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MATHEMATICS
U. M. MUKHTAROV
INVESTIGATION OF A NONLINEAR SINGULAR INTEGRAL EQUATION WITH A SPECIAL CAUCHY KERNEL IN THE CLASS \(H^k_{\alpha-\delta,\delta}(a,b)\)
(Presented by Academician N. I. Muskhelishvili on 11 VII 1967)
In the present note we investigate the equation
\[ U(x)=\lambda (x-a)^\alpha \int_a^b \frac{f[x,s,U(s)]}{(s-a)^\alpha(s-x)}\,ds \tag{1} \]
in the space \(H^k_{\alpha-\delta,\delta}(a,b)\), where \(0<\alpha<1\).
Definition. \(U(x)\in H^k_{\alpha-\delta,\delta}(a,b)\) if the function \(U(x)\) is defined for \(a\le x<b\) and satisfies the conditions:
\[ |U(x)|\le K(x-a)^\delta/(b-x)^{\alpha-\delta}, \]
\[ |U(x+\Delta x)-U(x)|\le K|\Delta x|^\delta/(b-x-\Delta x)^\alpha, \]
where \(0<\delta<\alpha<1,\; 0<|\Delta x|\le \min\{|x-a|/4,\; |b-x|/4\},\; K>0\).
For the exposition of the main result we state, without proof, several lemmas.
Lemma 1. If the function \(\varphi(x,s)\) is defined for \(a\le x\le b,\; a\le s<b\) and satisfies the conditions
\[ |\varphi(x,s)|\le L(s-a)^\delta/(b-s)^{\alpha-\delta}, \]
\[ |\varphi(x+\Delta x,s+\Delta s)-U(x,s)| \le L(|\Delta x|^{\delta_1}+|\Delta s|^\delta)/(b-s-\Delta s)^\alpha, \]
then the function
\[ W(x,s)=(s-a)^{-\alpha}\varphi(x,s) \]
for all \(a\le x\le b,\; a<s<b\) and \(0<|\Delta s|\le \min\{|s-a|/4,\; |b-s|/4\}\) satisfies the conditions:
\[ |W(x,s)|\le LL_1/(s-a)^{\alpha-\delta}(b-s)^{\alpha-\delta}, \]
\[ |W(x+\Delta x,s+\Delta s)-W(x,s)| \le \]
\[ \le LL_1(|\Delta x|^{\delta_1}+|\Delta s|^\delta)/(s-a)^\alpha(b-s-\Delta s)^\alpha, \]
where \(0<\delta<\alpha<1,\; \delta<\delta_1,\; L_1\) is an absolute constant.
Lemma 2. If the function \(\varphi(x,s)\) is defined for \(a\le x\le b,\; a<s<b\) and satisfies the conditions:
\[ |\varphi(x_1,s)|\le B/(s-a)^{\alpha-\delta}(b-s)^{\alpha-\delta}, \]
\[ |\varphi(x+\Delta x,s+\Delta s)-\varphi(x,s)| \le \]
\[ \le B(|\Delta x|^{\delta_1}+|\Delta s|^\delta)/(s-a)^\alpha(b-s-\Delta s)^\alpha, \]
then the function
\[ V(x,s)=(s-a)^\alpha\varphi(x,s) \]
for all \(a\le x\le b,\; a\le s<b\) and \(0<|\Delta s|\le \min\{|s-a|/4,\; |b-s|/4\}\)
satisfies the conditions
\[
|V(x,s)|\le BB_1(s-a)^\delta/(b-s)^{\alpha-\delta},
\]
\[
|V(x+\Delta x,s+\Delta s)-V(x,s)|\le
BB_1(|\Delta x|^{\delta_1}+|\Delta s|^\delta)/(b-s-\Delta s)^\alpha,
\]
where \(B_1=\mathrm{const}\).
Lemma 3. If the function \(f(x,s,u)\) is defined for \(a\le x,\ s<b\), \(-\infty<u<+\infty\), and satisfies the conditions
\[ |f(x+\Delta x,s+\Delta s,u+\Delta u)-f(x,s,u)|\le M_2(|\Delta x|^{\delta_1}+|\Delta s|^\delta)/(b-s-\Delta s)^\alpha+M_2|\Delta u| \tag{2} \]
and the function
\[ \omega(x)=(x-a)^\alpha\int_a^b \frac{f(x,s,0)}{(s-a)^\alpha(s-x)}\,ds \]
belongs to \(H^{k_1}_{\alpha-\delta,\delta}(a,b)\), then for all \(a\le x<b\) and
\[
|\lambda|<K/(K_1+M_3L_1B_1L_2)
\]
the operator
\[ Au=\lambda(x-a)^\alpha\int_a^b \frac{f[x,s,U(s)]}{(s-a)^\alpha(s-x)}\,ds \tag{3} \]
maps the space \(H^k_{\alpha-\delta,\delta}(a,b)\) into itself, where
\[
M_3=\max\{KM_2,\ 2M_1+KM_2\};
\]
\(L_2\) is a constant independent of \(M_3,L_1,B_1\).
