Mathematics

Yu. A. KLYUKOV

ON A BOUNDARY-VALUE PROBLEM FOR AN ORDINARY DIFFERENTIAL EQUATION OF ORDER \(n\)

(Presented by Academician L. S. Pontryagin, 8 XII 1966)

Consider the boundary-value problem

\[ x^{(n)}=f(t,x,\ldots,x^{(n-1)}), \tag{1} \]

\[ x^{(\nu)}(0)=a_\nu,\qquad x^{(k)}(\tau)=b,\qquad \nu=0,1,\ldots,n-2,\quad 0\le k\le n-3, \tag{2} \]

under the assumption that the function \(f(t,x_0,\ldots,x_{n-1})\) is defined in the domain
\(D_\tau(0\le t\le \tau,\ -\infty<x_0,\ldots,x_{n-1}<\infty)\), is measurable in \(t\) for fixed
\(x_0,\ldots,x_{n-1}\), continuous in \(x_0,\ldots,x_{n-1}\) for fixed \(t\), and for any bounded domain
\(G\) in the space \(x_0,\ldots,x_{n-1}\) there exists a summable function \(g(t)\ge 0\) such that
\(|f(t,x_0,\ldots,x_{n-1})|\le g(t)\), \(0\le t\le \tau\), \((x_0,\ldots,x_{n-1})\in G\) (Carathéodory conditions). In addition, in proving certain assertions we shall assume that the function \(f\) satisfies a generalized local Lipschitz condition in the variables \(x_0,\ldots,x_{n-1}\), i.e., for each bounded domain \(G\) in the space \(x_0,\ldots,x_{n-1}\) there exists a summable function \(L(t)\ge 0\) such that

\[ \left|f(t,x'_0,\ldots,x'_{n-1})-f(t,x''_0,\ldots,x''_{n-1})\right| \le L(t)\sum_{k=0}^{n-1}|x'_k-x''_k|, \]

\[ 0\le t\le \tau,\qquad (x'_0,\ldots,x'_{n-1})\in G,\quad (x''_0,\ldots,x''_{n-1})\in G. \]

Theorem 1. Suppose the function \(f(t,x_0,\ldots,x_{n-1})\) satisfies the Carathéodory conditions and the following conditions:

\((A_n)\). There exists \(h>0\) such that

\[ f(t,x_0,\ldots,x_{n-2},0)x_{n-2}\ge 0 \quad\text{for } |x_{n-2}|\ge h; \]

\[ 0\le t\le \tau,\qquad -\infty<x_0,\ldots,x_{n-3}<\infty. \]

\((B_n^k)\). For every \(R>0\) there exist summable functions \(a_0(t)\ge 0\), \(a_1(t)\ge 0\), \(b_r(t)\ge 0\), \(0\le t\le \tau\), \(r=1,\ldots,m\), a continuous function \(b(z)\ge 0\), \(0\le z<\infty\), and numbers
\(0<\alpha_1<\alpha_2<\cdots<\alpha_m<1/(n-k-1)\) such that:

1) \(b(z)\to 0\) as \(z\to\infty\);

2) for any \(t_0\in[0,\tau]\)

\[ \left|\int_t^{t_0} b_r(s)\,ds\right| \le \beta_r(|t_0-t|)\,|t_0-t|^{(n-k-1)\alpha_r}, \qquad t\in[0,\tau], \]

where \(\beta_r(t)\ge 0\), \(r=1,\ldots,m\), \(0\le t\le \tau\), are continuous monotonically decreasing functions satisfying the condition \(\beta_r(t)\to 0\) as \(t\to 0\);

3) when \(|x_0|+\cdots+|x_k|\le R,\ 0\le t\le \tau\),

\[ |f(t,x_0,\ldots,x_{n-1})| \le a_0(t)+a_1(t)z+\sum_{r=1}^{m} b_r(t)z^{1+\alpha_r} +b(z)z^{(n-k)/(n-k-1)}, \]

where

\[ z=|x_{k+1}|^{\,n-k-1}+|x_{k+2}|^{(n-k-1)/2}+\cdots+|x_{k-1}|. \]

Then a solution of the problem (1), (2) exists.

The proof of the theorem is based on paper \((^1)\) and on the following lemmas.

Lemma 1. Let \(x(t)\), \(0\leq t\leq \tau\), be a solution of problem (1), (2), where the function \(f(t,x_0,\ldots,x_{n-1})\) satisfies condition \((A_n)\) and the generalized local Lipschitz condition with respect to \(x_{n-1}\).

Then there exists a constant \(M>0\), depending on \(h\) and on the constants \(a_\nu\), \(\nu=0,1,\ldots,n-2\), \(b\), \(\tau\), such that
\[ |x(t)|\leq M,\qquad 0\leq t\leq \tau . \]

We note that Lemma 1 ceases to be valid if the function \(f\) does not satisfy the generalized local Lipschitz condition with respect to \(x_{n-1}\).

Lemma 2. Let \(x(t)\), \(0\leq \tau_1\leq t\leq \tau_2\leq \tau\), be a solution of equation (1), where the function \(f\) satisfies condition \((B_n^k)\). Then for every \(M>0\) one can indicate an \(N>0\) such that if \(|x^{(k)}(t)|<M\), then \(|x^{(n-1)}(t)|<N\), \(\tau_1\leq t\leq \tau_2\).

