Picard–Lindelöf theorem

In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named after Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy. Let be a closed rectangle with , the interior of . Let be a function that is continuous in and Lipschitz continuous in . Then, there exists some ε > 0 such that the initial value problem has a unique solution on the interval . The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. By integrating both sides, any function satisfying the differential equation must also satisfy the integral equation A simple proof of existence of the solution is obtained by successive approximations. In this context, the method is known as Picard iteration. Set and It can then be shown, by using the Banach fixed-point theorem, that the sequence of "Picard iterates" φk is convergent and that the limit is a solution to the problem. An application of Grönwall's lemma to φ(t) − ψ(t), where φ and ψ are two solutions, shows that φ(t) = ψ(t), thus proving the global uniqueness (the local uniqueness is a consequence of the uniqueness of the Banach fixed point). See Newton's method of successive approximation for instruction. Let the solution to the equation with initial condition Starting with we iterate so that : and so on. Evidently, the functions are computing the Taylor series expansion of our known solution Since has poles at this converges toward a local solution only for not on all of . To understand uniqueness of solutions, consider the following examples. A differential equation can possess a stationary point. For example, for the equation dy/dt = ay (), the stationary solution is y(t) = 0, which is obtained for the initial condition y(0) = 0.