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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
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Outline
Talk Outline
• Genericity
• Metric Fixed Point Theory
• Renorming Theory
S. Phothi Genericity of FPP under renorming 2/ 28
Outline
Talk Outline
• Genericity
• Metric Fixed Point Theory
• Renorming Theory
S. Phothi Genericity of FPP under renorming 2/ 28
Outline
Talk Outline
• Genericity
• Metric Fixed Point Theory
• Renorming Theory
S. Phothi Genericity of FPP under renorming 2/ 28
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”.
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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”.
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
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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”.
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 .
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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 .
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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 .
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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 .
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Generic Property
Properties of porous sets
Porous sets ⇒ Nowhere dense sets
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Generic Property
Properties of porous sets
Porous sets ⇒ Nowhere dense sets⇓
Null Lebesgue measure sets
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Generic Property
Properties of porous sets
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 .
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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.
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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.
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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 .
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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.
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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 .
Remark on Banach Contraction Principle
Banach theorem fails when k = 1 (i.e., T is non-expansive).
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Metric Fixed Point Theorem
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.
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Metric Fixed Point Theorem
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.
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Metric Fixed Point Theorem
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.
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.
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Metric Fixed Point Theorem
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.
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.
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Metric Fixed Point Theorem
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.
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.
(D.E. Alspach, 1981) There is a weakly compact convexsubset of L1([0, 1]) which fails to have fixed point fornon-expansive mapping.
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Metric Fixed Point Theorem
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.
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Metric Fixed Point Theorem
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.
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Metric Fixed Point Theorem
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.
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Metric Fixed Point Theorem
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.
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Metric Fixed Point Theorem
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.
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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 .
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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 .
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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).
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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).
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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
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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
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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
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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?
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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?
If X can be renormed to have the FPP (or the w-FPP). Howmany renormings of X do satisfy the FPP (the w-FPP)?
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Renorming theory and metric fixed point 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?
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Renorming theory and metric fixed point 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?
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.
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Renorming theory and metric fixed point 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?
J. Partington (1981)
Every renorming of ∞/c0 contains an isometriccopy of ∞ (so it fails to have the w-FPP).
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Renorming theory and metric fixed point theory
Problems
Let X be a reflexive Banach space. Is it possible to renorm X sothat the resultant space has the FPP?
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Renorming theory and metric fixed point theory
Problems
Let X be a reflexive Banach space. Is it possible to renorm X sothat the resultant space has the FPP?
Day-James-Swaminathan, V. Zizler (1971)
Every separable Banach space admits anequivalent uniformly convex in every direction(UCED) norm
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Renorming theory and metric fixed point theory
Problems
Let X be a reflexive Banach space. Is it possible to renorm X sothat the resultant space has the FPP?
The w-FPP renormability of separable spaces
Every separable space can be renormed to have the w-FPP.
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Renormability on non-separable spaces
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Renormability on non-separable spaces
D. Kutzarova and S.L. Troyanski (1982)
There are reflexive spaces without equivalent normswhich are UCED.
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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.
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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.
Main tool
For any WCG Banach space X , there exist a set Γ and a boundedone-to-one linear operator J : X → c0(Γ).
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Renormability on non-separable spaces
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.
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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)?
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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.
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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.
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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.
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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.
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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.
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.
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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.
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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.
J. Garcıa-Falset (1997)
Let X be a Banach space such that R(X ) < 2.Then X has the w-FPP.
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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.
The w-FPP on c0(Γ) space
c0(Γ) enjoys the w-FPP because R(c0(Γ)) = 1.
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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.
T. Domınguez Benavides and S.P. (2008)
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).
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Generic fixed point property on some classes of Banach
spaces
T. Domınguez Benavides and S.P. (2010)
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.
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Generic fixed point property on some classes of Banach
spaces
T. Domınguez Benavides and S.P. (2010)
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.
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The w-FPP renorming and the generic w-FPP on a space
embedded into Y satisfying R(Y ) < 2
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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 .
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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 .
T. Domınguez Benavides and S.P. (2010)
There exists an equivalent norm in X such that X endowed with thenew norm satisfies the w-FPP.
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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 .
T. Domınguez Benavides and S.P. (2010)
There exists an equivalent norm in X such that X endowed with thenew norm satisfies the w-FPP.
T. Domınguez Benavides and S.P. (2010)
There exists a residual subset R in P such that every q ∈ R, thespace (X , q) has the w-FPP.
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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.
T. Domınguez Benavides and S.P. (2010)
Assume that K (m) = ∅. Then, there exists a norm | · | equivalent tothe supremum norm · ∞ such that R(C (K ), | · |) ≤
√4−m−1.
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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.
T. Domınguez Benavides and S.P. (2010)
Assume that K (m) = ∅. Then, there exists a norm | · | equivalent tothe supremum norm · ∞ such that R(C (K ), | · |) ≤
√4−m−1.
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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.
T. Domınguez Benavides and S.P. (2010)
Assume that K (m) = ∅. Then, there exists a norm | · | equivalent tothe supremum norm · ∞ such that R(C (K ), | · |) ≤
√4−m−1.
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The w-FPP renorming on a space embedded into a space
of continuous functions
T. Domınguez Benavides and S.P. (2010)
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.
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The w-FPP renorming on a space embedded into a space
of continuous functions
T. Domınguez Benavides and S.P. (2010)
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).
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Generic w-FPP on spaces of continuous functions
T. Domınguez Benavides and S.P. (2010)
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.
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Remarks
1 A natural question would be to study if the word “almost”can be removed from our generic results.
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Remarks
1 A natural question would be to study if the word “almost”can be removed from our generic results.
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.
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Remarks
1 A natural question would be to study if the word “almost”can be removed from our generic results.
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).
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Remarks
1 A natural question would be to study if the word “almost”can be removed from our generic results.
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).
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.
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Remarks
1 A natural question would be to study if the word “almost”can be removed from our generic results.
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).
3 It would be also interesting to determine those non-reflexiveBanach spaces, such that our results hold for them.
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Remarks
1 A natural question would be to study if the word “almost”can be removed from our generic results.
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).
3 It would be also interesting to determine those non-reflexiveBanach spaces, such that our results hold for them.
Example
Do our results hold for 1?
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Remarks
1 A natural question would be to study if the word “almost”can be removed from our generic results.
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).
3 It would be also interesting to determine those non-reflexiveBanach spaces, such that our results hold for them.
4 Generic results can be useful to obtain standard results.
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Remarks
1 A natural question would be to study if the word “almost”can be removed from our generic results.
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).
3 It would be also interesting to determine those non-reflexiveBanach spaces, such that our results hold for them.
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.
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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.
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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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The End
Thank you very much
Hot balloons festival, Chiang Mai, Thailand.
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