UDC 517.948.33

UNIQUENESS THEOREMS FOR A SYSTEM OF MULTIDIMENSIONAL VOLTERRA INTEGRAL EQUATIONS

N. V. KASATKINA

Consider the system of integral equations

\[ x(t)= \sum_{\substack{1\le i_1<\cdots<i_k\le n\\ 1\le k\le n}} \int_{a_{i_1}}^{t_{i_1}}\cdots \int_{a_{i_k}}^{t_{i_k}} K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k}, \]
\[ x(t_1,\ldots,t_{i_1-1},s_{i_1},t_{i_1+1},\ldots,t_{i_2-1},s_{i_2},t_{i_2+1},\ldots,t_n)) \,ds_{i_1}\cdots ds_{i_k}+f(t). \tag{1} \]

Here \(t=(t_1,\ldots,t_n)\), \(s_{i_1\ldots i_k}=(s_{i_1},\ldots,s_{i_k})\), \(x(t)\) is a continuous \(m\)-dimensional vector function \(x(t)=\{x^1(t),\ldots,x^m(t)\}\). In what follows it is assumed that

1. the \(m\)-dimensional vector functions \(K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x)\) \((1\le i_1\le \cdots i_k\le n,\ 1\le k\le n)\) satisfy, for \(a_j\le s_j\le t_j<b_j\) \((j=1,2,\ldots,n)\), \(\|x\|<c^*\) \((b_j<\infty,\ c\le \infty)\), the conditions \((K)\).

a) \(K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x)\) is continuous in \(x\) for all \(t\) and almost all \(s_{i_1\ldots i_k}\), and is measurable in \(s_{i_1\ldots i_k}\) for all \(t\) and \(x\).

b) Whatever positive number \(\gamma<c\) is taken, there exist functions summable with respect to \(s_{i_1\ldots i_k}\),

\[ \mu_{\gamma,i_1\ldots i_k}(t,s_{i_1\ldots i_k}) \quad \text{and} \quad \nu_{\gamma,i_1\ldots i_k}(t^1,t,s_{i_1\ldots i_k}), \]

such that

\[ \sup_{\|x\|\le \gamma} \|K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x)\| \le \mu_{\gamma,i_1\ldots i_k}(t,s_{i_1\ldots i_k}), \]

\[ \sup_{\|x\|\le \gamma} \|K_{i_1\ldots i_k}(t^1,s_{i_1\ldots i_k},x) - K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x)\| \le \nu_{\gamma,i_1\ldots i_k}(t^1,t,s_{i_1\ldots i_k}) \]

and for every \(\varepsilon>0\) one can indicate a \(\delta>0\) such that from \(\|t^1-t\|<\delta\) there follows the inequality

\[ \int_{\Gamma_1}\cdots \int \mu_{\gamma,i_1\ldots i_k}(t^1,s_{i_1\ldots i_k}) \,ds_{i_1}\cdots ds_{i_k} + \]

\[ + \left\{ \int_{\Gamma_2}\cdots \int \nu_{\gamma,i_1\ldots i_k}(t^1,t,s_{i_1\ldots i_k}) \,ds_{i_1}\cdots ds_{i_k} \right\} <\varepsilon. \]

Here \(\Gamma_i\) \((i=1,2)\) are sets defined in the following way: if \(A(\tau_{i_1\ldots i_k})\) is the parallelepiped \([a_{i_1},\tau_{i_1};\ldots; a_{i_k},\tau_{i_k}]\), then

\[ \Gamma_2=A(t^1_{i_1\ldots i_k})\cap A(t_{i_1\ldots i_k}),\qquad \Gamma_1=\bigl[A(t^1_{i_1\ldots i_k})\cup A(t_{i_1\ldots i_k})\bigr]\setminus \Gamma_2. \]

\[ \text{*) By } \|x\| \text{ is meant some norm in } R_m,\ \text{for example } \|x\|=\sum_{j=1}^{m}|x^j|. \]

1. \(f(t)=\{f^1(t),\ldots,f^m(t)\}\) is continuous for \(a_j\leq t_j<b_j\) \((j=1,2,\ldots,n)\), \(\|f(a)\|<c\).

