An infinite algebraic system in the irregular case

We obtain sufficient conditions for the nontrivial solvability of systems of the form $$ \phi _i = b_i + \lambda _i \sum\limits_{j = 0}^\infty {a_{i - j} \phi _j ,i \in \mathbb{Z}_ + \underline{\underline {def}} \{ 0,1,2...,n,...\} ,} $$ and of the corresponding homogeneous systems. It is assumed that the sequences b = (b ₀, b ₁, b ₂, …) and λ = (λ ₀, λ ₁, λ ₂, …) and the Toeplitz matrix A = (a _i?j ) satisfy the conditions $$ \begin{gathered} a_j \geqslant 0,j \in \mathbb{Z},\sum\limits_{j = - \infty }^\infty {a_j = 1,} \sum\limits_{j = - \infty }^\infty {|j|a_j < \infty ,\sum\limits_{j = - \infty }^\infty {ja_j < 0,} } \hfill \\ b_j \geqslant 0,j \in \mathbb{Z},\sum\limits_{j = 0}^\infty {b_j = \infty ,} 1 \leqslant \lambda _i \leqslant \left( {\sum\limits_{j = - \infty }^i {a_j } } \right)^{ - 1} ,i \in \mathbb{Z}_ + . \hfill \\ \end{gathered} $$ . Under these conditions, we construct bounded solutions of homogeneous and inhomogeneous systems of the form indicated above.     

Convolution equations and nonlinear functional equations

The survey is devoted to applications of nonlinear integral equations to linear convolution equations, their discrete analogues, and also the connection of these equations with problems of radiative transfer, in particular, with the Ambartsumyan equations.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 22, pp. 175–244, 1984.     

Nontrivial solvability of the homogeneous Wiener–Hopf multiple integral equation in the conservative case and the Peierls equation

Theoretical and Mathematical Physics - We describe the process of constructing a positive solution of the homogeneous Wiener–Hopf integral equation in an octant in a special $$($$...     

Factorization of Wiener-Hopf conservative integral operators

Translated from Matematicheskie Zametki, Vol. 46, No. 1, pp. 3–10, July, 1989.     

The Wiener--Hopf Integral Equation in the Supercritical Case

We consider the scalar homogeneous equation $S(x) = \int_0^\infty {K(x - t)S(t)dt,{\text{ }}x \in \mathbb{R}^ + \equiv (0,\infty ),}$ with symmetric kernel $K:K( - x) = K(x),{\text{ }}x \in \mathbb{R}_1$ satisfying the conditions $0 \leqslant K \in L_1 (\mathbb{R}^ + ) \cap C^{\left( 2 \right)} (\mathbb{R}^ + )$ , $\int_0^\infty {K(t)dt > \frac{1}{2}} $ , $K' \leqslant 0{\text{ }}and 0 \leqslant K'' \downarrow {\text{ }}on \mathbb{R}^ + $ . We prove the existence of a real solution S of the equation given above with asymptotic behavior $S(x) = O(x){\text{ as }}x \to + \infty $ .     

On a conservative integral equation with two kernels

We study the solvability of the integral equation

On the Volterra Factorization of the Wiener–Hopf Integral Operator

Mathematical Notes - The problem of the factorization of the Wiener–Hopf integral operator in the form of the product of the upper and lower Volterra operators is considered. Conditions for...