Math 2.0

Ordinary differential equations Theorem

Picard–Lindelöf theorem in integral form

Builds on Picard–Lindelöf theoremIntegral equation form of an initial value problemPicard operator

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 such that α(t)=x0+∫t0tf(s,α(s))dsfor each t∈J.\alpha(t) = x_0 + \int_{t_0}^{t} f\left(s, \alpha(s)\right) ds \quad \text{for each } t \in J.

Remarks

The curve α\alpha is the unique fixed point of the Picard operator on the curves with values in BB. By the integral equation form, this is the same curve as the solution of the Picard–Lindelöf theorem.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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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…statementPicard–Lindelöf theorem in integral form, statement, no evidence levelPicard–Lindelöf theor…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  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
  13. Picard–Lindelöf theorem in integral form statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem, Proof sketch
About this page
Tag00BT70ND50
URIm2://00BT70ND50
Kindstatement, theorem
WritingW0 stub
MSC34A12
StatusThis copy of the site has no build data. Run m2 status 00BT70ND50.
UsesPicard–Lindelöf theorem, Integral equation form of an initial value problem, Picard operator

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