Below we shall use the following

Theorem. (Theorem on inequalities). Let the vector-functions \(K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x)\) be nondecreasing in \(x\). Then

1) there exists an unextendable solution \(\overline u(t)\) \(\{\underline u(t)\}\) such that \(\overline u(t)\) \(\{\underline u(t)\}\) is an upper (lower)\(^*\) solution of system (1);

2) if a continuous vector-function \(z(t)\), for \(a_j\leq t_j<b_j\) \((j=1,\ldots,n)\), satisfies the inequality

\[ \varphi(t)=z(t)-\sum \int_{a_{i_1}}^{t_{i_1}}\cdots\int_{a_{i_k}}^{t_{i_k}} K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},z)\,ds_{i_1}\cdots ds_{i_k} \]
\[ -f(t)\leq 0 \quad (\varphi(t)\geq 0), \quad \text{then } z(t)\leq \overline u(t)\quad (z(t)\geq \underline u(t))^{**} \]

for those \(t\) for which \(\overline u(t)\) \((\underline u(t))\) is defined.

In the case of continuous \(K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x)\) the theorem was proved in [5, 6]. Under condition (K) the proof is carried out analogously.

As is known (see, for example, [2, 7, 9]), theorems on inequalities make it possible to investigate various questions of qualitative theory. Below we use the theorem on inequalities to prove uniqueness criteria analogous to the criteria considered in [3, 8].

Theorem 1. Let the following conditions be satisfied:

1. There exist numbers \(\beta_\mu\) and a number \(\alpha\), \(\beta_\mu>-1\), \(0\leq\alpha<1\), such that for \(a_j\leq s_j\leq t_j\leq a_j+\varepsilon\) \((j=1,2,\ldots,n)\), \(\varepsilon>0\), and arbitrary \(x_1,x_2\),

\[ \left\|K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x_1)- K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x_2)\right\|\leq \]
\[ \leq P \prod_{\substack{\mu=i_j\\ j=1,2,\ldots,k}} (s_\mu-a_\mu)^{\beta_\mu} \prod_{\substack{\mu\ne i_j\\ j=1,2,\ldots,k}} (t_\mu-a_\mu)^{\beta_\mu+1} \|x_1-x_2\|^\alpha . \tag{2} \]

1. For \(a_j\leq s_j\leq t_j\leq a_j+\varepsilon\) \((j=1,2,\ldots,n)\),

\[ \|x_1-x_2\|\leq L \prod_{\substack{\mu=i_j\\ j=1,2,\ldots,k}} (s_\mu-a_\mu)^{\frac{1+\beta_\mu}{1-\alpha}} \prod_{\substack{\mu\ne i_j\\ j=1,2,\ldots,k}} (t_\mu-a_\mu)^{\frac{1+\beta_\mu}{1-\alpha}} \equiv M(t,s), \tag{3} \]

\[ L=\mathrm{const}=P^{\frac{1}{1-\alpha}} \left( \sum_{(i_1\ldots i_k)} \prod_{\substack{\mu=i_j\\ j=1,2,\ldots,k}} \frac{1}{1+\beta_\mu} \right)^{\frac{1}{1-\alpha}} \]

the inequality

\[ \left\|K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x_1)- K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x_2)\right\| \leq \frac{k}{(s_{i_1}-a_{i_1})\ldots(s_{i_k}-a_{i_k})} \|x_1-x_2\|, \tag{4} \]

is fulfilled.

\(^*\) The notions of an unextendable upper (lower) solution for a system of integral equations are defined in the same way as for ordinary differential equations (see, for example, [1, 5]).

\(^ {**}\) \(z(t)\leq \overline u(t)\) \((z(t)\geq \underline u(t))\) means \(z^j(t)\leq \overline u^j(t)\) \((z^j(t)\geq \underline u^j(t))\) \((j=1,2,\ldots,n)\).

where

\[ k=\operatorname{const}<\left(\sum_{(i_1\ldots i_k)} \prod_{\substack{\mu=i_j\\ j=1,2,\ldots,k}} \frac{1-\alpha}{1+\beta_\mu}\right)^{-1}. \tag{5} \]

Then the system (1) has a unique solution in a neighborhood of the point \(a\).

