On the convergence of the three-step iteration process in the class of quasi-contractive operators.
Let X be a nonempty set of cardinality at most and T be a selfmap of X. Our main theorem says that if each periodic point of T is a fixed point under T, and T has a fixed point, then there exist a metric d on X and a lower semicontinuous map ϕ :X→ ℝ ₊ such that d(x,Tx) ≤ ϕ(x) - ϕ(Tx) for all x∈ X, and (X,d) is separable. Assuming CH (the Continuum Hypothesis), we deduce that (X,d) is compact.
We study the covering dimension of the fixed point set of lower semicontinuous multifunctions of which many values can be non-closed or non-convex. An application to variational inequalities is presented.
We have given several proofs on the existence of the price equilibrium --- via variational inequality --- via degree theory and via Brouwer's theorems.
In this paper we prove some approximate fixed point theorems which extend, in a broad sense, analogous results obtained by Brânzei, Morgan, Scalzo and Tijs in 2003. By assuming also the weak demiclosedness property we state two fixed point theorems. Moreover, we study the existence of ɛ-Nash equilibria.