Abstract

A classical tool in nonlinear analysis is the notion of an approximating curve, whereby a particular solution to a nonuniquely solvable problem is obtained as the limit of the solutions to uniquely solvable perturbed problems. We introduce and analyze new types of approximating curves for nonexpansive fixed point problems and monotone inclusion problems in Hilbert spaces. The solution attained by these curves solves a strictly monotone variational inequality over the original solution set. Various special cases are discussed.

Author information

Contact details are reproduced from the original publication and may be historical.

Patrick L. Combettes

Laboratoire Jacques-Louis Lions, Université P. et M. Curie / Paris 6, 75005 Paris, France

plc@math.jussieu.fr

Sever A. Hirstoaga

Laboratoire Jacques-Louis Lions, Université P. et M. Curie / Paris 6, 75005 Paris, France

hirstoag@ann.jussieu.fr

Keywords

  • Approximating curve
  • monotone operator
  • nonexpansive operator
  • Tikhonov regularization
  • viscosity solution
  • variational inequality

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Suggested citation

P. L. Combettes, S. A. Hirstoaga. “Approximating Curves for Nonexpansive and Monotone Operators.” Journal of Convex Analysis 13 (2006), No. 3/4, 633–646. https://doi.org/10.68381/jca13049

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Published by Heldermann Verlag, 2006. Rights now held by Banach Press LLC.