Reports of the Academy of Sciences of the USSR
1970. Volume 192, No. 5

UDC 517.917

MATHEMATICS

I. T. Kiguradze

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

(Presented by Academician I. N. Vekua on 9 XII 1969)

In the present note, sufficient conditions are given for the solvability of the boundary value problem

[
u^{(n)}=f(t,u,u',\ldots,u^{(n-1)}),
\tag{1}
]

[
u^{(i-1)}(a)=u_{0i}\quad (i=1,2,\ldots,n-1),\qquad
u^{(m-1)}(b)=u_0,
\tag{2}
]

where (1\le m\le n-1), (-\infty<a<b<+\infty), (-\infty<u_{0i},u_0<+\infty) ((i=1,2,\ldots,n-1)), and the function (f(t,x_1,\ldots,x_n)) is defined in the domain (a<t<b), (-\infty<x_1,\ldots,x_n<+\infty), is measurable with respect to (t), continuous with respect to (x_1,\ldots,x_n), and

[
f^*(t;\rho)=\sup{|f(t,x_1,\ldots,x_n)|:\ |x_k|\le \rho\ (k=1,2,\ldots,n)}\in
]

[
\in L(a+\delta,b-\delta)
]

for any (\rho\in(0,+\infty)) and (\delta\in(0,b-a/2)).

In contrast to the cases considered by other authors (see, for example, ((^1,^2))), below it is assumed that the function (f(t,x_1,\ldots,x_n)), having singularities at (t=a) and (t=b), is in general not summable with respect to (t) on the interval (a<t<b).

For what follows it is convenient to introduce the following

Definition. Let (\omega(t,x_1,\ldots,x_k)) be a nonnegative function defined in the domain (t_1<t<t_2,\ -\infty<x_1,\ldots,x_n<+\infty). We shall say that it belongs to the set (B_j^k(t_1,t_2)), where (j\in{1,2}), if for every (\rho\in(0,+\infty)) there exists a nonnegative function (\varphi_\rho(t)), continuous on the interval (t_1<t<t_2), such that ((t-t_j)^{k-1}\varphi_\rho(t)\in L(t_1,t_2)) and (|v^{(k)}(t)|\le \varphi_\rho(t)) for (\min{t_0,t_j}<t<\max{t_0,t_j}), whatever the number (t_0\in[t_1,t_2]) and the function (v(t)), absolutely continuous together with (v^{(i)}(t)) ((i=1,2,\ldots,k)) on the interval (t_1\le t\le t_2) and satisfying the conditions

[
|v^{(i-1)}(t)|\le \rho |t-t_j|^{1-i}\quad (i=1,2,\ldots,k),
]

[
v^{(k+1)}(t)\operatorname{sign}[(t_j-t)v^{(k)}(t)]
\le
\omega(t,v'(t),\ldots,v^{(k)}(t))
\quad \text{for } t_1<t<t_2,
]

[
|v^{(k)}(t_0)|<\rho .
]

By (D_r(t_1,t_2)) below we denote the set

[
D_r(t_1,t_2)=
]

[
={t,x_1,\ldots,x_n):\ t_1<t<t_2,\ |x_k|\le r\ (k=1,\ldots,m),
]

[
|x_k|\le r(b-t)^{m-k}\ (k=m+1,\ldots,n-1),\ |x_n|<+\infty}.
]

Theorem 1. If

[
f(t,x_1,\ldots,x_{n-1},0)\operatorname{sign}x_{n-1}\ge 0
\quad \text{for } a<t<b,
]

[
-\infty<x_1,\ldots,x_{n-2}<+\infty,\qquad
r_0\le |x_{n-1}|<+\infty,
]

[
f(t,x_1,\ldots,x_n)\operatorname{sign}x_n
\ge
-\omega_1(t,x_n)
\quad \text{for } (t,x_1,\ldots,x_n)\in D_r(a,\beta),
\tag{3}
]

[
f(t,x_1,\ldots,x_n)\operatorname{sign}x_n
\le
\omega_2(t,x_{m+1},\ldots,x_n)
\quad \text{for } (t,x_1,\ldots,x_n)\in D(a,\beta),
]

where (a \leq \alpha < \beta \leq b),

[
r_0>0,\quad r=(n-m)!(n-1)(1+b-a)^{n-2}\times
]
[
\times \max{|u_{0i}|\ (i=1,2,\ldots,n-1),\ |u_0|,\ r_0},
]
[
\omega_1(t,x_1)\in B_1^1(a,\beta),\qquad
\omega_2(t,x_1,\ldots,x_{n-m})\in B_2^{\,n-m}(a,b),
\tag{4}
]

then problem (1)—(2) is solvable.

With a special choice of the functions (\omega_1(t,x_1)) and (\omega_2(t,x_1,\ldots,x_{n-m})), from Theorem 1 one can obtain a number of sufficient conditions for the solvability of problem (1)—(2). We give some of them.

Theorem 2. Let conditions (3) and (4) be satisfied,

[
f(t,x_1,\ldots,x_n)\operatorname{sign}x_n
\geq -h_1(t)(1+|x_n|)^{\lambda_1}
\quad\text{for }(t,x_1,\ldots,x_n)\in D_r(a,\beta),
\tag{5}
]

[
f(t,x_1,\ldots,x_n)\operatorname{sign}x_n
\leq h_2(t)(1+|x_n|)^{\lambda_2}
\quad\text{for }(t,x_1,\ldots,x_n)\in D_r(a,\beta),
\tag{6}
]

where (a\leq\alpha<\beta\leq b), and (\lambda_1) and (h_1(t)) satisfy one of the following two conditions:

1) (\lambda_1<1,\quad h_1(t)\geq0,\quad h_1(t)\in L(t_0,\beta)) for every (t_0\in(a,\beta)), and
[
\left[\int_t^\beta h_1(\tau)\,d\tau\right]^{\frac{1}{1-\lambda_1}}\in L(a,\beta);
]

2) (1\leq\lambda_1\leq2) and
[
(1+|\ln(t-a)|)^{-1}h_1(t)\in L^{p_1}(a,\beta),
]

where
[
p_1=\frac{1}{2-\lambda_1}\quad\text{if }\lambda_1<2,\qquad
p_1=+\infty\quad\text{if }\lambda_1=2,
]

and (\lambda_2) and (h_2(t)) satisfy one of the following two conditions:

1) (\lambda_2<1,\quad h_2(t)\geq0,\quad h_2(t)\in L(a,t_0)) for every (t_0\in(a,b)), and
[
(b-t)^{n-m-1}\left[\int_a^t h_2(\tau)\,d\tau\right]^{1/(1-\lambda_2)}\in L(a,b);
]

2) (1\leq\lambda_2\leq2) and
[
(b-t)^{(n-m-1)(1-\lambda_2)}(1+|\ln(b-t)|)^{-1}h_2(t)\in L^{p_2}(a,b),
]

where
[
p_2=\frac{1}{2-\lambda_2}\quad\text{if }\lambda_2<2,\qquad
p_2=+\infty\quad\text{if }\lambda_2=2.
]

Then problem (1)—(2) is solvable.

Theorem 3. Let conditions (3), (4), and (5) be satisfied, where (\lambda_1) and (h_1(t)) satisfy the conditions of Theorem 2. Suppose further that

[
f(t,x_1,\ldots,x_n)\operatorname{sign}x_n
\leq h_{20}(t)+
]
[
+\sum_{k=1}^{n-m} h_{2k}(t)(1+|x_{m+k}|)^{(n-m+1)/(k-1/kp_{2k})}
\quad\text{for }(t,x_1,\ldots,x_n)\in D_r(a,b),
]

where
[
a<\alpha<b,\qquad 1\leq p_{2k}<+\infty\quad (k=1,2,\ldots,n-m),
]
[
(b-t)^{n-m}h_{20}(t)\in L(a,b),\qquad
h_{2k}(t)\in L^{p_{2k}}(a,b)\quad (k=1,2,\ldots,n-m).
]

Then problem (1)—(2) is solvable.

Theorem 4. Let conditions (3), (4), and (6) be satisfied, where (a<\alpha1), the function (h_2(t)) is positive, (h_2(t)\in L(a,b)), and

[
\int_a^b (b-t)^{n-m-1}
\left[\int_t^b h_2(\tau)\,d\tau\right]^{1-(1-\lambda_2)}\,dt=+\infty.
]

Suppose, further,

[
f(t,x_1,\ldots,x_n)\operatorname{sign} x_n \geq -h_1(t)\bigl(1+|x_n|\bigr)^{\lambda_1}
\quad \text{for } (t,x,\ldots,x_n)\in D_r(a,b),
]

where (\lambda_1) and (h_1(t)) satisfy the conditions of Theorem 2 for any (\beta\in(a,b)). Then problem (1)—(2) is solvable.

Theorem 5. Suppose that conditions (3), (4), and (5) are satisfied, where (a<\beta1), the function (h_1(t)) is positive, (h_1(t)\in L(a,\beta)), and

[
\int_a^\beta
\left[
\int_a^t h_1(\tau)\,d\tau
\right]^{1/(1-\lambda_1)} dt
=+\infty .
]

Suppose, further,

[
f(t,x_1,\ldots,x_n)\operatorname{sign} x_n \leq h_2(t)\bigl(1+|x_n|\bigr)^{\lambda_2}
\quad \text{for } (t,x_1,\ldots,x_n)\in D_r(a,b),
]

where (\lambda_2) and (h_2(t)) satisfy the conditions of Theorem 2 for any (\alpha\in(a,b)). Then problem (1)—(2) is solvable.

Institute of Applied Mathematics
of Tbilisi State University

Received
3 XII 1969

CITED LITERATURE

1. L. Ya. Lepin, A. D. Myshkis, DAN, 169, No. 1, 16 (1966).
2. Yu. A. Klokov, DAN, 176, No. 3, 512 (1967).