Continuous evolution of equations and inclusions involving set-valued contraction mappings with applications to generalized fractal transforms

Abstract

Let T be a set-valued contraction mapping on a general Banach space $\mathcal{B}$. In the first part of this paper we introduce the evolution inclusion $\dot x + x \in Tx$ and study the convergence of solutions to this inclusion toward fixed points of T. Two cases are examined: (i) T has a fixed point $\bar y \in \mathcal{B}$ in the usual sense, i.e., $\bar y = T \bar y$ and (ii) T has a fixed point in the sense of inclusions, i.e., $\bar y \in T \bar y$. In the second part we extend this analysis to the case of set-valued evolution equations taking the form $\dot x + x = Tx$. We also provide some applications to generalized fractal transforms.

Keywords

• Set-valued evolution inclusions, set-valued evolution equations,
• contractive set-valued functions, fixed points