Existence/Uniqueness of solutions to quasi-Lipschitz ODEs - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T08:48:53Z http://mathoverflow.net/feeds/question/55289 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/55289/existence-uniqueness-of-solutions-to-quasi-lipschitz-odes Existence/Uniqueness of solutions to quasi-Lipschitz ODEs Ricky Demer 2011-02-13T05:30:39Z 2011-10-02T09:48:24Z <p>Would the <a href="http://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem#Picard.E2.80.93Lindel.C3.B6f_theorem" rel="nofollow">Picard–Lindelöf theorem</a> still be true if the requirement that f be Lipschitz continuous in y was replaced with the requirement that f be <a href="http://en.wikipedia.org/wiki/Modulus_of_continuity" rel="nofollow">almost Lipschitz</a> in y?</p> <p>If not, are there any moduli of uniform continuity weaker than Lipschitz continuity that it is known suffice, or results indicating that there can't be any?</p> http://mathoverflow.net/questions/55289/existence-uniqueness-of-solutions-to-quasi-lipschitz-odes/55310#55310 Answer by Andrey Rekalo for Existence/Uniqueness of solutions to quasi-Lipschitz ODEs Andrey Rekalo 2011-02-13T12:41:11Z 2011-02-14T00:53:32Z <p>Yes. This follows from the classical uniqueness theorem due to Osgood (the original <a href="http://www.springerlink.com/content/u51r2q4808563853/" rel="nofollow">paper</a> appeared in 1898).</p> <blockquote> <p><strong>Osgood's Criterion.</strong> Let $\omega(t,u)=\phi(t)\psi(u)$ where $\phi(t)\geq 0$ is continuous on the interval $(0,a)$ and $\psi(u)$ is continuous on $\mathbb R_{+}$, $\psi(0)=0$, $\psi(u)>0$ for $u>0$, and $$\int_{0}^{\epsilon}\phi(t)dt&lt;\infty,\qquad \int_{0}^{\epsilon}\frac{du}{\psi (u)}=\infty$$ for some $\epsilon>0$. Suppose that the mapping $f:[0,a]\times B_R(x_0)\to \mathbb R^d$ satisfies the condition $$||f(t,x_1)-f(t,x_2)||\leq\omega (t,||x_1-x_2||)$$ for any $t\in(0,a]$ and any $x_1,x_2\in B_R(x_0)$. Then the initial value problem $$\dot{x}=f(t,x),\qquad x(0)=x_0$$ has at most one solution on the interval $[0,\delta]$ with some $\delta>0$.</p> </blockquote> <p>Osgood's theorem allows for the mappings $f(t,x)$ which are discontinuous at $t=0$. (Actually, the condition that $\phi(t)$ is continuous on $(0,a)$ can be replaced with an assumption of mere integrability.) Of course, the existence of a local solution is implied by the Peano theorem under the additional assumption that $f$ is continuous in $(t,x)$.</p> <p>Moreover, Wintner showed that Osgood's uniqueness condition implies the convergence of successive Picard iterations to a local solution on a sufficiently small interval (A. Wintner, <a href="http://www.jstor.org/pss/2371736" rel="nofollow">"On the Convergence of Successive Approximations"</a>, <em>Amer. Journal of Math.</em> Vol. 68 (1946), pp. 13-19).</p> http://mathoverflow.net/questions/55289/existence-uniqueness-of-solutions-to-quasi-lipschitz-odes/76976#76976 Answer by prp for Existence/Uniqueness of solutions to quasi-Lipschitz ODEs prp 2011-10-02T09:48:24Z 2011-10-02T09:48:24Z <p>Osgood's theorem allows for the mappings f(t,x) which are discontinuous at t=0. (Actually, the condition that ϕ(t) is continuous on (0,a) can be replaced with an assumption of mere integrability.) Of course, the existence of a local solution is implied by the Peano theorem under the additional assumption that f is continuous in (t,x).</p>