Math 2.0

Ordinary differential equations Existence, uniqueness and dependence on data Theorem

Picard–Lindelöf theorem

Builds on Picard–Lindelöf hypothesesInitial value problemAn iterate of the Picard operator is a contractionIntegral equation form of an initial value problem

Statement

Let J=[tmin⁡,tmax⁡]J = [t_{\min}, t_{\max}] and t0∈Jt_0 \in J, let x0∈ℝnx_0 \in \mathbb{R}^n, and let f:J×ℝn→ℝnf : J \times \mathbb{R}^n \to \mathbb{R}^n satisfy the Picard–Lindelöf hypotheses at (t0,x0)(t_0, x_0) with constants a,L,Ka, L, K. Let BB be the closed ball of radius aa around x0x_0. Then there is exactly one continuous curve α:J→B\alpha : J \to B with α(t0)=x0\alpha(t_0) = x_0 and with this property. For each t∈Jt \in J, α\alpha has the derivative f(t,α(t))f(t, \alpha(t)) at tt within JJ.

Remarks

The proof uses the integral equation form and the fact that an iterate of the Picard operator is a contraction. The limit of Picard iteration is the solution. A common form: let ff be continuous on a neighbourhood of (t0,x0)(t_0, x_0) and locally Lipschitz in xx. Then the initial value problem has a unique solution on [t0−ε,t0+ε][t_0 - \varepsilon, t_0 + \varepsilon] for some ε>0\varepsilon > 0. The theorem is local. For x′=1+x2x' = 1 + x^2, x(0)=0x(0) = 0, the solution tan⁡t\tan t exists only for ∣t∣<π/2|t| < \pi/2. Without the Lipschitz condition, solutions exist by the Peano existence theorem, but they need not be unique. See the non-uniqueness example.

History

The theorem is named after Émile Picard and Ernst Lindelöf. It is also called the Cauchy–Lipschitz theorem, after Augustin-Louis Cauchy and Rudolf Lipschitz.

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Differential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionInitial value problem, definition, no evidence levelInitial value problemdefinitionPicard–Lindelöf hypotheses, definition, no evidence levelPicard–Lindelöf hypot…definitionPicard operator, definition, no evidence levelPicard operatordefinitionIntegral equation form of an initial value problem, statement, no evidence levelIntegral equation for…statementAn iterate of the Picard operator is a contraction, statement, no evidence levelAn iterate of the Pic…statementPicard–Lindelöf theorem, statement, no evidence levelPicard–Lindelöf theor…statement
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  1. Differential equation concept
  2. Function concept
  3. Interval concept
  4. Ordinary differential equation definition
  5. Lipschitz condition definition
  6. Solution of an ordinary differential equation definition
  7. Initial value problem definition
  8. Picard–Lindelöf hypotheses definition
  9. Picard operator definition
  10. Integral equation form of an initial value problem statement
  11. An iterate of the Picard operator is a contraction statement
  12. Picard–Lindelöf theorem statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem, Theorem
About this page
Tag0P6KRD5B7A
URIm2://0P6KRD5B7A
Aliasescauchy-lipschitz-theorem, existence-and-uniqueness-theorem
Kindstatement, theorem
WritingW1 stated
MSC34A12
StatusThis copy of the site has no build data. Run m2 status 0P6KRD5B7A.
UsesPicard–Lindelöf hypotheses, Initial value problem, An iterate of the Picard operator is a contraction, Integral equation form of an initial value problem
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