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. Peano first published the theorem in with an incorrect proof. The solution need not be unique: one and the same initial value x 0 , y 0 may give rise to many different solutions z. 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 ordinary differential equation.
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Preliminaries; Basics from linear algebra and real analysis like concepts of dependence, independence, basis, Rank-Nullity theorem, determinants and eigenvalues, remarks on Jordan decomposition theorem - convergence, uniform convergence, fixed point theorems, Lipschitz continuity, etc.
First and second order linear equations; Examples, A systematic procedure to solve first order and development of the concept integrating factor, Second order homogeneous and non-homogeneous equations, Wronskian, methods of solving.
General Existence and Uniqueness theory; Picard's iteration, Peano's exisentce theory, Existence via Arzela Ascoli theorem, non-uniqueness, continuous dependence. Linear systems; Understanding linear system via linear algebra, stability of Linear systems, Explicit phase portrait in 2D linear with constant coefficients. Periodic Solutions; Stability, Floquet theory, particular case o second order equations-Hill's equation.
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Peano existence theorem
Ordinary Differential Equations/Peano's theorem
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