Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular

Because of Minty's classical correspondence between firmly nonexpansive mappings and maximally monotone operators, the notion of a firmly nonexpansive mapping has proven to be of basic importance in fixed point theory, monotone operator theory, and convex optimization. In this note, we show that if finitely many firmly nonexpansive mappings defined on a real Hilbert space are given and each of these mappings is asymptotically regular, which is equivalent to saying that they have or "almost have" fixed points, then the same is true for their composition. This significantly generalizes the result by Bauschke from 2003 for the case of projectors (nearest point mappings). The proof resides in a Hilbert product space and it relies upon the Brezis-Haraux range approximation result. By working in a suitably scaled Hilbert product space, we also establish the asymptotic regularity of convex combinations.2010 Mathematics Subject Classification: Primary 47H05, 47H09; Secondary 47H10, 90C25.

[1]  G. Minty.  Monotone (nonlinear) operators in Hilbert space , 1962 .

[2]  C. W. Groetsch,et al.  Generalized inverses of linear operators , 1977 .

[3]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[4]  A. Iusem,et al.  Set-valued mappings and enlargements of monotone operators , 2008 .

[5]  Simeon Reich,et al.  A limit theorem for projections , 1983 .

[6]  E. Beckenbach.  CONVEX FUNCTIONS , 2007 .

[7]  K. Deimling.  Nonlinear functional analysis , 1985 .

[8]  D. Varberg.  Convex Functions , 1973 .

[9]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[10]  H. Brezis,et al.  Image d’une somme d’operateurs monotones et applications , 1976 .

[11]  S. Simons.  Minimax and monotonicity , 1998 .

[12]  S. Simons.  From Hahn-Banach to monotonicity , 2008 .

[13]  H. Brezis.  Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert , 1973 .

[14]  E. Zeidler.  Nonlinear Functional Analysis and Its Applications: II/ A: Linear Monotone Operators , 1989 .

[15]  Nonlinear functional analysis and its applications, part I: Fixed-point theorems , 1991 .

[16]  Heinz H. Bauschke.  The composition of projections onto closed convex sets in Hilbert space is asymptotically regular , 2002, Proceedings of the American Mathematical Society.

[17]  Dimitri P. Bertsekas,et al.  On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators , 1992, Math. Program..

[18]  Heinz H. Bauschke,et al.  Near equality, near convexity, sums of maximally monotone operators, and averages of firmly nonexpansive mappings , 2011, Math. Program..

[19]  S. Reich,et al.  Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings , 1984 .

[20]  Itai Shafrir,et al.  The asymptotic behavior of firmly nonexpansive mappings , 1987 .

[21]  Simeon Reich,et al.  On the asymptotic behavior of nonlinear semigroups and the range of accretive operators II , 1981 .

[22]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[23]  W. A. Kirk,et al.  Topics in Metric Fixed Point Theory , 1990 .