On the system of the Abel equations on the plane
Abstract
We find all of continuous, homeomorphic and Ck solutions of the system of the Abel equations
ϕ(f(x)) = ϕ(x)+a
ϕ(g(x)) = ϕ(x)+b for x∈ℝ2,
where a,b are linearly independent vectors and f,g are commutable orientation preserving homeomorphisms of the plane onto itself satisfying some condition which is equivalent to the fact that there exists a homeomorphic solution of the system above.
References
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2. R. Duda, Wprowadzenie do topologii, cz. I. Topologia ogólna, Biblioteka Mat. 61, PWN, Warszawa, 1986.
3. R. Engelking, K. Sieklucki, Topology. A Geometric Approach, Sigma Ser. Pure Math. 4, Heldermann, Berlin 1992.
4. M. Kuczma, On the Schröder equation, Rozprawy Mat. 34 (1963).
5. Z. Leśniak, On homeomorphic and diffeomorphic solutions of the Abel equation on the plane, Ann. Polon. Math. 58 (1993), 7-18.
6. M.H.A. Newman, Elements of the Topology of Plane Sets of Points, Cambridge University Press, London 1951.
7. H. Whitney, Analytic extension of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63-89.
LeśniakZ. (1995). On the system of the Abel equations on the plane. Annales Mathematicae Silesianae, 9, 105-122. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14211
Zbigniew Leśniak
Instytut Matematyki, Wyższa Szkoła Pedagogiczna im. Komisji Edukacji Narodowej w Krakowie Poland
Instytut Matematyki, Wyższa Szkoła Pedagogiczna im. Komisji Edukacji Narodowej w Krakowie Poland
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