# Peano existence theorem

Peano existence theorem

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.

The Peano theorem can be compared with another existence result in the same context, the Picard–Lindelöf theorem. The Picard–Lindelöf theorem both assumes more and concludes more. It requires Lipschitz continuity, while the Peano theorem requires only continuity; but it proves both existence and uniqueness where the Peano theorem proves only the existence of solutions. To illustrate, consider the ODE

:$y\text{'} = leftvert y ightvert^\left\{frac\left\{1\right\}\left\{2$ on the domain $left \left[0, 1 ight\right]$.

According to the Peano theorem, this equation has solutions, but the Picard-Lindelöf theorem does not apply since the right hand side is not Lipschitz continuous in any neighbourhood containing 0. Thus we can conclude existence but not uniqueness. It turns out that this ODE can have multiple possible solutions for some initial values - for example, for $y\left(0\right)=0$, we can have $y\left(x\right)=0$ or $y\left(x\right)=x^2/4$.

History

Peano first published the theorem in 1886 with an incorrect proof. In 1890 he published a new correct proof using successive approximations.

Theorem

Let "D" be an open and simply connected subset of R &times; R with :$fcolon D o mathbb\left\{R\right\}$ a continuous function and:$fleft\left(x,y\left(x\right) ight\right) = y\text{'}\left(x\right)$ a continuous, explicit first-order differential equation defined on "D", then an initial value problem:$yleft\left(x_0 ight\right) = y_0$for "f" with $\left(x_0, y_0\right) in D$has a local solution:$zcolon I o mathbb\left\{R\right\}$where $I$ is a neighbourhood of "x"0,such that $fleft\left(x,z\left(x\right) ight\right)=z\text{'}\left(x\right)$ for all $x in I$.

References

* G. Peano, "Sull’integrabilità delle equazioni differenziali del primo ordine", Atti Accad. Sci. Torino, 21 (1886) 677–685.
* G. Peano, "Demonstration de l’intégrabilité des équations différentielles ordinaires", Mathematische Annalen, 37 (1890) 182–228.
* W. F. Osgood, "Beweis der Existenz einer Lösung der Differentialgleichung dy/dx = f(x, y) ohne Hinzunahme der Cauchy-Lipschitzchen Bedingung", Monatsheft Mathematik,9 (1898) 331–345.
* E.A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations", McGraw-Hill, 1955.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Giuseppe Peano — Infobox Scientist name = Giuseppe Peano image width = 220px birth date = birth date|1858|8|27 birth place = Spinetta, Piedmont, Italy death date = death date and age|1932|4|20|1858|8|27 residence = Italy citizenship = Italian field = Mathematics… …   Wikipedia

• Picard–Lindelöf theorem — In mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard s existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to certain initial value problems.The… …   Wikipedia

• Arzelà–Ascoli theorem — In mathematics, the Arzelà–Ascoli theorem of functional analysis gives necessary and sufficient conditions to decide whether every subsequence of a given sequence of real valued continuous functions defined on a closed and bounded interval has a… …   Wikipedia

• Peano axioms — In mathematical logic, the Peano axioms, also known as the Dedekind Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used… …   Wikipedia

• Theorem — The Pythagorean theorem has at least 370 known proofs[1] In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements …   Wikipedia

• Théorème de Cauchy-Peano-Arzelà — Pour les articles homonymes, voir Cauchy. Le théorème de Cauchy Peano Arzelà (en) est un théorème d analyse, la branche des mathématiques qui est constituée du calcul différentiel et intégral et des domaines associés …   Wikipédia en Français

• Prime number theorem — PNT redirects here. For other uses, see PNT (disambiguation). In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are… …   Wikipedia

• Löwenheim–Skolem theorem — In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ. The… …   Wikipedia

• Kruskal's tree theorem — In mathematics, Kruskal s tree theorem states that the set of finite trees over a well quasi ordered set of labels is itself well quasi ordered (under homeomorphic embedding). The theorem was proved byharvs|txt=yes|year= 1960 |authorlink=Joseph… …   Wikipedia

• Robertson–Seymour theorem — In graph theory, the Robertson–Seymour theorem (also called the graph minor theorem[1]) states that the undirected graphs, partially ordered by the graph minor relationship, form a well quasi ordering.[2] Equivalently, every family of graphs that …   Wikipedia