- Peano existence theorem
In

mathematics , specifically in the study ofordinary differential equation s, the**Peano existence theorem**,**Peano theorem**or**Cauchy-Peano theorem**, named afterGiuseppe Peano andAugustin Louis Cauchy , is a fundamentaltheorem which guarantees theexistence of solutions to certaininitial value problem s.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:$y\text{'}\; =\; leftvert\; y\; ightvert^\{frac\{1\}\{2$ on the domain $left\; [0,\; 1\; ight]$.

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 $y(0)=0$, we can have $y(x)=0$ or $y(x)=x^2/4$.

**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**×**R**with :$fcolon\; D\; o\; mathbb\{R\}$ a continuous function and:$fleft(x,y(x)\; ight)\; =\; y\text{'}(x)$ a continuous, explicitfirst-order differential equation defined on "D", then an initial value problem:$yleft(x\_0\; ight)\; =\; y\_0$for "f" with $(x\_0,\; y\_0)\; in\; D$has a local solution:$zcolon\; I\; o\; mathbb\{R\}$where $I$ is a neighbourhood of "x"_{0},such that $fleft(x,z(x)\; ight)=z\text{'}(x)$ 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.

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