Property(K^*) Implies R(X) \leq 1 + \frac{\displaystyle 1}{\displaystyle 1 + r_{X^*}(1)}
aa r X i v : . [ m a t h . F A ] F e b PROPERTY( K ∗ ) IMPLIES R ( X ) r X ∗ (1) TIM DALBY
Abstract.
It is shown that if the dual of a Banach space satisfies Property( K ∗ )then R ( X ) r X ∗ (1) < r X ∗ ( c ) is Opial’s modulus for X ∗ . Thus X has the weak fixed point property. Introduction
The Banach space moduli of R ( X ) , r X ( c ) were introduced within the context of fixedpoint theory. This area of research looks at whether, in an infinite dimensionalreal Banach space, every nonexpansive mappings on every weak compact convexnonempty set has a fixed point. If this is so then the space is said to have theweak fixed point property (w-FPP). It can be shown that this propery is separablydetermined, so in this paper the Banach space is assumed to be separable.If the sets are not necessarily weak compact but closed then the property is calledthe fixed point property (FPP).Another aspect of this topic is that it involves weakly convergent sequences andusually a translation is employed so that the sequence being studied is weaklyconvergent to zero. Inherent in most results is the interplay between weak nullsequences and the norm. Much more background can be found in Goebel and Kirk[7] and Kirk and Sims [8].The effect of the dual space, X ∗ , on the outcomes in X have also been studied. Seefor example [3] and [4]. This paper continues along this line.Definitions and notation will be presented in the next section.In [3] Dalby showed that if X is a separable Banach space where the dual, X ∗ ,has Property( K ∗ ) and the nonstrict *Opial condition then R ( X ) <
2. This lastcondition implies X has the w-FPP. For the latter result see [6].In [1] Benavides showed that if X is a reflexive Banach space and if there exist c ∈ (0 ,
1) where r X ∗ ( c ) > R ( a, X ) max (cid:26) ac , a + 11 + r X ∗ ( c ) (cid:27) Date : February 22, 2021.2010
Mathematics Subject Classification.
Key words and phrases. weak fixed point property, property( K ∗ ), Opial’s condition, Opial’smodulus. where a > . This in turn implies R ( a, X ) < a if r X ∗ (1) > . Here r ( c ) is Opial’smodulus. Earlier in that paper it was shown that R ( a, X ) < a for some a > X has the w-FPP. Because of reflexivity, in this case X has the FPP.Recall from [5] that R ( a, X ) < a for some a > R (1 , X ) < R ( X ) < R (1 , X ) < . So the result in [1] can be rewritten as: If X is a reflexive Banach space where r X ∗ (1) > X has the FPP. But Dalby proved in [2] that r X (1) > X having Property( K ). This proof can readily be transferred to X ∗ . That is r X ∗ (1) > X ∗ having Property( K ∗ ). Hence the result in [1] canbe again rewritten as: If X is a reflexive Banach space where X ∗ has Property( K ∗ )then X has the FPP.So the results of Benavides and Dalby are similar.In this paper the requirement that X be reflexive has been changed to X beingseparable and the condition that X ∗ has the nonstrict *Opial property has beenremoved. 2. Definitions
Definition 2.1.
Sims, [11]A Banach space X has property( K ) if there exists K ∈ [0 ,
1) such that whenever x n ⇀ , lim n →∞ k x n k = 1 and lim inf n →∞ k x n − x k k x k K. If the sequence is in B X ∗ and is weak* convergent to zero then the property is calledProperty( K ∗ ). Definition 2.2.
Opial [10]A Banach space has Opial’s condition if x n ⇀ x = 0 implies lim sup n k x n k < lim sup n k x n − x k . The condition remains the same if both the lim sups are replaced by lim infs.If the < is replaced by then the condition is called nonstrict Opial.In X ∗ the conditions are the same except that the sequence is weak* null.Later a modulus was introduced to gauge the strength of Opial’s condition and astronger version of the condition was defined. Definition 2.3.
Lin, Tan and Xu, [9]Opial’s modulus is r X ( c ) = inf { lim inf n →∞ k x n − x k − c > , k x k > c, x n ⇀ n →∞ k x n k > } .X is said to have uniform Opial’s condition if r X ( c ) > c > . See [9] formore details.
ROPERTY( K ∗ ) IMPLIES R ( X ) r X ∗ (1) 3 Result
Proposition 3.1.
Let X be a separable Banach space with X ∗ having Property( K ∗ ).Then R ( X ) r X ∗ (1) . Proof.
Let X be a separable Banach space such that X ∗ has Property( K ∗ ). Then r X ∗ (1) > . Also, let x n ∈ B X for all n, x n ⇀ x ∈ B X . Then for each n > x ∗ n ∈ X ∗ such that k x ∗ n k = 1 and x ∗ n ( x n + x ) = k x n + x k . Using theproperty that X is separable we can assume, without loss of generality, that x ∗ n ∗ ⇀x ∗ where k x ∗ k n →∞ k x n + x k = lim inf n →∞ x ∗ n ( x n + x )= lim inf n →∞ x ∗ n ( x n ) + x ∗ ( x ) lim inf n →∞ k x ∗ n kk x n k + k x ∗ kk x k = lim inf n →∞ k x n k + k x ∗ kk x k k x ∗ k . † The proof now splits into four cases.
