Locally uniform Opial conditions.
The authors consider several refinements of Opial's condition [Bull. Am. Math. Soc. 73, 591--597 (1967; Zbl 0179.19902)] that have appeared in the literature. A Banach space \(X\) satisfies a locally uniform Opial condition if, for every \(c>0\) and every weakly null sequence \(\{x_n\}\) in \(X\) with \(\limsup_n\| x_n \| \geq 1 \), there exists \(r>0\) such that \(\limsup_n \| x+x_n \| \geq 1+r\) for every \(x\in X\) with \(\| x \| \geq c\). A similar definition with the limit superiors replaced by limit inferiors could be considered and in [\textit{P. K. Lin, K. K. Tan} and \textit{H. K. Xu}, Nonlinear Anal., Theory Methods Appl. 24, 929--946 (1995; Zbl 0865.47040)] it is asserted that the definitions would be equivalent. In the article under review, the authors provide an example to show that, in general, the two notions are distinct. The authors also note that, for Banach spaces with separable duals, the limit inferior locally uniform Opial condition is equivalent to the uniform Opial property defined by \textit{S. Prus} in [Nonlinear Anal., Theory Methods Appl. 18, 697--704 (1992; Zbl 0786.46023)], but, in general, these properties also differ.
scientific article; zbMATH DE number 1910234