Nonexpansive mappings. Existence of fixed points

Some open problems in Metric FixedPoint Theory

Enrique Llorens Fuster

Universidad de Valencia

Valencia November 16, 2012

Enrique Llorens Fuster Some open problems in Metric Fixed Point Theory

The most general

(FPP), (WFPP)

A Banach space (X , ‖ · ‖) is said to have the fixed point propertyfor nonexpansive mappings ((FPP) for short) provided that everynonexpansive selfmapping of every nonempty, closed, convex,bounded subset C of X has a fixed point. If the same holds forevery nonempty, weakly compact convex subset C of X we saythat this space has the weak fixed point property for nonexpansivemappings ((WFPP)).

The Banach space (X , ‖ · ‖) is said to have the RNP (resp. KMP)if every nonempty bounded closed convex set K in X has a denting(resp. extreme) point.

PROBLEM 0 If a Banach spaces has the Radon-Nicodymproperty, then does it have (WFPP)?

Let K be a nonempty weakly compact convex subset of X . LetT : K → K be a nonexpansive mapping. We may suppose that Kis minimal for T . For r ∈ (0, 1) let Tr : K → K given byTr (x) := rx + (1− r)T (x). Then Tr is also nonexpansive. If T isfixed point free, then so is Tr , and we can obtain a minimal weaklycompact convex Tr invariant subset Kr of K . If X has RNP thenthere exists a denting point xr ∈ Kr .

Nonexpansive mappings. Existence of fixed points

The most famous

PROBLEM I Does every reflexive Banach space have (FPP)?

PROBLEM II Does every superreflexive Banach space have (FPP)?

PROBLEM III Does every Banach space which is isomorphic to `p(1 < p <∞) have (FPP)?

Enrique Llorens Fuster Some open problems in Metric Fixed Point Theory

It is well known that, for 1 < p <∞, there exists bp > 1 suchthat, if ‖ · ‖ is an equivalent norm on `p such that‖ · ‖p ≤ ‖ · ‖ ≤ bp‖ · ‖p, then the space (`p, ‖ · ‖) enjoys (FPP).The case p = 2 has been largely studied (Eva’s talk).

PROBLEM IV To improve the nowadays known values for bp.

Regarding the case p = 2, it its known that, given ε > 0 thereexists a uniformly convex norm ‖ · ‖uc such that for every x ∈ `2,‖x‖uc ≤ ‖x‖ ≤ (1 + ε)‖x‖uc .

PROBLEM V Given an equivalent renorming of `2, and givenb < BEva does exist a Hilbert norm, say ‖ · ‖h on `2 such that‖ · ‖h ≤ ‖ · ‖ ≤ b‖ · ‖h ?

It is well known that, for 1 < p <∞, there exists bp > 1 suchthat, if ‖ · ‖ is an equivalent norm on `p such that‖ · ‖p ≤ ‖ · ‖ ≤ bp‖ · ‖p, then the space (`p, ‖ · ‖) enjoys (FPP).The case p = 2 has been largely studied (Eva’s talk).

PROBLEM IV To improve the nowadays known values for bp.

Regarding the case p = 2, it its known that, given ε > 0 thereexists a uniformly convex norm ‖ · ‖uc such that for every x ∈ `2,‖x‖uc ≤ ‖x‖ ≤ (1 + ε)‖x‖uc .

PROBLEM V Given an equivalent renorming of `2, and givenb < BEva does exist a Hilbert norm, say ‖ · ‖h on `2 such that‖ · ‖h ≤ ‖ · ‖ ≤ b‖ · ‖h ?

PROBLEM VI Can c0 be equivalently renormed so that theresulting space has (FPP)?

It is known that (c0, ‖.‖∞) enjoys (WFPP).

Theorem (Borwein y Sims, 1984)

If either d(c0,X ) < 2 or d(c ,X ) < 2 then X has (WFPP).

Indeed, if d(c0,X ) < 2 then X has (OC).

PROBLEM VII Is this bound 2 sharp?

PROBLEM VIII Can be c0 renormed such that the resulting spacefails (WFPP)?

PROBLEM VI Can c0 be equivalently renormed so that theresulting space has (FPP)?

It is known that (c0, ‖.‖∞) enjoys (WFPP).

Theorem (Borwein y Sims, 1984)

If either d(c0,X ) < 2 or d(c ,X ) < 2 then X has (WFPP).

Indeed, if d(c0,X ) < 2 then X has (OC).

PROBLEM VII Is this bound 2 sharp?

PROBLEM VIII Can be c0 renormed such that the resulting spacefails (WFPP)?

A Banach space is said to have the non strict Opial property(NSO) provided that(xn ⇀ x ⇒ ∀y ∈ X , lim inf ‖xn − x‖ ≤ lim inf ‖xn − y‖).

