N. V. GOVOROV

AN INHOMOGENEOUS RIEMANN BOUNDARY-VALUE PROBLEM WITH INFINITE INDEX

(Presented by Academician A. A. Dorodnitsyn, 5 VI 1964)

MATHEMATICS

1°. Let, in the complex \(z\)-plane, a domain \(D\) be given whose boundary \(L\) is the ray \(1 \le t \le \infty\). We consider in the domain \(D\) the Riemann boundary-value problem with infinite index

\[ \Phi^{+}(t)=G(t)\Phi^{-}(t)+g(t) \tag{1} \]

under the following assumptions:

1) \(\arg G(t)=2\pi \varphi(t)t^\rho,\quad -1<\varphi(1)\le 0,\quad \varphi(\infty)\ne 0,\quad 0<\rho<\infty,\)
\tag{2}

where \(\varphi(t)\in H(\mu)\), i.e.

\[ |\varphi(t_1)-\varphi(t_2)|<A\left|\frac{1}{t_1}-\frac{1}{t_2}\right|^\mu,\quad t_k\in L,\quad A,\mu=\text{const}, \tag{3} \]

and

\[ \frac{\rho}{\rho+1}<\mu\le 1; \tag{4} \]

2) \(\ln |G(t)|,\ g(t)\in H(\lambda)\quad (0<\lambda\le 1);\)
\tag{5}

3) \(g(1)=g(\infty)=0.\)
\tag{6}

The case \(g(t)\equiv 0\) (the homogeneous problem) was considered in \((^{5,6})\). The inhomogeneous problem was solved in \((^5)\) for the case \(0<\rho<{}^{1}/_{2},\ \mu>\rho,\ \varphi(\infty)>0\).

2°. Definition 1. Let a set of points \(\{z_n\}\) be given in the domain \(D\), with \(\lim\limits_{n\to\infty} z_n=\infty\). Then, if for all \(\psi_1,\psi_2\) such that

\[ -2\pi<\psi_1<\psi_2<2\pi,\quad |\psi_1-\psi_2|<2\pi,\quad \psi_k\in N, \]

where \(N\) is at most countable, for some \(\sigma>0\) there exists the limit

\[ \lim_{r\to\infty}\frac{1}{r^\sigma} \sum_{\substack{\psi_1<\arg z_n\le \psi_2\\ |z_n|\le r}} \arg z_n = \nu(\psi_1,\psi_2)\ne\infty \tag{7} \]

and the asymptotic estimate

\[ \sum_{\substack{-\pi<\arg z_n\le \pi\\ |z_n|\le r}} |\arg z_n|<Kr^\sigma \quad (K>0,\ K=\text{const}) \]

holds, then we shall say that the set \(\{z_n\}\) has, in the domain \(D\), an argument density. In doing so we adopt, by definition,

\[ \nu(\psi,\psi)=0;\qquad \nu(\psi_1,\psi_2)=-\nu(\psi_2,\psi_1),\quad \text{if } \psi_1>\psi_2, \]

\[ \nu(\psi_1,\psi_2)=\nu(\psi_1,+0)+\nu(+0,\psi_2),\quad \text{if } |\psi_1-\psi_2|>2\pi. \]

By the argument density we shall mean the function \(\nu(\psi)=\nu(\psi_0,\psi)\), determined up to an additive constant for fixed \(\psi_0\), extended at the points of the set \(N\) by right-continuity.

The concept of argument density is closely connected with the concept of angular density introduced by B. Ya. Levin ((²), p. 118). We note that the quantity
\(\sum_{|z_n|\le r}\sin \arg z_n\), related to the sum in (7), was considered by R. Nevanlinna (³).

