Genericity of the fixed point property under renorming › functanalysis › meetingsold › LaManga › pdfs › Phothi.pdf · Metric Fixed Point Theorem Monographs relating to Metric

Genericity of the fixed point property under

renorming

Supaluk Phothi, Chiangmai University(Thailand)Universidad de Sevilla, Sevilla

La Manga del Mar Menor, MurciaApril 19 - 21, 2012

S. Phothi Genericity of FPP under renorming 1/ 28

Outline

Talk Outline

• Genericity

• Metric Fixed Point Theory

• Renorming Theory


Outline

Talk Outline

• Genericity




Outline

Talk Outline

• Genericity




Generic Property

Generic property

A property P is said to be generic in a set A if all elements in A

satisfy P except those in a “negligible set”.


Generic Property

Generic property



What are negligible sets?

Space Negligible setCardinality Countable sets

Measure space (X ,Σ, µ) Null µ-measurable setsTopological spaces First Baire category sets

Metric spaces σ-Porous sets


Generic Property

Generic property



Porous sets

Let (X , d) be a complete metric space. A subset E ⊂ X is porousin (X , d) if there exist α ∈ (0, 1) and r0 > 0 such that for eachr ∈ (0, r0] and each x ∈ X , there exists y ∈ X for which

Bd(y ,αr) ⊂ Bd(x , r)\E .


Generic Property

Nowhere dense sets

A subset E ⊂ X is nowhere dense in if for each x ∈ X and eachr > o, there are a point y ∈ X and a number s > 0 such that

B(y , s) ⊆ B(x , r)\E .


Generic Property

Nowhere dense sets


B(y , s) ⊆ B(x , r)\E .


Generic Property

Nowhere dense sets


B(y , s) ⊆ B(x , r)\E .


Generic Property

Properties of porous sets

Porous sets ⇒ Nowhere dense sets


Generic Property


Porous sets ⇒ Nowhere dense sets⇓

Null Lebesgue measure sets


Generic Property


Porous sets ⇒ Nowhere dense sets⇓

Null Lebesgue measure sets

Porous and σ-porous sets

A subset Y ⊂ X is said to be σ-porous if it is a countable union ofporous subsets of X .


Example of Generic properties

Peano-Cauchy Theorem

Let Ω be a subset of Rn+1, f : Ω → Rn a continuousfunction and (t0, x0) a point in Ω. The I.V.P.x = f (t, x) ; x(t0) = x0 has a solution

W. Orlicz (1932)

For almost all function f ∈ C (Ω;Rn)(in the sense of the Baire category) the above problemhas exactly one solution.


Example of Generic properties

Peano-Cauchy Theorem

Let Ω be a subset of Rn+1, f : Ω → Rn a continuousfunction and (t0, x0) a point in Ω. The I.V.P.x = f (t, x) ; x(t0) = x0 has a solution

W. Orlicz (1932)

For almost all function f ∈ C (Ω;Rn)(in the sense of the Baire category) the above problemhas exactly one solution.


Metric Fixed Point Theorem

Banach Contraction Principle (1922)

Let (M, d) be a complete metric space andT : M → M be a contraction mapping. Then T hasa unique fixed point in M. Moreover, for any x0 ∈ M

the sequence of iterates x0,T (x0),T 2(x0),T 3(x0), ...converges to a fixed point of T .






Contraction mappings

A mapping T is called contraction if there exists k ∈ [0, 1) suchthat d(Tx ,Ty) ≤ kd(x , y) for all x , y ∈ M.






Remark on Banach Contraction Principle

Banach theorem fails when k = 1 (i.e., T is non-expansive).



F. Browder - D. Gohde (1965)

Let X be a Banach space, C a closed convex boundedsubset of X . Assume that T : C → C is a non-expansive mapping. Then T has a fixed point if X isa uniformly convex space.





The failure of Browder - Gohde Theorem

Assume that B is the closed unit ball of c0. The mappingT : B → B defined by T (x1, x2, ...) = (1, x1, x2, ...) is a fixed pointfree isometry.





Remarks on the previous example

In this case, the non-existence of fixed points is due to thefact that the set B is not weakly compact.







(B. Maurey, 1981)Every non-expansive mapping T definedfrom a weakly compact convex subset C of c0 into C has afixed point.







(B. Maurey, 1981)Every non-expansive mapping T definedfrom a weakly compact convex subset C of c0 into C has afixed point.

(D.E. Alspach, 1981) There is a weakly compact convexsubset of L1([0, 1]) which fails to have fixed point fornon-expansive mapping.



