FEDERAL RULES OF EVIDENCE: 801-03, 901

Existence, uniqueness, and regularity Cauchy-Lipschitz theorem I = [0;T]; d: positive integer; f : I RdRd Suppose that f 2C0(I Rd) DEFINITION:Lipschitz condition There exists a constant C f 0 s t ,for any x 1; 2 2Rd and any t 2I, ( ) jf(t;x 1) f(t;x 2)j C f jx 1 x 2j: f satis es a Lipschitz condition on I C f: Lipschitz constant for f

Existence and Uniqueness 1 Lipschitz Conditions

Existence and Uniqueness 1 Lipschitz Conditions We now turn our attention to the general initial value problem dy dt = f(t;y) y(t0) = y0; where f is a diﬁerentiable function We would like to know when we have existence of a unique solution for given initial data One condition on f which guarantees this is the following

The Existence and Uniqueness Theorem (of the solution a first

the Existence and Uniqueness Theorem, therefore, a continuous and differentiable solution of this initial value problem is guaranteed to exist uniquely on any interval containing t 0 = 2 π but not containing any of the discontinuities The largest such intervals is (3 π/2, 5 π/2) It is the interval of validity of this problem Indeed, the

EXISTENCE LOCALE ET UNICITE DE SOLUTIONS DE L’´ EQUATION D

Dans cette Note, nous introduisons la condition aux limites (1 4), et nous montrons l’existence et l’unicite´ de solutions de (1 1)–(1 4) dans le cas de la dimension d’espace 2 (c-a`-dM et (u,ρ,p) inde´pendants de

13 Initial Conditions; Initial-Value Problems

The questions of existence and uniqueness of solutions will be addressed in the speciﬁc cases of interest to us A general treatment of existence and uniqueness of solutions of initial-value problems is beyond the scope of this course Exercises 1 3 1 (a) Show that each member of the one-parameter family of functions y = Ce5x

Existence and stability of steady-state solutions of reaction

d ∈ (d∗(τ)− (τ), ∗(τ)), where d∗(τ) is the bifurcation point and (τ) is a small constant depending on τ So our results here again conﬁrm the nonoccurrence of Hopf bifurcation for the diﬀusion kernel case and weak distribution kernel as indicated in [9,50] The results in this paper take an entirely diﬀerent