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Uniform asymptotic normal structure, the uniform semi-Opial property and fixed points of asymptotically regular uniformly Lipschitzian semigroups. I. (English) Zbl 0973.47042

The authors introduce the notions of uniform asymptotic normal structure and of uniform semi-Opial properties for Banach spaces, aiming at fixed point results for uniformly Lipschitzian nonlinear semi-groups. Such notions are defined using estimates relating the asymptotic Chebyshev radius and the asymptotic diameter of weakly compact regular sequences in the Banach space.
In this first part of the work the authors prove some results concerning those spaces that satisfy the properties introduced. Given a Banach space \(X\), denote by \(AN(X)\), \(w\)-\(AN(X)\) and by \(w\)-\(SOC(X)\) respectively the asymptotic normal structure coefficient, the asymptotic normal structure coefficient with respect to the weak topology and the semi-Opial coefficient with respect to the weak topology. The authors prove that \(AN(X)>1\) implies that \(X\) is reflexive; moreover, they observe that \(w\)-\(AN(X)>1\) implies the weak fixed point property for non expansive mappings.
As to the infiniteness of the above constants, the authors prove that \(AN(X)=+\infty\) iff \(X\) is finite dimensional, and that \(w\)-\(SOC(X)= +\infty\) iff \(w\)-\(AN(X)=+ \infty\) iff \(X\) is a Schur space.
In the last part of the paper the authors compute \(w\)-\(SOC(x)\) when \(X=X^p_\beta\), \(p>1\) and \(1<\beta <+\infty\), which is the space of \(p\)-summable sequences endowed with the norm \(\|x \|= \max\{\|x\|_\infty, \beta^{-1} \|x\|_p\}\); they prove that \(w\)-\(SOC(X^p_\beta) =\max\{1,\min \{2^{1/p}, \beta^{-1} 4^{1/p}\}\}\).

MSC:

47H10 Fixed-point theorems
46B20 Geometry and structure of normed linear spaces
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