We shall call the following function the indicator of a function \(f(z)\), regular and of order \(\sigma>0\) in the angle \(\alpha<\arg z<\beta\) ((⁵), p. 1247; (⁴), p. 209):

\[ h_f(\theta)=\varlimsup_{r\to\infty} r^{-\sigma}\ln\left|f\left(re^{i\theta}\right)\right| \qquad (\alpha<\theta<\beta). \]

Definition 2. A function \(f(z)\), regular and of order \(\sigma>0\) in the angle \(\alpha<\arg z<\beta\), continuous for \(\alpha\le \arg z\le \beta\) (for \(z\ne\infty\)), is called a function of completely regular growth in the closed angle \(\alpha\le \arg z\le \beta\), if the function \(r^{-\sigma}\ln |f(re^{i\theta})|\) tends to \(h_f(\theta)\) uniformly for \(\alpha\le \theta\le \beta\), as \(r\) tends to \(+\infty\), except for values from a set \(E\), common to all \(\theta\), of zero relative measure ((²), pp. 127, 182).

We shall call \(f(z)\) a function of completely regular growth (of order \(\sigma>0\)) in the open angle \(\alpha<\arg z<\beta\), if it has completely regular growth in each angle \(\alpha+\varepsilon\le \arg z\le \beta-\varepsilon\) (\(\varepsilon>0\)) and if, asymptotically,

\[ \sup_{|z|\le r}|f(z)|<\exp(Kr^\sigma)\qquad (K>0). \]

Denote by \(B_\sigma\) the class of functions regular, of finite order \(\sigma>0\), and of completely regular growth in the domain \(D\).

3°. Consider problem (1) for the case of a plus-infinite index, i.e., when \(\varphi(\infty)>0\). We shall solve it in the class \(B_\sigma\), where \(0<\sigma<\min(\rho,\tfrac12)\). (For \(\sigma>\min(\rho,\tfrac12)\) there are no solutions.) Denote by \(\Psi(z)\) the solution of the corresponding homogeneous problem (1):

\[ \Psi^{+}(t)=-G(t)\Psi^{-}(t). \tag{8} \]

Theorem 1. Let \(\Phi_0(z)\) be some bounded solution of problem (1) (with \(\varphi(\infty)>0\)) of order not less than \(\sigma_0\), \(0<\sigma<\sigma_0<\min(\rho,\tfrac12)\). Then the general solution of this problem in the class \(B_\sigma\) has the form

\[ \Phi(z)=\Phi_0(z)+\Psi(z), \tag{9} \]

where \(\Psi(z)\) is the general solution of the homogeneous problem (8) in the class \(B_\sigma\). Moreover, the indicators of the functions \(\Phi(z)\) and \(\Psi(z)\) in the interval \(0<\theta<2\pi\) coincide.

The general solution of the homogeneous problem was found in (⁶). It remains to find \(\Phi_0(z)\). Applying the method of F. D. Gakhov ((¹), p. 117), one can obtain:

\[ \Phi_0(z)=\frac{\Psi_0(z)}{2\pi i}\int_1^\infty \frac{g(x)\,dx}{\Psi_0^{+}(x)(x-z)} , \tag{10} \]

where \(\Psi_0(z)\) is some solution of the homogeneous problem (8). In order that the integral converge and that \(\Phi_0(z)\) be bounded and of order not less than \(\sigma_0\), it is sufficient to find such a \(\Psi_0(z)\) that the following conditions be satisfied:

a) \(\Psi_0(z)\) is bounded in \(D\);

b) in any angle \(\varepsilon\le \arg z\le 2\pi-\varepsilon\), for \(|z|\equiv r>r_\varepsilon\), the estimate

\[ \max_{\varepsilon\le \theta\le 2\pi-\varepsilon} \left|\Psi_0\left(re^{i\theta}\right)\right| < \exp\left(-K_\varepsilon r^{\sigma_0}\right) \qquad (K_\varepsilon>0); \]

c) \(g(t)/\Psi_0^{+}(t)\in H(\tau)\quad (0<\tau\le 1)\), \(\displaystyle \lim_{t\to\infty} g(t)/\Psi_0^{+}(t)=0\).

