The theory of some asymptotic fixed point theorems

In this paper we present the theory about some fixed point theorems for convex contraction mappings. We give some results on data dependence of fixed points, on sequences of operators and fixed points, on well-possedness of fixed point problem, on limit shadowing property and on Ulam-Hyers stability for equations of fixed points.

On the Theorems of Borsuk-Ulam and Ljusternik-Schnirelmann-Borsuk

Let p ≥ 3 be a prime number and m a positive integer, and let S be the sphere S(m-1)(p-1)-1. Let f:S→S be a map without fixed points and with fp = idS. We show that there exists an h: S→ℝm with h(x) ≠ h(f(x)) for all x ∈ S. From this we conclude that there exists a closed cover U1,…, U4m of S with Uinf(Ui) = Ø for i = 1,…, 4m. We apply these results to Borsuk-Ulam and Ljusternik-Schnirelmann-Borsuk theorems in the framework of the sectional category and to a problem in asymptotic fixed point theory.