Modulus of convexity, characteristic of convexity and fixed point theorems

Abstract

Let C be a bounded closed convex subset of a Banach space E and let T be a nonexpansive mapping from C into itself. Browder [2] and Gohde [10] showed that if E is uniformly convex then T has a fixed point, while Kirk [13] proved that if E is reflexive and if C has normal structure then T has a fixed point. On the other hand, Goebel [7] defined the characteristic ε 0 of convexity of E and showed that E is uniformly convex if and only if ε o =O, if ε o <l then E has normal structure and if ε o <2 then E is reflexive. Also, Bynum [3] defined the normal structure coefficient N(E) of E, and then Maluta [17] and Bae [1] proved that if NiE)' 1^ then E is reflexive and has normal structure. Using these coefficients, Goebel and Kirk [8], Goebel, Kirk and Thele [9] and Casini and Maluta [4] proved the fixed point theorems for uniformly ^-lipschitzian mappings. (For the results on Hubert space, see [5], [12], [14].) But it seems natural to define these coefficients for a convex set, since for any Banach space E, a nonexpansive mapping has a fixed point if C is weakly compact and has normal structure.In this paper, we introduce the modulus δ(C, ε) of convexity, the characteristic e o (C) of convexity and the constant N(C) of uniformity of normal structure for a convex subset C of a Banach space and prove some results similar to [3], [7], [11], [17]. For example, we show that if N(C)<1 then C is boundedly weakly compact. Further, by using these coefficients, we prove three fixed point theorems. All of these proofs are given by explicitly constructing a sequence which converges to a fixed point. We first show a fixed point theorem for nonexpansive semigroups. Secondly, we obtain a fixed point theorem for uniformly ^-lipschitzian semigroups on C under k<γ, where γ is determined by the modulus of convexity of C. Also, using our results, we evaluate γ as l<τ^l+(l -ε o (C))/2. Finally, we prove that Casini and Maluta's result [4] is valid under more general semigroups. § 2. Preliminaries.

Fixed Point Theorems for Lipspchitzian Semigroups

“…Lifschitz [10], Downing and Ray [4] and Ishihara and Takahashi [8] proved that in a Hilbert space a uniformly k-lipschitzian semi-group with k < y/l has a common fixed point. Also Casini and Maluta [3] and Ishihara and Takahashi [9] proved that a uniformly ^-lipschitzian semigroup in a Banach space E has a common fixed point if k < N{E)~XI 2 , where N(E) is the constant of uniformity of normal structure. In these results, except [7], domain U of semigroups were assumed to be closed and convex.…”

“…Next, by a method similar to that of the proof of Theorem 1, we prove a fixed point theorem in a Banach space. An important lemma is a result proved in[9], which we state here as:…”

Some inequalities concerning the James constant in Banach spaces

“…is due to Ishihara and Takahashi [9]. A Banach space X is uniformly nonsquare if and only if δ X ( ) > 0 for some 0 < < 2 (or equivalently J (X) < 2).…”

Monotonicity characteristic of Köthe–Bochner spaces

“…We start with the characteristic of convexity (called also convexity coefficient). So far this characteristic has been studied for some Köthe spaces (see [11,17,19,21,24] or [25]) and its relation to the fixed point theory has been observed (see [13,28,30] or [42]). Besides L.L.…”