UDC 517.43

MATHEMATICS

Kh. M. Salpagarov

INTEGRAL INEQUALITIES WITH OPERATORS OF VOLTERRA TYPE

(Presented by Academician A. A. Dorodnitsyn, February 1, 1967)

By \(E^+\) we shall denote a linear partially ordered set of nonnegative elements \(u, v, \psi, \ldots\), and by \(K^+\) the set of operators \(L, l, L_i\) \((i = 1, \ldots, n)\) of Volterra type that map the set \(E^+\) into itself, for which the series

\[ R_i(\lambda_i) = (I - \lambda_i L_i)^{-1} = \sum_{k=0}^{\infty} \lambda_i^k L_i^k \]

converge for every value of the complex parameter \(\lambda_i\). We shall assume that the set \(K^+\), together with any two operators, contains their sum, product, and the operator \(T_i(\lambda) = L_i R_i(\lambda)\), if \(L_i \in K^+\) \((\lambda \geq 0)\).

We shall be interested in the following properties of the operators. Multiplying the equality

\[ L_i = (\lambda - \alpha)^{-1}[(I - \alpha L_i) - (I - \lambda L_i)] \qquad (\lambda \ne \alpha) \]

by \(R_i(\alpha)R_i(\lambda)\), we easily arrive at the relation

\[ T_i(\alpha)R_i(\lambda) = (\lambda - \alpha)^{-1}[R_i(\lambda) - R_i(\alpha)], \]

whence, for \(\lambda > \alpha \geq 0\), in turn follows the inequality

\[ T_i(\alpha)R_i(\lambda) \leq (\lambda - \alpha)^{-1}[R_i(\lambda) - I] < (\lambda - \alpha)^{-1}R_i(\lambda). \]

Using this inequality, by the method of complete mathematical induction one can show that

\[ T_i^m(\alpha)R_i(\lambda) < (\lambda - \alpha)^{-m}[R_i(\lambda) - I]. \tag{1} \]

Let now the operators \(L_i\) satisfy the relation \(L_iL_k \leq L_kL_i\) for \(i > k\).

Then, if \(\alpha_i \geq 0\), \(\alpha_k \geq 0\), \(i > k\), then

\[ T_i(\alpha_i)R_k(\alpha_k) \leq R_k(\alpha_k)T_i(\alpha_i). \tag{2} \]

Consider the operator \(R = R_1(\lambda_1)\cdots R_n(\lambda_n)\). If \(\lambda_i > \alpha_i \geq 0\), using inequalities (1) and (2), one can obtain that

\[ T_i^m R \leq (\lambda_i - \alpha_i)^{-m}(R - I). \tag{3} \]

In what follows we shall assume that the nonnegative real numbers \(\lambda_i\) and \(\alpha_{ij}\) \((i = 1, \ldots, n;\ j = 1, \ldots, p)\) satisfy the inequality \(\lambda_i > \alpha_{ij}\), and \(m_{ij}\) will denote integers.

For simplicity put \(T_{ij} = T_i(\alpha_{ij})\). Starting from inequality (3), by the method of complete mathematical induction one can finally prove the inequality

\[ \prod_{i=1}^{n}\prod_{j=1}^{p} T_{ij}^{m_{ij}} R \leq \prod_{i=1}^{n}\prod_{j=1}^{p}(\lambda_i - \alpha_{ij})^{-m_{ij}}(R - I), \tag{4} \]

Let us introduce into consideration the operator

\[ L=\sum_{m=1}^{N} a(m_{11},\ldots,m_{ij},\ldots,m_{np}) \prod_{i=1}^{n}\prod_{j=1}^{p} T_{ij}^{m_{ij}}, \]

where, respectively, \(m=\sum_{i=1}^{n}\sum_{j=1}^{p} m_{ij}\) are natural numbers, and the coefficients \(a\) are nonnegative numbers. The equation

\[ F(\lambda)\equiv \sum_{m=1}^{N} a(m_{11},\ldots,m_{np}) \prod_{i=1}^{n}\prod_{j=1}^{p}(\lambda_i-a_{ij})^{-m_{ij}}=1, \tag{5} \]

where \(\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_n)\), will be called characteristic for the equation \(u=\psi+Lu\). Obviously, \(L\in K^+\).

Let us formulate the main inequality.

