On the Hammerstein equation in the space of functions of bounded $\varphi $-variation in the plane

Luis Azócar, Hugo Leiva, Jesús Matute, and Nelson Merentes

Address:
Luis Azócar, Área de Matemáticas, Universidad Nacional Abierta, Caracas, Venezuela
Hugo Leiva, Dpto. de Matemáticas, Universidad de Los Andes, La Hechicera, Mérida 5101, Venezuela
Corresponding author: Jesús Matute, Dpto. de Matemáticas, Universidad de Los Andes, La Hechicera, Mérida 5101, Venezuela
Nelson Merentes, Escuela de Matemáticas, Universidad Central de Venezuela, Caracas, Venezuela

E-mail:
azocar@yahoo.com
hleiva@ula.ve
jmatute@ula.ve
nmerucv@gmail.com

Abstract: 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.

AMSclassification: primary 45G10.

Keywords: existence and uniqueness of solutions of the Hammerstein integral equation in the plane, $\varphi$-bounded total variation norm on a rectangle.

DOI: 10.5817/AM2013-1-51