ON THE STRUCTURE OF FIXED-POINT SETS OF NONEXPANSIVE MAPPINGS - PDF document

ON THE STRUCTURE OF FIXED-POINT SETS OF NONEXPANSIVE MAPPINGS Andrzej Wi´ snicki Maria Curie-Sk� lodowska University, Lublin, Poland NonStandard Methods and Applications in Mathematics Pisa, May 25-31, 2006 Banach’s Contraction Principle. Let ( M, ρ ) be a complete metric space and T : M → M a contraction: ρ ( T x, T y ) ≤ kρ ( x, y ) for some k < 1 and every x, y ∈ M . Then T has a unique fixed point: T x 0 = x 0 . 1

Definition. A mapping T : M → M is called nonexpansive if ρ ( T x, T y ) ≤ ρ ( x, y ) for every x, y ∈ M . Our standard assumptions: C - a bounded closed and convex sub- set of a Banach space X , T : C → C - nonexpansive: � T x − T y � ≤ � x − y � . Example: X = l 1 , � � ( x n ) ∈ l 1 : x n ≥ 0 , � x � = 1 C = , T x = T ( x 1 , x 2 , ... ) = (0 , x 1 , x 2 , ... ) . Then T : C → C is an isometry with- out fixed points. 2

Definition. We say that a Banach space X has the fixed point property (FPP) if every nonexpansive mapping T : C → C (defined on a closed con- vex bounded set C ) has a fixed point: T x = x . The first existence results were ob- tained by F. Browder, D. G¨ ohde and W. A. Kirk in 1965. Problem: • Does reflexivity imply FPP? • Does FPP imply reflexivity? Theorem (Maurey [1981], Dowling, Lennard [1997]). Let X = L 1 [0 , 1] and Y be a (closed) subspace of X . Then Y is reflexive iff Y has FPP. 3

Let T : C → C be nonexpansive, fix x 0 ∈ C , and put � � T n x = 1 1 − 1 T x, x ∈ C. nx 0 + n Then T n : C → C is a contraction and, by Banach’s contraction princi- ple, there exists x n ∈ C such that T n x n = x n . Consequently, we obtain the so-called approximate fixed point sequence ( x n ) for T : n →∞ � T x n − x n � = 0 . lim 4

Question: • Let T, S : C → C be commuting, nonexpansive mappings: T ◦ S = S ◦ T . Does there exist a sequence ( x n ) such that n →∞ � T x n − x n � = lim lim n →∞ � Sx n − x n � = 0? Nonstandard reformulation: Consider ∗ T, ∗ S : ∗ C → ∗ C and de- fine nonexpansive mappings T , � � S : � C → � C by putting T ( ◦ x ) = ◦ ( ∗ T x ) , � S ( ◦ x ) = ◦ ( ∗ T x ) , � where C = ◦ ( ∗ C ) = { ◦ x : x ∈ ∗ C } � 5

and ◦ x = { y ∈ ∗ E : � x − y � ∗ ≈ 0 } denotes the (generalized) standard part of x ). Question: x ∈ � C such that � • Does there exist ˆ T ˆ x = � S ˆ x = ˆ x ? Definition. F ix T is said to be a non- expansive retract of C if there exists a nonexpansive mapping r : C → F ix T such that rx = x for every x ∈ F ix T . 6

Theorem ([2003]). Suppose T, S : C → C are commuting nonexpansive mappings and F ix � T is a nonexpansive retract of � x ∈ � C . Then there exists ˆ C such that � x = � T ˆ S ˆ x = ˆ x. and, consequently, n →∞ � T x n − x n � = lim lim n →∞ � Sx n − x n � = 0 for some ( x n ). Proof sketch: the mixture of Bruck’s ideas [1973] and (iterated) nonstandard techniques. Let r : � C → Fix � T be a nonexpansive retraction onto Fix � T . 7

C → � r : � � Fix � • By transfer, � T is a non- expansive retraction in the (double) nonstandard hull � � X . • If x ∈ Fix � T , then � T ◦ � S x = � S ◦ � T x = � � S x and hence � � Fix � ⊂ Fix � S T T . � � By transfer, � � ⊂ � � Fix � Fix � T . S T r ) x = x , then x ∈ � • If ( � � Fix � S ◦ � T , rx = x , (since � r is a retraction), and � consequently ( � r ) x = � � � S ◦ � Sx = x . (Bruck’s argument). • Hence � T ∩ Fix � S = Fix ( � Fix � � � S ◦ � r ) � = ∅ , (it follows from ℵ 1 -saturation and the existence of an approximate fixed 8

point sequence: C → � � r : � � � Fix � S ◦ � T is a nonexpansive and neocontinu- ous mapping defined on a neocom- pact set � � C ). • But S ⊃ � Fix � T ∩ Fix � T ∩ Fix � � � Fix � � S � = ∅ and consequently � � � � � � � � � � � � T x n − x n S x n − x n lim � = lim � = 0 . n →∞ n →∞ for some sequence ( x n ) in � C . • By neocompactness again, Fix � T ∩ Fix � S � = ∅ . 9

Question: • If T : C → C is a nonexpansive mapping, is then F ix � T a nonexpan- sive retract of � C ? (Note that F ix T need not be a non- expansive retract of C but a mapping T : � � C → � C is much more regular). Theorem ([2006]). For any (at most) countable set A ⊂ Fix � T there exists a nonexpansive mapping r : � C → Fix � T such that rx = x for x ∈ A . Proof sketch: 10

∗ C , ω ∈ ∗ N \ N , and con- • Fix x ∈ sider an (internal) mapping T x : ∗ C → ∗ C defined by � � T x z = 1 1 − 1 ∗ T z , z ∈ ∗ C . ω x + ω • By transfer of the Banach Contrac- tion Principle, there exists exactly ∗ C such that one point, say, F ω x ∈ T x F ω x = F ω x . This defines a map- ∗ C → ∗ C which is *- ping F ω : nonexpansive. Moreover � � T x F ω x = F ω x = 1 1 − 1 ∗ T F ω x ω x + ω for x ∈ ∗ C . • Hence � ∗ T F ω x − F ω x � ∗ ≤ 1 ω diam C and � F ω x − x � ∗ ≤ ( ω − 1) � ∗ T x − x � ∗ . 11

• Put ◦ x = ◦ ( F ω x ) , x ∈ ∗ C . r ω and notice that r ω : � C → Fix � T is a well-defined nonexpansive mapping. • By ℵ 1 -saturation, for any countable set A ⊂ Fix � T there exists ω ∈ ∗ N \ N such that rx = x for x ∈ A. (The argument is not very easy in the language of Banach space ultrapro- ducts). For more details: A. Wi´ snicki, On fixed-point sets of nonex- pansive mappings in nonstandard hulls and Banach space ultrapowers , Nonlinear Anal., to appear. 12