Lemma 4. If the function \(f(x,s,u)\) is defined for \(a\le x,\ s<b\), \(-\infty<u<+\infty\), and satisfies condition (2), and \(U(x)\in H^k_{\alpha-\delta,\delta}(a,b)\), then for
\[
\delta_1\ge \delta+2\alpha,\qquad
p>\max\{1/(1-\alpha-\delta),\,1/(\delta_1-\delta-2\alpha)\}
\]
the operator (3) is continuous in the sense of the metric \(L_p(\rho_1)\), where
\[
\rho_1(s)=(b-s)^{(\alpha+\delta)p},\qquad 0<\alpha+\delta<1.
\]
We note that the set \(H^k_{\alpha-\delta,\delta}(a,b)\) is a closed, convex, compact subset of the space \(L_p(\rho_1)\) for any \(p>1\). Consequently, on the basis of the generalized Schauder principle there exists a fixed point, i.e., the following holds.
Theorem 1. If the function \(f(x,s,u)\) satisfies the conditions of Lemmas 3 and 4, then there exists a number
\[ \bar\lambda=K/(K_1+M_3L_1B_1L_2), \]
such that for \(|\lambda|<\bar\lambda\) equation (1) has at least one solution
\[
U(x)\in H^k_{\alpha-\delta,\delta}(a,b).
\]
One can also show the uniqueness of the solution of equation (1).
If the function \(f(x,s,u)\) satisfies the conditions of Lemma 3 and, in addition, the function
\[ K(x,s,u)=f(x,s,u)-f(s,s,u) \]
satisfies the condition
\[ |K(x,s,u)-K(x,s,v)|\le C|x-s|^{\alpha_1}|u-v| \tag{4} \]
for all \(a\le x,s<b\), \(-\infty<u<\infty\), \(0<1-\alpha_1+\alpha+\delta<1\), then the operator (3) acts from \(L_p(\rho_1)\) into \(L_p(\rho_1)\) and satisfies the Lipschitz condition
\[ \|Au-Av\|_{L_p(\rho_1)}\le |\lambda|(RC+M_2F)\|u-v\|_{L_p(\rho_1)}, \]
where
\[ 1<q<q_0;\qquad 1/p+1/q=1;\qquad 0<(1-\alpha_1+\alpha+\delta)q_0<1; \]
\[ R=\left\{\int_a^b s(x)\left[\int_a^b s^{-q/p}(s)|x-s|^{q(\alpha_1-1)}\,ds\right]^{p/q}dx\right\}^{1/p}; \qquad \rho(x)=(b-x)^{(\alpha+\delta)p}(x-a)^{\alpha p}. \]
\(F\) is a constant independent of \(U(s)\).
Since \(H_{\alpha-\delta,\delta}^{k}(a,b)\) is complete in the sense of the metric \(L_p(\rho_1)\), the following is true.
Theorem 2. If the function \(f(x,s,u)\) satisfies the conditions of Theorem 1 and condition (4), then, for
\[ |\lambda|<\bar{\lambda}_{0}=\min\{K/(K_1+M_3L_1B_1L_2),\;1/(RC+M_2F)\}, \]
equation (1) has a unique solution \(U^*(x)\in H_{\alpha-\delta,\delta}^{k}(a,b)\). It can be found by the method of successive Picard approximations. The successive approximations will converge in the sense of the metric of the space \(L_p(\rho_1)\); moreover, if
\[ U_n(x)=\lambda (x-a)^\alpha \int_a^b \frac{f[x,s,U_{n-1}(s)]}{(s-a)^\alpha(s-x)}\,ds, \qquad U_0(x)\in H_{\alpha-\delta,\delta}^{k}(a,b), \]
then
\[ \rho_{L_p(\rho_1)}(U_n,U_0^*)< \frac{\lambda_0^n}{1-\lambda_0}\,\gamma, \]
where \(\gamma\) is a constant independent of \(n\), \(0<\lambda_0=|\lambda|(RC+M_2F)<1\).
Moreover, if (2) is taken into account, then the successive approximations also converge in the sense of the metric \(\rho_{\delta'}\), and the estimate
\[ \rho_{\delta'}(U_n,U_0^*)\le l(K_2)^{1/(1+\delta p)} \frac{\lambda_0^{n\delta p/(1+\delta p)}}{(1-\lambda_0)^{\delta p/(1+\delta p)}} \gamma^{\delta p/(1+\delta p)} + \]
\[ +(2l)^{(\delta-\delta')/\delta}(K_2)^{(1+\delta'p)/(1+\delta p)} \frac{\lambda_0^{(\delta-\delta')pn/(1+\delta p)}}{(1-\lambda_0)^{(\delta-\delta')p/(1+\delta p)}} \gamma^{(\delta-\delta')p/(1+\delta p)} \]
holds, where \(K_2=Kl_1\); \(l_1=\mathrm{const}\); \(l\) is a constant independent of \(K_2\).
If \(U(x,\lambda_1)\) and \(U(x,\lambda_2)\) are solutions of equations (1) corresponding to the parameters \(\lambda_1\) and \(\lambda_2\) \((|\lambda_1|<\lambda_0,\ |\lambda_2|<\lambda_0)\), and the function \(f(x,s,u)\) satisfies the conditions of Theorem 2, then it can be proved that the solution depends continuously on \(\lambda\).
Dagestan State University
named after V. I. Lenin
Received
20 VI 1967
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