The Fixed Point Property

Let X be a Banach space and C a closed bounded convex subset ofX . We say that the space X enjoys the fixed point property (FPP)if every non-expansive mapping T : C → C has a fixed point.

The Weak Fixed Point Property

X is said to have the weak fixed point property (w-FPP) if everyweakly compact convex subset C of X and every non-expansivemapping T : C → C has a fixed point.



The Fixed Point Property

Let X be a Banach space and C a closed bounded convex subset ofX . We say that the space X enjoys the fixed point property (FPP)if every non-expansive mapping T : C → C has a fixed point.

The Weak Fixed Point Property

X is said to have the weak fixed point property (w-FPP) if everyweakly compact convex subset C of X and every non-expansivemapping T : C → C has a fixed point.



Geometrical properties implying the FPP and the w-FPP

• Uniform convexity (UC)

• Uniform smoothness (US)

• Nearly uniform convexity (NUC)

• Uniform convexity in every direction (UCED)

Remark

No characterization of the FPP in terms of some other geometricalproperties is known.



Geometrical properties implying the FPP and the w-FPP

• Uniform convexity (UC)

• Uniform smoothness (US)

• Nearly uniform convexity (NUC)

• Uniform convexity in every direction (UCED)

Remark

No characterization of the FPP in terms of some other geometricalproperties is known.



Monographs relating to Metric Fixed Point Theory

K. Goebel, and W.A. Kirk.Topics in metric fixed point theory.

Cambridge Studies in Advanced Mathematics, 28. CambridgeUniversity Press, Cambridge, 1990.

Edited by W.A. Kirk and B. Sims.Handbook of metric fixed point theory..Kluwer Academic Publishers, Dordrecht, 2001.

K. Goebel, and S. Reich.Uniform convexity, hyperbolic geometry, and nonexpansive

mappings.

Monographs and Textbooks in Pure and Applied Mathematics,83. Marcel Dekker, Inc., New York, 1984.


Genericity and metric fixed point theory

G. Vidossich (1974)

Let X be a Banach space and C be a bounded closed and convexsubset of X . Denote by A = A : C → C : A is non-expansiveendowed with a metric h(A,B) = supAx − Bx : x ∈ C. Thenthe subset F0 of all F ∈ A which have a unique fixed point isresidual in A.

F.S. De Blasi and J. Myjak (1989)

There is a subset F1 ⊂ A such thatthe complement A\F1 is σ-porous inA and for each A ∈ F1 the followingproperty holds: There exists a uniquexA ∈ C for which AxA = xA andAnx → xA as n → ∞ uniformly on C .


Genericity and metric fixed point theory

G. Vidossich (1974)

Let X be a Banach space and C be a bounded closed and convexsubset of X . Denote by A = A : C → C : A is non-expansiveendowed with a metric h(A,B) = supAx − Bx : x ∈ C. Thenthe subset F0 of all F ∈ A which have a unique fixed point isresidual in A.

F.S. De Blasi and J. Myjak (1989)

There is a subset F1 ⊂ A such thatthe complement A\F1 is σ-porous inA and for each A ∈ F1 the followingproperty holds: There exists a uniquexA ∈ C for which AxA = xA andAnx → xA as n → ∞ uniformly on C .


Renorming theory

The objective of renorming

Renorming method is an attempt to find an equivalent norm ona Banach space which satisfies (or which does not satisfy) somespecific properties.

Monographs relating to Remorming Theory

R. Deville, G. Godefroy, and V. ZizlerSmoothness and Renormings in Banach Spaces (1993).

M. Fabian, P. Habala, P. Hajek, V. Montesinos Santalucıa,J. Pelant, and V. ZizlerFunctional Analysis and Infinite-dimensional Geometry (2001).

G. GodefroyRenormings of Banach spaces

Handbook of the geometry of Banach spaces (2001).


Renorming theory

The objective of renorming

Renorming method is an attempt to find an equivalent norm ona Banach space which satisfies (or which does not satisfy) somespecific properties.

Monographs relating to Remorming Theory

R. Deville, G. Godefroy, and V. ZizlerSmoothness and Renormings in Banach Spaces (1993).

M. Fabian, P. Habala, P. Hajek, V. Montesinos Santalucıa,J. Pelant, and V. ZizlerFunctional Analysis and Infinite-dimensional Geometry (2001).

G. GodefroyRenormings of Banach spaces

Handbook of the geometry of Banach spaces (2001).


Renorming theory and Metric fixed point theory

Remark.

The FPP and the w-FPP are not an isometric property.

Non-isometric property of the FPP and the w-FPP

• (P.K. Lin, 2008) The space 1 can be renormed to have theFPP

• (D. Van Dulst, 1982) The space L1([0, 1]) can be renormed tohave normal structure and so the w-FPP



Remark.







