Virtually stable maps and their fixed point sets

Abstract

We introduce the concept of virtually stable selfmaps of Hausdorff spaces, which generalizes virtually nonexpansive selfmaps of metric spaces introduced in the previous work by the first author, and explore various properties of their convergence sets and fixed point sets. We also prove that the fixed point set of a virtually stable selfmap satisfying a certain kind of homogeneity is always star-convex.

Fixed points of multi-valued Subrahmanyan contractions

“…In general, the selection f ∞ : Graph(F) → Fix(F) of F ∞ is not continuous. For the continuity of the operator f ∞ in the single-valued case, see [12].…”

“…It is an open question to present a fixed point theory (see [17] and [9]) for multi-valued Feng-Liu-Subrahmanyan contractions in complete metric spaces. Another open problem is to obtain topological properties of the fixed point set for multi-valued Subrahmanyan contractions, following the approach given in [12] for virtually nonexpansive operators. For the case of nonexpansive operators, we refer to the papers of Bruck [19,20].…”

Generalized Virtually Stable Maps and Their Associated Sequences

“…Clearly, a uniformly virtually stable self-map is always virtually stable, and a continuously (uniformly) virtually stable self-map as defined in [1] is also (uniformly) virtually stable according to our new definition. Moreover, if is uniformly virtually stable with respect to ( ), it is immediately uniformly virtually stable with respect to any subsequence of ( ).…”

“…Proof. (1) The fact that gcd( ) = gcd⟨ ⟩ follows directly from the definition. Now, let = min{gcd{ : ≤ } : ∈ N}.…”

“…Virtual stability of a self-map was first introduced in [1], where it was proved to unify various (continuous) types of nonexpansiveness in fixed-point theory. The main feature of a virtually stable self-map is that its fixed-point set is always a retract of its convergence set, and this fact immediately allows us to connect topological structures (e.g., connectedness and contractibility) of the fixed-point set to those of the convergence set.…”