Proof. Suppose that the system (1) has, in some neighborhood of the point \(a\), two solutions: \(x_1(t)\) and \(x_2(t)\). We shall show that the estimate (3) holds for \(x_1(t)\) and \(x_2(t)\). From (2), by virtue of the theorem on inequalities, it follows that the estimate (3) is valid if the equation

\[ y(t)=\sum \int_{a_{i_1}}^{t_{i_1}}\cdots \int_{a_{i_k}}^{t_{i_k}} P \prod_{\substack{\mu=i_j\\ j=1,2,\ldots,k}} (s_\mu-a_\mu)^{\beta_\mu} \prod_{\substack{\mu\ne i_j\\ j=1,2,\ldots,k}} (t_\mu-a_\mu)^{\beta_\mu+1} \times \]

\[ \times y^\alpha(\ldots,t_{i_j-1},s_{i_j},t_{i_j+1},\ldots)\, ds_{i_1}\cdots ds_{i_k}=V(y) \tag{6} \]

has an upper solution

\[ \overline{y}(t)\leq L\prod_{\mu=1}^n (t_\mu-a_\mu)^{\frac{1+\beta_\mu}{1-\alpha}} . \tag{7} \]

Since \(\overline{y}(a)=0\), by continuity of the function \(\overline{y}(t)\), in some neighborhood of the point \(a\) we have \(\overline{y}(t)<1\). Obviously, in this neighborhood \(\overline{y}(t)\leq y_r(t)\), where

\[ y_0(t)=1,\qquad y_{r+1}(t)=V(y_r). \tag{8} \]

It is easy to compute that

\[ \overline{y}(t)\leq L_r\prod_{\mu=1}^n (t_\mu-a_\mu)^{(1+\beta_\mu)(1+\alpha+\ldots+\alpha^r)}, \tag{9} \]

where

\[ L_r=P^{1+\alpha+\ldots+\alpha^r} \left(\sum \prod_{\substack{\mu=i_j\\ j=1,2,\ldots,k}} \frac{1}{1+\beta_\mu} \right)^{1+\alpha+\ldots+\alpha^r}. \]

Passing to the limit in the right-hand side of inequality (9), we obtain (7). Put

\[ \overline{K}_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},y)= \]

\[ = \begin{cases} 0, & \text{for } -1\leq y\leq 0,\\[6pt] k\left[\displaystyle\prod_{\substack{\mu=i_j\\ j=1,2,\ldots,k}} (s_\mu-a_\mu)\right]^{-1}\min\{y;\,M(t,s)\}, & \text{for } 0\leq y\leq 1. \end{cases} \tag{10} \]

It is easy to see that the functions \(\overline{K}_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},y)\) satisfy, in a neighborhood of the point \(a\), the conditions \((K)\) and monotonicity with respect to \(y\). Moreover, by virtue of the definition of \(\overline{K}_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},y)\) and (4),

\[ \bigl\|K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x_1) - K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x_2)\bigr\| \leq \overline{K}_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},\|x_1-x_2\|) \]

for \(\|x_1-x_2\|\leq M(t,s)\).

From the theorem on inequalities it follows that the solution of system (1) is unique in a neighborhood of the point \(a\), if the equation

\[ y(t)=\sum \int_{a_{i_1}}^{t_{i_1}}\cdots \int_{a_{i_k}}^{t_{i_k}} \overline{K}_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},y)\,ds_{i_1}\cdots ds_{i_k} \tag{11} \]

has, in a neighborhood of the point \(a\), the upper solution \(y(t)\equiv 0\).

For equation (11), just as for equation (6), by virtue of (7) and (10) we obtain that in some neighborhood of the point \(a\), for any \(r\), the inequality

\[ \overline{y}(t)\leq k^r L \left(\sum_{\substack{\mu=i_j\\ j=1,2,\ldots,k}} \prod \frac{1-\alpha}{1+\beta_\mu}\right)^r \prod_{\mu=1}^{n}(t_\mu-a_\mu)^{\frac{1+\beta}{1-\alpha_\mu}} \tag{12} \]

is satisfied.

Passing to the limit in the right-hand side of inequality (12), by virtue of (5) we obtain that in some neighborhood of the point \(a\)

\[ \overline{y}(t)\leq 0. \tag{13} \]

Since zero is a solution of equation (11) in some neighborhood of the point \(a\), it follows from (13) that \(\overline{y}(t)\equiv 0\), which proves the theorem.