Remark.






Interesting problems related to genericity, metric fixed

point theory and renorming theory

Conjectures.

Let X be a Banach space. Is it possible to renorm X so thatthe resultant space has the FPP or the w-FPP?


Interesting problems related to genericity, metric fixed

point theory and renorming theory

Conjectures.


If X can be renormed to have the FPP (or the w-FPP). Howmany renormings of X do satisfy the FPP (the w-FPP)?


Renorming theory and metric fixed point theory

Conjectures.




Conjectures.


P. Dowling, C. Lennard and B. Turett (2003)

Every renorming of c0(Γ) when Γ is uncountablecontains an asymptotically isometric copy of c0and so it fails to have the FPP.



Conjectures.


J. Partington (1981)

Every renorming of ∞/c0 contains an isometriccopy of ∞ (so it fails to have the w-FPP).



Problems

Let X be a reflexive Banach space. Is it possible to renorm X sothat the resultant space has the FPP?



Problems


Day-James-Swaminathan, V. Zizler (1971)

Every separable Banach space admits anequivalent uniformly convex in every direction(UCED) norm



Problems


The w-FPP renormability of separable spaces

Every separable space can be renormed to have the w-FPP.


Renormability on non-separable spaces



D. Kutzarova and S.L. Troyanski (1982)

There are reflexive spaces without equivalent normswhich are UCED.





Amir-Lindenstrauss (1968)

Every WCG Banach space admits an equivalent strictlyconvex norm.





Amir-Lindenstrauss (1968)

Every WCG Banach space admits an equivalent strictlyconvex norm.

Main tool

For any WCG Banach space X , there exist a set Γ and a boundedone-to-one linear operator J : X → c0(Γ).



T. Domınguez Benavides (2009)Assume that X is a Banach spacesuch that there exists a boundedone-one linear operator from X intoc0(Γ). Then, X has an equivalentnorm such that every non-expansivemapping T for the new norm definedfrom a convex weakly compact set Cinto C has a fixed point.


Genericity concerning with renorming theory and metric

fixed point theory

Conjectures.

If a Banach space X can be renormed to have the FPP (orthe w-FPP). How many renormings of X do satisfy the FPP(the w-FPP)?


Genericity concerning with renorm theory and metric fixed

point theory

The space of all renormings of a Banach space

Let P be the set of all equivalent norms on a Banach space (X , ·).Define the metric ρ on P in the following way:

ρ(p, q) = sup|p(x)− q(x)| : x ∈ BX.


Genericity concerning with renorm theory and metric fixed

point theory

The space of all renormings of a Banach space

Let P be the set of all equivalent norms on a Banach space (X , ·).Define the metric ρ on P in the following way:

ρ(p, q) = sup|p(x)− q(x)| : x ∈ BX.

Remark

(P, ρ) is a Baire space.


Generic fixed point property on separable spaces

M. Fabian, L. Zajıcek and V. Zizler (1982)

If (X , · ) is UCED then, there exists aresidual subset R (in fact a dense-Gδ) of P,such that for all p ∈ R, the space (X , p)is UCED.





Generic FPP on renormings of separable spaces

If X is a separable Banach space then, almost all renormings of X(in the sense of Baire category) have the w-FPP.





T. Domınguez Benavides and S.P. (2008)

Almost all renormings of a separable Banach space satisfy thew-FPP except those in a σ-porous set.


Generic fixed point property on some classes of Banach

spaces

The Banach space coefficient R(·)The coefficient R(X ) is defined by

R(X ) := suplim infn→∞

xn + x

where the supremum is taken over all weakly null sequences (xn) ofthe unit ball and all points x of the unit ball.



spaces



xn + x


J. Garcıa-Falset (1997)

Let X be a Banach space such that R(X ) < 2.Then X has the w-FPP.



spaces



xn + x


The w-FPP on c0(Γ) space

c0(Γ) enjoys the w-FPP because R(c0(Γ)) = 1.



spaces



xn + x



Let X be a Banach space such that R(X ) < 2. Then, there existsa σ-porous subset R of P such that for every norm p ∈ P \ R, wehave R(X , p) < 2 (and so (X , p) has the w-FPP).



spaces


Let X be a Banach space such that for some set Γ there exists aone-to-one linear continuous mapping J : X → c0(Γ). Then, thereexists a residual subset R in P such that for every q ∈ R, everyq-non-expansive mapping T defined from a weakly compact convexsubset C of X into C has a fixed point.



spaces


Let X be a Banach space such that for some set Γ there exists aone-to-one linear continuous mapping J : X → c0(Γ). Then, thereexists a residual subset R in P such that for every q ∈ R, everyq-non-expansive mapping T defined from a weakly compact convexsubset C of X into C has a fixed point.

