In this paper we prove that for each \(1< p, \tilde{p} < \infty\), the Banach space \((l^{\tilde{p}}, \left\|\cdot\right\|_{\tilde{p}})\) can be equivalently renormed in such a way that the Banach space \((l^{\tilde{p}},\left\|\cdot\right\|_{L,\alpha,\beta,p,\tilde{p}})\) is LUR and has a diametrically complete set with empty interior. This result extends the Maluta theorem about existence of such a set in \(l^2\) with the Day norm. We also show that the Banach space \((l^{\tilde{p}},\left\|\cdot\right\|_{L,\alpha,\beta,p,\tilde{p}})\) has the weak fixed point property for nonexpansive mappings.
In this paper we prove the following general theorem. Let (E, • E) be a uniformly convex Banach space, and let C be a bounded, closed and convex subset of E. Assume that C has nonempty interior and is locally uniformly rotund. Let F be a commutative nonexpansive semigroup acting on C. If F has no fixed point in the interior of C, then there exists a unique point x on the boundary of C such that each orbit of F converges in norm to x. We also establish analogous results for semigroups and mappings which are asymptotically nonexpansive in the intermediate sense.
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