Let C be a closed, bounded, convex subset of a uniformly convex Banach space, and let $\{T_s\}$ be an asymptotic nonexpansive semigroup of nonlinear mappings acting within C. Consider the implicit iteration process defined by the sequence of equations:

$$ \begin{align*} x_{k+1} = c_k T_{s_{k+1}}(x_{k+1}) + (1 - c_k) x_k,\end{align*} $$

where each $c_k \in (0,1)$ and the initial point $x_0 \in C$ is arbitrarily chosen. In this context, we investigate the conditions under which the sequence $\{x_k\}$ converges, either weakly or strongly, to a common fixed point of the semigroup $\{T_s\}$. We also touch upon the question of the stability of such processes.

We study the existence of fixed points for contraction multivalued mappings in modular metric spaces endowed with a graph. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. This paper can be seen as a generalization of Nadler and Edelstein’s fixed point theorems to modular metric spaces endowed with a graph.

This paper studies the existence and the order structure of strong Berge equilibrium, a refinement of Nash equilibrium, for games with strategic complementarities à la strong Berge. It is shown that the equilibrium set is a nonempty complete lattice. Moreover, we provide a monotone comparative statics result such that the greatest and the lowest equilibria are increasing.

In this paper we present new fixed point theorems for inward and weakly inward type maps between Fréchet spaces. We also discuss Kakutani–Mönch and contractive type maps.

Kakutani's Theorem states that every point convex and use multifunction ϕ defined on a compact and convex set in a Euclidean space has at least one fixed point. Some necessary conditions are given here which ϕ must satisfy if c is the unique fixed point of ϕ. It is e.g. shown that if the width of ϕ(c) is greater than zero, then ϕ cannot be lsc at c, and if in addition c lies on the boundary of ϕ(c), then there exists a sequence {xk} which converges to c and for which the width of the sets ϕ(xk) converges to zero. If the width of ϕ(c) is zero, then the width of ϕ(xk) converges to zero whenever the sequence {xk} converges to c, but in this case ϕ can be lsc at c.