Geodesic
A differential equation related to the $\mathbf{l}^{p}$-norms
Jacek Bojarski1 ; Tomasz Małolepszy1 ; Janusz Matkowski2
1Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra Podgórna 50 65-246 Zielona Góra, Poland
2Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra Podgórna 50 65-246 Zielona Góra, Poland and Institute of Mathematics Silesian University Bankowa 14 40-007, Katowice, Poland
Annales Polonici Mathematici, Tome 101 (2011) no. 3, pp. 251-265
Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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Let $p\in (1,\infty )$. The question of existence of a curve in $\mathbb{R}% _{+}^{2}$ starting at $(0,0)$ and such that at every point $(x,y)$ of this curve, the $\mathbf{l}^{p}$-distance of the points $(x,y)$ and $(0,0)$ is equal to the Euclidean length of the arc of this curve between these points is considered. This problem reduces to a nonlinear differential equation. The existence and uniqueness of solutions is proved and nonelementary explicit solutions are given.
DOI : 10.4064/ap101-3-5
Keywords: infty question existence curve mathbb starting every point curve mathbf distance points equal euclidean length arc curve between these points considered problem reduces nonlinear differential equation existence uniqueness solutions proved nonelementary explicit solutions given

Jacek Bojarski  1   ; Tomasz Małolepszy  1   ; Janusz Matkowski  2

1 Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra Podgórna 50 65-246 Zielona Góra, Poland
2 Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra Podgórna 50 65-246 Zielona Góra, Poland and Institute of Mathematics Silesian University Bankowa 14 40-007, Katowice, Poland 
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Jacek Bojarski; Tomasz Małolepszy; Janusz Matkowski. A differential equation related to the $\mathbf{l}^{p}$-norms. Annales Polonici Mathematici, Tome 101 (2011) no. 3, pp. 251-265. doi: 10.4064/ap101-3-5

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