Peano existence theorem

Peano existence theorem Differential equations Navier–Stokes differential equations used to simulate airflow around an obstruction Scope show Fields Classification show Types show Relation to processes Solution show Existence and uniqueness show General topics show Solution methods People show List vte In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems.

Contents 1 History 2 Theorem 3 Proof 4 Related theorems 5 Notes 6 References History Peano first published the theorem in 1886 with an incorrect proof.[1] In 1890 he published a new correct proof using successive approximations.[2] Theorem Let {displaystyle D} be an open subset of {displaystyle mathbb {R} times mathbb {R} } with {displaystyle fcolon Dto mathbb {R} } a continuous function and {displaystyle y'(x)=fleft(x,y(x)right)} a continuous, explicit first-order differential equation defined on D, then every initial value problem {displaystyle yleft(x_{0}right)=y_{0}} for f with {displaystyle (x_{0},y_{0})in D} has a local solution {displaystyle zcolon Ito mathbb {R} } where {displaystyle I} is a neighbourhood of {displaystyle x_{0}} in {displaystyle mathbb {R} } , such that {displaystyle z'(x)=fleft(x,z(x)right)} for all {displaystyle xin I} .[3] The solution need not be unique: one and the same initial value {displaystyle (x_{0},y_{0})} may give rise to many different solutions {displaystyle z} .

Proof By replacing {displaystyle y} with {displaystyle y-y_{0}} , {displaystyle x} with {displaystyle x-x_{0}} , we may assume {displaystyle x_{0}=y_{0}=0} . As {displaystyle D} is open there is a rectangle {displaystyle R=[-x_{1},x_{1}]times [-y_{1},y_{1}]subset D} .

Because {displaystyle R} is compact and {displaystyle f} is continuous, we have {displaystyle textstyle sup _{R}|f|leq C

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