doi: 0.55/umcsmath-205-0008 ANNALES UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA LUBLIN POLONIA VOL. LXVIII, NO. 2, 204 SECTIO A 85 9 ŁUKASZ PIASECKI Renormings of c 0 and the minimal displacement problem Abstract. The aim of this paper is to show that for every Banach space (X, ) containing asymptotically isometric copy of the space c 0 there is a bounded, closed and convex set C X with the Chebyshev radius r(c) = such that for every there exists a -contractive mapping T : C C with x Tx > for any x C.. Introduction and Preliminaries. Let C be a nonempty, bounded, closed and convex subset of an infinitely dimensional real Banach space (X, ). The Chebyshev radius of C relative to itself is the number r(c) = inf x y. sup y C x C We say that a mapping T : C C satisfies the Lipschitz condition with a constant or is -lipschitzian, if for all x, y C, Tx Ty x y. The smallest constant for which the above inequality holds is called the Lipschitz constant for T and it is denoted by (T ). ByL() we denote the class of all -lipschitzian mappings T : C C. A mapping T : C C is called -contractive if for all x, y C, x y, we have Tx Ty < x y. 2000 Mathematics Subject Classification. 47H09, 47H0. Key words and phrases. Minimal displacement, asymptotically isometric copies of c 0, lipschitzian mappings, -contractive mappings, renormings.
86 Ł. Piaseci The minimal displacement problem has been raised by Goebel in 973, see [6]. Standard situation is the following. For any -lipschitzian mapping T : C C the minimal displacement of T is the number given by d(t )=inf{ x Tx : x C}. It is nown that for every (for the proof see for example [8]) ( d(t ) ) r(c). For any set C we define the function ϕ C :[, ) [0,r(C)) by ϕ C () =sup{d(t ):T L()}. Consequently, for every, ( ϕ C () ) r(c). The function ϕ C is called the characteristic of minimal displacement of C. If C is the closed unit ball B X, then we write ψ X instead of ϕ BX. We also define the characteristic of minimal displacement of the whole space X as ϕ X () =sup{ϕ C () :C X, r(c) =}. Hence, for every >, ψ X () ϕ X (). The minimal displacement problem is the tas to find or evaluate functions ϕ and ψ for concrete sets or spaces. Obviously this problem is matterless in the case of compact set C because, in view of the celebrated Schauder s fixed point theorem, we have ϕ C () =0for any >. If C is noncompact then by the theorem of Sternfeld and Lin [2] we get ϕ C () > 0 for all >. Hence we additionally assume that C is noncompact and we restrict our attention to the class of lipschitzian mappings with (T ). The set C for which ϕ C () = ( ) r(c) for every > is called extremal (with respect to the minimal displacement problem). There are examples of spaces having extremal balls. Among them are spaces of continuous functions C[a, b], bounded continuous functions BC(R), sequences converging to zero c 0, all of them endowed with the standard uniform norm (see [7]). Recently the present author [3] proved that also the space c of converging sequences with the sup norm has extremal balls. It is still unnown if the space l of all bounded sequences with the sup norm has extremal balls. Very recently Bolibo [] proved that { (3 2 2)( ) for 2+ 2, ψ l () 2 for >2+ 2.
