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

Annales Polonici Mathematici (1993)

- Volume: 58, Issue: 1, page 7-18
- ISSN: 0066-2216

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topZbigniew Leśniak. "On homeomorphic and diffeomorphic solutions of the Abel equation on the plane." Annales Polonici Mathematici 58.1 (1993): 7-18. <http://eudml.org/doc/262502>.

@article{ZbigniewLeśniak1993,

abstract = {
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.
},

author = {Zbigniew Leśniak},

journal = {Annales Polonici Mathematici},

keywords = {functional Abel equation; free mapping; homeomorphic solutions; diffeomorphic solutions; Abel equation},

language = {eng},

number = {1},

pages = {7-18},

title = {On homeomorphic and diffeomorphic solutions of the Abel equation on the plane},

url = {http://eudml.org/doc/262502},

volume = {58},

year = {1993},

}

TY - JOUR

AU - Zbigniew Leśniak

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

JO - Annales Polonici Mathematici

PY - 1993

VL - 58

IS - 1

SP - 7

EP - 18

AB -
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.

LA - eng

KW - functional Abel equation; free mapping; homeomorphic solutions; diffeomorphic solutions; Abel equation

UR - http://eudml.org/doc/262502

ER -

## References

top- [1] S. A. Andrea, On homeomorphisms of the plane which have no fixed points, Abh. Math. Sem. Hamburg 30 (1967), 61-74. Zbl0156.43704
- [2] S. A. Andrea, The plane is not compactly generated by a free mapping, Trans. Amer. Math. Soc. 151 (1970), 481-498. Zbl0205.54002
- [3] R. Engelking and K. Sieklucki, Topology. A Geometric Approach, Sigma Ser. Pure Math. 4, Heldermann, Berlin 1992.
- [4] T. Homma and H. Terasaka, On the structure of the plane translation of Brouwer, Osaka Math. J. 5 (1953), 233-266. Zbl0051.14701
- [5] M. Kuczma, On the Schröder equation, Rozprawy Mat. 34 (1963). Zbl0121.33703
- [6] R. Sikorski, Advanced Calculus. Functions of Several Variables, Monograf. Mat. 52, PWN, Warszawa 1969. Zbl0182.37901
- [7] M. C. Zdun, On continuous iteration groups of fixed-point free mapping in ℝ² space, in: Proc. European Conference on Iteration Theory, Batschuns 1989, World Scientific, Singapore 1991, 362-368.

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