Full Text
Kh. Sh. Mukhtarov
Investigation of a Class of Nonlinear Singular Equations with Hilbert Kernel
(Presented by Academician I. N. Vekua, 11 VII 1963)
In the present article we study the equation
\[ u(x)=\lambda \frac{a(x)}{2\pi}\int_{-\pi}^{+\pi} f(s,u(s))\operatorname{ctg}\frac{s-x}{2}\,ds+b(x) \tag{1} \]
in the space \(H_k(\varphi)\), which is a generalization of the Hölder space \(H_{k,\delta}\).
Let \(\varphi(\sigma)\) be a continuous monotonically increasing function satisfying the condition: there exists a constant \(c>1\) such that
\[ 1<\lim_{\sigma\to 0}\frac{\varphi(c\sigma)}{\varphi(\sigma)} \leq \overline{\lim_{\sigma\to 0}}\frac{\varphi(c\sigma)}{\varphi(\sigma)}<c, \qquad \varphi(0)=0. \]
Definition. A periodic function \(u(x)\) of period \(2\pi\), defined on \([-\pi,\pi]\), is said to belong to the space \(H_k(\varphi)\) if it satisfies the conditions:
\[ |u(x)|\leq k,\qquad |u(x+\Delta x)-u(x)|\leq k\varphi(|\Delta x|), \]
where \(k=\mathrm{const}\).
We note that an equation of the form (1) was studied in papers \((^{1,2})\) in the space \(H_{k,\delta}\). It is not hard to see that \(H_{k,\delta}\in H_k(\varphi)\), and \(H_k(\varphi)\) is a complete compact metric space in the sense of the metric \(C(-\pi,\pi)\).
In \(H_k(\varphi)\) we introduce two metrics. If \(u(x),v(x)\in H_k(\varphi)\), then
\[ \rho_{C(-\pi,\pi)}(u,v)=\max |u(x)-v(x)|, \tag{2} \]
\[ \rho_{L_2(-\pi,\pi)}(u,v)= \left[\int_{-\pi}^{\pi}|u(s)-v(s)|^2\,ds\right]^{1/2}. \tag{3} \]
Since
\[ \rho_{L_2(-\pi,\pi)}(u,v)\leq \sqrt{2\pi}\,\rho_{C(-\pi,\pi)}(u,v), \]
then, by virtue of Lemma 3 \((^3)\), \(H_k(\varphi)\) is complete with respect to the metric (3).
We first give several lemmas.
Lemma 1. If a sequence \(\{u_n(x)\}\in H_k(\varphi)\) converges to \(u_0(x)\in H_k(\varphi)\) in the sense of the metric (3), then the same sequence also converges in the sense of the metric (2).
Proof. Let
\[ a_n=\int_{-\pi}^{\pi}|u_n(x)-u_0(x)|^2\,dx,\qquad \lim_{n\to\infty}a_n=0. \]
Without loss of generality, let \(a_n < \pi^2\). Introduce the following sets:
\[ E_n(x_0)= \begin{cases} [x_0,\, x_0+\sqrt{a_n}], & \text{if } -\pi \leq x_0 \leq 0,\\ [x_0-\sqrt{a_n},\, x_0], & \text{if } 0<x_0 \leq \pi. \end{cases} \]
Obviously, \(E_n(x_0)\subset[-\pi,\pi]\).
Applying the theorem on the mean value of an integral, we obtain
\[ a_n \geq \int_{E_n(x_0)} |u_n(x)-u_0(x)|^2\,dx = |u_n(\xi_n)-u_0(\xi_n)|^2 \operatorname{mes} E_n(x_0). \]
Hence it follows that
\[ |u_n(\xi_n)-u_0(\xi_n)| \leq \sqrt[4]{a_n}. \]
Since \(\xi_n\in E_n(x_0)\), we have \(|\xi_n-x_0|\leq \sqrt{a_n}\) and, consequently,
\(\varphi(|\xi_n-x_0|)\leq \varphi(\sqrt{a_n})\).
Taking the above into account, we have
\[ |u_n(x_0)-u_0(x_0)| \leq \]
\[ \leq |u_n(x_0)-u_n(\xi_n)|+|u_n(\xi_n)-u_0(\xi_n)| +|u_0(x_0)-u_0(\xi_n)| \leq \]
\[ \leq 2k\varphi(|\xi_n-x_0|)+\sqrt[4]{a_n} \leq 2k\varphi(\sqrt{a_n})+\sqrt[4]{a_n}. \]
Thus, for any \(x\in[-\pi,\pi]\) we have
\[ |u_n(x)-u_0(x)| \leq 2k\varphi(\sqrt{a_n})+\sqrt[4]{a_n}. \tag{4} \]
If in inequality (4) we pass to the limit as \(n\to\infty\) and take into account that \(\varphi(0)=0\), we obtain the assertion of Lemma 1.
Lemma 2. If \(u(x)\in H_k(\varphi)\), then
\[ V(x)=-\frac{1}{2\pi}\int_{-\pi}^{+\pi} u(s)\operatorname{ctg}\frac{s-x}{2}\,ds \]
satisfies the conditions
\[ |V(x)|\leq lk,\qquad |V(x+\Delta x)-V(x)|\leq lk\varphi(|\Delta x|), \]
where \(l\) is a constant independent of \(k\).
The assertion of Lemma 2 follows from work (4).
Lemma 3. If \(f(s,u)\) is periodic in \(s\) with period \(2\pi\), defined for \(-\pi\leq s\leq \pi\), \(-k\leq u\leq k\), and satisfies the conditions
\[ |f(s_1,u_1)-f(s_2,u_2)|\leq l_1\varphi(|s_1-s_2|)+l_2|u_1-u_2|, \tag{5} \]
\[ f(s,0)\in H_{k_0}(\varphi) \tag{6} \]
and if \(a(x)\in H_{l_3}^{*}(\varphi)\), \(b(x)\in H_{k_1}(\varphi)\) \((k_1<k)\), then the operator
\[ Bu=\frac{\lambda a(x)}{2\pi}\int_{-\pi}^{+\pi} f(s,u(s))\operatorname{ctg}\frac{s-x}{2}\,ds+b(x) \]
for
\[ |\lambda|<\lambda_0=\frac{k-k_1}{2ll_3(2l_1+l_2k+k_0)} \]
acts from \(H_k(\varphi)\) into \(H_k(\varphi)\) and satisfies the Lipschitz condition, i.e.
\[ \rho_{L_2(-\pi,\pi)}(Bu,Bv)\leq |\lambda|\, l_3l_2\, \rho_{L_2(-\pi,\pi)}(u,v). \]
Hence it follows
Lemma 4. Under the conditions of Lemma 3, if
\[ |\lambda|<\min\left\{\lambda_0,\frac{1}{l_3l_2}\right\}, \]
equation (1) has a unique solution \(u_0(x)\in H_k(\varphi)\), whose successive approximations converge in the sense of the metric (3).
If Lemma 1 is taken into account, then these successive approximations converge in the sense of the metric (2).
Thus, we have proved
Theorem. If \(f(s,u)\) satisfies conditions (5), (6), and \(a(x)\in H_{l_3}(\varphi)\), \(b(x)\in H_{k_1}(\varphi)\) \((k_1<k)\), then the nonlinear singular integral equation (1) has a unique solution in \(H_k(\varphi)\) (for small \(|\lambda|\)), and this solution can be found by the method of successive approximations; the successive approximations will converge uniformly.
Dagestan State University
named after V. I. Lenin
Received
6 VII 1963
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