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A class of integral equations of convolution type. (English. Russian original) Zbl 1155.45002
Sb. Math. 198, No. 7, 949-966 (2007); translation from Mat. Sb. 198, No. 7, 45-62 (2007).
The existence of a non-trivial solution \(B\in L_\infty(\mathbb{R})\) of the integral equation
\[ B(x)=\int_{-\infty}^\infty \lambda(t)K(x-t)B(t)dt,\;x\in\mathbb{R}, \tag{1} \] is studied under the conditions:
\[ \begin{gathered} 0\leq K\in L_1(\mathbb{R}),\;\int_{-\infty}^\infty K(x)\,dx=1,\;\int_{-\infty}^\infty x^2 K(x)\,dx<+\infty;\tag{2}\\ \lambda\text{ is measurable on }\mathbb{R},\;0\leq\lambda\leq 1\text{ and } \lambda\not\equiv 0\text{ on }\mathbb{R}.\tag{3} \end{gathered} \] Main result: If conditions (2)–(3) hold, \(\nu:=\int_{-\infty}^\infty xK(x)\,dx\neq 0\) and for some \(\tau\in\mathbb{R}\),
\[ 1-\lambda(x)\in L_1(\tau,+\infty)\;(\text{if }\nu<0) \quad\text{or}\quad 1-\lambda(x)\in L_1(-\infty,\tau)\;(\text{if }\nu>0), \tag{4} \] then equation (1) has a non-trivial bounded solution \(B\). If conditions (2)–(3) hold, \(\lambda\) increases on \(\mathbb{R}\) and equation (1) has a non-trivial bounded solution \(B\), then \(\nu\neq 0\) and (4) is fulfilled.
The existence of the limits \(B(\pm\infty)=\lim_{x\to\pm\infty}B(x)\) and their relation to the first-order moment \(\nu\) are also studied.

45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45M05 Asymptotics of solutions to integral equations
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