,\ldots,x_n), \]

where \(g_i(t)\) is summable, and \(N_i(t,x_1,\ldots,x_n)\) is nondecreasing with respect to \(x_k\), \(k=1,\ldots,n\). Suppose further that \(f_i(t,x_1,\ldots,x_n)\) satisfies the conditions of [4]. Then system (17) has solutions \(u_1(t)\) and \(u_2(t)\) in the sense of [4] such that every solution \(u(t)\) of system (17) in the sense of [4] satisfies the inequalities: \(u_1(t)\leq u(t)\leq u_2(t)\). This is a refinement of Vikto-

rovsky [4] follows from Theorem 2, since system (17) reduces to the system of integral equations

\[ x(t)=\int_a^t K(t,s)N(s,x(s))\,ds, \tag{18} \]

where

\[ N(t,x)=\{N_i(t,x_1,\ldots,x_n)\}, \]

and \(K(t,s)\) is a diagonal matrix with elements \(K_{ii}=e^{\int_s^t g_i(\tau)\,d\tau}>0\). The subintegral expression (18) is monotone. In view of the remark to Theorem 2, there exists an interval on which there exist upper \(u_2(t)\) and lower \(u_1(t)\) solutions of (18). It is easy to show that \(u_2(t)\) and \(u_1(t)\) are solutions of system (17) in the sense of Viktorovsky and satisfy the inequalities

\[ u_1(t)\leq u(t)\leq u_2(t), \]

where \(u(t)\) is any solution of system (17) in the sense of Viktorovsky.

In the discussion of the present work at the Izhevsk seminar, various critical remarks were made concerning the definition of solution introduced above. Some of the objections mentioned lose their force if the solution is defined as follows.

A vector function \(x(t)\) is a solution of system (1) if

\[ x(t)=\int_a^t \varphi(t,s)\,ds+f(t), \]

where \(\varphi(t,s)\) is a vector function continuous in \(t\) and satisfying the inequalities

\[ m_x\{K(t,s,x(s))\}\leq \varphi(t,s)\leq M_x\{K(t,s,x(s))\} \]

for almost all \(s\in[a,b)\).

With this definition of solution, the theorems formulated above remain valid.

I take this opportunity to express my gratitude to N. V. Azbelev and Z. B. Tsalyuk for suggesting the topic and for valuable advice in developing it, and to the Izhevsk seminar for its interest in the work.

References

1. Azbelev N. V., Tsalyuk Z. B. Matematicheskii sb., 56 (93): 3, 1962, pp. 325–341.
2. Azbelev N. V., Tsalyuk Z. B. Doklady AN SSSR, 156, No. 2, 239–242, 1964.
3. Filippov A. F. Matematicheskii sb. 51 (93): 1, 1960, pp. 99–128.
4. Viktorovsky E. E. Matematicheskii sb. 34 (76): 2, 1954, pp. 213–249.
5. Li Mun Su. Trudy Izhevskogo matematicheskogo seminara, issue 1, 1963, pp. 18–19.
6. Li Mun Su. Doklady Tret’ei sibirskoi konferentsii po matematike i mekhanike, 1964, pp. 127–128.
7. Li Mun Su. Doklady Tret’ei sibirskoi konferentsii po matematike i mekhanike, 1964, pp. 128–129.
8. Li Mun Su. Differentsial’nye uravneniya, 1, No. 3, 387–392, 1965.
9. Samarov A. B. Doklady Tret’ei sibirskoi konferentsii po matematike i mekhanike, 1964, p. 149.
10. Bokshtein M. F. Uchenye zapiski MGU, mathematics, issue 15, 1939, pp. 13–72.

Received by the editors
January 29, 1965

Izhevsk Mechanical
Institute