Abstract Generated abstract
The paper studies a generalized class of convolution type integral equations on the real line with bounded measurable coefficients having generalized limits at infinity, in a range of Banach function spaces including Lebesgue, bounded measurable, and continuous function spaces. By decomposing the operator into a paired convolution operator and completely continuous perturbations, the authors establish necessary and sufficient conditions for the Noether property and give an index formula in terms of the Fourier transforms of the kernels and limiting coefficient values. These results are then applied, through a logarithmic change of variables, to integral equations on a finite interval with homogeneous kernels in weighted spaces, yielding corresponding Noether criteria and index formulas via Mellin transforms. Examples illustrate Fredholm behavior and index computation for particular homogeneous and logarithmic kernel equations.
Full Text
UDC 517.948.32
MATHEMATICS
N. K. KARAPETYANTS, S. G. SAMKO
ON ONE CLASS OF INTEGRAL EQUATIONS OF CONVOLUTION TYPE AND ITS APPLICATION
(Presented by Academician N. I. Muskhelishvili on 30 I 1970)
The following generalization of integral equations of convolution type is considered:
\[ H\varphi \equiv a_0(t)\varphi(t)+ \sum_{j=1}^{n} a_j(t)\int_{-\infty}^{\infty} b_j(\tau)h_j(t-\tau)\varphi(\tau)\,d\tau = f(t),\qquad -\infty<t<\infty . \tag{1} \]
Here and everywhere below \(h_j(t)\in {\mathcal L}_1(-\infty,\infty)\); \(a_j(t)\) and \(b_j(t)\) are bounded measurable functions, with \(|a_0(t)|\ge c>0\); \(\varphi(t), f(t)\in E^*\).
Various special cases of equation (1) (mainly cases of piecewise-constant coefficients) have been studied, for example, in \((^{2-6})\). The general equation of the form (1) with \(b_j(t)\equiv 1\), \(j=1,2,\ldots,n\), was investigated by L. S. Rakovshchik \((^1)\) under the assumption that \(\varphi(t), f(t)\in {\mathcal L}_p(-\infty,\infty)\), \(p>1\).
In this note the results of \((^1)\) are generalized to the operator \(H\) in the case of any of the spaces \(E\). The results obtained are applied to the study of one class of integral equations of the form
\[ K\psi \equiv \psi(x)+\sum_{j=1}^{n} c_j(x)\int_{0}^{a} d_j(y)k_j(x,y)\psi(y)\,dy = g(x),\qquad 0<x<a, \tag{2} \]
where \(k_j(x,y)\) are homogeneous functions of arbitrary orders \(\alpha_j\); \(c_j(x)\), \(d_j(x)\) belong to certain subclasses of bounded measurable functions. Necessary and sufficient conditions will be found for the Noether property of the operator \(K\) in a series of weighted spaces \(E_\beta\), and the index will be computed. We note that equation (2) with \(\alpha_j=-1\), \(c_j(x), d_j(x)\in C(0,a)\) belongs to the class of equations studied by L. G. Mikhailov in \((^{12})\).
No. 1. Let us introduce the spaces \(M_0^{\mathrm{sup}}\), \(M_0^{\mathrm{mes}}\), \(M^{\mathrm{sup}}\), \(M^{\mathrm{mes}}\) of bounded measurable functions:
\[ M^{\mathrm{sup}}=M_0^{\mathrm{sup}}\oplus X,\qquad M^{\mathrm{mes}}=M_0^{\mathrm{mes}}\oplus X, \]
\[ M_0^{\mathrm{sup}}=\{\varphi(t)\mid \varphi(t)\in M;\ \lim_{n\to\infty}\sup_{|t|>n}|\varphi(t)|=0\}, \]
\[ M_0^{\mathrm{mes}}=\{\varphi(t)\mid \varphi(t)\in M;\ \lim_{n\to\infty}\operatorname{mes}(t:|\varphi(t)|>\varepsilon,\ |t|>n;\ \varepsilon>0)=0\}, \]
where \(X\) is the two-dimensional space with basis \(\theta(x)\), \(\theta(-x)\), \(\theta(x)=\frac12(1+\operatorname{sign}x)\). Obviously, \(\varphi(t)\in M^{\mathrm{sup}}\) (or \(M^{\mathrm{mes}}\)) if and only if there exist constants \(c_{\pm}\) such that \(\theta(\pm t)[\varphi(t)-c_{\pm}]\in M_0^{\mathrm{sup}}\) (or \(M_0^{\mathrm{mes}}\), respectively). The constants \(c_{\pm}\) will be called the (generalized) values of the bounded measurable function at infinity: \(c_{\pm}=\varphi(\pm\infty)\) (cf. \((^1)\)).
\[ \text{*} \]
We note that to the series \(E\) of Banach spaces \({\mathcal L}_p, M, M^u, M^c, C, C^0\) \((^7)\) we add the subspaces important for us, \(M^{\mathrm{sup}}\), \(M^{\mathrm{mes}}\), of bounded measurable functions having (in a certain sense) values at infinity.
