Continuous evolution of equations and inclusions involving set-valued contraction mappings with applications to generalized fractal transforms (Q2877628)

Motivated by the theory of iterated function systems, the authors of this interesting paper study the asymptotic behavior of solutions to continuous evolution equations and inclusions governed by strictly contractive set-valued mappings \(T\) in general Banach spaces. They first use continuous single-valued selections and study single-valued solutions, and then analyze the behavior of set-valued solutions. In both cases they provide sufficient conditions for the convergence of trajectories to fixed points of \(T\). An application to generalized fractal transforms is also presented. In this connection, note that the existence of continuous selections and fixed points is guaranteed by the classical theorems of \textit{E. Michael} [Ann. Math. (2) 63, 361--382 (1956; Zbl 0071.15902)] and \textit{S. B. Nadler jun.} [Pac. J. Math. 30, 475--488 (1969; Zbl 0187.45002)], respectively. Note also that earlier results regarding the asymptotic behavior of solutions to evolution inclusions governed by monotone and accretive operators in Hilbert and Banach spaces can be found, for instance, in the papers by \textit{A. Pazy} [J. Anal. Math. 34, 1--35 (1978; Zbl 0399.47057)], the reviewer [Nonlinear Anal., Theory Methods Appl. 2, 85--92 (1978; Zbl 0375.47032)], \textit{O. Nevanlinna} and \textit{S. Reich} [Isr. J. Math. 32, 44--58 (1979; Zbl 0427.47049)] and by \textit{M. M. Israel jun.} and the reviewer [J. Math. Anal. Appl. 83, 43--53 (1981; Zbl 0508.47060)].