On Semi-Uniform Kadec-Klee Banach Spaces

and Applied Analysis 3 We now introduce a property lying between U-space and semi-KK. Definition 1.8. We say that a Banach space X is semi-uniform Kadec-Klee if for every ε > 0 there exists a δ > 0 such that semi-UKK : {xn} ⊂ SX xn ⇀ x 〈 xn − x, fn 〉 ≥ ε, for some {fn } ⊂ SX∗ satisfying fn ∈ ∇xn ∀n ⎫ ⎪ ⎪⎬ ⎪ ⎪⎭ ⇒ ‖x‖ ≤ 1 − δ. 1.10 In this paper, we prove that semi-UKK property is a nice generalization of U-space and semi-KK property. Moreover, every semi-UKK space has weak normal structure. We also give a characterization of the direct sum of finitely many Banach spaces which is semi-KK and semi-UKK. We use such a characterization to construct a Banach space which is semiUKK but not UKK. Finally we give a remark concerning the uniformly alternative convexity or smoothness introduced by Kadets et al. 7 . 2. Results 2.1. Some Implications For a sequence {xn} ⊂ SX and {fn} ⊂ SX∗ satisfying fn ∈ ∇xn for all n, we let sep{fn}{xn} inf {〈 xn − xm, fn 〉 : n < m } . 2.1 It is clear that sep{fn}{xn} ≤ sep{xn}. Theorem 2.1. A Banach space X is semi-UKK if and only if for every ε > 0 there exists a δ > 0 such that {xn} ⊂ SX xn ⇀ x sep{fn}{xn} ≥ ε, for some { fn } ⊂ SX∗ satisfying fn ∈ ∇xn , ∀n ⎫ ⎪ ⎪⎬ ⎪ ⎪⎭ ⇒ ‖x‖ ≤ 1 − δ. 2.2 The following theorem shows that our new property is well placed. Theorem 2.2. The following implication diagram holds: UC ⇒ UKK ⇒ KK ⇓ ⇓ ⇓ U-space ⇒ semi-UKK ⇒ semi-KK. 2.3 Remark 2.3. The implication U -space ⇒ semi-UKK strengthens the result of Vlasov. In fact, it was proved by Vlasov 5, Theorem 7 that every uniformly smooth Banach space is semiKK and by Lau 4, Corollary 2.5 that every uniformly smooth Banach space is a U-space. 4 Abstract and Applied Analysis 2.2. Sufficient Conditions for Weak Normal Structure Recall that a Banach space X has weak normal structure normal structure, resp. if for every weakly compact bounded and closed, resp. convex subset C of X containing more than one point there exists a point x0 ∈ C such that sup{‖x0 − z‖ : z ∈ C} < diamC see 8 . It is clear that normal structure and weak normal structure coincide whenever the space is reflexive. It was Kirk 9 who proved that if a Banach space X has weak normal structure, then every nonexpansive self-mapping defined on a weakly compact convex subset of X has a fixed point. In this subsection, we present a new and wider class of Banach spaces with weak normal structure. Lemma 2.4 Bollobás 10 . Let X be a Banach space, and let 0 < ε < 1. Given z ∈ BX and h ∈ SX∗ with 1 − 〈z, h〉 < ε2/4, then there exist y ∈ SX and g ∈ ∇y such that ‖y − z‖ < ε and ‖g − h‖ < ε. Theorem 2.5. If a Banach space X has the following property: there are two constants 0 < ε < 1 and 0 < δ < 1 such that {xn} ⊂ SX xn ⇀ x 〈 xn − x, fn 〉 ≥ ε, for some {fn } ⊂ SX∗ satisfying fn ∈ ∇xn , ∀n ⎫ ⎪ ⎪⎬ ⎪ ⎪⎭ ⇒ ‖x‖ ≤ 1 − δ, 2.4 then X has weak normal structure. Proof. Suppose that X does not have weak normal structure. Then there exists a sequence {xn} in X such that the following properties are satisfied see 11 : i diam{xn} 1; ii xn ⇀ 0; iii ‖xn − x‖ → 1 for all x ∈ co{xn}. In particular, since 0 ∈ co{xn}, we have ‖xn‖ → 1. We now show that for each 0 < ε < 1 and 0 < δ < 1, there are an element z ∈ X and sequences {zn} ⊂ SX and {fn} ⊂ SX∗ such that i zn ⇀ z; ii 〈zn − z, fn〉 ≥ ε and 〈zn, fn〉 1 for all n; iii ‖z‖ > 1 − δ. To see this, let 0 < δ < 1 and 0 < ε < 1 be given. We may assume that ‖x1‖ > 1 − δ. For each n, let gn ∈ ∇xn− 1/2 x1 . This implies 〈xn − 1/2 x1, gn〉 ‖xn − 1/2 x1‖ → 1. We observe that