We note the following properties of the classes \(M_0^{\sup}, M_0^{\mathrm{mes}}, M^{\sup}, M^{\mathrm{mes}}\):
1) they are complete in the norm \(M\); 2) functions from \(M^{\mathrm{mes}}\) are continuous in measure on the whole axis; 3) \(\varphi(t)\in M_0^{\sup}\) if and only if \(\varphi(t)\) has a majorant from \(C^0\): \(|\varphi(t)|\le a(t)\in C^0\); 4) functions of bounded variation (on the closed axis) belong to \(M^{\sup}\); 5) \(M\cap \mathscr L_p\subset M_0^{\mathrm{mes}}\) for any \(p\ge 1\).
Everywhere below we shall assume that \(E=\mathscr L_1,\ \mathscr L_p,\ M,\ M^{\sup},\ M^{\mathrm{mes}},\ M_0^{\sup},\ M_0^{\mathrm{mes}},\ M^u,\ M^c,\ C,\ C^0\), where the spaces \(E=M^u,\ M^c,\ C,\ C^0\) of continuous functions will be understood in the following sense: \(E=E_+\oplus E_-\) (see (8)).
Lemma 1. If \(h(t)\in\mathscr L_1\), then the convolution operator \(A\varphi=h*\varphi\) acts from \(M_0^{\mathrm{mes}}\) into \(C\) and from \(M_0^{\mathrm{mes}}\) into \(C^0\).
No. 2. Put, without loss of generality, \(a_0(t)\equiv 1\). Let \(a_j(t), b_j(t)\in M^{\mathrm{mes}}\). Represent the operator \(H\) in the form
\[ H\varphi=\Pi\varphi+T_1\varphi+T_2\varphi, \tag{3} \]
where
\[ \Pi\varphi= \begin{cases} \displaystyle \varphi(t)+\int_{-\infty}^{\infty}\sum_{j=1}^{n} a_j(\infty)b_j(\infty)h_j(t-\tau)\varphi(\tau)\,d\tau, & t>0,\\[1.2em] \displaystyle \varphi(t)+\int_{-\infty}^{\infty}\sum_{j=1}^{n} a_j(-\infty)b_j(-\infty)h_j(t-\tau)\varphi(\tau)\,d\tau, & t<0. \end{cases} \]
is a paired operator, and
\[ T_1\varphi=\sum_{j=1}^{n}\tilde a_j(h_j*b_j\varphi)+ \sum_{j=1}^{n}(a_j-\tilde a_j)(h_j*\tilde b_j\varphi), \]
\[ \tilde a_j(t)=a_j(t)-a_j(\infty)\theta(t)-a_j(-\infty)\theta(-t);\qquad \tilde b_j(t)=b_j(t)-b_j(\infty)\theta(t)- \]
\[ {}-b_j(-\infty)\theta(-t), \]
\[ T_2\varphi= \sum_{j=1}^{n}(b_j(\infty)-b_j(-\infty))a_j(-\infty)\theta(-t) \int_{-\infty}^{\infty}\theta(\tau)h_j(t-\tau)\varphi(\tau)\,d\tau- \]
\[ {}-\sum_{j=1}^{n}(b_j(\infty)-b_j(-\infty))a_j(\infty)\theta(t) \int_{-\infty}^{\infty}\theta(-\tau)h_j(t-\tau)\varphi(\tau)\,d\tau. \]
Operators of the form \(T_2\) are completely continuous \(({}^{9,10})\) in all spaces \(E\). For a broad class of coefficients this is also true for the operator \(T_1\). Namely, the following holds.
Lemma 2. The operator \(T\varphi=a(h*b\varphi)\), \(h(t)\in\mathscr L_1\), \(a(t), b(t)\in M\), is completely continuous in the space \(E\), if:
1) either
\[ a(t)\in \begin{cases} M_0^{\mathrm{mes}} & \text{for } E=\mathscr L_p\ (p\ge 1),\\ M_0^{\sup} & \text{for } E=M,\ M_0^{\sup},\ M_0^{\mathrm{mes}},\ M^{\sup},\ M^{\mathrm{mes}},\\ C_0 & \text{for } E=M^u,\ M^c,\ C,\ C^0; \end{cases} \tag{4} \]
2) or
\[ b(t)\in \begin{cases} M_0^{\sup} & \text{for } E=\mathscr L_1,\\ M_0^{\mathrm{mes}} & \text{for the remaining } E; \end{cases} \tag{5} \]
\[ a(t)\in C,\quad \text{if } E=M^u,\ M^c,\ C,\ C^0. \]
For \(b(t)\equiv 1,\ a(t)\in M_0^{\mathrm{mes}},\ E=\mathscr L_p\ (p>1)\), the assertion of Lemma 2 was obtained in \(({}^{1})\).
