A differential equation related to the $\mathbf{l}^{p}$-norms

Tom 101 / 2011

Jacek Bojarski, Tomasz Małolepszy, Janusz Matkowski Annales Polonici Mathematici 101 (2011), 251-265 MSC: Primary 46B20, 46B45; Secondary 34A34. DOI: 10.4064/ap101-3-5

Streszczenie

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.

Autorzy

  • Jacek BojarskiFaculty of Mathematics, Computer Science
    and Econometrics
    University of Zielona Góra
    Podgórna 50
    65-246 Zielona Góra, Poland
    e-mail
  • Tomasz MałolepszyFaculty of Mathematics, Computer Science
    and Econometrics
    University of Zielona Góra
    Podgórna 50
    65-246 Zielona Góra, Poland
    e-mail
  • Janusz MatkowskiFaculty 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
    e-mail

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