Electron. J. Diff. Equ., Vol. 2014 (2014), No. 142, pp. 1-17.

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

Herb Kunze, Davide La Torre, Franklin Mendivil, Edward R. Vrscay

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.

Submitted July 16, 2013. Published June 18, 2014.
Math Subject Classifications: 34A60, 28A80.
Key Words: Set-valued evolution inclusions, set-valued evolution equations, contractive set-valued functions, fixed points.

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Herb Kunze
Department of Mathematics and Statistics
University of Guelph, Guelph, Ontario, Canada
email: hkunze@uoguelph.ca
Davide La Torre
Department of Economics, Management, and Quantitative Methods
University of Milan, Milan, Italy.
email: davide.latorre@unimi.it, davide.latorre@kustar.ac.ae
Franklin Mendivil
Department of Mathematics and Statistics, Acadia University
Wolfville, Nova Scotia, Canada
email: franklin.mendivil@acadiau.ca
Edward R. Vrscay
Department of Applied Mathematics, Faculty of Mathematics
University of Waterloo, Waterloo, Ontario, Canada
email: ervrscay@uwaterloo.ca

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