# zbMATH — the first resource for mathematics

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