Abstract Generated abstract
This note studies a singular boundary value problem for an nth order ordinary differential equation with initial conditions at one endpoint up to order n minus 2 and one additional derivative condition at the other endpoint. The coefficient function is allowed to have nonintegrable singularities at both endpoints, and the paper introduces classes of comparison functions used to control the growth of solutions and their derivatives near the singularities. The main contribution is a set of sufficient conditions ensuring solvability of the problem, formulated through one-sided sign and growth inequalities for the nonlinear term. Several corollaries specialize these conditions to power-type estimates with weighted integrability assumptions.
Full Text
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 \(a<\alpha<b\), \(\lambda_2>1\), 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 \(a<\beta<b\), \(\lambda_1>1\), 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
- L. Ya. Lepin, A. D. Myshkis, DAN, 169, No. 1, 16 (1966).
- Yu. A. Klokov, DAN, 176, No. 3, 512 (1967).