Picard–Lindelöf theorem
In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem, Picard's existence theorem, Cauchy–Lipschitz theorem, or existence and uniqueness theorem gives a set of conditions under which an initial value problem has a unique solution. The theorem is named after Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy. Theorem. Let formula_1be a closed rectangle with formula_2. Let formula_3 be a function that is continuous in formula_4 and Lipschitz continuous in formula_5. Then, there exists some such that the initial value problem formula_6 has a unique solution formula_7 on the interval formula_8. Note that  formula_9 is often instead required to be open but even under such an assumption, the proof only uses a closed rectangle within formula_9. Proof sketch. The proof relies on transforming the differential equation, and applying fixed-point theory. 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" is convergent and  that the limit is a solution to the problem. An application of Grönwall's lemma to , where and are two solutions, shows that , 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. Example of Picard iteration. Let formula_14 the solution to the equation formula_15 with initial condition formula_16 Starting with formula_17 we iterate so that formula_19: and so on. Evidently, the functions are computing the Taylor series expansion of our known solution formula_23 Since formula_24 has poles at formula_25 this converges  toward a local solution only for formula_26 not on all of formula_27. Example of non-uniqueness. To understand uniqueness of solutions, consider the following examples. A differential equation can possess a stationary point. For example, for the equation (formula_28), the stationary solution is , which is obtained for the initial condition . Beginning with another initial condition , the solution "y"("t") tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. However, for an equation in which the stationary solution is reached after a "finite" time, the  uniqueness fails. This happens for example for the equation , which has at least two solutions corresponding to the initial condition such as: or so the previous state of the system is not uniquely determined by its state after "t" = 0. The uniqueness theorem does not apply because the function has an infinite slope at and therefore is not Lipschitz continuous, violating the hypothesis of the theorem. Detailed proof. Let where: This is the compact cylinder where    is defined. Let this is, the supremum of (the absolute values of) the slopes of the function. Finally, let "L" be the Lipschitz  constant of with respect to the second variable. We will proceed to apply the Banach fixed-point theorem using the metric on formula_33 induced by the uniform norm We define an operator between two function spaces of continuous functions, Picard's operator, as follows: defined by: We must show that this operator maps a complete non-empty metric space "X" into itself and also is a contraction mapping. We first show that, given certain restrictions on formula_37, formula_38 takes formula_39 into itself in the space of continuous functions with the uniform norm. Here, formula_39 is a closed ball in the  space of continuous (and bounded) functions "centered" at the constant function formula_41. Hence we need to show that implies where formula_44 is some number in formula_45 where the maximum is achieved. The last inequality in the chain is true if we impose the requirement formula_46. Now let's prove that this operator is a contraction mapping. Given two functions formula_47, in order to apply the Banach fixed-point theorem we require for some formula_49. So let formula_4 be such that Then using the definition of formula_38, This is a contraction if formula_54 We have established that the Picard's operator  is a contraction on the Banach spaces with the metric induced by the uniform norm. This allows us to apply the Banach fixed-point theorem to conclude that the operator has a unique fixed point. In particular, there is a unique function such that . This function is the unique solution of the initial value problem, valid on the interval "I" where "a" satisfies the condition Optimization of the solution's interval. Nevertheless, there is a corollary of the Banach fixed-point theorem: if an operator "T" is a contraction for some "n" in N, then "T" has a unique fixed  point. Before applying this theorem to the Picard operator, recall the following: "Proof." Induction on "m". For the base of the induction () we have already seen this, so suppose the inequality holds for , then we have: formula_57 By taking a supremum over formula_58 we see that formula_59. This inequality assures that for some large "m", formula_60 and hence Γ will be a contraction. So by the previous corollary Γ will have a unique fixed point. Finally, we have been able to optimize the interval of the solution by taking . In the end, this result shows the interval of definition of  the solution does not depend on the Lipschitz constant of the field, but only on the interval of definition of the field and its maximum absolute value. Other existence theorems. The Picard–Lindelöf theorem shows that the solution exists and that it is unique. The Peano existence theorem shows only existence, not uniqueness, but it assumes only that is continuous in , instead of Lipschitz continuous. For example, the right-hand side of the equation with initial condition is continuous but not Lipschitz continuous. Indeed, rather than being unique, this equation has three solutions: Even more general is Carathéodory's existence  theorem, which proves existence (in a more general sense) under weaker conditions on . Although these conditions are only sufficient, there also exist necessary and sufficient conditions for the solution of an initial value problem to be unique, such as Okamura's theorem.