Full Text
UDC 517.948.32+517.544
MATHEMATICS
A. I. GUSEINOV, Kh. Sh. MUKHTAROV
ON A METHOD FOR INVESTIGATING NONLINEAR SINGULAR EQUATIONS
(Presented by Academician I. N. Vekua, 25 IX 1965)
In the present note a new method is proposed for investigating the equation
\[ u(x)=\lambda\int_a^b \frac{f[x,s,u(s)]}{s-x}\,ds \tag{1} \]
in the Guseinov class \(H_{\alpha,\beta,\delta}^{K}\) (1).
We first give several functional inequalities for functions from the Hölder class \(H,\delta\) and \(H_{\alpha,\beta,\delta}^{K}\) \((0<\alpha+\delta,\ \beta+\delta,\ \delta,\ \alpha,\ \beta<1)\), which were obtained by one of the authors of the present note.
\(1^\circ\). Let \(u(x)\in H_{\alpha,\beta,\delta}^{K}\). Then \(W(x)=u(x)\psi(x)\) belongs to the class \(H_{l_1,K,\delta}^{0}\), where \(\psi(x)=(x-a)^{\alpha+\delta}(b-x)^{\beta+\delta}\), and \(H_{l_1,K,\delta}^{0}\) is the Hölder class of functions that vanish at the endpoints of the interval \((a,b)\),
\[ l_1=\left(\frac54\right)^{\alpha+\delta} +4^{-\alpha}\left(\frac54\right)^{\beta+\delta}(b-a)^\delta +4^{-\beta}(b-a)^\delta . \]
\(2^\circ\). If \(u(x)\in H_{K,\delta}^{0}\), then \(W(x)=u(x)(x-a)^{-\alpha-\delta}(b-x)^{-\beta-\delta}\) belongs to the class \(H_{\alpha,\beta,\delta}^{l_2K}\); \(l_2\) is a constant independent of \(K\).
\(3^\circ\). If \(u(x)\in H_{K,\delta}\), then
\[ \max_{a\le x\le b}|u(x)| \le lK^{1/(1+\delta p)} \left\{\int_a^b |u(x)|^p\,dx\right\}^{\delta/(1+\delta p)}, \qquad p>0; \]
\[ l= \begin{cases} \dfrac{(b-a)^{p\delta^2/(1+\delta p)}}{2^\delta} -\dfrac{\sqrt[p]{2}}{(b-a)^{\delta/(1+\delta p)}}, & \text{if } b-a<\dfrac{2^{(1+\delta p)/\delta p}}{(\delta p)^{1/\delta}}=\delta_0,\\[1.2em] (\delta p)^{1/(1+\delta p)} +\left(\dfrac{1}{\delta p}\right)^{\delta p/(1+\delta p)}, & \text{if } b-a\ge \delta_0. \end{cases} \]
From \(1^\circ\) and \(3^\circ\) it follows:
\(4^\circ\). If \(u(x)\in H_{\alpha,\beta,\delta}^{K}\), then
\[ \sup_{a<x<b}\{|u(x)|\psi(x)\} \le l(l_1K)^{1/(1+\delta)} \left\{\int_a^b \rho(x)|u(x)|^p\,dx\right\}^{\delta/(1+\delta p)}, \]
where \(\rho(x)=[\psi(x)]^p\).
Since \(\rho(x)\leq (b-a)^{\bar\gamma}\rho_1(x)\), \(\rho_1(x)=(x-a)^\gamma(b-x)^{\gamma'}\), \(\bar\gamma=(\alpha+\delta)p+(\beta+\delta)p-\gamma-\gamma'\), \(\varphi-1<\gamma\leq \alpha p+\delta p\), \(\beta p-1<\gamma'\leq \beta p+\delta p\), from \(4^\circ\) we obtain:
\(5^\circ.\) If \(u(x)\in H_{\alpha,\beta,\delta}^{K}\), then
\[ \sup_{a<x<b}\{|u(x)|\psi(x)\}\leq l_3 l(l,K)^{1/(1+\delta p)} \left\{\int_a^b \rho_1(x)|u(x)|^p\,dx\right\}^{\delta/(1+\delta p)}, \]
\[ l_3=(b-a)^{\gamma\delta/(1+\delta p)}. \]
\(6^\circ.\) If \(u(x)\in H_{K,\delta}\), then
\[ \sup_{a\leq x,\,y\leq b} \frac{|u(x)-u(y)|}{|x-y|^{\delta'}} \leq 2^{1-\delta'/\delta}K^{\delta'/\delta} \{\max |u(x)|\}^{(\delta-\delta')/\delta}, \qquad 0<\delta'<\delta . \]
From \(1^\circ\) and \(6^\circ\) it follows
\(7^\circ.\) For \(u(x)\in H_{\alpha,\beta,\delta}^{K}\) one has
