Solutions of Hammerstein equations in the space (Λ_1,Λ_2)BV(I^b_a)

Abstract

In this paper, we study the form of Hammerstein integral equations

$u(x)=v(x)+\lambda\int_{I^{b}_{a}}k(x,y)f(y,u(y))dy,(\lambda\in\mathbb{R})$

and Volterra Hammerstein integral equations in the condition of two-variables. Show the definition of $(\Lambda_{1},\Lambda_{2})$ bounded variation, write as $(\Lambda_{1},\Lambda_{2})BV(I^{b}_{a})$. If $v$, $k$ are $(\Lambda^{(1)},\Lambda^{(2)})BV(I^{b}_{a};\mathbb{R})$ functions and $f$ is a locally Lipschitz function, there exists a number $\rho>0$ such that when $|\lambda|<\rho$, Hammerstein integral equations has a unique solution. Give the proof and extend.