Systems of Wiener--Hopf integral equations, and nonlinear factorization equations

N. B. Engibaryan ; L. G. Arabadzhyan

Sbornik. Mathematics, Tome 52 (1985) no. 1, pp. 181-208

Résumé

Systems of Wiener–Hopf integral equations \begin{equation} f(x)=g(x)+\int_0^\infty T(x-t)f(t)\,dt \end{equation} and corresponding nonlinear factorization equations \begin{align} U(x)=T(x)+\int_0^\infty V(t)U(x+t)\,dt, \nonumber \\ V(x)=T(-x)+\int_0^\infty V(x+t)U(t)\,dt,\qquad x>0, \end{align} are studied. It is assumed that $T$ is a matrix-valued function with nonnegative components from $L_1(-\infty,\infty)$, with $\mu=r(A)\leqslant1$, where $\displaystyle A=\int_{-\infty}^\infty T(x)\,dx$, and $r(A)$ is the spectral radius of the matrix $A$. The conservative case $\mu=1$, to which major attention is given, falls outside the general theory of Wiener–Hopf integral equations, since the symbol of equation (1) degenerates. A number of results have been obtained about the properties of the solution of the factorization equation (2), and about the existence, asymptotics and other properties of the solution of the homogeneous and nonhomogeneous conservative equation (1). Bibliography: 21 titles.