\[ \sup_{a<x,\,y<b} \left\{ \frac{|u(x)\psi(x)-u(y)\psi(y)|}{|x-y|^{\delta'}} \right\} \leq \]
\[ \leq 2^{(\delta-\delta')/\delta}(l,K)^{\delta'/\delta} \left\{\sup_{a<x<b}|u(x)|\psi(x)\right\}^{(\delta-\delta')/\delta}. \]
\(8^\circ.\) If \(u(x)\in H_{\alpha,\beta,\delta}^{K}\), then
\[ \sup_{\substack{a<x,\,y<b\\ |y-x|\leq \sigma(x)}} \left\{ \frac{|u(x)-u(y)|}{|x-y|^{\delta'}} (x-a)^{\alpha+\delta}(b-y)^{\beta+\delta} \right\} \leq \]
\[ \leq l_4K^{\delta'/\delta} \left\{\sup_{a<x<b}|u(x)|\psi(x)\right\}^{(\delta-\delta')/\delta}, \]
\[ l_4=\left\{(4/3)^{\alpha+\delta}+(5/4)^{\beta+\delta}\right\}^{(\delta-\delta')/\delta}, \qquad \min\left\{\frac{x-a}{4},\,\frac{b-x}{4}\right\}=\sigma(x). \]
From inequalities \(5^\circ\) and \(8^\circ\) it follows that
\[ \|u\|_1\leq \widetilde f_1\left(\|u\|_{L_p(\rho_1)}\right), \tag{*} \]
where
\[ \widetilde f_1(x)= l_3l(l,K)^{1/(1+\delta p)}x^{\delta p/(1+\delta p)} + l_4\{l_3l(l,K)^{1/(1+\delta p)}\}^{(\delta-\delta')/\delta} x^{p(\delta-\delta')/(1+\delta p)}, \]
\[ \|u\|_1=\sup_{a<x<b}\{|u(x)|\psi(x)\}+ \]
\[ +\sup_{\substack{a<x<b\\ |\Delta x|<\sigma(x)}} \left\{ \frac{|u(x+\Delta x)-u(x)|}{|\Delta x|^{\delta'}} (x-a)^{\alpha+\delta}(b-x-\Delta x)^{\beta+\delta} \right\}, \]
\[ \|u\|_{L_p(\rho_1)} = \left\{ \int_a^b \rho_1(x)|u(x)|^p\,dx \right\}^{1/p}. \]
From inequalities \(5^\circ\) and \(7^\circ\) it follows that
\[ \|u\|_2\leq f_2\left(\|u\|_{L_p(\rho_1)}\right), \tag{**} \]
where
\[ f_1(x)= l_3l(l,K)^{1/(1+\delta p)}x^{\delta p/(1+\delta p)} + 2^{(\delta-\delta')/\delta}(l,K)^{\delta'/\delta} \times \]
\[ \times \{l_3l(l,K)^{1/(1+\delta p)}\}^{(\delta-\delta')/\delta} x^{p(\delta-\delta')/(1+\delta p)}, \]
\[ \|u\|_2=\sup_{a<x<b}\{|u(x)|\psi(x)\}+ \]
\[ +\sup_{\substack{a<x<b\\ |\Delta x|<\sigma(x)}} \left\{ \frac{|u(x+\Delta x)\psi(x+\Delta x)-u(x)\psi(x)|}{|\Delta x|^{\delta'}} \right\}. \]
Theorem 1. Let \(f(x,s,u)\) be defined for \(a<s,\ s<b,\ -\infty<u<+\infty\) and satisfy the conditions
\[ \left| f(x+\Delta x,s+\Delta s,u+\Delta u)-f(x,s,u)\right|\leq \frac{A\{|\Delta x|^{\delta'}+|\Delta s|^\delta\}}{(s-a)^{\alpha+\delta}(b-s-\Delta s)^{\beta+\delta}} +B|\Delta u|, \tag{2} \]
\[ \int_a^b \frac{f(x,s,0)}{s-x}\,ds \in H_{\alpha,\beta,\delta}^{K_0}, \qquad 0<\delta<\delta_1\leq 1 . \tag{3} \]
Suppose, further, that \(F(x,s,u)=f(x,s,u)-f(s,s,u)\) satisfies the condition
\[ |F(x,s,u)-F(x,s,v)|\leq g(x,s)|u-v|, \tag{4} \]
where \(g(x,s)\) is a nonnegative function such that the integral
\[ R_1=\left\{ \int_a^b \rho_1(x) \left[ \int_a^b \rho_1^{-q/p}(s) \left|\frac{g(x,s)}{s-x}\right|^q ds \right]^{1/p} \right\} \tag{5} \]
converges for some \(p>1,\ 1/p+1/q=1,\ \gamma<1,\ \gamma'<1\). Then the operator
\[ Mu=\lambda\int_a^b \frac{f[x,s,u(s)]}{s-x}\,ds \]
for small \(\lambda\) maps \(H_{\alpha,\beta,\delta}^{K}\) into \(H_{\alpha,\beta,\delta}^{K}\) and is a contraction operator in the space \(H_{\alpha,\beta,\delta}^{K}\) with metric
