$$\alpha $$-Firmly nonexpansive operators on metric spaces

We extend to p-uniformly convex spaces tools from the analysis of fixed point iterations in linear spaces. This study is restricted to an appropriate generalization of single-valued, pointwise averaged mappings. Our main contribution is establishing a calculus for these mappings in p-uniformly convex spaces, showing in particular how the property is preserved under compositions and convex combinations. This is of central importance to splitting algorithms that are built by such convex combinations and compositions, and reduces the convergence analysis to simply verifying that the individual components have the required regularity pointwise at fixed points of the splitting algorithms. Our convergence analysis differs from what can be found in the previous literature in that the regularity assumptions are only with respect to fixed points. Indeed we show that, if the fixed point mapping is pointwise nonexpansive at all cluster points, then these cluster points are in fact fixed points, and convergence of the sequence follows. Additionally, we provide a quantitative convergence analysis built on the notion of gauge metric subregularity, which we show is necessary for quantifiable convergence estimates. This allows one for the first time to prove convergence of a tremendous variety of splitting algorithms in spaces with curvature bounded from above.

Convergence of proximal splitting algorithms in $\operatorname{CAT}(\kappa)$ spaces and beyond

In the setting of $\operatorname{CAT}(\kappa)$ CAT ( κ ) spaces, common fixed point iterations built from prox mappings (e.g. prox-prox, Krasnoselsky–Mann relaxations, nonlinear projected-gradients) converge locally linearly under the assumption of linear metric subregularity. Linear metric subregularity is in any case necessary for linearly convergent fixed point sequences, so the result is tight. To show this, we develop a theory of fixed point mappings that violate the usual assumptions of nonexpansiveness and firm nonexpansiveness in p-uniformly convex spaces.

Iterative methods for monotone nonexpansive mappings in uniformly convex spaces

We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space (X, ∣·∣, ⪯), T:C → C a monotone 1-Lipschitz mapping and x ⪯ T(x), then the sequence of averages 1 n ∑ i = 0 n − 1 T i ( x ) $ \frac{1}{n}\sum\nolimits_{i=0}^{n-1}T^{i}(x) $ converges weakly to a fixed point of T. As a consequence, it is shown that the sequence of Picard’s iteration {T n (x)} also converges weakly to a fixed point of T. The results are new even in a Hilbert space. The Krasnosel’skiĭ-Mann and the Halpern iteration schemes are studied as well.

Strict and uniform convexity

Ways to classify and measure convexity of balls are described. Properties like strict convexity, uniform convexity, and squareness are discussed. The main tool, the modulus of convexity of a space, is studied. In the case of uniformly convex spaces, nearest point projections and asymptotic centres of sequences are presented.

ORTHOGONALITY AND PARALLELISM OF OPERATORS ON VARIOUS BANACH SPACES

We present some properties of orthogonality and relate them with support disjoint and norm inequalities in $p$-Schatten ideals. In addition, we investigate the problem of characterization of norm-parallelism for bounded linear operators. We consider the characterization of the norm-parallelism problem in $p$-Schatten ideals and locally uniformly convex spaces. Later on, we study the case when an operator is norm-parallel to the identity operator. Finally, we give some equivalence assertions about the norm-parallelism of compact operators. Some applications and generalizations are discussed for certain operators.

A Generalization of Uniformly Extremely Convex Banach Spaces

We discuss a new class of Banach spaces which are the generalization of uniformly extremely convex spaces introduced by Wulede and Ha. We prove that the new class of Banach spaces lies strictly between either the classes ofk-uniformly rotund spaces andk-strongly convex spaces or classes of fullyk-convex spaces andk-strongly convex spaces and has no inclusive relation with the class of locallyk-uniformly convex spaces. We obtain in addition some characterizations and properties of this new class of Banach spaces. In particular, our results contain the main results of Wulede and Ha.