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On conjugacy equation in dimension one

Krzysztof Ciepliński, Zbigniew Leśniak (2013)

Banach Center Publications

In this paper, recent results on the existence and uniqueness of (continuous and homeomorphic) solutions φ of the equation φ ∘ f = g ∘ φ (f and g are given self-maps of an interval or the circle) are surveyed. Some applications of these results as well as the outcomes concerning systems of such equations are also presented.

On continuous composition operators

Wilhelmina Smajdor (2010)

Annales Polonici Mathematici

Let I ⊂ ℝ be an interval, Y be a normed linear space and Z be a Banach space. We investigate the Banach space Lip₂(I,Z) of all functions ψ: I → Z such that M ψ : = s u p | | [ r , s , t ; ψ ] | | : r < s < t , r , s , t I < , where [r,s,t;ψ]:= ((s-r)ψ(t)+(t-s)ψ(r)-(t-r)ψ(s))/((t-r)(t-s)(s-r)). We show that ψ ∈ Lip₂(I,Z) if and only if ψ is differentiable and its derivative ψ’ is Lipschitzian. Suppose the composition operator N generated by h: I × Y → Z, (Nφ)(t):= h(t,φ(t)), maps the set (I,Y) of all affine functions φ: I → Y into Lip₂(I,Z). We prove that if N is...

On convex and *-concave multifunctions

Bożena Piątek (2005)

Annales Polonici Mathematici

A continuous multifunction F:[a,b] → clb(Y) is *-concave if and only if the inclusion 1 / ( t - s ) s t F ( x ) d x ( F ( s ) * + F ( t ) ) / 2 holds for every s,t ∈ [a,b], s < t.

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