Renormings of c 0 and the minimal displacement problem 87 The same estimate holds for the space of summable functions (equivalent classes) L (0, ) equipped with the standard norm (see [2]) as well as few other spaces (see [9]). In the case of space l of all summable sequences with the classical norm we have: { 2+ 3 4 ( ) for 3+2 3, ψ l () + +3 for >3+2 3. Nevertheless, the subset { } S + = {x n } n= : x n 0, x n = l is extremal and ϕ S +() = ( ) r(s + )=2 ( ) for every > (for the proof see [7]). In this paper we deal with a problem of existence of extremal sets in spaces containing isomorphic copies of c 0. Obviously, for every such space X we have ϕ X () = as an immediate consequence of the following theorem by James [0], its stronger version states: Theorem. (James s Distortion Theorem, stronger version). A Banach space X contains an isomorphic copy of c 0 if and only if, for every null sequence {ɛ n } n= in (0, ), there exists a sequence {x n} n= in X such that ( ɛ )sup t n n n= n= t n x n ( + ɛ )sup t n n holds for all {t n } n= c 0 and for all =, 2,... However, it is not nown if all isomorphic copies of c 0 contain an extremal subset. We shall prove that the answer is affirmative in the case of spaces containing an asymptotically isometric copies of c 0. This class of spaces has been introduced and widely studied by Dowling, Lennard and Turett (see [], Chapter 9). Let us recall that a Banach space X is said to contain an asymptotically isometric copy of c 0 if for every null sequence {ɛ n } n= in (0, ), there exists a sequence {x n } n= in X such that sup( ɛ n ) t n t n x n sup t n, n n n= for all {t n } n= c 0. Dowling, Lennard and Turett proved the following theorems. Theorem.2 (see [3] or []). If a Banach space X contains an asymptotically isometric copy of c 0, then X fails the fixed point property for nonexpansive (and even contractive) mappings on bounded, closed and convex subsets of X.
88 Ł. Piaseci Theorem.3 (see [5] or []). If Y is a closed infinite dimensional subspace of (c 0, ), then Y contains an asymptotically isometric copy of c 0. Theorem.4 (see [5]). Let Γ be an uncountable set. Then every renorming of c 0 (Γ) contains an asymptotically isometric copy of c 0. Let us recall that a mapping T : C C is said to be asymptotically nonexpansive if T n x T n y n x y for all x, y C and for all n =, 2,..., where { n } n= is a sequence of real numbers with lim n n =. Now we are ready to cite the following theorem. Theorem.5 (see [4] or []). If a Banach space X contains an isomorphic copy of c 0, then there exists a bounded, closed, convex subset C of X and an asymptotically nonexpansive mapping T : C C without a fixed point. In particular, c 0 cannot be renormed to have the fixed point property for asymptotically nonexpansive mappings. Remar.6 (see [5] or []). There is an isomorphic copy of c 0 which does not contain any asymptotically isometric copy of c 0. 2. Main result. Theorem 2.. If a Banach space X contains an asymptotically isometric copy of c 0, then there exists a bounded, closed and convex subset C of X with r(c) =such that for every there exists a -contractive mapping T : C C with x Tx > for every x C. Proof. Let {λ i } be a strictly decreasing sequence in (, 3 2 ) converging to. Then there is a null sequence {ɛ i } in (0, ) such that λ i+ < ( ɛ i )λ i for i =, 2,... By assumption there exists a sequence {x i } in X such that sup( ɛ i ) t i t i x i sup t i i i for every {t i } c 0. Define y i = λ i x i for i =, 2,... and { } C = t i y i : {t i } c 0, 0 t i for i =, 2,.... It is clear that C is a bounded, closed and convex subset of X.
Renormings of c 0 and the minimal displacement problem 89 We claim that r(c) =. Fix w = t iy i C and let {z n } n=2 be a sequence of elements in C defined by n z n = t i y i + y n + t i y i. Then i=n+ w z n = (t n )y n ( ɛ n )( t n )λ n λ n+ ( t n ). Letting n,wegetr(w, C) := sup{ w x : x C} for any w C and consequently r(c). Now let {z n } n= be a sequence in C given by n z n = 2 y i. Then for every w = t iy i C we have n ( ) z n w = 2 t i y i + ( t i y i ) i=n+ n ( ) = 2 t i λ i x i + ( t i λ i x i ) i=n+ { } sup 2 t λ,..., 2 t n λ n,t n+ λ n+,t n+2 λ n+2,... { sup 2 λ,..., } 2 λ n,λ n+,λ n+2,... { sup 2 3 2,..., 2 3 } 2,λ n+,λ n+2,... { } 3 =sup 4,λ n+ = λ n+. Hence r(z n,c) λ n+. Letting n,wegetr(c). Finally r(c) =. To construct desired mapping T we shall need the function α :[0, ) [0, ] defined by { t if 0 t, α(t) = if t>. It is clear that the function α satisfies the Lipschitz condition with the constant, that is, for all s, t [0, ) we have α(t) α(s) t s.
