Abstract
The following problem often occurs in functional analysis and optimization theory, as well as in other fields of pure and applied mathematics: given a nonempty, closed and convex subset K of a Banach space X and an operator T:K→K, do the sequences iteratively generated in K by the rule x k+1=Tx k converge to a fixed point of T no matter how the initial point x 0∈K is chosen? It is well known that this indeed happens, in some sense, for “standard” classes of operators (e.g., certain nonexpansive operators and operators of contractive type). In this chapter we show that the question asked above has an affirmative answer even if the operator T is not contractive in any standard sense, but is relatively nonexpansive with respect to a Bregman distance induced by a convex function f. More precisely, we prove that in appropriate complete metric spaces of operators which are relatively nonexpansive with respect to Bregman distances there exists a subset which is a countable intersection of open and everywhere dense sets such that for any operator belonging to this subset, all its orbits converge strongly.
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References
Browder, F. E. (1976). Proceedings of symposia in pure mathematics: Vol. 18. Nonlinear operators and nonlinear equations of evolution in Banach spaces (Part 2). Providence: Am. Math. Soc.
Butnariu, D., & Iusem, A. N. (1997). In Contemporary mathematics: Vol. 204. Recent developments in optimization theory and nonlinear analysis (pp. 61–91).
Butnariu, D., & Iusem, A. N. (2000). Totally convex functions for fixed points computation and infinite dimensional optimization. Dordrecht: Kluwer Academic.
Butnariu, D., Censor, Y., & Reich, S. (1997). Computational Optimization and Applications, 8, 21–39.
Butnariu, D., Reich, S., & Zaslavski, A. J. (1999). Numerical Functional Analysis and Optimization, 20, 629–650.
Butnariu, D., Iusem, A. N., & Burachik, R. S. (2000). Computational Optimization and Applications, 15, 269–307.
Butnariu, D., Iusem, A. N., & Resmerita, E. (2000). Journal of Convex Analysis, 7, 319–334.
Butnariu, D., Reich, S., & Zaslavski, A. J. (2001). Journal of Applied Analysis, 7, 151–174.
Censor, Y., & Lent, A. (1981). Journal of Optimization Theory and Applications, 34, 321–353.
Censor, Y., & Reich, S. (1996). Optimization, 37, 323–339.
Censor, Y., & Zenios, S. A. (1997). Parallel optimization. New York: Oxford University Press.
Goebel, K., & Reich, S. (1984). Uniform convexity, hyperbolic geometry, and nonexpansive mappings. New York: Dekker.
Reich, S. (1996). In Theory and applications of nonlinear operators of accretive and monotone type (pp. 313–318). New York: Dekker.
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Reich, S., Zaslavski, A.J. (2014). Relatively Nonexpansive Operators with Respect to Bregman Distances. In: Genericity in Nonlinear Analysis. Developments in Mathematics, vol 34. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9533-8_5
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DOI: https://doi.org/10.1007/978-1-4614-9533-8_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-9532-1
Online ISBN: 978-1-4614-9533-8
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