Let p ∈ (1,∞). The question of existence of a curve in ℝ₊² starting at (0,0) and such that at every point (x,y) of this curve, the -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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap101-3-5,
author = {Jacek Bojarski and Tomasz Ma\l olepszy and Janusz Matkowski},
title = {A differential equation related to the $l^{p}$-norms},
journal = {Annales Polonici Mathematici},
volume = {101},
year = {2011},
pages = {251-265},
zbl = {1269.46014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap101-3-5}
}
Jacek Bojarski; Tomasz Małolepszy; Janusz Matkowski. A differential equation related to the $l^{p}$-norms. Annales Polonici Mathematici, Tome 101 (2011) pp. 251-265. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap101-3-5/