Adriana Nicolae

Fixed point theorems for singlevalued and multivalued generalized contractions in metric spaces endowed with a graph

Abstract The purpose of this paper is to present some fixed point results for self-generalized (singlevalued and multivalued) contractions in ordered metric spaces and in metric spaces endowed with a graph. Our theorems generalize and extend some recent results in the literature.

What Do `Convexities' Imply on Hadamard Manifolds?

Various results based on some convexity assumptions (involving the exponential map along with affine maps, geodesics and convex hulls) have been recently established on Hadamard manifolds. In this paper, we prove that these conditions are mutually equivalent and they hold, if and only if the Hadamard manifold is isometric to the Euclidean space. In this way, we show that some results in the literature obtained on Hadamard manifolds are actually nothing but their well-known Euclidean counterparts.

The Asymptotic Behavior of the Composition of Firmly Nonexpansive Mappings

In this paper, we provide, in the setting of geodesic spaces, a unified treatment of some convex minimization problems, which allows for a better understanding and, in some cases, improvement of results proved recently in this direction. For this purpose, we analyze the asymptotic behavior of compositions of finitely many firmly nonexpansive mappings focusing on asymptotic regularity and convergence results.

Quantitative results on Fejér monotone sequences

We provide in a unified way quantitative forms of strong convergence results for numerous iterative procedures which satisfy a general type of Fejer monotonicity where the convergence uses the compactness of the underlying set. These quantitative versions are in the form of explicit rates of so-called metastability in the sense of T. Tao. Our approach covers examples ranging from the proximal point algorithm for maximal monotone operators to various fixed point iterations (x_n) for firmly nonexpansive, asymptotically nonexpansive, strictly pseudo-contractive and other types of mappings. Many of the results hold in a general metric setting with some convexity structure added (so-called W-hyperbolic spaces). Sometimes uniform convexity is assumed still covering the important class of CAT(0)-spaces due to Gromov.

Effective results on nonlinear ergodic averages in CAT(κ) spaces

In this paper we apply proof mining techniques to compute, in the setting of CAT

Generalized Asymptotic Pointwise Contractions and Nonexpansive Mappings Involving Orbits

(\kappa)

Best Proximity Pair Results for Relatively Nonexpansive Mappings in Geodesic Spaces

spaces (with

Quantitative asymptotic regularity results for the composition of two mappings

\kappa >0

Chebyshev sets in geodesic spaces

), effective and highly uniform rates of asymptotic regularity and metastability for a nonlinear generalization of the ergodic averages, known as the Halpern iteration. In this way, we obtain a uniform quantitative version of a nonlinear extension of the classical von Neumann mean ergodic theorem.

A proof-theoretic bound extraction theorem for CAT(κ)-spaces

We give fixed point results for classes of mappings that generalize pointwise contractions, asymptotic contractions, asymptotic pointwise contractions, and nonexpansive and asymptotic nonexpansive mappings. We consider the case of metric spaces and, in particular, CAT spaces. We also study the well-posedness of these fixed point problems.