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Convolutions equations and nonlinear functional equations. (Russian) Zbl 0568.45004
Itogi Nauki Tekh., Ser. Mat. Anal. 22, 175-244 (1984).
The solution of the celebrated Wiener-Hopf equation \((1)\quad f(x)- \int^{\infty}_{0}K(x-y)f(y)dy=y(x),\) \(x\in {\mathbb{R}}^+\) with \(K\in L_ 1({\mathbb{R}})\), \(f,g\in L_ p({\mathbb{R}}^+)\) (1\(\leq p\leq \infty)\) is equivalent to the factorization \((2)\quad 1-F_+K(\lambda)=[1- F_+V_+(\lambda)][1-F_-V_-(\lambda)]\) with \(V_{\pm}\in L_ 1({\mathbb{R}}^+),\quad F_{\pm}\phi (\lambda)=\int^{\infty}_{0}e^{\pm i\lambda t}\phi (t)dt.\) Equation (2) can be rewritten also in the form of the nonlinear equation \((3^{\pm})\quad V_{\pm}(x)=K(\pm x)+\int^{\infty}_{0}V_{\pm}(y)V_{\pm}(x+t)dt,\) \(x\in {\mathbb{R}}^+\); in the symmetric case \(K(\pm x)\equiv K(x)\) there remains only one equation (3) for \(V_{\pm}(x)\equiv V(x)\). If the kernel function \(K_{\pm}(x)\) of (1) is represented in the form \(K_{\pm}(x)=\int^{b}_{a}e^{-xs}d\sigma_{\pm}(s),\) \(x\in {\mathbb{R}}^+\), \(0\leq a<b<\infty\), with non-decreasing functions \(\sigma_{\pm}(s)\), and we look for the solution of \((3^{\pm})\) in the following form \(V_{\pm}(x)=\int^{b}_{a}e^{- ixy}\phi_{\pm}(s)d\sigma_{\pm}(s),\) then \((3^{\pm})\) is equivalent to the nonlinear Chandrasekharan-Ambartsumyan’s equations \((4^{\pm})\quad \phi_{\pm}(s)=1+\phi_{\pm}(s)\int^{b}_{a}(\phi_{\mp}(r)/(s+r))d\;sigma_{\mp}(r).\) Due to the classical works of N. Wiener, E. Hopf, V. Fok, M. Krejn, V. Ambartsumyan, K. Chandrasekharan and others almost everything about the solutions of (1-4) is known. The paper is devoted to a survey of (mostly their own) results on the equations (1-4) and related topics.
Reviewer’s remarks: 1) the equality \(\| K\|_ p=\mu =\int^{\infty}_{-\infty}| K(x)| dx\) holds for \(p=1\) and for \(p>1\), K(x)\(\geq 0\) only (cf. p. 176). 2) Conditions (0.6) are not equivalent to \(\mu <1\) (only under the assumption (0.10); cf. p. 179).
Reviewer: R.Dudučava

MSC:
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45G10 Other nonlinear integral equations
47Gxx Integral, integro-differential, and pseudodifferential operators
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators