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MATHEMATICS
A. B. NERSESYAN
EXPANSION IN EIGENFUNCTIONS OF CERTAIN NON-SELF-ADJOINT BOUNDARY-VALUE PROBLEMS
(Presented by Academician I. N. Vekua on 29 VI 1960)
1°. Let \(a(x)\) and \(b(x)\) be certain complex-valued functions of bounded variation on the interval \([0,l]\) \((0<l<+\infty)\), continuous at the endpoints of this interval and such that the Riemann–Stieltjes integral exists
\[ \int_0^l a(x)\,db(x). \tag{1} \]
Further, let the real functions \(\Delta_i(x)\) \((0\leq x\leq l,\ i=1,2,\ldots,m)\) be differentiable and satisfy the conditions
\[ \Delta_i(x)\geq 0,\qquad \Delta_i(l)<l,\qquad \Delta_i'(x)\leq \theta<1 \quad (0\leq x\leq l;\ i=1,2,\ldots,m), \tag{2} \]
and let the complex-valued functions \(q_i(x)\in L_1(0,l)\) be identically zero, respectively, for \(x-\Delta_i(x)<0\) \((i=1,2,\ldots,m)\). Finally, let the complex-valued kernel \(K(x,t)\) \((0\leq t\leq x\leq l)\) satisfy the condition
\[ \int_0^l \int_t^l |K(x,t)|\,dx\,dt<+\infty. \tag{3} \]
Denote
\[ \varphi_i(x)\equiv x-\Delta_i(x)\quad (0\leq x\leq l;\ i=1,2,\ldots,m). \tag{4} \]
From conditions (2) it follows that there exist functions
\[ \psi_i(x)=\varphi_i^{-1}(x)\equiv x+\Delta_i^*(x),\qquad \Delta_i^*(x)\geq 0 \tag{5} \]
\[ (\varphi_i(0)\leq x\leq \varphi_i(l);\ i=1,2,\ldots,m). \]
In the class of functions of bounded variation on \([0,l]\), consider the following nonhomogeneous eigenvalue problems:
Problem (A).
\[ \frac{d}{dx}[y(x)-a(x)] +\sum_{i=1}^m q_i(x)y(x-\Delta_i(x)) +\int_0^x K(x,t)y(t)\,dt =\lambda y(x), \tag{6} \]
\[ y(0)=\alpha, \tag{7} \]
\[ \beta y(l)+\int_0^l y(x)\,db(x)=A_0. \tag{8} \]
Problem (A*).
\[ -\frac{d}{dx}[z(x)+b(x)] +\sum_{i=1}^m q_i^*(x)z(x+\Delta_i^*(x)) +\int_x^l K(t,x)z(t)\,dt =\lambda z(x), \tag{6*} \]
\[ \alpha z(0)+\int_0^l z(x)\,da(x)=A_0, \tag{7*} \]
\[ z(l)=\beta, \tag{8*} \]
where \(\alpha,\beta\), and \(A_0\) are certain complex constants and
\[ q_i^*(x)= \begin{cases} 0, & \text{for } \varphi_i(l)<x<l,\\ q_i(\psi_i(x))\psi_i'(x), & \text{for } 0\le x\le \varphi_i(l). \end{cases} \qquad (i=1,2,\ldots,m) \tag{9} \]
Denote by \(y(x,\lambda)\) the solution of problem (6), (7), and by \(z(x,\lambda)\) the solution of problem \((6^*)\), \((8^*)\). These solutions exist, are unique* and are entire functions of \(\lambda\).
It is established that:
- The eigenvalues of problems \((A)\) and \((A^*)\) coincide and are the \(A_0\)-points of the entire function
\[ \omega(\lambda)=\beta y(l,\lambda)+\int_0^l y(x,\lambda)\,db(x). \tag{10} \]
- For arbitrary \(\lambda\) and \(\mu\) the relation
\[ \int_0^l y(x,\lambda)z(x,\mu)\,dx = \frac{\omega(\lambda)-\omega(\mu)}{\lambda-\mu} \tag{11} \]
holds.
