Abstract Generated abstract
This paper studies a nonlinear singular integral equation with a shift on a finite interval, under Lipschitz-type and modulus-of-continuity assumptions on the shift function and kernel. Building on previously established existence and uniqueness in a Hölder-type class for sufficiently small parameter values, it proves that the solution depends continuously on the shift function. The argument compares two equations with nearby shifts, derives estimates for the difference of the corresponding singular integral operators, and combines an \(L_p\) bound with a uniform norm estimate to obtain an explicit modulus of continuity for the difference of solutions. The paper also notes related stability with respect to the kernel and Hölder dependence on the parameter.
Full Text
G. M. MAGOMEDOV
CONTINUOUS DEPENDENCE OF SOLUTIONS OF A SINGULAR INTEGRAL EQUATION WITH SHIFT
(Presented by Academician I. N. Vekua, January 18, 1965)
In the present paper we study the solution of the singular integral equation
\[ u(x)=\int_a^b \frac{K[x,s,u(s)]}{s-\alpha(x)}\,ds . \tag{1} \]
The function \(y=\alpha(x)\) maps the segment \([a,b]\) one-to-one onto itself with preservation of orientation; moreover,
\[ 0<m\leq \left|\frac{\alpha(x_2)-\alpha(x_1)}{x_2-x_1}\right|\leq M<\infty \tag{2} \]
for all \(a\leq x_1,x_2\leq b\). Hence \(\alpha(x)\) is a monotone function, \(\alpha(a)=a\), \(\alpha(b)=b\), and there exists an inverse function \(x=\beta(y)\) to \(\alpha(x)\), mapping \([a,b]\) onto itself with preservation of orientation and
\[ 0<\frac{1}{M}\leq \left|\frac{\beta(y_2)-\beta(y_1)}{y_2-y_1}\right|\leq \frac{1}{m}<\infty . \tag{2¹} \]
The function \(K(x,s,u)\) \((a\leq x,s\leq b,\ |u|\leq N)\) satisfies the conditions
\[ |K(x+\Delta x,s+\Delta s,u+\Delta u)-K(x,s,u)|\leq \]
\[ \leq N_1\psi(|\Delta x|)+N_2\varphi(|\Delta s|)+N_3|\Delta u|; \tag{3} \]
\[ K(a,a,u)\equiv K(b,b,u)\equiv 0; \tag{4} \]
\[ |g(x,s,u)-g(x,s,v)|\leq N_4\varphi_1(|x-s|)|u-v|, \tag{5} \]
where \(g(x,s,u)=K(x,s,u)-K(s,s,u)\); the functions \(\psi(t)\), \(\varphi(t)\), \(\varphi_1(t)\) belong to the space \(\Phi\) for \(0<t\leq b-a\). (For the space \(\Phi\) and the classes \(H_n(\varphi)\), see works \((^{1,3})\).) The author has proved the existence and uniqueness of a solution of equation (1) in the class \(H_n(\varphi)\), provided the functions \(\alpha(x)\) and \(K(x,s,u)\) satisfy conditions (2), (3), (4), (5).
Theorem. If \(K(x,s,u)\) and \(\alpha(x)\) satisfy conditions (2), (3), (4), (5), then the solution of equation (1), for sufficiently small \(|\lambda|\), depends continuously on the function \(\alpha(x)\).
We give a scheme of the proof of the theorem. Let \(u_2(x)\) and \(u_1(x)\) be solutions of equations (1) corresponding to \(\alpha_2(x)\) and \(\alpha_1(x)\), and let
\[ \max_{a\leq x\leq b}|\alpha_2(x)-\alpha_1(x)|<\varepsilon . \]
Set \(\alpha_1(x)=y\) \((a\leq x\leq b)\). Then \(x=\beta(y)\) \((a\leq y\leq b)\);
\[ \alpha_2(x)=\alpha_1(x)+[\alpha_2(x)-\alpha_1(x)]=y+\gamma(y), \]
\[ \gamma(a)=\gamma(b)=0,\qquad |\gamma(y)|<\varepsilon \quad (a\leq y\leq b). \]
The function
\[ K^*(y,s,u)=K[\beta(y),s,u] \]
satisfies conditions (3), (4), (5), only \(N_1\) and \(N_4\) change.
