Carathéodory existence theorem
E547409
The Carathéodory existence theorem is a result in the theory of ordinary differential equations that guarantees the existence (and sometimes uniqueness) of solutions under weaker regularity conditions on the right-hand side than those required by classical theorems like Picard–Lindelöf.
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf | mathematical theorem ⓘ |
| allows | discontinuous right-hand side in time variable ⓘ |
| appearsIn |
advanced textbooks on ordinary differential equations
ⓘ
monographs on differential equations with discontinuous right-hand sides ⓘ |
| appliesTo | initial value problems for ordinary differential equations ⓘ |
| assumes |
Carathéodory conditions on the right-hand side
ⓘ
continuity of f in the state variable for almost every time ⓘ local integrable bound on f ⓘ measurability of f in the time variable ⓘ |
| comparedTo | Picard–Lindelöf theorem NERFINISHED ⓘ |
| concerns |
Carathéodory-type right-hand side f(t,x)
ⓘ
differential equation x'(t)=f(t,x(t)) ⓘ |
| concludes |
existence of an absolutely continuous solution
ⓘ
solution satisfies the differential equation almost everywhere ⓘ |
| context |
measurable dependence on time
ⓘ
non-Lipschitz right-hand sides ⓘ |
| ensures | local solvability under Carathéodory conditions ⓘ |
| field |
mathematical analysis
ⓘ
ordinary differential equations ⓘ |
| generalizes |
Picard–Lindelöf existence theorem
NERFINISHED
ⓘ
classical existence theorems for ODEs ⓘ |
| guarantees | existence of solutions to ordinary differential equations ⓘ |
| hasVersion |
global existence version under growth conditions
ⓘ
local existence version ⓘ |
| implies | existence of Carathéodory solutions ⓘ |
| mayGuarantee |
local existence of solutions
ⓘ
maximal interval of existence for solutions ⓘ |
| namedAfter | Constantin Carathéodory NERFINISHED ⓘ |
| relatedTo |
Filippov theory of differential equations with discontinuous right-hand sides
NERFINISHED
ⓘ
Peano existence theorem NERFINISHED ⓘ |
| reliesOn |
Lebesgue integration
NERFINISHED
ⓘ
absolute continuity of functions ⓘ |
| requires |
for each compact set in state space, an integrable majorant of f
ⓘ
local boundedness of f by an integrable function ⓘ |
| sometimesGuarantees | uniqueness of solutions under additional conditions ⓘ |
| statedFor |
systems of ordinary differential equations
ⓘ
vector-valued unknown functions ⓘ |
| strengthens | pure measurability assumptions by continuity in state variable ⓘ |
| topicIn | theory of initial value problems ⓘ |
| typeOfSolution | Carathéodory solution NERFINISHED ⓘ |
| usedIn |
control theory
ⓘ
differential inclusions and related generalizations ⓘ measure differential equations ⓘ nonlinear dynamical systems ⓘ |
| usesWeakerConditionsThan | Picard–Lindelöf theorem NERFINISHED ⓘ |
Referenced by (1)
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