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A class of integral equations of convolution type
L. G. Arabadzhyanab,
A. S. Khachatryanb a Institute of Mathematics, National Academy of Sciences of Armenia
b Armenian State Teachers' Training University named after Khachatur Abovian
Abstract:
Conditions (both necessary and sufficient)
for the existence of a non-trivial bounded solution
$B$ of the integral equation
$$
B(x)=\int_{-\infty}^{+\infty}\lambda(t)K(x-t)B(t)\,dt,\qquad x\in \mathbb R^1,
$$
are obtained for
fixed functions
$K$ and
$\lambda$ satisfying the following conditions:
\begin{gather*}
0\le K\in L_1(\mathbb R^1),
\qquad
\int_{-\infty}^\infty K(t)\,dt=1,
\\
\int_{-\infty}^\infty t^2K(t)\,dt<\infty,
\qquad
\nu\stackrel{\mathrm{def}}{=}\int_{-\infty}^{+\infty}tK(t)\,dt\ne0,
\\
0\le\lambda(x)\le1,
\qquad
x\in \mathbb R^1,
\qquad
\lambda\not\equiv0.
\end{gather*}
The existence of the limits
$B(\pm\infty)=\lim_{x\to\pm\infty}B(x)$ is proved and a relation
between these limits, the first-order moment
$\nu$, and
the integral norm of
$B$ is found.
Bibliography: 9 titles.
UDC:
517.968.2
MSC: Primary
45E10; Secondary
47G10 Received: 26.12.2005 and 02.10.2006
DOI:
10.4213/sm1483
Fulltext:
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