UDC 517.917

MATHEMATICS

I. T. Kiguradze

ON A SINGULAR NICOLETTI PROBLEM

(Presented by Academician I. N. Vekua on 11 XI 1968)

In the present note we consider the boundary-value problem

\[ dx_i/dt=f_i(t,x_1,\ldots,x_n)\qquad (i=1,2,\ldots,n); \tag{1} \]

\[ x_i(t_i)=0\qquad (i=1,2,\ldots,n), \tag{2} \]

where \(f_i(t,x_1,\ldots,x_n)\) \((i=1,2,\ldots,n)\) are real functions defined in the domain \(R_{a-b}\bigl(a\le t\le b,\ -\infty<x_1,\ldots,x_n<+\infty\bigr)\), \(-\infty<a<b<+\infty\), and \(t_i\in[a,b]\) \((i=1,2,\ldots,n)\). By a solution of this problem we mean absolutely continuous functions \(x_1(t),\ldots,x_n(t)\) on \([a,b]\) satisfying the system (1) for almost all \(t\in[a,b]\) and the boundary conditions (2).

Problem (1)—(2) was posed by O. Nicoletti \((^4)\). Various sufficient conditions for the existence and uniqueness of its solution were given in \((^{1-7})\). In contrast to the cases considered by other authors, in the present note it is assumed that the functions \(f_i(t,x_1,\ldots,x_n)\) \((i=1,2,\ldots,n)\), having singularities at \(t=t_i\) \((i=1,2,\ldots,n)\), in general are not summable with respect to \(t\) on the interval \([a,b]\).

Below we adopt the following notation:

The notation \(\varphi(t,x_1,\ldots,x_m)\in K(a,b)\), where \(m\) is an arbitrary natural number, means that the function \(\varphi(t,x_1,\ldots,x_m)\), for every \(r\in(0,+\infty)\), satisfies the Carathéodory conditions in the domain \(a\le t\le b,\ -r\le x_1,\ldots,x_m\le r\), i.e., \(\varphi(t,x_1,\ldots,x_m)\) is measurable in \(t\) on the interval \([a,b]\) for all \(x_i\in[-r,r]\) \((i=1,2,\ldots,m)\), is continuous in \(x_1,\ldots,x_m\) in the domain \(-r\le x_1,\ldots,x_m\le r\) for almost all \(t\in[a,b]\), and

\[ \sup_{|x_i|\le r\ (i=1,2,\ldots,m)} |\varphi(t,x_1,\ldots,x_m)|\in L(a,b). \]

By \(\overline{K}(a,b;t_0)\), where \(t_0\in[a,b]\), is denoted the set of all functions belonging to the set \(K(\alpha,\beta)\) for all \(\alpha\) and \(\beta\) satisfying \(a\le\alpha<\beta\le b,\ t_0\in[\alpha,\beta]\).

\[ l_k=(k-1)^{1/k}\left(\frac{k}{\pi}\sin\frac{\pi}{k}\right)^{-1}. \]

Theorem 1. Let

\[ f_i(t,x_1,\ldots,x_n)\in \overline{K}(a,b;t_i)\qquad (i=1,2,\ldots,n) \tag{3} \]

and let the inequalities

\[ f_i(t,x_1,\ldots,x_n)\,\operatorname{sign}\bigl[(t-t_i)x_i\bigr] \le g_i(t,|x_1|,\ldots,|x_n|) \qquad (i=1,2,\ldots,n), \]

hold in the domain \(R_{ab}\), where the functions \(g_i(t,x_1,\ldots,x_n)\in K(a,b)\) \((i=1,2,\ldots,n)\) are nonnegative and nondecreasing in each \(x_j,\ 1\le j\le n\). Moreover, suppose that there exists a positive number \(M\) such that

\[ \rho_i(t)\le M\quad \text{for } a\le t\le b\quad (i=1,2,\ldots,n) \]

whatever absolutely continuous functions \(\rho_i(t)\) \((i=1,2,\ldots,n)\) on the interval \([a,b]\) may be that satisfy on this interval the conditions

\[ \rho_i(t_i)=0,\qquad 0\le \rho_i'(t)\operatorname{sign}(t-t_i)\le g_i(t,\rho_1(t),\ldots,\rho_n(t)) \]

\[ (i=1,2,\ldots,n). \tag{4} \]

Then problem (1)—(2) has at least one solution.

