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

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