Abstract Generated abstract
This note studies a nonlinear singular integral equation with Cauchy kernel in a Hölder-type class of functions that vanish at the endpoints of a finite interval with prescribed power behavior. Under a Lipschitz condition on the nonlinearity, the authors prove mapping and continuity properties for the associated integral operator in weighted metrics, then apply Schauder’s fixed point principle to obtain existence for sufficiently small parameter values. They further establish uniqueness by a contraction estimate in a weighted Lebesgue space and show that successive approximations converge first in that space and then in the weighted uniform metric. The results are also indicated to extend to more general equations with dependence on the variables in the kernel and on an auxiliary singular integral expression.
Full Text
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).