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On homeomorphic and diffeomorphic solutions of the Abel equation on the plane

Zbigniew Leśniak — 1993

Annales Polonici Mathematici

We consider the Abel equation φ[f(x)] = φ(x) + a on the plane ℝ², where f is a free mapping (i.e. f is an orientation preserving homeomorphism of the plane onto itself with no fixed points). We find all its homeomorphic and diffeomorphic solutions φ having positive Jacobian. Moreover, we give some conditions which are equivalent to f being conjugate to a translation.

On conjugacy equation in dimension one

Krzysztof CieplińskiZbigniew 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.

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