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    Picard–Lindelöf theorem - Wikipedia

    • 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, R… 展开

    Proof sketch

    A standard proof relies on transforming the differential equation into an integral equation, then applying the
    Integrating … 展开

    Example of non-uniqueness

    To understand uniqueness of solutions, contrast the following two examples of first order ordinary differential equations for y(t). Both differential equations will possess a single stationary point y = 0.
    First, the homog… 展开

    Detailed proof

    Let
    where:
    This is the compact cylinder where  f  is defined.
    Let L be the Lipschitz constant of  f  with respect to the se… 展开

    Optimization of the solution's interval

    We wish to remove the dependence of the interval Ia on L. To this end, 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 t… 展开

    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  f  is continuous in y, instead of Lipschitz continu… 展开

    Global existence of solution

    The Picard–Lindelöf theorem ensures that solutions to initial value problems exist uniquely within a local interval , possibly dependent on each solution. The behavior of solutions beyond this local interval can vary depending … 展开

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    Proof sketch
    Example of non-uniqueness
    Detailed proof
    Optimization of the solution's interval
    Other existence theorems
    Global existence of solution