aa r X i v : . [ m a t h . F A ] F e b D ( X ) < OR ˆ D ( X ) < IMPLY PROPERTY( K ) TIM DALBY
Abstract.
Two new Banach space moduli, that involve weak convergent se-quences, are introduced. It is shown that if either one of these moduli are strictlyless than 1 then the Banach space has Property( K ). Introduction
A Banach space, X , has the weak fixed point property, w-FPP, if every nonexpan-sive mapping, T , on every weak compact convex nonempty subset, C , has a fixedpoint. The past forty or so years has seen a number of Banach space propertiesshown to imply the w-FPP. Some such properties are weak normal structure, Opial’scondition, Property( K ) and Property( M ). Here two new moduli are introduced andare linked to one of these properties, Property( K ). More information on the w-FPPand associated Banach space properties and moduli can be found in [3].The key definitions and terminology are below. Definition 1.1.
Sims, [6]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. Definition 1.2.
Opial [5]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.Later a modulus was introduced to gauge the strength of Opial’s condition and astronger version of the condition was defined.
Definition 1.3.
Lin, Tan and Xu, [4]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 > } . Date : February 9, 2021.2010
Mathematics Subject Classification.
Key words and phrases. weak fixed point property, property( K ), Opial condition, Opial’smodulus. X is said to have uniform Opial’s condition if r X ( c ) > c > . See [4] formore details.There is a direct link between Opial’s modulus and Property( K ). Dalby proved in[1] that r X (1) > X having Property( K ). This will be used in thenext section.The two new moduli are defined next. Definition 1.4.
Let X be a Banach space. Let D ( X ) = sup { lim inf n →∞ k x n − x k : x n ⇀ x, k x n k = 1 for all n } and let ˆ D ( X ) = sup {k x k : x n ⇀ x, k x n k = 1 for all n } . So 0 D ( X ) D ( X ) . Some values for D ( X ) are D ( ℓ ) = 0 , D ( c ) = 1 and D ( ℓ p ) = 2 /p . The reason that these two moduli are introduced is that in [2] Dalby showed thatif in the dual, X ∗ , a certain weak* convergent sequence, ( w ∗ n ) , satisfies either one oftwo properties then X satisfied the w-FPP. Let w ∗ n ∗ ⇀ w ∗ where k w ∗ k w ∗ is ‘deep’ within the dual unit ball or w ∗ n − w ∗ eventually ‘deep’ within the dualunit ball then X has the w-FPP. So D ( X ∗ ) < D ( X ∗ ) < Results
Proposition 2.1.
Let X be a separable Banach space. If D ( X ) < then r X (1) > . That is, X has Property( K ).Proof. Let x n ⇀ , lim inf n →∞ k x n k > k x k > . Using the lower semi-continuity of the norm, lim inf n →∞ k x n + x k > k x k > . Bytaking subsequences if necessary we may assume that k x n + x k 6 = 0 for all n. Now (cid:13)(cid:13)(cid:13)(cid:13) x n + x k x n + x k (cid:13)(cid:13)(cid:13)(cid:13) = 1 for all n, x n + x k x n + x k ⇀ x lim inf n →∞ k x n + x k . For ease of reading let α = lim inf n →∞ k x n + x k . Then ( X ) < D ( X ) < K ) 3 lim inf n →∞ (cid:13)(cid:13)(cid:13)(cid:13) x n + x k x n + x k − xα (cid:13)(cid:13)(cid:13)(cid:13) = lim inf n →∞ k x n + x k lim inf n →∞ (cid:13)(cid:13)(cid:13) x n + x − k x n + x k xα (cid:13)(cid:13)(cid:13) = 1 α lim inf n →∞ (cid:13)(cid:13)(cid:13) x n + x − k x n + x k xα (cid:13)(cid:13)(cid:13) = 1 α lim inf n →∞ (cid:13)(cid:13)(cid:13)(cid:13) x n − (cid:18) k x n + x k α − (cid:19) x (cid:13)(cid:13)(cid:13)(cid:13) > α (cid:12)(cid:12)(cid:12)(cid:12) lim inf n →∞ k x n k − lim inf n →∞ (cid:12)(cid:12)(cid:12)(cid:12) k x n + x k α − (cid:12)(cid:12)(cid:12)(cid:12) k x k (cid:12)(cid:12)(cid:12)(cid:12) = 1 α (cid:12)(cid:12)(cid:12) lim inf n →∞ k x n k + (cid:12)(cid:12)(cid:12) αα − (cid:12)(cid:12)(cid:12) k x k (cid:17) = 1 α lim inf n →∞ k x n k > α = 1lim inf n →∞ k x n + x k . We have D ( X ) > lim inf n →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) x n + x k x n + x k − x lim inf n →∞ k x n + x k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) > n →∞ k x n + x k . † So lim inf n →∞ k x n + x k > D ( X ) . This means that r X (1) + 1 > D ( X ) or r X (1) > D ( X ) − > . (cid:3) A second way to prove this proposition is via a contradiction as shown below.
