Common fixed points of commuting holomorphic mappings

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 of Weakly Commuting Mappings and Set - Valued Mappings

ABS]RACT. Our main theorem establishes the uniqueness of the common fixed point of tw set-valued mappings and of two single-valued mappings defined on a complete metric space, under a contractive condition and a weak commutativity concept. This improves a theorem of the second author. Let (X,d) be a complete metric space and let B(X) be the set of all nonempty, bounded subsets of X As in [I], l...

Fixed Points of Commuting Holomorphic Mappings Other than the Wolff Point

Let ∆ be the unit disc of C and let f, g ∈ Hol(∆,∆) be such that f ◦ g = g ◦ f . For A > 1, let FixA(f) := {p ∈ ∂∆ | limr→1 f(rp) = p, limr→1 |f ′(rp)| ≤ A}. We study the behavior of g on FixA(f). In particular, we prove that g(FixA(f)) ⊆ FixA(f). As a consequence, besides conditions for FixA(f)∩FixA(g) 6= ∅, we prove a conjecture of C. Cowen in case f and g are univalent mappings.

Commuting semigroups of holomorphic mappings

Let S1 = {Ft}t≥0 and S2 = {Gt}t≥0 be two continuous semigroups of holomorphic self-mappings of the unit disk ∆ = {z : |z| < 1} generated by f and g, respectively. We present conditions on the behavior of f (or g) in a neighborhood of a fixed point of S1 (or S2), under which the commutativity of two elements, say, F1 and G1 of the semigroups implies that the semigroups commute, i.e., Ft◦Gs = Gs◦...