To each zero \(\lambda_n\) of the function \(\omega(\lambda)-A_0\) we assign two systems of functions:
\[ \left.\frac{\partial^j y(x,\lambda)}{\partial \lambda^j}\right|_{\lambda=\lambda_n} \qquad (j=0,1,\ldots,p_n-1); \tag{12} \]
\[ \left. \sum_{k=0}^{p_n-j-1} \frac{b_{p_n-j-k-1}^{(n)}}{k!\,j!} \frac{\partial^k z(x,\lambda)}{\partial \lambda^k} \right|_{\lambda=\lambda_n} \qquad (j=0,1,\ldots,p_n-1), \tag{12*} \]
where \(p_n\) is the multiplicity of the zero \(\lambda\) and
\[ b_k^{(n)} = \left. \frac{1}{k!}\frac{d}{d\lambda^k} \left\{ \frac{(\lambda-\lambda_n)^{p_n}}{\omega(\lambda)-A_0} \right\} \right|_{\lambda=\lambda_n} \qquad (k=0,1,\ldots,p_n-1). \tag{13} \]
Denote
\[ |\lambda_0|=\min_{\omega(\lambda_n)=A_0}|\lambda| \qquad (\omega(\lambda_0)=A_0); \tag{14} \]
\[ y_0(x)=y(x,\lambda_0),\qquad z_0(x)= \left. \sum_{k=0}^{p_0-1} \frac{b_{p_0-k-1}^{(0)}}{k!} \frac{\partial^k z(x,\lambda)}{\partial \lambda^k} \right|_{\lambda=\lambda_0}. \tag{15} \]
Number all the remaining functions of the form (12) and \((12^*)\) in the order of nondecreasing \(|\lambda_n|\), so that each two \(j\)-th functions have identical numbers: for \(\operatorname{Im}\lambda_n\ge 0\), positive numbers, and for \(\operatorname{Im}\lambda_n<0\), negative numbers. We denote the resulting system of functions by
\[ \{y_k(x),z_k(x)\}\qquad (k=0,\pm1,\pm2,\ldots). \tag{16} \]
* For definiteness, all functions of bounded variation will be assumed continuous from the left.
According to Theorem 1 of paper \((^1)\), it follows from formula (11) that the system (16) is biorthogonal on \([0,l]\) in the sense
\[ \int_0^l y_k(x)z_p(x)\,dx= \begin{cases} 0, & k\ne p,\\ 1, & k=p. \end{cases} \tag{17} \]
Thus, it is natural to call problems (A) and \((A^*)\) mutually adjoint.
\(2^\circ\). In this subsection we give formulations of some results on expansion with respect to the system (16), when the functions \(q_i(x)/\Delta_j'(x)\) \((i,j=1,2,\ldots,m)\) have bounded variation on \([0,l]\), the kernel \(K(x,t)\) has bounded variation on \([0,l]\) in one of its arguments, uniformly with respect to the other argument (for \(t>x\) we put \(K(x,t)\equiv0\)), and
\[ \alpha\beta A_0\ne0. \tag{18} \]
In this case all eigenvalues of problems (A) and \((A^*)\) lie in a certain strip
\[
\sigma_1\le \operatorname{Re}\lambda\le\sigma_2
\]
\((\sigma_1\le \log |A_0|/|\alpha\beta|\le\sigma_2)\). Let
\[ a(x)=a_1(x)+a_2(x)+a_3(x), \]
\[ b(x)=b_1(x)+b_2(x)+b_3(x), \tag{19} \]
where \(a_1(x)\) and \(b_1(x)\) are absolutely continuous functions; \(a_2(x)\) and \(b_2(x)\) are jump functions; \(a_3(x)\) and \(b_3(x)\) are singular functions.
