an:01047579
Zbl 0882.45002
Engibaryan, N. B.
Convolutions equations containing singular probability distributions
EN
Izv. Math. 60, No. 2, 251-279 (1996); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 60, No. 2, 21-48 (1996).
00042634
1996
j
45E10 47B35 47A68
convolution equations; Wiener-Hopf operator; singular probability distribution; nonlinear factorization; existence
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.
K.Georgiev (Rostov-na-Donu)