Pseudo-convergence in normed linear spaces (Q1206743)

A bounded sequence \(\{x_n\}\) in a Banach space \(X\) is said to pseudo-converge to a point \(x_0\), called the pseudo-limit, if \(x_0\) minimizes the function \(f_S(x)= \limsup \| y_m-x\|\), \(x\in X\), for every subsequence \(S= \{y_m\}\) of \(\{x_n\}\). If the pseudo-limit is unique, we call the sequence \(\{x_n\}\) uniquely pseudo-convergent. This notion of convergence arose in connection with fixed point theory of multivalued nonexpansive mappings. The basic idea is that if \(x_n\) is a bounded sequence of approximate fixed points of a multivalued nonexpansive mapping \(T\), then there may exist a uniquely pseudo-convergent subsequence of \(x_n\) whose pseudo-limit is a fixed point of \(T\). In this paper we characterize pseudo-convergence in certain Banach spaces. A main result is that, in a space with a uniformly Gateaux differentiable norm, a sequence \(\{x_n\}\) pseudo-converges to \(x\) if and only if \(J(x_n- x)\) converges weak*ly to 0, where \(J\) is a duality map. We also consider other related types of convergence.