Abstract
We introduce a new geometric coefficient which is related Garcia-Falset coefficient and weak star fixed point property. The Garcia-Falset coefficient that was introduced by Benavides in (Houst. J. Math. 22:835-849, 1996) is calculated in this paper for Musielak-Orlicz sequence spaces equipped with the Luxemburg norm. Specifically, in reflexive Banach spaces, the new geometric coefficient and the Garcia-Falset coefficient are the same.
Highlights
Introduction and preliminaries Throughout this paperX is a Banach space which is assumed not to have the Schur property, i.e., X has a weakly convergent sequence that is not norm convergent
A Banach space X is said to have the WFPP whenever it satisfies the above condition from the definition of FPP with ‘weakly compact’ in place of ‘bounded closed’
We introduce a new geometric coefficient that is a related García-Falset coefficient and a weak∗ fixed point property, written R∗(X∗, a)
Summary
The idea of Theorem . is similar to Corollary . in [ ]. Proof Suppose that X does not have an absolutely continuous norm. Ε , there exists n ∈ N such that x (i)ei ≤ ( + ε )ε In this way, we get by induction a sequence {ni} of natural numbers such that ni+. Proof Let d = sup cx : x ∈ with x = a and N(x) being finite. By the definition of cx there exists n ∈ N such that sup cx,y > : I cx,y y cx,y. By the definition of the supremum, there exists y ∈ S( ) with N(y ) > n such that cx,y > d In such a way, we can prove by induction that there exist a sequence {yk}∞ k= ⊂
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