On common fixed and periodic points of commuting functions

Commuting Functions with No Common Fixed Points)

Introduction. Let/and g be continuous functions mapping the unit interval / into itself which commute under functional composition, that is, f(g(x)) = g(f(x)) for all x in /. In 1954 Eldon Dyer asked whether/and g must always have a common fixed point, meaning a point z in / for which f(z) = z=g(z). A. L. Shields posed the same question independently in 1955, as did Lester Dubins in 1956. The p...

On Common Fixed Points, Periodic Points, and Recurrent Points of Continuous Functions

It is known that two commuting continuous functions on an interval need not have a common fixed point. However, it is not known if such two functions have a common periodic point. We had conjectured that two commuting continuous functions on an interval will typically have disjoint sets of periodic points. In this paper, we first prove that S is a nowhere dense subset of [0,1] if and only if {f...

Common Fixed Points of Two Commuting Mappings in Complete Metric Spaces

Common Fixed Points and Invariant Approximations for Cq-commuting Generalized nonexpansive mappings

Some common fixed point theorems for Cq-commuting generalized nonexpansive mappings have been proved in metric spaces. As applications, invariant approximation results are also obtained. The results proved in the paper extend and generalize several known results including those of M. Abbas and J.K. Kim [Bull. Korean Math. Soc. 44(2007) 537-545], I. Beg, N. Shahzad and M. Iqbal [Approx. Theory A...

On Common Fixed Points

Some fixed point theorems based on an asymptotic regularity condition have been obtained, which generalize the previously well-known results. 2000 Mathematics Subject Classification: 47H10, 54H25.