Proof.
Assume that D ( X ) < r X (1) > . Then r X (1) = 0 . Given ǫ > x n ) in X where x n ⇀ , lim inf n →∞ k x n k > x ∈ X, k x k > n →∞ k x n + x k < ǫ. Therefore 1 k x k lim inf n →∞ k x n + x k < ǫ. So apart from the last inequalitythe set up is the same as in the previous proof and this proof follows the samepathway. So now jumping to a line above, the one labeled with † , D ( X ) > n →∞ k x n + x k >
11 + ǫ .
Letting ǫ → D ( X ) > D ( X ) < . TIM DALBY
So the desired contradiction is arrived at. (cid:3)
Next is the second moduli’s turn.
Proposition 2.2.
Let X be a separable Banach space. If ˆ D ( X ) < then r X (1) > . That is, X has Property( K ).Proof. Let x n ⇀ , lim inf n →∞ k x n k > k x k > . Now x n + x ⇀ x so lim inf n →∞ k x n + x k > k x k > . Without loss of generality wemay assume k x n + x k 6 = 0 for all n. Then (cid:13)(cid:13)(cid:13)(cid:13) x n + x k x n + x k (cid:13)(cid:13)(cid:13)(cid:13) = 1 for all n, x n + x k x n + x k ⇀ x lim inf n →∞ k x n + x k . Hence 1 > ˆ D ( X ) > k x k lim inf n →∞ k x n + x k leading to k x k lim inf n →∞ k x n + x k ˆ D ( X )lim inf n →∞ k x n + x k > k x k ˆ D ( X ) > D ( X )Thus r X (1) + 1 > D ( X ) r X (1) > D ( X ) − > . (cid:3) A second way to prove this proposition is by finding a value of K for Property( K ). Proof.
Let x n ⇀ , k x n k = 1 for all n and lim inf n →∞ k x n − x k . If lim inf n →∞ k x n − x k = 0 then because k x k lim inf n →∞ k x n − x k we have x = 0and K can be taken as zero.So assume lim inf n →∞ k x n − x k > k x n − x k 6 = 0 for all n. Using the same argument as in the previous proof ( X ) < D ( X ) < K ) 5 k x k lim inf n →∞ k x n − x k ˆ D ( X ) ˆ D ( X ) < . So K can be taken as ˆ D ( X ) . (cid:3) References [1] T. Dalby,
Relationships between properties that imply the weak fixed point property , J. Math.Anal. Appl. (2001), 578-589.[2] T. Dalby,
Property( K ∗ ) implies the weak fixed point property , arXiv preprint arXiv:2007.00942(2020).[3] W. A. Kirk and B. Sims (ed.), Handbook of metric fixed point theory , Kluwer Academic Pub-lishers, Dordrecht, 2001.[4] P.-K. Lin, K.-K.Tan and H.-K. Xu,
Demiclosedness principle and asymptotic behavior forasymptotically nonexpansive mappings , Nonlinear Anal. (1995), 929-946.[5] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansivemappings , Bull. Amer. Math. Soc. (1967), 591-597.[6] B. Sims, A class of spaces with weak normal structure , Bull. Austral. Math. Soc. (1994),523-528. Email address ::