We note that condition \((B_n^k)\) is sharp and in the general case cannot be improved. As functions \(b_r(t)\) satisfying condition \((B_n^k)\), one may take, for example, functions of the form
\[ b_r(t)=\sum_{s=1}^{p}\frac{b_{r,s}(t)} {|t-t_{r,s}|^{1-(n-k-1)\alpha_r}},\qquad t_{r,s}\in[0,\tau], \]
where \(b_{r,s}(t)\geq 0\), \(s=1,\ldots,p\), \(0\leq t\leq \tau\), are continuous functions, and \(b_{r,s}(t_{r,s})=0\). Lemma 2 strengthens the corresponding results of papers \((^2,^3)\).

Theorem 2. If the function \(f(t,x_0,\ldots,x_{n-1})\) is nondecreasing in \(x_0,\ldots,x_{n-2}\) and satisfies the generalized local Lipschitz condition in \(x_0,\ldots,x_{n-1}\), then problem (1), (2) cannot have two solutions.

Remark. The assertion of Theorem 1 remains valid if conditions (2) are replaced by the somewhat more general ones:
\[ x^{(\nu)}(0)=\varphi_\nu\bigl(x^{(n-1)}(0)\bigr),\qquad x^{(k)}(\tau)=-\varphi\bigl(x^{(n-1)}(\tau)\bigr) \tag{3} \]
\[ \nu=0,1,\ldots,n-2;\quad 0\leq k\leq n-3, \]
where \(\varphi_\nu(s)\) and \(\varphi(s)\), \(-\infty<s<\infty\), are continuous functions, bounded above for \(s\leq 0\) and bounded below for \(s\geq 0\). If \(\varphi_\nu(s)\) and \(\varphi(s)\) are nondecreasing functions of \(s\) and the conditions of Theorem 2 are fulfilled, then problem (1), (3) cannot have two solutions.

Theorem 3. Consider the boundary conditions
\[ x^{(\nu)}(0)=a_\nu,\qquad x^{(n-2)}(\tau)=b,\qquad \nu=0,\ldots,n-2 . \tag{4} \]

A solution of problem (1), (4) exists if the function \(f(t,x_0,\ldots,x_{n-1})\) satisfies condition \((A_n)\) of Theorem 1 and condition \((B_n^{\,n-2})\). For every \(R>0\) there exist summable functions \(a_0(t)\geq 0\), \(a_1(t)\geq 0\), \(b_r(t)\geq 0\), \(r=1,\ldots,m\), \(0\leq t\leq \tau\), a continuous positive nondecreasing function \(b(z)>0\), \(0\leq z<\infty\), and constants \(a_k>0\), \(0<\alpha_1<\cdots<\alpha_n=1\), such that
\[ |f(t,x_0,\ldots,x_{n-1})| \leq a_0(t)+a_1(t)|x_{n-1}|+\sum_{r=1}^{m} b_r(t)|x_{n-1}|^{1+\alpha_r} + \]
\[ {}+b(|x_{n-1}|)x_{n-1}^{2}, \]
\[ 0\leq t\leq \tau,\qquad |x_0|+\cdots+|x_{n-2}|<R,\qquad -\infty<x_{n-1}<\infty, \]
and, moreover, for every \(t_0\in[0,\tau]\)
\[ \int_{t}^{t_0} \frac{ds}{ \left|\sum_{r=1}^{m}\int_{s}^{t_0} b_r(u)\,du\right|^{1/\alpha_r} + \left|\int_{s}^{t_0} b(|\dot v(u)|)\,du\right|} =\infty, \]
where \(\dot v(t)\) is a solution of the equation
\[ \ddot v=b(|\dot v|)\dot v^{\,2}, \]
satisfying the condition
\[ \dot v(t)\to\infty\quad \text{as } t\to t_0 . \]

We note that condition \((\mathrm{B}_n^{\,n-2})\) is also sharp and in the general case cannot be improved. As functions \(b_r(t)\), \(0 \leq t \leq \tau\), satisfying condition \((\mathrm{B}_n^{\,n-2})\), one may take, for example, functions of the form

\[ b_r(t)=\sum_{s=1}^{p}\frac{b_{r,s}(t)}{|t-t_{r,s}|^{1-\alpha_r}}\,|\ln |t-t_{r,s}||^{\beta_r},\quad t_{r,s}\in[0,\tau], \]

where \(b_{r,s}(t)\) are continuous functions and \(\beta_r \leq \alpha_r\). In this case, as the function \(b(z)\) one may take, for example, the function \(b(z)=c_1+c_2\ln(1+|z|)\), where \(c_1 \geq 0\) and \(c_2 \geq 0\) are arbitrary constants.

Riga Institute of Civil Aviation Engineers
named after the Lenin Komsomol

Received
8 XII 1966

REFERENCES

1. R. Conti, Ann. mat. pura ed appl., 57, 49 (1962).
2. Yu. A. Klokov, Boundary-value problems with a condition at infinity for equations of mathematical physics, Riga, 1963.
3. A. Ya. Lepin, A. D. Myshkis, Differential Equations, 1, No. 9, 1260 (1965).