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Convolutions equations containing singular probability distributions. (English. Russian original) Zbl 0882.45002
Izv. Math. 60, No. 2, 251-279 (1996); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 60, No. 2, 21-48 (1996).
Let $$V_C[a,b]$$, $$-\infty\leq a<b\leq\infty$$, be the Banach space of continuous functions of bounded variation. The article is devoted to equations of the form $\varphi(x)= g(x)- \int_0^\infty \varphi(t) dT(x-t),$ where $$T$$ is in $$V_C(R)$$ and contains a singular component. In particular the following classes of operators are considered: $$\Omega_C^+= \{U^+\varphi(x)= -\int_0^x \varphi(t) dT(x-t): T\in V_C[0,\infty]\}$$, $$\Omega_C^-= \{U^+\varphi(x)= \int_x^\infty \varphi(t) dT(t-x): T\in V_C^+\}$$, $$\Omega_C= \{T_\varphi(x)= - \int_0^\infty \varphi(t) dT(x-t): T\in V_C(R)\}$$.
The author introduces and studies nonlinear factorization equations for $$T$$, i.e. equations of the form $$I-T= (I-U^-)(I-U^+)$$, where $$T$$ is a given operator in $$\Omega_C$$ and $$U\pm$$ are operators in $$\Omega_C^\pm$$ to find. Factorization is constructed in the case when $$T(-\infty)= 0$$, $$T(x)\uparrow$$ in $$x$$, and $$T(+\infty)= \mu\leq 1$$. With the aid of this factorization, existence theorems are proved for homogeneous $$(g=0)$$ and non-homogeneous equations in the singular case $$\mu=1$$. Asymptotic and other properties of the solutions of formal Volterra equations corresponding to $$T(x)=0$$ for $$x\leq 0$$ are also investigated.

##### MSC:
 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
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