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common fixed points of commutative applications

<= f(x)- d(x, g(x))

Exercise Let f and g be two continuous maps from I = [0,1] in I. Assume that fog = gof.
i) Show that there exists an x belonging to I such that f (x) = g (x).
ii) Assume that f is increasing on I, show that f and g have a common fixed point.
iii) Assume that f is affine, ie, d, f (x) = ax + b, (a and b two constants in IR), show that f and g have common fixed point
iv) Assume that f is contractive; show that the set of fixed points of f is convex; deduce that f and g admit a fixed point Common

Notes.

The classical results on common fixed point of a family
any applications are the theorems of [Markoff-Kakutani] and [Kakutani]. The first theorem concerns the common fixed point of a family of affine and commutative stabilizing a compact convex spaces evtlcs (= topological vector spaces separated locally convex), the second concerns the common fixed point of a family group of affine bijections and equicontinuous (not commutative) stabilizing a compact convex spaces evtlcs.Rappelons the Markoff-Kakutani theorem is used to redemontrer the theorem of Hahn-Banch.Reciproquement, [Dirk Wener] has sought to demonstrate the converse, in his paper, he presented an elegant demonstration of the existence of the fixed point of any affine continuous stabilizing a convex compact in the spaces evtlcs, and this, using the strong version of the theorem of Hahn-Banach separartion. Note that
[Marr] showed a result (similar) Markoff-Kakutani theorem for nonexpansive in any Banach space, this result is improved by [Belluci -Kirk].
Rappleons also a family of nonexpansive commutative, while stabilizing convex bounded and closed in a uniformly convex Banach space admits a common fixed point, the demonstration of this result is almost immédiate.Voir, for example, [Deimling ] the same result is valid in the context of Banach spaces reflexive strictly convex normal structure.
The literature is relatively high on a test sujet.Pour synthesis of fixed point theorems [Darbo] and theorem [Markoff-Kakutani], see [Hajji-Hanebaly]
References contained in the message 2 / 02/2007