Theorem 2. Suppose there exist numbers

\[ 0<\alpha_j<1,\quad \gamma_{i_1\ldots i_k},\quad g_\mu,\quad \alpha_\mu\gamma_{i_1\ldots i_k}<g_\mu<1, \]

such that in the domain \(a_j\leq s_j\leq t_j<b_j\) \((j=1,2,\ldots,n)\), \(\|x\|<c\), the following conditions are fulfilled:

1. The vector-functions \(K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x)\) \((1\leq i_1\leq\cdots\leq i_k\leq n,\ 1\leq k\leq n)\) and \(f(t)\) are either all nonnegative or all nonpositive, and moreover

\[ N\prod_{\mu=1}^{n}(t_\mu-a_\mu)^{\alpha_\mu}\leq \|f(t)\|, \tag{14} \]

\[ M_{i_1\ldots i_k} \prod_{\substack{\mu=i_j\\ j=1,2,\ldots,k}} (s_\mu-a_\mu)^{\alpha_\mu-1} \prod_{\substack{\mu\ne i_j\\ j=1,2,\ldots,k}} (t_\mu-a_\mu)^{\alpha_\mu} \leq \|K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x)\|, \tag{15} \]

where \(N,\ M_{i_1\ldots i_k}\) are constants, \(N,\ M_{i_1\ldots i_k}\geq 0\).

2. For \(a_j\leq s_j\leq t_j<b_j\) \((j=1,2,\ldots,n)\) and arbitrary \(x_1,x_2\)

\[ \|K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x_1) - K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x_2)\| \leq \]

\[ \leq L\max_{p=1,2}\bigl(\|x_p\|^{-\gamma_{i_1\ldots i_k}}\bigr) \prod_{\substack{\mu\ne i_j\\ j=1,2,\ldots,k}} (t_\mu-a_\mu)^{g_\mu}\, \|x_1-x_2\|, \]

where \(L\) is a constant.

Then the solution of system (1) is unique for \(a_j\leq t_j<b_j\) \((j=1,2,\ldots,n)\).

Proof. Suppose that system (1) has, for \(a_j\leq t_j<b_j\) \((j=1,2,\ldots,n)\), two solutions \(x_1(t)\) and \(x_2(t)\). By virtue of (14) and (15), for any solution of system (1), when \(a_j\leq t_j<b_j\) \((j=1,2,\ldots,n)\), the inequality

\[ \|x(t)\|=\sum \int_{a_{i_1}}^{t_{i_1}}\cdots \int_{a_{i_k}}^{t_{i_k}} \|K_{i_1\ldots i_k}(t,s_{i_1\ldots i_k},x)\|\,ds_{i_1}\cdots ds_{i_k}+ \]

\[ + \|f(t)\| \geqslant N_1 \prod_{\mu=1}^{n} (t_\mu-a_\mu)^{\alpha_\mu}, \tag{16} \]

in which

\[ N_1=\sum M_{i_1\ldots i_k}\prod_{\mu\in(i_1\ldots i_k)} \frac{1}{\alpha_\mu}+N. \]

From condition (2), by virtue of the theorem on inequalities and (16), we have
\(\|x_1(t)-x_2(t)\|\leqslant \bar y(t)\), where \(\bar y(t)\) is the upper solution of the equation

\[ \begin{aligned} y(t)=\sum \int_{a_{i_1}}^{t_{i_1}}\cdots \int_{a_{i_k}}^{t_{i_k}} &N_1L \prod_{\substack{\mu\ne i_j\\ j=1,2,\ldots,k}} (t_\mu-a_\mu)^{g_\mu-\gamma_{i_1\ldots i_k}\alpha_\mu} \\ &\times \prod_{\substack{\mu=i_j\\ j=1,2,\ldots,k}} (s_\mu-a_\mu)^{-\gamma_{i_1\ldots i_k}\alpha_\mu} \,y(\ldots,t_{i_j-1},s_{i_j},t_{i_j+1},\ldots)\, ds_{i_1}\cdots ds_{i_k}. \end{aligned} \tag{17} \]

It is not difficult to see that \(\bar y(t)\equiv 0\) for
\(a_j\leqslant t_j<b_j\) \((j=1,2,\ldots,n)\), which proves the theorem.

Consider the problem

\[ \frac{\partial^m u(t)}{\partial t_1\cdots dt_m}=f(t_1,\ldots,t_m,u), \]

\[ u(t)\big|_{t_i=a_i}=\varphi_i(\ldots,t_{i_1-1},t_{i_1+1},\ldots). \tag{18} \]

This problem can obviously be reduced to a system of integral equations (1). Applying to this system the results formulated above, we obtain uniqueness theorems for equation (18) (cf., for example, [4, 10]).

Taking this opportunity, I thank N. V. Azbelev and Z. B. Tsalyuk for valuable advice and comments, and the participants of the Izhevsk seminar for their attention to the work.

References

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Received by the editors
December 21, 1964

Udmurt State
Pedagogical Institute