A new measure of weak noncompactness is introduced. A logarithmic convexity-type result on the behaviour of this measure applied to bounded linear operators under real interpolation is proved. In particular, it gives a new proof of the theorem showing that if at least one of the operators T : Ai -» Bi, i = 0,1 is weakly compact, then so is T : A g

B e ,p for all 0 < 6 < 1 and 1 < p < oo.

We introduce and study the class of nearly uniformly noncreasy Banach spaces. It is proved that they have the weak fixed point property. A stability result for this property is obtained.

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We show that infinite dimensional geometric moduli introduced by Milman are strongly related to nearly uniform convexity and nearly uniform smoothness. An application of those moduli to fixed point theory is given.

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Abstract. Let H be an at least two-dimensional real Hilbert space with the unit sphere S H . For α ∈ [−1, 1] and n ∈ S H we define an (α, n)-spherical cap by Sα,n = {x ∈ S H : x, n ≥ α}. We show that the distance between the set of contractions T : Sα,n → Sα,n and the identity mapping is positive iff α < 0. We also study the fixed point property and the minimal displacement problem in this setting for nonexpansive mappings.

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In 2015, Goebel and Bolibok defined the initial trend coefficient of a mapping and the class of initially nonexpansive mappings. They proved that the fixed point property for nonexpansive mappings implies the fixed point property for initially nonexpansive mappings. We generalize the above concepts and prove an analogous fixed point theorem. We also study the initial trend coefficient more deeply.