PROBLEM IX Does the following implication hold?(NSO) ⇒ (WFPP).

µ(X ) := inf{r > 0 : lim sup ‖xn + x‖ ≤ r lim sup ‖xn − x‖ : xn ⇀ 0X x ∈ X},

Theorem (Fetter and Gamboa, 2010)

If X reflexive and µ(X ) = 1 then X has (FPP).

PROBLEM X Does the following implication hold?µ(X ) = 1⇒ X has (WFPP).

µ(X ) := inf{r > 0 : lim sup ‖xn+x‖ ≤ r lim sup ‖xn−x‖ : xn ⇀ 0 x ∈ X}.Since 1 ≤ µ(X ) ≤ 3, and reflexive Banach spaces with µ(X ) = 1have (FPP), an obvious question arises

PROBLEM XI. Does µ(X ) < b ∈ (1, 3] imply that X enjoy(WFPP)?

Notice that µ(c) = 3 and c has (WFPP).

(A. Jimenez Melado, E.Ll.F. and S. Saejung, 2006).If X is a non-Hilbert space,

µ(X ) <√

1CNJ(X )−1 ⇒ X is (Refl .) + (NS), where CNJ(X ) is the

Jordan-von Neumann constant of X .µ(X ) < 1

J(X )−1 ⇒ X is (Refl .) + (NS). Here J(X ) is the Jamesconstant of X .Indeed,µ(X ) < 1

J(X )−1 ⇒ X satisties (DL), where (DL) is the

Domınguez-Lorenzo condition, stronger than (NS). A. Kaewkhao(2007) From these results the following question arises

PROBLEM XII. In reflexive spaces, does µ(X ) < b ∈ (1, 3] implythat X enjoy (FPP)?

(UR) Banach spaces are reflexive and they have (FPP). But for(LUR), (MLUR), (H) and (R) is not known even in the reflexivecase. It would be an interesting task a systematic study of theseproperties regarding its relationships with (WFPP) and othersufficient conditions for (WFPP), as, for instance (OC) and (PSz).(LUR) neither implies nor is implied by (NS).

PROBLEM XIII Does the following implication hold?(H) ⇒ (WFPP).

PROBLEM XIV. Does the following implication hold?(MLUR)⇒(WFPP).

PROBLEM XV. Does the following implication hold?(LUR)⇒(WFPP).

P(n,X ) := sup{r > 0 : BX contains n disjoint balls with radius r}.= sup{r > 0 : ∃x1, . . . , xn ∈ X : ‖xi − xj‖ ≥ 2r y ‖xi‖ ≤ 1− r}.P(1,X ) = 1, and 1

3 ≤ P(n,X ) ≤ 12 if n > 1 and dim(X ) = +∞.

Definition (Kottman, 1970)

(X , ‖ · ‖) is said to be P-convex if P(n,X ) < 12 for some n ∈ N.

The notion of P-convex space has been introduced by Kottman asan evaluation of the efficiency of the tightest packing of balls ofequal size in the unit ball of X . Kottman proved that the conditionis weaker at the same time than uniform convexity and uniformsmoothness, but still guarantees (super)reflexivity: moreover hecharacterized the dual property, called F -convexity.(UC )⇒ (P − Cvx)⇒ (SpRfl)

Kottman (1970) showed that if δX(

12

)> 0 then P(3,X ) < 1

2 andhence X is P-convex.

Theorem (O. Muniz 2010)

If δX (1) > 0 then P(3,X ) < 12 and hence X is P-convex.

1 Finite dimensional spaces are P-convex. For β ≥ 1,Eβ := (`2, |.|β) where |x |β := max{‖x‖2, β‖x‖∞}, isP-convex. (Maluta, 2012; Helga 2012).

2 There are P convex Banach spaces which fail the mostimportant sufficient conditions for (FPP).

Definition

For ε ∈ (0, 1), a convex subset A of BX is said to be an ε-flat ifA ∩ (1− ε)BX = ∅. A collection D of ε-flats is said to becomplemented (c.c.) if for any different A,B ∈ D then A ∪ Bcontains a couple of antipodal points.

For n ∈ N, F (n,X ) := inf{ε : BX contains a c.c. with n ε− flats}.

Definition

X is F -convex if F (n,X ) > 0 for some n ∈ N.

1 X is P-convex iff X ∗ is F -convex; X is F -convex iff X ∗ isP-convex.

2 Every uniformly convex Banach space is both P-convex andF -convex. (Hence every uniformly smooth Banach space isboth P-convex and F -convex).

Theorem (Satit Saejung (2008))

Every F -convex Banach space has uniformly normal structure.

Thus, if X is P-convex, then X ∗ has (UNS), and hence (FPP).

Amir and Franchetti (1984), defined the following.

