Approximation of infinite dimensional fractals generated by integral equations
The authors present an interesting research result: For a Fredholm integral equation \[ u(x)= f(x)+\lambda \int^b_a K(x, y)u(y)\,dy,\quad f\in C([a,b]), \quad K\in C([a,b]\times[a,b]), \] consider \(K^i:[a,b]\times[a,b]\to\mathbb{R}\) and \(f^i:[a,b]\to \mathbb{R}\), \(1\leq i\leq h\). Then define contractions \(T^i: u\to v^i\) by \[ T^i(u):= v^i(x)= f^i(x)+\lambda \int^b_a K^i(x,y) u(y)\,dy. \] Let \(\{T^i: 1\leq i\leq h\}\) be an iterated function system, and \[ F(B)= \bigcup^h_{i=1} T^i(B). \] Then the attractor \(A\) generated by this IFS, has the property \(H(A,A_n)\to 0\), where \(H\) is the Hausdorff metric of \(A\) and \(A_n\).
scientific article; zbMATH DE number 5710747