We shall have
\[ u_2[\beta(y)]-u_1[\beta(y)] =\lambda \int_a^b \frac{K^*[y,s,u_2(s)]}{s-y-\gamma(y)}\,ds -\lambda \int_a^b \frac{K^*[y,s,u_1(s)]}{s-y}\,ds = \]
\[ =\lambda(Bu_2-Bu_1)+\lambda I^*, \tag{6} \]
where
\[ Bu=\int_a^b \frac{K^*[y,s,u(s)]}{s-y}\,ds, \]
\[ I^*=\int_a^b \frac{K^*[y+\gamma(y),s,u_2(s)]}{s-y-\gamma(y)}\,ds -\int_a^b \frac{K[y,s,u_2(s)]}{s-y}\,ds+ \]
\[ +\int_a^b \frac{K^*[y,s,u_2(s)]-K^*[y+\gamma(y),s,u_2(s)]}{s-y-\gamma(y)}\,ds =\widetilde{B}u_2-Bu_1+I. \tag{7} \]
If we make the substitution \(y=\alpha(x)\), we obtain
\[ I=\int_a^b \frac{f(x,s)-f[x,\alpha_2(x)]}{s-\alpha_2(x)}\,ds +f[x,\alpha_2(x)]\int_a^b \frac{ds}{s-\alpha_2(x)}\,ds =I_1+I_2, \]
where \(f(x,s)=K^*[a_1(x),s,u_2(s)]-K^*[a_2(x),s,u_2(s)]\).
The function \(f(x,s)\) satisfies the conditions
\[ |f(x+\Delta x,s+\Delta s)-f(x,s)| \le N_5\{\psi(|\Delta x|)+\varphi(|\Delta s|)\}; \tag{3'} \]
\[ f(a,a)=f(b,b)=0; \tag{4'} \]
\[ |f(x,s)|\le N_1'\psi[|\alpha_2(x)-\alpha_1(x)|]\le N_1'\psi(\varepsilon) \tag{8} \]
for \(a\le x,\ x+\Delta x,\ s,\ s+\Delta s\le b\).
Using (8), \((3')\), and the properties of the functions of the space \(\Phi\), we obtain
\[ |I_1|\le \{2N_1'\psi(\varepsilon)\}^{1-\mu}N_5^\mu \int_a^b \frac{[\varphi(|s-\alpha_2(x)|)]^\mu}{|s-\alpha_2(x)|}\,ds \le M_1[\psi(\varepsilon)]^{1-\mu}, \]
\[ |I_2|\le M_2[\psi(\varepsilon)]^{1-\mu}, \]
\[ |I|\le M_3[\psi(\varepsilon)]^{1-\mu}, \]
where \(0<\mu<1\) is arbitrary.
In proving the existence of a solution of equation (1), the estimate obtained is
\[ |\widetilde{B}u_2-Bu_1|\le M_4[\gamma(y)]\le M_4\varphi(\varepsilon). \]
Combining the last two inequalities, according to (7), we have
\[ |I^*|\le M_5[\varphi(\varepsilon)]^{1-\mu}. \]
According to (6), we have
\[ |u_2(\beta(y))-u_1[\beta(y)]| \le |\lambda|\,|Bu_2-Bu_1|+|\lambda|M_5[\varphi(\varepsilon)]^{1-\mu}. \]
Hence it follows that
\[ \|u_2-u_1\|_{L_p} \le |\lambda|\,\|Bu_2-Bu_1\|_{L_p} +|\lambda|M_5[\varphi(\varepsilon)]^{1-\mu}(b-a)^{1/p}. \]
In the proof of the uniqueness theorem for the solution of equation (1) (analogously to (2)), one obtains the inequality
\[ \left\|B u_2 - B u_1\right\|_{L_p} \leqslant c \left\|u_2 - u_1\right\|_{L_p}, \]
where \(c>0\) is some constant.
If \(\lambda\) is chosen so that \(|\lambda|c<1\) (this is the condition for uniqueness of the solution of equation (1)), we obtain
\[ \left\|u_2-u_1\right\|_{L_p}\leqslant M_6[\varphi(\varepsilon)]^{1-\mu}. \tag{9} \]
Kh. Sh. Mukhtarov has established that for any \(u(x)\in H_N(\varphi)\) one has
\[ \max_{a\leqslant x\leqslant b}|u(x)| \leqslant D N^{\alpha/p}\left\{\left\|u\right\|_{L_p}\right\}^{(p-\alpha)/p^2}, \tag{10} \]
where \(p\geqslant 1\) is arbitrary, and \(0<\alpha<p\) is a certain definite number; \(D\) is a constant independent of \(N\).
If (10) is applied to inequality (9), we finally obtain
\[ \max_{a\leqslant x\leqslant b}|u_2(x)-u_1(x)| \leqslant M_7[\varphi(\varepsilon)]^{(p-\alpha)(1-\mu)/p^2}, \]
where \(M_7\) is a constant depending only on \(N_1, N_2, N_3; N_0, \varphi(t), \varphi_1(t), m, M; (b-a)\).
In conclusion we report that, for equation (1), the author has proved the continuous dependence of the solutions on \(K(x,s,u)\) (stability of solutions), and also that the solution of equation (1) with respect to the parameter \(\lambda\) (for small \(\lambda\)) belongs to the Hölder class.
Dagestan State University
named after V. I. Lenin
Received
12 I 1965
References
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