Definition

For ε > 0 and n ≥ 2, X is said to be Q(n, ε)-convex if cannot befound x1, x2, ..., xn ∈ BX such that:‖xk − (x1 + x2 + ...+ xk−1)‖ > k − ε for k = 1, 2, ..., n. X inQ − convex if it is Q(n, ε)-convex for some n ∈ N, and some ε > 0.

Very recently Papini defined:

Definition

X if Q∞-convex for some ε > 0 cannot be found a sequence (xn)in BX such that ‖xn −

∑n−11 xi‖ > n − ε for every n ∈ N.

Definition

A collection D of ε-flats is called jointly complemented (c.j.c.) iffor every different A,B ∈ D the sets A∩B y A∩−B are nonempty.

For n ∈ N letE (n,X ) = inf{ε : BX contains a c.j.c. of n ε− flats}.

Definition (Naidu y Sastry, 1978)

A Banach space X is said to be E -convex if E (n,X ) > 0 for somepositive integer n.

Theorem (Saejung, 2008)

If X is E -convex it has the (WORTH) property then it has normalstructure.

Definition

A collection D of ε-flats is called jointly complemented (c.j.c.) iffor every different A,B ∈ D the sets A∩B y A∩−B are nonempty.

For n ∈ N letE (n,X ) = inf{ε : BX contains a c.j.c. of n ε− flats}.

Definition (Naidu y Sastry, 1978)

A Banach space X is said to be E -convex if E (n,X ) > 0 for somepositive integer n.

Theorem (Saejung, 2008)

If X is E -convex it has the (WORTH) property then it has normalstructure.

A subset A of a Banach X is said to be simmetricaly ε separated ifthe distance between two different points of A ∪ −A is greater orequal to ε.

Definition

A Banach is said to be O-convex if there exist a positive integer nand ε > 0 such that BX does not contain ε-separated subsets ofcardinality n.

1 O-convex spaces are superreflexive.2 X is O-convex iff X ∗ is E -convexo; X is E -convex iff X ∗ is

O-convex.3 If ε0(X ) < 2 then X is both O-convex and E -convex. The

converse is not true.

A subset A of a Banach X is said to be simmetricaly ε separated ifthe distance between two different points of A ∪ −A is greater orequal to ε.

Definition

A Banach is said to be O-convex if there exist a positive integer nand ε > 0 such that BX does not contain ε-separated subsets ofcardinality n.

1 O-convex spaces are superreflexive.2 X is O-convex iff X ∗ is E -convexo; X is E -convex iff X ∗ is

O-convex.3 If ε0(X ) < 2 then X is both O-convex and E -convex. The

converse is not true.

Saejung asked

PROBLEM XVI Does P-convexity imply (FPP)?

Also it is open the following

PROBLEM XVII Does O-convexity imply (FPP)?

Saejung also posed the same problem for the E -convex Banachspaces. It was solved in 2008.

Theorem (Dowling, Randrianantoanina y Turett,2008)

If X is E -convex then X enjoys (FPP).

Very recently, E.Ll.F and O. Muniz have shown that E convexBanach spaces in fact satisfy a stronger geometrical property, theso called Prus-Sczcepanick condition.

P.L. Papini (2010) posed the following.

PROBLEM XVIII Does Q∞-convexity imply (WFPP)?

Of course an easier problem is

PROBLEM XIX Does Q-convexity imply (WFPP)?

Uniformly non-octahedral Banach spaces

Notice that if ε0(X ) < 2 then X is both O-convex and E -convex.Jimenez Melado (1999) defined the following modulus. Lets2(X ) := sup{ε ∈ [0, 2] : ∃x1, x2, x3 ∈ BX : ‖xi − xj‖ ≥ ε, 1 ≤ i 6= j ≤ k}.δX : [0, s2(X ))→ [0, 1]

δX (ε) =

inf{

1− ‖x1+x2+x3‖3 : x1, x2, x3 ∈ BX , ‖xi − xj‖ ≥ ε, 1 ≤ i 6= j ≤ k

}.

Uniformly non-octahedral Banach spaces

Notice that if ε0(X ) < 2 then X is both O-convex and E -convex.Jimenez Melado (1999) defined the following modulus. Lets2(X ) := sup{ε ∈ [0, 2] : ∃x1, x2, x3 ∈ BX : ‖xi − xj‖ ≥ ε, 1 ≤ i 6= j ≤ k}.δX : [0, s2(X ))→ [0, 1]

δX (ε) =

inf{

1− ‖x1+x2+x3‖3 : x1, x2, x3 ∈ BX , ‖xi − xj‖ ≥ ε, 1 ≤ i 6= j ≤ k

}.

Uniformly non-octahedral Banach spaces

There exist Banach spaces X with ε0(X ) = 2 and ε0(X ) < 2.Since ε0(X ) < 2⇒ (FPP) the following problem naturally arises.