Corollary 1. Suppose that conditions (3) are satisfied and, in the domain \(R_{ab}\), the inequalities
\[ f_i(t,x_1,\ldots,x_n)\operatorname{sign}\bigl[(t-t_i)x_i\bigr]\leq \]
\[ \leq h_i(t,|x_1|,\ldots,|x_{i-1}|)\omega_i(|x_i|) \qquad (i=1,2,\ldots,n), \]
hold, where \(h_1(t,x_0)\equiv h_1(t)\in L(a,b)\), \(h_i(t,x_1,\ldots,x_{i-1})\in K(a,b)\) \((i=2,\ldots,n)\), and \(\omega_i(x)\) \((i=1,2,\ldots,n)\) are continuous, positive, nondecreasing functions on the interval \([0,+\infty)\) satisfying the conditions
\[ \int_0^{+\infty}\frac{dt}{\omega_i(t)}=+\infty \qquad (i=1,2,\ldots,n). \]
Then problem (1)—(2) has at least one solution.

Corollary 2. Suppose that conditions (3) are satisfied and, in the domain \(R_{ab}\), the inequalities
\[ f_i(t,x_1,\ldots,x_n)\operatorname{sign}\bigl[(t-t_i)x_i\bigr]\leq \]
\[ \leq g_i\left(t,\sum_{j=1}^n |x_j|\right)|x_i| +h_i\left(t,\sum_{j=1}^n |x_j|\right) \qquad (i=1,2,\ldots,n), \]
hold; the functions \(g_i(t,x)\in K(a,b)\) and \(h_i(t,x)\in K(a,b)\) \((i=1,2,\ldots,n)\) are nonnegative and nondecreasing in \(x\), and
\[ \max_{a\leq t\leq b}\sum_{i=1}^n \left| \int_{t_i}^{t} \left( h_i(\tau,x)\exp\left\{\left|\int_{\tau}^{t} g_i(s,x)\,ds\right|\right\} \right)d\tau \right|<x \qquad \text{for } x>M, \]
where \(M\) is some positive constant. Then problem (1)—(2) has at least one solution.

Corollary 3. Suppose that conditions (3) are satisfied and, in the domain \(R_{ab}\), the inequalities
\[ f_i(t,x_1,\ldots,x_n)\operatorname{sign}\bigl[(t-t_i)x_i\bigr] \leq g_i\left(t,\sum_{j=1}^n |x_j|^k\right) \qquad (i=1,2,\ldots,n), \tag{5} \]
hold, \(1<k<+\infty\), and the functions \(g_i(t,x)\in K(a,b)\) \((i=1,2,\ldots,n)\) are nonnegative, nondecreasing in \(x\), and
\[ \int_a^b g(t,x)\,dt<l_k \qquad \text{for } x>M, \tag{6} \]
where \(M\) is some positive constant, and
\[ g(t,x)= \sup_{x\exp(-l_k)\leq y\leq x} \left\{ \sum_{i=1}^n \left[ \frac{g_i(t,y^k)}{y} \right]^k \right\}^{1/k}. \]
Then problem (1)—(2) has at least one solution.

Condition (6) is satisfied, for example, if
\[ \left\{ \sum_{i=1}^n \bigl[g_i(t,|x|^k)\bigr]^k \right\}^{1/k} \leq g(t)|x|+h(t,|x|), \]
\[ g(t)\geq 0 \quad \text{for } a\leq t\leq b,\qquad \int_a^b g(t)\,dt<l_k, \]
and the function \(h(t,x)\in K(a,b)\) is nonnegative, nondecreasing in \(x\), and
\[ \lim_{x\to+\infty}\frac{1}{x}\int_a^b h(t,x)\,dt=0. \]
If, in addition, one assumes that \(f_i(t,x_1,\ldots,x_n)\in K(a,b)\) \((i=1,2,\ldots,n)\),

$k=2$ and, instead of (5), the inequalities

\[ \left| f_i(t,x_1,\ldots,x_n)\right|\leqslant g_i\left(t,\sum_{j=1}^{n}|x_j|^2\right)\quad (i=1,2,\ldots,n), \]

hold, then we obtain the theorem of Lasota—Olech (${}^2$).