One of the simplest functions subject to conditions a)—c) has the form*

\[ \Psi_0(z)=\exp\left[\frac{z}{2\pi i}\int_0^\infty \frac{\ln G(x)-2\pi i n_{\Psi_0}(x)}{x(x-z)}\,dx\right] \prod_{n=1}^{\infty} \frac{\left(1-\frac{z}{r_n}e^{-ir_n^{-\rho}}\right)\left(1+\frac{z}{s_n}\right)} {\left(1-\frac{z}{r_n}\right)\left(1-\frac{z}{s_n}\right)}, \tag{11} \]

where

\[ s_n=\left(\frac{2n-1}{2\cos\sigma_0\pi}\right)^{1/\sigma_0},\quad 1\le r_1\le r_2\le\cdots, \]

and the number \(n_1(r)\) of points \(r_n\) on the interval \([1,r]\) is determined by the equality

\[ n_1(r)=\max\left\{\left[\max_{1\le x\le r}\left\{\varphi(x)x^\rho-x^{\sigma_0}+\frac12\right\}\right],0\right\}^{**}, \]

while \(n_{\Psi_0}(r)\) is the total number of points \(s_n,r_n\in[0,r]\) ((6), p. 16).

Let us formulate the final result.

Theorem 2. The general solution of the nonhomogeneous problem (1) in the class \(B_\sigma\), \(0<\sigma<\min(\rho,1/2)\), has the form

\[ \Phi(z)=\frac{\Psi_0(z)}{2\pi i}\int_1^\infty \frac{g(x)\,dx}{\Psi_0^+(x)(x-z)} + \]

\[ +cz^m\exp\left[\frac{z}{2\pi i}\int_0^\infty \frac{\ln G(x)-2\pi i\widetilde n(x)}{x(x-z)}\,dx\right] \times\prod_{n=1}^{\infty}\frac{1-z/z_n}{1-z/|z_n|}, \tag{12} \]

where \(\Psi_0(z)\) is defined by formula (11), \(\widetilde n(r)\) is the number of points \(z_n\) in \(0<|z|\le r\), and the following conditions are satisfied:

I. The set \(\{z_n\}\) has angular \(\sigma\)-density \(\nu(\psi)\).

II. There exists the finite limit

\[ \lim_{r\to\infty}\frac{\sigma^2}{r^\sigma} \int_0^r\frac{dt}{t}\int_0^t \frac{\frac{1}{2\pi}\arg G(x)-\widetilde n(x)}{x}\,dx =\gamma\ge 0, \]

where, if \(\gamma=\gamma(\sigma)=0\), then \(\gamma(\sigma-\varepsilon)=+\infty\) for any \(\varepsilon>0\).

III. The integral

\[ \int_0^\infty \frac{1}{x^2}\left[\frac{1}{2\pi}\arg G(x)-\widetilde n(x)\right]\,dx \]

converges.

IV. For \(t>t_\infty,\ t\ne |z_n|\), the estimate

\[ t\int_0^\infty \frac{\frac{1}{2\pi}\arg G(x)-\widetilde n(x)} {x(x-t)}\,dx +m\ln t+\sum_{n=1}^{\infty} \ln\left|\frac{z_n-t}{|z_n|-t}\right|<C_\Phi=\mathrm{const} \]

holds.

In this case the indicator of the solution is expressed by the formula \((0<\theta<2\pi)\)

\[ h_\Phi(\theta)= \frac{\pi}{\sin\pi\sigma} \left[ \int_{-\pi}^{\pi}\alpha(\psi,\theta)\,d\nu(\psi) -\gamma\cos\sigma(\theta-\pi) \right], \]

where it is set that

\[ \alpha(\psi,\theta)= \begin{cases} \psi^{-1}\,[\cos\sigma(|\theta-\psi|-\pi)-\cos\sigma(\theta-\pi)], & 0<\psi\le\pi,\\ \psi^{-1}\,[\cos\sigma(|\theta-\psi-2\pi|-\pi)-\cos\sigma(\theta-\pi)], & -\pi\le\psi<0,\\ \sigma\sin\sigma(\theta-\pi), & \psi=0. \end{cases} \]

Corollary. Problem (1) has in the class \(B_\sigma\), \(0<\sigma<\min(\rho,1/2)\), an infinite set of linearly independent solutions.

