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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.

##### MSC:
 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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