Generic w-FPP on reflexive spaces

Almost all renormings of a reflexive Banach space have the FPP.


The w-FPP renorming and the generic w-FPP on a space

embedded into Y satisfying R(Y ) < 2




Let (X , · X ) and (Y , · Y ) be Banach spaces. Assumethat R(Y ) < 2 and there exists a one-to-one linear continuousmapping J : X → Y .






There exists an equivalent norm in X such that X endowed with thenew norm satisfies the w-FPP.






There exists an equivalent norm in X such that X endowed with thenew norm satisfies the w-FPP.


There exists a residual subset R in P such that every q ∈ R, thespace (X , q) has the w-FPP.


The w-FPP renorming on spaces of continuous functions

Let K be a scattered compact topological space. Denote byK

(1) the set of all accumulation points of K . If α is an ordinalnumber, we define the αth-derived set by transfinite induction:

K(0) = K K

(α+1) = (K (α))(1) K(λ) =

α<λ

K(α)

where λ is a limit ordinal.


Assume that K (m) = ∅. Then, there exists a norm | · | equivalent tothe supremum norm · ∞ such that R(C (K ), | · |) ≤

√4−m−1.





K(0) = K K

(α+1) = (K (α))(1) K(λ) =

α<λ

K(α)




√4−m−1.





K(0) = K K

(α+1) = (K (α))(1) K(λ) =

α<λ

K(α)




√4−m−1.


The w-FPP renorming on a space embedded into a space

of continuous functions


Let X be a Banach space which can be continuously embedded in(C (K ), · ∞) for some compact set K such that K (ω) = ∅. Then,X can be renormed to satisfy the w-FPP.


The w-FPP renorming on a space embedded into a space

of continuous functions


Let X be a Banach space which can be continuously embedded in(C (K ), · ∞) for some compact set K such that K (ω) = ∅. Then,X can be renormed to satisfy the w-FPP.

K. Ciesielski and R. Pol (1984)

Ciesielski-Pol set K is a (non-metrizable) compactset which satisfies K (3) = ∅.However, there is no bounded linear injective mapfrom C (K ) to any c0(Γ).


Generic w-FPP on spaces of continuous functions


Let X be a Banach space which can be continuously embedded in(C (K ), · ∞) for some compact set K such that K (ω) = ∅ and letP be the set of all equivalent norms on X equipped with the metricρ. Then, there exists a σ-porous set A ⊂ P such that if p ∈ P \ Athe space (X , p) satisfies the w-FPP.


Remarks

1 A natural question would be to study if the word “almost”can be removed from our generic results.


Remarks


Example

• It is unknown if any Banach space isomorphic to a Hilbertspace satisfies the FPP.

• It is unknown if there exists a reflexive Banach space whichdoes not have the FPP.


Remarks


2 It would be interesting to determine other properties on theset of all equivalent norms which do not satisfy the FPP orthe w-FPP (if non-empty).


Remarks



Example

• There are some results proving that several properties of aBanach space X implying the FPP are stable.

• If H is a Hilbert space and X is a renorming of H such thatρ(X ,H) < .36..., then X satisfies the FPP.


Remarks



3 It would be also interesting to determine those non-reflexiveBanach spaces, such that our results hold for them.


Remarks




Example

Do our results hold for 1?


Remarks




4 Generic results can be useful to obtain standard results.


Remarks




4 Generic results can be useful to obtain standard results.

Example

Let X be a reflexive Banach space. Then, there exists an equivalentnorm p of X such that (X , p) and (X ∗, p∗) satisfy the FPP.


Publications

T. Domınguez Benavides and S. P.Porosity of the fixed point property under renorming.Fixed point theory and its applications, 29-41, YokahomaPubl. (2008).

T. Domınguez Benavides and S. P.Genericity of the fixed point property for reflexive spaces underrenormings.Contemporary Mathematics. 513 (2010), 143-155.


Publications

T. Domınguez Benavides and S. P.The fixed point property under renorming in some classes ofBanach spaces.Nonlinear Anal. 72 (2010), no 3-4, 1409-1416.

T. Domınguez Benavides and S. P.Genericity of the fixed point property under renorming in someclasses of Banach spaces.Fixed point theory and its applications, 55-69, YokahomaPubl. (2010).


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Genericity of the fixed point property under renorming › functanalysis › meetingsold › LaManga › pdfs › Phothi.pdf · Metric Fixed Point Theorem Monographs relating to Metric