By virtue of the complete continuity of the operators \(T_1, T_2\) from (3), we obtain the following theorem:
Theorem 1. Let \(h_j(t)\in\mathscr L_1,\ j=1,2,\ldots,n\), and let \(a_j(t),\ j=0,1,\ldots,n\), and \(b_j(t),\ j=1,2,\ldots,n\), satisfy respectively the condi-
problems (4), (5), in which the index 0 is omitted*. Then the necessary and sufficient condition for the Noether property of the operator \(H\) in \(E\) has the form
\[ \sigma(\lambda)_{\pm}=a_0(\pm\infty)+\sum_{j=1}^{n} a_j(\pm\infty)b_j(\pm\infty)\mathcal H_j(\lambda)\ne 0,\qquad -\infty\le \lambda\le \infty, \]
where
\[ \mathcal H_j(\lambda)=\int_{-\infty}^{\infty} h_j(t)e^{it\lambda}\,dt. \]
Under these conditions
\[ \varkappa_E(H)=-\frac{1}{2\pi}\Delta\left[\arg\frac{\sigma(\lambda)^+}{\sigma(\lambda)^-}\right]_{-\infty}^{\infty}. \]
Using one result from the theory of linear operators ((\(^{5}\), Lemma 1), we arrive at the following theorem:
Theorem 2. The operator \(H\) has in all spaces \(E\) the same zeros and the same solvability conditions**, if one of the following conditions is fulfilled:
1) \(a_j(t)\in M^{\mathrm{mes}},\quad b_j(t)\in M^{\mathrm{mes}},\quad E=\mathscr L_p(p>1);\)
2) \(a_j(t)\in M^{\mathrm{mes}},\quad b_j(t)\in M^{\mathrm{sup}},\quad E=\mathscr L_p(p\ge 1);\)
3) \(a_j(t)\in M^{\mathrm{sup}},\quad b_j(t)\in M^{\mathrm{sup}},\quad E=\mathscr L_p(p\ge 1),\quad M,\ M_0^{\mathrm{mes}},\)
\(M_0^{\mathrm{sup}},\ M^{\mathrm{mes}},\ M^{\mathrm{sup}};\)
4) \(a_j(H\in M^{\mathrm{sup}},\quad b_j(t)\in M^{\mathrm{mes}},\quad E=\mathscr L_p(p>1),\quad M,\ M_0^{\mathrm{mes}},\)
\(M_0^{\mathrm{sup}},\ M^{\mathrm{mes}},\ M^{\mathrm{sup}};\)
5) \(a_j(t)\in C,\quad b_j(t)M^{\mathrm{mes}};\quad E\)—arbitrary, except, perhaps, \(\mathscr L_1;\)
6) \(a_j(t)\in C,\quad b_j(t)\in M^{\mathrm{sup}},\quad E\)—arbitrary.
No. 3. In equation (2) we shall assume (cf. (\(^{11}\))) that there exists a number \(\beta\) such that
\[ \int_{0}^{\infty} |k_j(1,y)|\,y^{-\beta}\,dy<\infty,\qquad j=1,2,\ldots,n. \tag{6} \]
Using a well-known device ((\(^{13}\), p. 396), we reduce equation (2), by means of the transformation \(x=ae^{-t},\ y=ae^{-\tau},\ \varphi(t)=e^{-\beta t}\psi(x),\ f(t)=e^{-\beta t}g(x)\), to an equation on the semiaxis of the form (1). In this case
\[ h_j(t)=a^{1+\alpha_j}e^{(1-\beta)t}k_j(1,e^t)\in \mathscr L_1, \]
\[ a_j(t)=\theta(t)c_j(ae^{-t})e^{-(1+\alpha_j)t},\qquad b_j(t)=\theta(t)d_j(ae^{-t}). \]
The indicated transformation establishes an isometry between the space \(E(0,\infty)\) and the weighted space \(E_\beta=E_\beta(0,a)=\{\psi\mid e^{-\beta t}\psi(ae^{-t})=\varphi(t)\in E(0,\infty)\},\ \|\psi\|_{E_\beta}=\|\varphi\|_{E_+}\). Obviously,
\[ E_\beta= \begin{cases} x^{-\beta}E(0,a), & E\ne \mathscr L_p,\\ x^{-\beta+1/p}E(0,a), & E=\mathscr L_p, \end{cases} \]
and if \(U\) is the operator realizing the isometry of \(E_\beta\) onto \(E_+\), then \(K=U^{-1}HU\).