Denote
\[ A=\bigvee_0^l(a_2+a_3),\qquad B=\bigvee_0^l(b_2+b_3). \tag{20} \]
Theorem 1. Suppose
\[ |\beta|A+|\alpha|B+AB\le \min\{|A_0|,|\alpha\beta|\}. \tag{21} \]
Then, if \(f(x)\in L_2(0,l)\), then
\[ \lim_{n\to\infty}\int_0^l \left|f(x)-\sum_{k=-n}^{n}y_k(x)\int_0^l f(t)z_k(t)\,dt\right|^2dx = \]
\[ = \lim_{n\to\infty}\int_0^l \left|f(x)-\sum_{k=-n}^{n}z_k(x)\int_0^l f(t)y_k(t)\,dt\right|^2dx=0. \tag{22} \]
Before formulating propositions on pointwise convergence and on equiconvergence of expansions with respect to the system (16), we introduce some sets characterized by the function \(a(x)\).
By \(C\{a(x)\}\) we denote the set of all points of discontinuity of the function \(a(x)\) on the interval \((0,l)\). A set \(e\in C\{a(x)\}\) will be called a set of type \(\widetilde C\{a(x)\}\) if for every \(\varepsilon>0\) there exists \(\delta=\delta(\varepsilon)>0\) such that \(|a(x)-a(t)|\le\varepsilon\), if and only if \(x\in e\), \(t\in(0,l)\), and \(|x-t|\le\delta\). Obviously, every closed set \(e\in C\{a(x)\}\) is a set of type \(\widetilde C\{a(x)\}\).
Next, denote by \(L\{a(x)\}\) the set of all those points \(x\in C\{a(x)\}\) for which there exist constants \(0<\delta<\min(x,l-x)\) and \(M(x)<+\infty\) such that \(|a'(t)|\le M(x)\) for \(t\in(x-\delta(x),x+\delta(x))\).
Finally, a set \(e\in L\{a(x)\}\) will be called a set of type \(\widetilde L\{a(x)\}\) if, for \(x\in e\), \(\delta(x)\) and \(M(x)\) may be taken independent of \(x\).
Let us note that each function of bounded variation \(a(x)\) obviously determines a nonempty set \(C\{a(x)\}\), whereas a nonempty set \(L\{a(x)\}\) may fail to exist.
Theorem 2. Let the functions \(a(x)\) and \(b(x)\) satisfy condition (21). Then:
1) If \(f(x)\) is a function of bounded variation on \([0,l]\), then for \(x \in C\{a(x)\}\)
\[ \frac{1}{2}[f(x+0)+f(x-0)] = \sum_{k=-\infty}^{\infty} y_k(x)\int_0^l f(t)z_k(t)\,dt, \tag{23} \]
where the series on the right converges uniformly on every set of the type \(\widetilde{C}\{a(x)\}\).
2) If
\[ f(x)\log\frac{x}{1+x}\in L_1(0,l), \tag{24} \]
then for \(x \in L\{a(x)\}\)
\[ \lim_{n\to\infty} \left\{ \sum_{k=-n}^{n} y_k(x)\int_0^l f(t)z_k(t)\,dt - \frac{1}{\pi}\int_0^l f(t)\frac{\sin \frac{2\pi n}{l}(x-t)}{x-t}\,dt \right\}=0, \tag{25} \]
where the limit is attained uniformly on every set of the type \(\widetilde{L}\{a(x)\}\).
An analogous theorem is also valid for expansions with respect to the system \(\{z_k(x)\}\) \((k=0,\pm1,\ldots)\).
Let us also note that, when condition (21) is fulfilled, all sufficiently large (in modulus) zeros of the function \(\omega(\lambda)-A_0\) are simple. However, one can give examples where no zero of the function \(\omega(\lambda)-A_0\) is simple, but Theorems 1 and 2 remain valid.
In conclusion I express my gratitude to Academician of the Academy of Sciences of the Armenian SSR M. M. Dzhrbashyan, under whose guidance the present work was carried out.
Institute of Mathematics and Mechanics
Academy of Sciences of the Armenian SSR
Received
23 VI 1960
REFERENCES
- M. M. Dzhrbashyan, A. B. Nersesyan, Izv. AN ArmSSR, 12, No. 5 (1959).