Theorem 1. If an element \(u\) satisfies the inequality \(u\leqslant \psi+Lu\) \((u,\psi\in E^+)\), then

\[ u\leqslant R\psi=R_1(\lambda_1)\cdots R_n(\lambda_n)\psi\equiv v(\lambda), \tag{6} \]

where \(\lambda_1,\ldots,\lambda_n\) satisfy the characteristic equation (5).

To prove the assertion of the theorem, consider the expression \(\psi+Lv\). From inequality (4), taking into account that \(\lambda\) satisfy relation (5), it is easy to obtain the inequalities

\[ \psi+Lv=\psi+LR\psi\leqslant \psi+F(\lambda)(R-I)\psi =\psi+(R-I)\psi=R\psi=v, \]

i.e., \(v\geqslant \psi+Lv\). Now the assertion of the theorem follows from the following proposition.

Lemma. If \(u,v\) satisfy the inequalities \(u\leqslant \psi+Lu,\ v\geqslant \psi+Lv\), where \(L\in K^+\), then \(u\leqslant v\).

The assertion of the lemma is proved as follows: subtracting the first inequality from the second, we obtain \(v-u\geqslant L(u-v)\). Hence \(v-u=\psi_1+L(v-u)\) \((\psi_1\in E^+)\), and \(v-u=(I-L)^{-1}\psi_1\geqslant 0\).

Remark 1. The characteristic equation (5) for \(n>1\) has an infinite set of solutions. To each solution there corresponds its own estimate of the form (6). Denote \(v_*=\inf_{F(\lambda)=1} v(\lambda)\). It can be proved: if \(u\) satisfies the condition of Theorem 1, then \(u\leqslant v_*\).

For operators of a special form one can also obtain the reverse inequality.

Theorem 2. If \(v\) satisfies the inequality \(v\geqslant \psi+\sum_{k=1}^{n} a_k L_k v\), then

\[ v\geqslant R_n(a_n)\cdots R_1(a_1)\psi. \]

Corollary. From Theorems 1 and 2 there follows the assertion: if the operator \(l\) is defined by the rule

\[ lu=\sum_{k=1}^{n} a_k L_k u, \]

then

\[ R_n(a_n)\cdots R_1(a_1)\leqslant (I-l)^{-1}\leqslant R_1(\lambda_1)\cdots R_n(\lambda_n), \]

where the nonnegative numbers \(\lambda_1,\ldots,\lambda_n\) satisfy the relation

\[ \lambda_1^{-1}a_1+\cdots+\lambda_n^{-1}a_n=1. \tag{7} \]

Examples. If

\[ L_k u=\int_{0}^{t_k}\alpha_k(s_k)\, u(t_1,\ldots,t_{k-1},\tau_k,t_{k+1},\ldots,t_n)\,d\tau_k, \tag{8} \]

then

\[ T_k(\lambda_k)\psi= \int_{0}^{t_k}\alpha_k \exp\left\{\lambda_k\int_{s_k}^{t_k}\alpha_k\,d\tau_k\right\} \psi(\ldots,t_{k-1},s_k,t_{k+1},\ldots)\,ds_k. \tag{9} \]

and \(R_k(\lambda_k)=I+\lambda_k T_k(\lambda_k)\), where the operators \(L_k\) \((k=1,\ldots,n)\) commute pairwise.

From Theorems 1 and 2 one can derive the following inequalities.

1. If the function \(u(t)\) satisfies the inequality
\[ u(t)\leq \psi(t)\int_0^t +\alpha u\,d\tau, \]
then
\[ u(t)\leq \psi(t)+\int_0^t \alpha \exp\left\{\int_s^t \alpha\,d\tau\right\}\psi\,d\tau . \]

This is the well-known Gronwall—Bellman inequality \((^1)\).