PROBLEM XX Does the following implication hold?ε0(X ) < 2⇒ (FPP).

1 ε0(X ) < 1⇒ X is Reflexive andithas (NS), and hence Xenjoys (FPP).

2 ε0(X ) < 43 ⇒ (FPP). [Eva Mazcunan, 2009]

Uniformly non-octahedral Banach spaces

Since ε0(X ) < 2 implies that X is both O-convex and E -convex,an obvious must is

PROBLEM XXI To investigate the relationships between conditionε0(X ) < 2 with O-convexity, E -convexity and Q-convexity.

Since δX (.) ≥ δX (.), and given that δX (1) > 0 implies that X isP-convex, the following question naturally arises.

PROBLEM XXII Does the following implication hold?δX (1) > 0⇒ X is P-convex.

Theorem (P.L. Papini, 2010)

If A1(X ) = 2 then X fails to be Q∞-convex (and hence it fails tobe Q convex).

Here A1(X ) := inf{

sup{‖x+y‖+‖x−y‖2 : y ∈ SX} : x ∈ SX

}is a

constant defined by Baronti, Casini and Papini in 2000. It is known

that for very Banach space X , A1(X ) ∈[

3+√

216 , 2

]. Moreover,

A1(c0) = A1(`∞) = 32 and A1(`1) = 2.

PROBLEM XXIII A1 <32 ⇒ (WFPP)? Is there any link between

A1 and (NS)?

Orthogonal convexity

NotationsMβ(x , y) := B[x , 1+β

2 d(x , y)]⋂

B[y , 1+β2 d(x , y)]. If A is a

bounded subset of X , |A| := sup{‖x‖ : x ∈ A}.

D[(xn)] := lim supm lim supn(‖xm − xn‖).

Orthogonal convexity

Mβ(x , y) := B[x , 1+β2 d(x , y)]

⋂B[y , 1+β

2 d(x , y)],|A| = sup{‖x‖ : x ∈ A}. D[(xn)] := lim supn(lim supm ‖ xn− xm ‖).

Orthogonal convexity (A. Jimenez-Melado, 1988).

(X , ‖.‖) is orthogonally convex (OC) if for every weakly nullsequence (xn) with D[(xn)] > 0, there exists β > 0 such that

lim supn

(lim supm

∣∣Mβ(xn, xm)∣∣) < D[(xn)].

(OC) Banach spaces have the (WFPP).

(UC )⇒ (OC )⇒ (FPP)

Orthogonal convexity

Mβ(x , y) := B[x , 1+β2 d(x , y)]

⋂B[y , 1+β

2 d(x , y)],|A| = sup{‖x‖ : x ∈ A}. D[(xn)] := lim supn(lim supm ‖ xn− xm ‖).

Orthogonal convexity (A. Jimenez-Melado, 1988).

(X , ‖.‖) is orthogonally convex (OC) if for every weakly nullsequence (xn) with D[(xn)] > 0, there exists β > 0 such that

lim supn

(lim supm

∣∣Mβ(xn, xm)∣∣) < D[(xn)].

(OC) Banach spaces have the (WFPP).

(UC )⇒ (OC )⇒ (FPP)Some nonreflexive spaces, as c0 and the James space, and manyothers have (OC) and (OC) Banach spaces have the Banach-Saksproperty.

PROBLEM XXIV ε0(X ) < 1⇒ X is OC. (Carlos’ talk) Is thebound 1 sharp?

More on the constants

PROBLEM XXV Are the respective bounds for CNJ(X ), CZ (X ),J(X ) y DW (X ) sharp?

PROBLEM XXVI Give sufficient conditions for (NS) in terms ofA1(X ), A2(X ) and T (X ).

More on the constants

ε0(X ) < 1⇒ X has (UNS) (Goebel, 1976). (Sharp).

CNJ(X ) < 1+√

32 ⇒ X has (UNS) (Saejung, 2006).

CZ (X ) < 1+√

32 ⇒ X has (UNS) (Gao, Saejung, 2009).

J(X ) < 1+√

52 ⇒ X has (UNS) (Saejung, 2003).

DW (X ) <(3 + 2

√2) 1

3 +(3− 2

√2) 1

3 ⇒ X has (NS)Jimenez- E.Ll.F.-Mazcunan, 2006).

For A2(X ), T (X ), no analogous results are known.

PROBLEM XXV Are the respective bounds for CNJ(X ), CZ (X ),J(X ) y DW (X ) sharp?

PROBLEM XXVI Give sufficient conditions for (NS) in terms ofA1(X ), A2(X ) and T (X ).

More on the constants

PROBLEM XXV Are the respective bounds for CNJ(X ), CZ (X ),J(X ) y DW (X ) sharp?

PROBLEM XXVI Give sufficient conditions for (NS) in terms ofA1(X ), A2(X ) and T (X ).