Reports of the Academy of Sciences of the USSR
1964. Volume 156, No. 3

MATHEMATICS

A. I. Guseinov, Kh. Sh. Mukhtarov

Investigation of a Class of Nonlinear Singular Integral Equations with Cauchy Kernel in the Class of Functions Vanishing at the Endpoints

(Presented by Academician I. N. Vekua, January 11, 1964)

In the paper (¹) we established existence and uniqueness theorems for a bounded solution of the equation

\[ u(x)=\lambda \int_a^b \frac{K[x,s,u(s)]}{s-x}\,ds \tag{1} \]

in the Hölder space \(H_{K,\delta}\) (even if \(K(x,s,u)\) has power growth). The elements of the space \(H_{K,\delta}\) on the segment \([a,b]\) satisfy the conditions

\[ |u(x)|\le K,\qquad |u(x+\Delta x)-u(x)|\le K|\Delta x|^\delta, \]

where \(K=\mathrm{const}\), \(0<\delta<1\).

The purpose of the present note is to prove the theorem of existence and uniqueness of the solution of the equation

\[ u(x)=\lambda q(x)\int_a^b \frac{f[u(s)]}{s-x}\,dS \tag{2} \]

in the class \(H^0_{M,\delta}\), whose elements on the segment \([a,b]\) satisfy the conditions

\[ |u(x)|\le Ml(x), \tag{3} \]

\[ |u(x+\Delta x)-u(x)|\le M|\Delta x|^\delta, \tag{4} \]

where \(M=\mathrm{const}\), \(l(x)=(x-a)^\delta(b-x)^\delta\), \(0<\delta<1\), \(q(x)=(x-a)^{\delta_1}\times (b-x)^{\delta_1}\), \(0<\delta<\delta_1<1\).

In \(H^0_{M,\delta}\) the metric is introduced:

\[ \rho_{C(l_1)}(u,v)=\max_{x\in[a,b]} l_1(x)|u(x)-v(x)|, \]

\[ l_1(x)=[(x-a)(b-x)]^{-\delta'},\qquad 0<\delta'<\delta. \tag{5} \]

We note that \(H^0_{M,\delta}\) is a closed, convex, and compact set in the sense of the metric \(C(l_1)\). In addition, \(H^0_{M,\delta}\subset \mathscr{L}_p(\rho)\), and \(H_{M,\delta}\) is complete in the sense of convergence in \(\mathscr{L}_p(\rho)\).

The space \(\mathscr{L}_p(\rho)\) consists of functions \(u(x)\) for which

\[ \int_a^b \rho(x)|u(x)|^p\,dx<+\infty, \]

where \(\rho(x)=[(x-a)(b-x)]^{-\delta'p}\), \(1<p<\delta'^{-1}\).

Lemma 1. If on the interval \([-M(b-a)^{2\delta},\, M(b-a)^{2\delta}]\) the function \(f(u)\) satisfies the Lipschitz condition

\[ |f(u_1)-f(u_2)| \le A|u_1-u_2|, \tag{6} \]

then the function

\[ w(x)=\int_a^b \frac{f[u(s)]-f(0)}{s-x}\,ds \tag{7} \]

for \(x\in[a,b]\), \(0<\Delta x<\min\left(\frac{|x-a|}{4},\,\frac{|x-b|}{4}\right)\), \(u(x)\in H^0_{M,\delta}\), satisfies the conditions

\[ |W(x)| \le MAL,\qquad |W(x+\Delta x)-W(x)| \le MAL|\Delta x|^\delta, \tag{8} \]

where \(L=\mathrm{const}\), independent of \(M\) and \(A\).

Lemma 2. Under condition (6), the operator

\[ Bu=q(x)\int_a^b \frac{f[u(s)]-f(0)}{s-x}\,ds \tag{9} \]

maps \(H^0_{M,\delta}\) into \(H^0_{M',\delta}\), where \(M'=MC\),

\[ C=\max\{AL(b-a)^{2(\delta_1-\delta)},\, [(b-a)^{2\delta_1-\delta}+(b-a)^{2\delta}]AL\}. \]

Lemma 3. Under condition (6), the operator \(B\) is continuous in the sense of the metric \(C(l_1)\).

Consider the operator

\[ Ku=\lambda q(x)\int_a^b \frac{f[u(s)]}{s-x}\,ds =\lambda Bu+\lambda f(0)\,q(x)\ln\frac{b-x}{x-a}. \tag{10} \]

Since

\[ \left|[(x-a)(b-x)]^{-\delta}\ln\frac{b-x}{x-a}\right|<L', \]

then, by Lemma 2, we have

\[ |v(x)|=|Ku|\le |\lambda|(MC+L'L'')\,l(x), \tag{11} \]

\[ |v(x+\Delta x)-v(x)|\le |\lambda|(MC+L''L''')|\Delta x|^\delta, \tag{12} \]

where \(L''=|f(0)|\), and \(L'''\) is the Hölder constant for the function \(q(x)\ln\frac{b-x}{x-a}\).

From the continuity of the operator \(B\) in the sense of \(C(l_1)\), it is easy to obtain the continuity of the operator \(K\) in the same sense.

Thus, applying Schauder’s principle, the following is established.