2. If the function \(u=u(t_1,\ldots,t_n)\) satisfies the inequality
\[ u\leq \psi+\sum_{k=1}^{n}L_k u, \]
then
\[ u\leq \prod_{k=1}^{n}(I+\lambda_k T_k)\psi = \psi+\sum_{k=1}^{n}\lambda_k T_k\psi+\cdots+\lambda_1\cdots\lambda_n T_1\cdots T_n\psi; \]
in particular, if \(\psi=C>0\) is constant, then
\[ u\leq C\exp\left\{\sum_{k=1}^{n}\lambda_k\int_0^{t_k}\alpha_k\,d\tau_k\right\}; \tag{10} \]
if
\[ \psi=\int_0^{t_1}\cdots\int_0^{t_n}\varphi\,d\tau_n\ldots d\tau_1, \]
then
\[ u\leq \int_0^{t_1}\cdots\int_0^{t_n} \exp\left\{\sum_{k=1}^{n}\lambda_k\int_{s_k}^{t_k}\alpha_k\,d\tau_k\right\} \varphi\,d\tau_n\ldots d\tau_1, \tag{11} \]
where the operators \(L_k, T_k=T_k(\lambda_k)\) are determined from relations (8) and (9), \(\psi\geq 0\), \(\lambda_1^{-1}+\cdots+\lambda_n^{-1}=1\).

These inequalities improve some inequalities of Wendroff \((^1)\) and inequalities from \((^2)\).

Remark. By varying the numbers \(\lambda_k\), instead of inequalities (10) and (11) one can obtain the sharper inequalities
\[ u\leq C\exp\left\{\sum_{k=1}^{n}\left[\int_0^{t_k}\alpha_k\,d\tau_k\right]^{1/2}\right\}^{2}, \]
\[ u\leq \int_0^{t_1}\cdots\int_0^{t_n} \exp\left\{\sum_{k=1}^{n}\left[\int_{s_k}^{t_k}\alpha_k\,d\tau_k\right]^{1/2}\right\}^{2} \varphi\,ds_n\ldots ds_1 . \]

3. Let now
\[ L_k u=\int_0^t e^{a_k(t-\tau)}u\,d\tau\qquad (k=1,\ldots,n). \]

Then the operators \(L_k\) commute among themselves and
\[ R_k(\lambda_k)\psi = \psi+\lambda_k\int_0^t e^{(\lambda_k+a_k)(t-\tau)}\psi(\tau)\,d\tau . \]

One can also find constants \(C_k\) \((k=1,\ldots,n)\) such that
\[ \prod_{k=1}^{n}R_k(\lambda_k)\psi \leq \psi+\sum_{k=1}^{n}C_k\int_0^t e^{(\lambda_k+a_k)(t-\tau)}\psi(\tau)\,d\tau . \tag{12} \]

Therefore, if the function \(u(t)\) satisfies the inequality

\[ u \leqslant \psi+\sum_{k=1}^{n} a_k \int_{0}^{t} e^{\alpha_k(t-\tau)}u\,d\tau, \]

where \(a_k \geqslant 0,\ \psi \geqslant 0\), then the function \(u\) will be less than the right-hand side of relation (12) for a corresponding choice of the constants \(C_k\). \((\lambda_k>0\) are determined from relation (7).)

1. From Theorems 1 and 2 one can obtain various inequalities also with matrix coefficients; for example, if

\[ u(t)\leqslant \psi(t)+A\int_{0}^{t} e^{P(t-\tau)}u\,d\tau +B\int_{0}^{t} e^{Q(t-\tau)}u\,d\tau, \]

then

\[ \begin{aligned} u(t) \leqslant {}& \psi(t)\lambda A\int_{0}^{t} e^{(\lambda A+P)(t-\tau)}\psi(\tau)\,d\tau +\mu B\int_{0}^{t} e^{(\mu B+Q)(t-\tau)}\psi(\tau)\,d\tau \\ &+\lambda\mu AB\int_{0}^{t} e^{(\lambda A+P)(t-s)} \int_{0}^{s} e^{(\mu B+Q)(s-\tau)}\psi(\tau)\,d\tau, \end{aligned} \]

where \(u,\psi\) are nonnegative vector functions; \(A,B,P,Q\) are square matrices with nonnegative components that are pairwise permutable with one another; \(\lambda>1,\ \mu>1\) satisfy the equality \(\lambda^{-1}+\mu^{-1}=1\).

These inequalities can be used to prove theorems on existence, uniqueness, continuous dependence, and stability of solutions of differential and integral equations in many variables.

Karachay-Cherkess State Pedagogical Institute

Received
30 I 1967

REFERENCES

¹ E. Beckenbach, R. Bellman, Inequalities, Moscow, 1965. ² Kh. M. Salpagarov, Proceedings of the XII Scientific Conference of Kyrgyz University, Frunze, 1964.