* We assume that \(G(t)=1,\ g(t)=0\) for \(0\le t<1\).

** \([a]\) denotes the integer part of the real number \(a\).

Remark. The convergence of the infinite product and of the second of the integrals in equality (12) follows respectively from I and III.

4°. Let us now consider problem (1) for minus-infinite index.

Theorem 3. Let assumptions (2)—(6) be satisfied, with \(\varphi(\infty)<0\), and let \(\widetilde{\Psi}(z)\) be meromorphic in the domain \(D\) and such that:

1) \(\widetilde{\Psi}^{+}(t)=G(t)\widetilde{\Psi}^{-}(t)\quad (1<t<\infty)\);

2) \(g(t)/\widetilde{\Psi}^{+}(t)\in H\);

3) \(|\widetilde{\Psi}^{\pm}(t)|<\mathrm{const}\quad (1\le t<\infty)\);

4) on some sequence of circles \(|z|=k_n\) \((k_n\to\infty)\)

\[ \frac{1}{M}<\sup_{n=1,2,\ldots}\ \max_{|z|=k_n}\bigl|\widetilde{\Psi}(z)\bigr|<M<\infty, \]

where \(M=\mathrm{const}\).

Then, for the solvability of problem (1) in the class of bounded functions, it is necessary and sufficient that at all poles \(z_n\) of the function \(\widetilde{\Psi}(z)\), the number of which is known to be infinite, the equalities

\[ \int_{1}^{\infty}\frac{g(x)\,dx}{\widetilde{\Psi}^{+}(x)(x-z_n)}=0 \qquad (n=1,2,\ldots) \tag{13} \]

hold.

If (13) is fulfilled, the solution is unique and is expressed by the formula

\[ \Phi(z)=\frac{\widetilde{\Psi}(z)}{2\pi i} \int_{1}^{\infty}\frac{g(x)\,dx}{\widetilde{\Psi}^{+}(x)(x-z)}. \tag{14} \]

One of the simplest functions subject to conditions 1)—4) has the form

\[ \widetilde{\Psi}(z)= \exp\left[ \frac{z}{2\pi i}\int_{1}^{\infty} \frac{\ln G(x)+2\pi i p(x)}{x(x-z)}\,dx \right] \prod_{n=1}^{\infty}\frac{1-z/|z_n|}{1-z/z_n}, \]

where \(p(r)\) denotes the number of poles \(z_n\) of the function \(\widetilde{\Psi}(z)\) in \(0<|z|\le r\), and

\[ z_n=r_n e^{i r_n^{-\rho}}, \qquad p(r)=\left[\max_{1\le x\le r}\left\{\frac12-\varphi(x)x^\rho\right\}\right]. \]

Remark. For \(0<\rho<1/2\), (14) can be simplified to the form

\[ \Phi(z)=\frac{X(z)}{2\pi i} \int_{1}^{\infty}\frac{g(x)\,dx}{X^{+}(x)(x-z)}, \qquad \text{where }\quad X(z)=\exp\left[ \frac{z}{2\pi i}\int_{1}^{\infty}\frac{\ln G(x)\,dx}{x(x-z)} \right] \]

(if (13) is fulfilled); in this case the integrals are certainly convergent.

In conclusion I express my deep gratitude to Prof. F. D. Gakhov, who supervised the present work, and also to A. A. Gol’dberg, who made valuable suggestions.

Novocherkassk
Polytechnic Institute

Received
2 VI 1964

CITED LITERATURE

¹ F. D. Gakhov, Boundary Value Problems, Moscow, 1963.
² B. Ya. Levin, Distribution of Zeros of Entire Functions, Moscow, 1956.
³ R. Nevanlinna, Acta Soc. Sci. Fenn., 50, No. 12 (1925).
⁴ E. Titchmarsh, Theory of Functions, Moscow–Leningrad, 1951.
⁵ N. V. Govorov, Dokl. Akad. Nauk SSSR, 154, No. 6, 1247 (1964).
⁶ N. V. Govorov, Izv. Akad. Nauk BSSR, No. 1, 12 (1964).