On the Hammerstein equation in the space of functions of bounded $\varphi $-variation in the plane
Luis Azócar, Hugo Leiva, Jesús Matute, Nelson Merentes · Archivum Mathematicum · 2013
In this paper we study existence and uniqueness of solutions for the Hammerstein equation \[ u(x) = v(x) + \lambda \int _{I_{a}^{b}} K(x,y) f\big (y,u(y)\big )\, dy\,, \quad x \in I_{a}^{b} := [a_{1},b_{1}] \times [a_{2},b_{2}]\,, \] in the space $BV_{\varphi }^{\mathbb{R}}(I_{a}^{b})$ of function of bounded total $\varphi -$variation in the sense of Riesz, where $ \lambda \in \mathbb{R} $, $ K \colon I_{a}^{b} \times I_{a}^{b} \rightarrow \mathbb{R} $ and $ f\colon I_{a}^{b} \times \mathbb{R} \rightarrow \mathbb{R}$ are suitable functions.