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Mathematics · unit version 1

Picard construction and uniqueness

Construct local differential-equation solutions in a complete function space, certify integral iterates, and prove the boundaries of uniqueness and continuation.

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Readiness and preparation

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Which hypotheses does the contraction theorem require?

The integral operator must preserve a complete domain before its shrinking-distance estimate can construct a solution.

Compare reasoning and find a refresher

A nonempty complete metric space, a self-map, and one Lipschitz constant q strictly below 1.

Review the contraction theorem

Why use uniform distance between candidate functions?

An integral estimate must control values over the entire time interval.

Compare reasoning and find a refresher

The supremum norm bounds every input simultaneously and makes continuous functions on a compact interval a complete space.

Review uniform Cauchy completeness

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Changed equations, inexact iterates and continuation

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Lesson sequence

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  1. Picard’s operator and invariant function balls

    Lesson details and objectives

    Construct a nonlinear initial-value solution by choosing a complete invariant function ball and proving an integral operator is a contraction.

    • Choose a time window satisfying domain, self-map and strict contraction inequalities separately.
    • Prove completeness of the closed continuous-function ball and all operator estimates.
    • Use the fundamental theorem of calculus to establish fixed-point/IVP equivalence and the precise local uniqueness class.

    6 practice tasks.

  2. Picard iterates and error certificates

    Lesson details and objectives

    Construct exact polynomial iterates, prove uniform error bounds and distinguish a failed certificate from a failed differential equation.

    • Compute integral Picard iterates without replacing the integrals by a time-stepping rule.
    • Derive and apply a priori, successive-iterate and residual bounds with their correct indices.
    • Choose a valid invariant rectangle and distinguish exact uniform certificates from finite numerical observations.

    6 practice tasks.

  3. State exactly where uniqueness and continuation hold

    Lesson details and objectives

    Compare solutions through their integral equations, continue across controlled intervals, and separate failed hypotheses from actual nonuniqueness.

    • Prove uniqueness on the selected rectangle and extend comparisons through shorter overlapping intervals.
    • Prove a finite-time continuation criterion using a compact subset of the equation domain.
    • Verify nonunique delayed solutions and a non-Lipschitz equation with forward uniqueness, distinguishing time direction.

    6 practice tasks.