We shall assume that in (2) \(\psi(x), g(x)\in E_\beta\). Applying the results obtained for the operator \(H\), we arrive at the following theorem:
Theorem 3. Let the kernels \(k_j(x,y)\) be homogeneous functions of orders \(\alpha_j,\ -\infty<\alpha_j<\infty,\ j=1,2,\ldots,n\), satisfying condition (6), and suppose also that \(d_j(x)\in C(0,a),\ x^{1+\alpha_j}c_j(x)\in C(0,a)\). Then the necessary and sufficient condition for the Noether property of the operator \(K\) in \(E_\beta\) has the form
\[ \sigma(\lambda)=1+\sum_{j=1}^{n} a^{1+\alpha_j}d_j(0)\lim_{x\to 0}\left[x^{1+\alpha_j}c_j(x)\right]K_j(i\lambda-\beta+1)\ne 0, \]
\[ -\infty\le \lambda\le \infty, \]
* That is, the functions \(\tilde a_j(t),\tilde b_j(t)\) satisfy the very same conditions (4), (5).
** The solvability conditions may be interpreted as orthogonality conditions to the zeros of the transposed operator, for example, in \(\mathscr L_p\) \((p>1)\).
where
\[ K_j(s)=\int_0^\infty k_j(1,x)x^{s-1}\,dx \]
is the Mellin transform of the function \(k_j(1,x)\). If this condition is satisfied,
\[ \varkappa_{E_\beta}(K)=-\frac{1}{2\pi}\Delta[\arg\sigma(\lambda)]_{-\infty}^{\infty}. \]
Let us note that the formulation of Theorem 3, for simplicity, is given for the case of continuous coefficients \(x^{1+\alpha_j}c_j(x)\), \(d_j(x)\). It is not difficult to obtain a theorem for more general coefficients, for example, for \(c_j(x)x^{1+\alpha_j}\), \(d_j(x)\in M^{\sup}(0,a)\), where
\[ M^{\sup}(0,a)=\{\psi(x)\mid \psi(x)\in M(0,a);\ \lim_{n\to\infty}\sup_{0<x<1/n}|\psi(x)-c|=0\},\quad c=\mathrm{const}. \]
Similarly to Theorem 3, from Theorem 2 there follows a theorem on the zeros of the operator \(K\), on whose formulation we shall not dwell. We also note that analogous theorems hold in the case \(a=\infty\).
As an example, consider the equation
\[ K_1\psi\equiv \psi(x)+\int_0^a\sum_{j=1}^n k_j(x,y)\psi(y)\,dy=f(x),\qquad 0<x<a, \tag{7} \]
where \(k_j(\lambda x,\lambda y)=\lambda^{\alpha_j}k_j(x,y)\). In the case \(n=1\) and \(\alpha_1=-1\) this equation was studied by L. G. Mikhailov \((^{11})\). Applying Theorem 3, we see that, for example, when \(\alpha_j>-1,\ j=1,2,\ldots,n\), equation (7) is Fredholm in all spaces \(E_\beta\): \(\varkappa_{E_\beta}(K_1)=0\) provided condition (6) is satisfied. It can be shown, using a theorem of L. Hörmander \((^{14}),\) p. 9, that the operator (7) is unbounded in any of the spaces \(E_\beta\) if \(\alpha_j<-1\) at least for one \(j,\ j=1,2,\ldots,n\). And, finally, we note that when \(n=1,\ \alpha_j=\alpha\ge -1\), the zeros of the operator (7) have the form \(\psi(x)=x^{1+\alpha-\beta}\psi_0(x)\), where \(\psi_0(x)\) is continuous and \(\psi(0)=0\).
As a second example, consider the equation
\[ K_2\psi\equiv \psi(x)+c(x)\int_0^1 d(y)\frac{\ln y-\ln x}{y-x}\psi(y)\,dy=g(x),\qquad 0<x<1, \]
in the spaces \(\mathscr L_p\) \((1<p<\infty)\). Let \(c(x), d(x)\in C(0,1)\) and
\[ v_p=\frac{1}{\pi}\sin\frac{\pi}{p}. \]
The necessary and sufficient condition for Noetherianity has the form
\[ c(0)d(0)>-\frac{1}{\pi^2}=-v_2^2 \]
for \(p=2\), and
\[ c(0)d(0)\ne -v_p^2 \]
for \(p\ne 2\). In this case \(\varkappa_{\mathscr L_p}(K_2)=0\) if \(c(0)d(0)>-v_p^2,\ 1<p<\infty\), and
\[ \varkappa_{\mathscr L_p}(K_2)=\operatorname{sign}(2-p) \]
if \(c(0)d(0)<-v_p^2,\ p\ne2\).
Rostov State University
Received
21 I 1970
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