90 Ł. Piaseci For arbitrary we define a mapping T : C C by ( ) T t i y i = y + α(t i )y i. Then for any w = t iy i and z = s iy i in C such that w z we have Tw Tz = (α(t i ) α(s i )) y i = (α(t i ) α(s i )) λ i x i sup λ i α(t i ) α(s i ) i 2 sup λ i t i s i i 2 < sup ( ɛ i )λ i t i s i,2,... (t i s i )λ i x i = w z. Hence the mapping T is -contractive. We claim that for every x C x Tx >. Indeed, suppose that there exists w = t iy i C such that w Tw, that is, w Tw = (t )y + (t i α(t i ))y i = (t )λ x + (t i α(t i ))λ i x i This implies that. ( ɛ )λ ( t ) and ( ɛ i )λ i t i α(t i ) for i 2. Hence t i for i =, 2,... But {t i} c 0, a contradiction.
Renormings of c 0 and the minimal displacement problem 9 Corollary 2.2. If a Banach space X contains an asymptotically isometric copy of c 0, then X contains an extremal subset. Corollary 2.3. If Y is a closed infinite dimensional subspace of (c 0, ), then Y contains an extremal subset. Corollary 2.4. Let Γ be uncountable set. Then every renorming of c 0 (Γ) contains an extremal subset. Acnowledgement. The author has been partially supported by MNiSW, grant NN20393737. References [] Bolibo, K., The minimal displacement problem in the space l, Cent. Eur. J. Math. 0 (202), 22 224. [2] Bolibo, K., Constructions of lipschitzian mappings with non zero minimal displacement in spaces L (0, ) and L 2 (0, ), Ann. Univ. Mariae Curie-Słodowsa Sec. A 50 (996), 25 3. [3] Dowling, P. N., Lennard C. J., Turett, B., Reflexivity and the fixed point property for nonexpansive maps, J. Math. Anal. Appl. 200 (996), 653 662. [4] Dowling, P. N., Lennard, C. J., Turett, B., Some fixed point results in l and c 0, Nonlinear Anal. 39 (2000), 929 936. [5] Dowling, P. N., Lennard C. J., Turett, B., Asymptotically isometric copies of c 0 in Banach spaces, J. Math. Anal. Appl. 29 (998), 377 39. [6] Goebel, K., On the minimal displacement of points under lipschitzian mappings, Pacific J. Math. 45 (973), 5 63. [7] Goebel, K., Concise Course on Fixed Point Theorems, Yoohama Publishers, Yoohama, 2002. [8] Goebel, K., Kir, W. A., Topics in metric fixed point theory, Cambridge University Press, Cambridge, 990. [9] Goebel, K., Marino, G., Muglia, L., Volpe, R., The retraction constant and minimal displacement characteristic of some Banach spaces, Nonlinear Anal. 67 (2007), 735 744. [0] James, R. C., Uniformly non-square Banach spaces, Ann. of Math. 80 (964), 542 550. [] Kir, W. A., Sims, B. (Eds.), Handboo of Metric Fixed Point Theory, Kluwer Academic Publishers, Dordrecht, 200. [2] Lin, P. K., Sternfeld, Y., Convex sets with the Lipschitz fixed point property are compact, Proc. Amer. Math. Soc. 93 (985), 633 639. [3] Piaseci, Ł., Retracting a ball onto a sphere in some Banach spaces, Nonlinear Anal. 74 (20), 396 399. Łuasz Piaseci Institute of Mathematics Maria Curie-Słodowsa University pl. M. Curie-Słodowsiej 20-03 Lublin Poland e-mail: piaseci@hetor.umcs.lublin.pl Received October, 202