Theorem 1. If on \([-M(b-a)^{2\delta},\, M(b-a)^{2\delta}]\) the function \(f(u)\) satisfies condition (6), then there exists a number

\[ \lambda_0=\min\left(\frac{M}{MC+L'L''},\,\frac{M}{MC+L''L'''}\right), \]

such that for \(|\lambda|<\lambda_0\) the nonlinear singular integral equation (2) has at least one solution

\[ u(x)\in H^0_{M,\delta}. \]

Above we noted that the space \(H^0_{M,\delta}\) is complete in the sense of the metric \(\mathcal L_p(\rho)\). On the other hand, for the operator \(K\) it is easy to prove the validity of the inequality

\[ \|Ku-Kv\|\le |\lambda|(b-a)^{2\delta_1}AF\|u-v\|_{\mathcal L_p(\rho)}, \tag{13} \]

where \(F\) is the norm of a certain linear singular operator in the sense of \(\mathcal L_p(\rho)\) (2). Consequently, the following is true.

Theorem 2. If \(f(u)\) satisfies the conditions of Theorem 1 and condition (13), then for

\[ |\lambda|<\min\left(\lambda_0,\ \frac{1}{(b-a)^{2\delta_1}AF}\right) \]

equation (2) has a unique solution \(\varphi(x)\) in the space \(H^0_{M,\delta}\). The successive approximations will converge in the metric of the space \(\mathscr L_p(\rho)\).

We shall now establish the nature of the convergence of the successive approximations. Let

\[ d_n=\left(\int_a^b \rho(x)\,|u_n(x)-\varphi(x)|^p\,dx\right)^{1/p}, \]

where

\[ u_n(x)=\lambda q(x)\int_a^b \frac{f[u_{n-1}(s)]}{s-x}\,ds, \]

\(\varphi(x)\) is the solution of equation (2). We have \(u_n(x)\in H^0_{M,\delta}\) and \(d_n\to 0\) as \(n\to 0\). Noting this, construct the set

\[ E_n(x_0)=\left[x_0-\frac{\sqrt{d_n}}{2},\ x_0+\frac{\sqrt{d_n}}{2}\right], \]

where \(x_0\) is an arbitrary point of the interval \((a,b)\). Note that \(E_n(x_0)\subset(a,b)\). The inequality

\[ \left| \frac{u_n(\xi_n)-\varphi(\xi_n)} {(\xi_n-a)^{\delta'}(b-\xi_n)^{\delta'}} \right| \le d_n^{\,1-1/2p}, \tag{14} \]

is easily established, where \(\xi_n\) is some point of \(E_n(x_0)\). Introduce the notation:

\[ u_n^*(x)=[(x-a)(b-x)]^{-\delta'}u_n(x), \qquad \varphi^*(x)=[(x-a)(b-x)]^{-\delta'}\varphi(x). \]

Since \(\varphi(x)\) and \(u_n(x)\) belong to \(H^0_{M,\delta}\), by virtue of (3) (see p. 25),

\[ [(x-a)(b-x)]^{-\delta'}u_n(x)=u_n^*(x)\in H_{M'',\,\delta-\delta'} . \]

We may now estimate the differences:

\[ \left|(x_0-a)^{-\delta'}(b-x_0)^{-\delta'}[u_n(x_0)-\varphi(x_0)]\right| \le \]

\[ \le |u_n^*(x_0)-u_n^*(\xi_n)| +|u_n^*(\xi_n)-\varphi^*(\xi_n)| +|\varphi^*(x_0)-\varphi^*(\xi_n)|, \]

or

\[ \left|[(x_0-a)(b-x_0)]^{-\delta'}[u_n(x_0)-\varphi(x_0)]\right| \le 2M''d_n^{(\delta-\delta')/2}+d_n^{\,1-1/2p}. \tag{15} \]

Since \(x_0\) is an arbitrary point of \([a,b]\), from (15) we obtain that the sequence \(\{u_n(x)\}\) converges to \(\varphi(x)\) in the metric \(C(l_1)\).

Thus the following has been proved.

Theorem 3. Convergence of a sequence of elements of \(H^0_{M,\delta}\) in the metric \(\mathscr L_p(\rho)\) entails convergence of the same sequence in the metric \(C(l_1)\).

From Theorems 2 and 3 it follows:

Theorem 4. Under the conditions of Theorem 2, the unique solution in \(H^0_{M,\delta}\) of equation (2) can be found by the method of successive approximations. The successive approximations converge in the metric \(C(l_1)\).

Remark. These results are valid for more general equations

\[ u(x)=\lambda q(x)\int_a^b \frac{f[x,s,u(s)]}{s-x}\,ds, \]

\[ u(x)=\lambda F[x,w(x)], \]

where

\[ w(x)=q(x)\int_a^b \frac{f[x,s,u(s)]}{s-x}\,ds. \]

Azerbaijan State University
named after S. M. Kirov

Received
6 I 1964

REFERENCES

¹ A. I. Guseinov, Kh. Sh. Mukhtarov, DAN, 146, No. 2 (1962). ² B. V. Khvedelidze, Tr. Tbilissk. matem. inst., 23 (1956). ³ A. A. Babaev, Uch. zap. Azerb. gos. univ. im. S. M. Kirova, ser. phys.-math. and chem. sciences, No. 1 (1961).