Case 1 x ∗ = From † we have lim inf n →∞ k x n + x k . This means that R ( X ) R ( X ) = 1 r X ∗ (1) . Case 2 x ∗ = , k x ∗ k lim inf n →∞ k x ∗ n − x ∗ k We have x ∗ n k x ∗ k − x ∗ k x ∗ k ∗ ⇀ n →∞ (cid:13)(cid:13)(cid:13)(cid:13) x ∗ n k x ∗ k − x ∗ k x ∗ k (cid:13)(cid:13)(cid:13)(cid:13) > . Thus lim inf n →∞ (cid:13)(cid:13)(cid:13)(cid:13) x ∗ n k x ∗ k − x ∗ k x ∗ k + x ∗ k x ∗ k (cid:13)(cid:13)(cid:13)(cid:13) > r X ∗ (1) + 11 = lim n →∞ k x ∗ n k > k x ∗ k ( r X (1) + 1) k x ∗ k r X ∗ (1) + 1 . From † we have lim inf n →∞ k x n + x k r X ∗ (1) + 1 . Therefore R ( X ) r X ∗ (1) + 1 . TIM DALBY
Case 3 x ∗ = , < lim inf n →∞ k x ∗ n − x ∗ k < k x ∗ k To make the presentation easier to read let a = lim inf n →∞ k x ∗ n − x ∗ k then a < k x ∗ k . Note that x ∗ n − x ∗ a ∗ ⇀ n →∞ (cid:13)(cid:13)(cid:13)(cid:13) x ∗ n − x ∗ a (cid:13)(cid:13)(cid:13)(cid:13) = 1 . Now lim inf n →∞ (cid:13)(cid:13)(cid:13)(cid:13) x ∗ n a − x ∗ a + x ∗ a (cid:13)(cid:13)(cid:13)(cid:13) = lim inf n →∞ k x ∗ n k a = 1 a . Using k x ∗ k a > , lim inf n →∞ (cid:13)(cid:13)(cid:13)(cid:13) x ∗ n a − x ∗ a + x ∗ a (cid:13)(cid:13)(cid:13)(cid:13) > r X ∗ ( k x ∗ k a ) + 1 > r X ∗ (1) + 1 since r X ∗ ( c ) is nondecreasing.Thus 1 > a ( r X ∗ (1) + 1) and 1 r X ∗ (1) + 1 > a = lim inf n →∞ k x ∗ n − x ∗ k . This leads tolim inf n →∞ k x n + x k = lim inf n →∞ x ∗ n ( x n + x )= lim inf n →∞ ( x ∗ n − x ∗ )( x n + x ) + lim inf n →∞ x ∗ ( x n + x )= lim inf n →∞ ( x ∗ n − x ∗ )( x n ) + x ∗ ( x ) lim inf n →∞ k x ∗ n − x ∗ kk x n k + k x k †† r X ∗ (1) + 1 + 1So R ( X ) r X ∗ (1) + 1 . Case 4 x ∗ = , lim inf n →∞ k x ∗ n − x ∗ k = From †† , lim inf n →∞ k x n + x k k x k but weak lower semicontinuity of the normmeans k x k lim inf n →∞ k x n + x k . Therefore lim inf n →∞ k x n + x k = k x k r X ∗ (1) + 1 leading to R ( X ) r X ∗ (1) + 1 . (cid:3) Note
Careful reading of the proof reveals that R ( X ) r X ∗ (1) is always true.So Opial’s modulus for X ∗ plays a role in the behaviour of weak null sequences in X. Property( K ∗ )’s role is ensure that r X ∗ (1) > R ( X ) < . ROPERTY( K ∗ ) IMPLIES R ( X ) r X ∗ (1) 5 Remark
Using R (1 , X ) RW (1 , X ) R ( X ) we have:Property( K ∗ ) implies RW (1 , X ) r X ∗ (1) + 1 < K ∗ ) implies R (1 , X ) r X ∗ (1) + 1 < . In [5] it was shown that RW (1 , X ) < RW ( a, X ) < a for some a > . Similarly, R (1 , X ) < R ( a, X ) < a for some a > . References [1] T. Dom´ınguez-Benavides,
A geometric coefficient implying the fixed point property and stabilityresults , Houston J. Math. (1996), 835-84[2] T. Dalby, Relationships between properties that imply the weak fixed point property , J. Math.Anal. Appl. (2001), 578-589.[3] T. Dalby,
The effect of the dual on a Banach space and the weak fixed point property , Bull.Austral. Math. Soc. (2003), 177-185.[4] T. Dalby, Property( K ∗ ) implies the weak fixed point property , arXiv preprint arXiv:2007.00942(2020).[5] T. Dalby, Properties of R ( X ) , R ( a, X ) and RW ( a, X ), arXiv preprint arXiv:2005.08492(2020).[6] J. Garc´ıa-Falset, The fixed point property in Banach spaces with the NUS-property , J. Math.Anal. Appl. (1997), 532-534.[7] K. Goebel and W. A. Kirk,
Topics in metric fixed point theory , Cambridge University Press,Cambridge, 1990.[8] W. A. Kirk and B. Sims (ed.),
Handbook of metric fixed point theory , Kluwer Academic Pub-lishers, Dordrecht, 2001.[9] P.-K. Lin, K.-K.Tan and H.-K. Xu,
Demiclosedness principle and asymptotic behavior forasymptotically nonexpansive mappings , Nonlinear Anal. (1995), 929-946.[10] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansivemappings , Bull. Amer. Math. Soc. (1967), 591-597.[11] B. Sims, A class of spaces with weak normal structure , Bull. Austral. Math. Soc. (1994),523-528. Email address ::