Let $K$ be a bounded, closed, and convex subset of a Banach space. For a Lipschitzian self-mapping $A$ of $K$, we denote by $\text{Lip}\left(A\right)$ its Lipschitz constant. In this paper, we establish a convergence property of infinite products of Lipschitzian self-mappings of $K$. We consider the set of all sequences ${\left\{A_t\right\}}_{t=1}^{\infty }$ of such self-mappings with the property ${\text{limsup}}_{t\to \infty }\text{Lip}\left(A_t\right)\le 1$. Endowing it with an appropriate topology, we establish a weak ergodic theorem for the infinite products corresponding to generic sequences in this space.

Let $f$ be an essential map of $S^{n-1}$ into itself (i.e., $f$ is not homotopic to a constant mapping) admitting an extension mapping the closed unit ball ${\overline{B}}^n$ into ${ℝ}^n$. Then, for every interior point $y$ of $B^n$, there exists a point $x$ in $f^{-1}\left(y\right)$ such that the image of no neighborhood of $x$ is contained in a coordinate half space with $y$ on its boundary. Under additional conditions, the image of a neighborhood of $x$ covers a neighborhood of $y$. Differential versions are valid for quasianalytic functions. These results are motivated by game-theoretic considerations.

It is shown that if $X$ is a Banach space and $C$ is a union of finitely many nonempty, pairwise disjoint, closed, and connected subsets $\left\{C_i:1\le i\le n\right\}$ of $X$, and each $C_i$ has the fixed-point property (FPP) for asymptotically regular nonexpansive mappings, then any asymptotically regular nonexpansive self-mapping of $C$ has a fixed point. We also generalize the Goebel-Schöneberg theorem to some Banach spaces with Opial's property.

We give an example of an unbounded, convex, and closed set $C$ in the Hilbert space $l^2$ with the following two properties: (i) $C$ has the approximate fixed-point property for nonexpansive mappings, (ii) $C$ is not contained in a block for every orthogonal basis in $l^2$.

We provide some new estimates of the smallest possible Lipschitz constant for retractions of the unit ball $B$ onto the unit sphere $S$ in infinite-dimensional Banach spaces.

The aim of this paper is to work with the measure of nonhyperconvexity in a similar way as J. Cano (1990) does with the index of nonconvexity. We apply this measure to obtain different extensions of the famous Schauder fixed-point theorem in hyperconvex spaces.

We study the topology of a subspace of the function space of continuous self-mappings of a given manifold: the subspace determined by maps having the least number of fixed points in its homotopy class. In the case that the manifold is a closed disk of finite dimension, we prove that this subspace is both globally and locally path connected. We also prove this result when the manifold is a sphere of dimension 1, 3, or 7.