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.

Show me the PDF file (337 KB), TEX file, and other files for this article.

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

Return to the EJDE web page