Theorem 2. Suppose that for $(t,x_1,\ldots,x_n)$, $(t,y_1,\ldots,y_n)\in R_{ab}$ the inequalities

\[ \begin{aligned} &[f_i(t,x_1,\ldots,x_n)-f_i(t,y_1,\ldots,y_n)]\operatorname{sign}[(t-t_i)(x_i-y_i)] \leqslant \\ &\qquad\leqslant g_i(t,|x_1-y_1|,\ldots,|x_n-y_n|) \quad (i=1,2,\ldots,n), \end{aligned} \]

hold, where the functions $g_i(t,x_1,\ldots,x_n)\in K(a,b)$ $(i=1,2,\ldots,n)$ are nonnegative and nondecreasing in each $x_j$, $1\leqslant j\leqslant n$. Moreover, there do not exist absolutely continuous functions $\rho_i(t)$ $(i=1,2,\ldots,n)$ on the interval $[a,b]$, among which at least one is not identically equal to zero and which would satisfy conditions (4). Then problem (1)—(2) has at most one solution.

Corollary 1. Suppose that for $(t,x_1,\ldots,x_n)$, $(t,y_1,\ldots,y_n)\in R_{ab}$ the inequalities

\[ \begin{aligned} &[f_i(t,x_1,\ldots,x_n)-f_i(t,y_1,\ldots,y_n)]\operatorname{sign}[(t-t_i)(x_i-y_i)] \leqslant \\ &\qquad\leqslant g_i(t)|x_i-y_i|+h_i(t)\sum_{j=1}^{n}|x_j-y_j| \quad (i=1,2,\ldots,n), \end{aligned} \]

hold, where the functions $g_i(t)\in L(a,b)$ and $h_i(t)\in L(a,b)$ $(i=1,2,\ldots,n)$ are nonnegative and

\[ \max_{a\leqslant t\leqslant b}\sum_{i=1}^{n} \left|\int_{t_i}^{t} g_i(\tau)\exp\left(\int_{\tau}^{t} h_i(s)\,ds\right)d\tau\right|<1. \]

Then problem (1)—(2) has at most one solution.

Corollary 2. Suppose that for $(t,x_1,\ldots,x_n)$, $(t,y_1,\ldots,y_n)\in R_{ab}$ the inequalities

\[ \begin{aligned} &[f_i(t,x_1,\ldots,x_n)-f_i(t,y_1,\ldots,y_n)]\operatorname{sign}[(t-t_i)(x_i-y_i)] \leqslant \\ &\qquad\leqslant g_i(t)\left\{\sum_{j=1}^{n}|x_j-y_j|^k\right\}^{1/k} \quad (i=1,2,\ldots,n), \end{aligned} \]

hold, where $1<k<+\infty$, and the functions $g_i(t)\in L(a,b)$ $(i=1,2,\ldots,n)$ are nonnegative and

\[ \int_a^b \left\{\sum_{i=1}^{n}[g_i(t)]^k\right\}^{1/k}\,dt<l_k. \]

Then problem (1)—(2) has at most one solution.

According to Corollaries 2 and 3 of Theorem 1, it is easy to conclude that if $f(t,0,\ldots,0)\in L(a,b)$, $f_i(t,x_1,\ldots,x_n)\in K(a,b;t_i)$ $(i=1,2,\ldots,n)$ and either the conditions of Corollary 1 or the conditions of Corollary 2 of Theorem 2 are satisfied, then problem (1)—(2) has one and only one solution.

Theorem 3. Suppose that

\[ f_i(t,|t-t_1|x_1,\ldots,|t-t_n|x_n)\in K(a,b;t_i), \]

\[ \operatorname*{vrai\,max}_{a\leqslant t\leqslant b}|f_i(t,0,\ldots,0)|<+\infty \quad (i=1,2,\ldots,n) \]

and, for $(t,x_1,\ldots,x_n)$, $(t,y_1,\ldots,y_n)\in R_{ab}$, the inequalities

\[ \begin{aligned} &[f_i(t,x_1,\ldots,x_n)-f_i(t,y_1,\ldots,y_n)]\operatorname{sign}[(t-t_i)(x_i-y_i)] \leqslant \\ &\qquad\leqslant \alpha_i\left|\frac{x_i-y_i}{t-t_i}\right| +\beta_i\sum_{j=1}^{n}\left|\frac{x_j-y_j}{t-t_j}\right| \quad (i=1,2,\ldots,n), \end{aligned} \]

where \(0 \leq \alpha_i, \beta_i < 1\) \((i=1,2,\ldots,n)\) and

\[ u \sum_{i=1}^{n} \frac{\beta_i}{1-\alpha_i} < 1 . \]

Then system (1) has one and only one solution \(x_1(t), \ldots, x_n(t)\), satisfying the boundary conditions

\[ \overline{\lim}_{t \to t_i} \left| \frac{x_i(t)}{t-t_i} \right| < +\infty \qquad (i=1,2,\ldots,n). \]

Tbilisi State University

Received
30 X 1968

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