\[ \rho_2(u,v)=\|u-v\|_{L_p(\rho_1)} \]
provided
\[ |\lambda|<\lambda_0=\min\{K/K',1/R_2\},\qquad K'=K_0+C_0(2A+KB), \]
\[ R_2=C_1B+R_1,\quad C_0 \text{ and } C_1 \text{ are certain constants.} \]
If one takes into account inequality \((*)\), then we have proved
Theorem 2. If \(f(x,s,u)\) satisfies the conditions of Theorem 1 and \(|\lambda|<\lambda_0\), then equation (1) has a unique solution \(u_0^*(x)\in H_{\alpha,\beta,\delta}^{\alpha K}\), and this solution can be found by the method of successive Picard approximations
\[ u_n(x)=\lambda\int_a^b \frac{f[x,s,u_{n-1}(s)]}{s-x}\,ds, \qquad u_0(x)\in H_{\alpha,\beta,\delta}^{K}. \]
The successive approximations converge in the sense of the metric
\[ \rho_3(u,v)=\|u-v\|_2, \]
and
\[ \rho_3(u_n,u_0^*)\leq f_1\left\{ \frac{R_3^n}{1-R_3}\rho_2(Mu_0,u_0) \right\}, \tag{6} \]
where \(R_3=|\lambda|R_2<1\).
If one takes into account that
\[ \rho_2(Mu_0,u_0)\leq KR_4,\quad R_4=2\int_a^b \widetilde{\rho}_1(x)\,dx,\qquad \widetilde{\rho}_1(x)=(x-a)^{\gamma-\alpha p}(b-x)^{\gamma'-\beta p}, \]
then, substituting its expression in place of \(f_1(x)\), we obtain
\[ \rho_3(u_n,u_0^*) \leq K\{\tilde l_4 R_3^{n\delta p/(1+\delta p)}+l_5 R_3^{np(\delta-\delta')/(1+\delta p)}\}, \tag{6'} \]
where
\[ \tilde l_4=l_3l_1\left(\frac{R_4}{1-R_3}\right)^{\delta p/(1+\delta p)}, \]
\[ l_5=2^{(\delta-\delta')/\delta}l_1^{(1+\delta' p)/(1+\delta p)}l_3^{(\delta-\delta')/\delta} \left(\frac{R_4}{1-R_3}\right)^{p(\delta-\delta')/(1+\delta p)}. \]
Inequalities of type \((6')\) can also be obtained in the sense of the metric
\[ \rho_4(u,v)=\|u-v\|_1. \]
It should be noted that inequality \((6')\) for nonlinear singular equations has been obtained here for the first time.
On the basis of the inequalities given above, a theorem is proved which characterizes the dependence of the solution of equation (1) on the parameter \(\lambda\).
Theorem 3. Suppose that \(f(x,s,u)\) satisfies all the conditions of Theorem 2. Suppose, further, that \(u(x,\lambda_1)\), \(u(x,\lambda_2)\) are solutions of equation (1), corresponding to the parameters \(\lambda_1,\lambda_2\) \((|\lambda_1|<\lambda_0,\ |\lambda_2|<\lambda_0)\). Then
\[ \sup_{a<x<b}\{|u(x,\lambda_1)-u(x,\lambda_2)|\psi(x)\} \leq R_6|\lambda_1-\lambda_2|^{\delta p/(1+\delta p)}, \]
\[ R_6=l(2l,K)^{1/(1+\delta p)} \left(\frac{R_5}{1-R_2|\lambda_2|}\right)^{\delta p/(1+\delta p)}, \tag{7} \]
\[ R_5^p=\int_a^b \rho_1(x) \left\{\int_a^b \frac{f[x,s,u(s,\lambda_1)]}{s-x}\,ds\right\}^p dx. \]
If in inequality (7) we set \(p=1/(1-\delta)\), taking \(\gamma=\alpha p-1+\varepsilon\), \(\gamma'=\beta p-1+\varepsilon_1\), \(0<\varepsilon,\varepsilon_1<1\), we obtain
\[ \sup_{a<x<b}\{|u(x,\lambda_1)-u(x,\lambda_2)|\psi(x)\} \leq R_6|\lambda_1-\lambda_2|^\delta \]
under the condition that integral (5) converges.
Dagestan State University
named after V. I. Lenin
Received
18 IX 1965
CITED LITERATURE
- A. I. Guseinov, Izv. AN SSSR, ser. matem., 12, 193 (1948).