A general Lipschitz uniqueness criterion for scalar ordinary differential equations

作者：Diblik Josef^*; Nowak Christine; Siegmund Stefan

来源：Electronic Journal of Qualitative Theory of Differential Equations, 2014, (34): 1-6.

DOI：10.14232/ejqtde.2014.1.34

• fundamental theory of ordinary differential equations
• initial value problems
• uniqueness
• Lipschitz type conditions

The classical Lipschitz-type criteria guarantee unique solvability of the scalar initial value problem x = f(t, x), x(t(0)) = x(0), by putting restrictions on |f(t, x) - f(t, y)| in dependence of |x - y|. Geometrically it means that the field differences are estimated in the direction of the x-axis. In 1989, Stettner and the second author could establish a generalized Lipschitz condition in both arguments by showing that the field differences can be measured in a suitably chosen direction v = (d(t), d(x)), provided that it does not coincide with the directional vector (1, f (t(0), x(0))).
Considering the vector v depending on t, a new general uniqueness result is derived and a short proof based on the implicit function theorem is developed. The advantage of the new criterion is shown by an example. A comparison with known results is given as well.

• 出版日期2014