A general Lipschitz uniqueness criterion for scalar ordinary differential equations

Notes on the existence and uniqueness of solutions of Stieltjes differential equations

“…Notably, an extensive collection of uniqueness criteria is available in the monograph [1], and research has continued to add and refine such conditions, cf. [3, 5]. So far, the results ensuring existence and uniqueness of solutions for Stieltjes differential equations cover Picard and Peano type theorems (see [8, Section 7]).…”

Integration by parts and by substitution unified, Green's Theorem and uniqueness for ODEs

“…Uniqueness for differential equations is an old subject far from being solved, see [1,11] and references therein, and that still sparks interest in researching [4,6,7,8,12]. In this section we show how our general version of the formula of change of variables yields new effective conditions for uniqueness.…”

“…Moreover, for all (t, x) ∈ U , t > 0, we have Notice that ϕ(x) > x for x > 0, and therefore we cannot deduce from (3.25) that a Nagumo condition is satisfied. Morevorer, it is important to note that f does not satisfy any local Lipschitz condition with respect to x or with respect to t, and therefore (3.23) falls outside the scope of recent uniqueness results such as those in [7,12].…”

A Lipschitz condition along a transversal foliation implies local uniqueness for ODEs

“…By taking either (u 1 , u 2 ) = (0, 1) or (u 1 , u 2 ) = (1, 0) this result covers both the classical Lipschitz uniqueness theorem and the previous alternative version. Moreover this result has been remarkably generalized in [8] by Diblík, Nowak and Siegmund by allowing the vector (u 1 , u 2 ) to depend on t.…”

“…Uniqueness for ODEs is an important and quite old subject, but still an active field of research [7][8][9], being Lipschitz uniqueness theorem the cornerstone on the topic. Besides the existence of many generalizations of that theorem, see [1,6,10], one recent and fruitful line of research has been the searching for alternative or weaker forms of the Lipschitz condition.…”