Rodrigo López Pouso
Published 2013-04-12Version 1
We present a new elementary proof of the existence of the least and the greatest solutions to initial value problems in the conditions of Peano's existence theorem. Our proof is based on a modification of Perron's method which allows us to obtain quite easily the greatest solution as the solution with biggest possible integral. In doing so, we simplify the usual proofs, technically overloaded with lower (upper) solutions and/or related differential inequalities. Moreover, those differential (and integral) inequalities, which are interesting in their own right, can be quickly proven by means of known techniques once we know that the greatest and the least solutions exist. Summing up, we revert the usual approach from differential inequalities to extreme solutions, getting a somewhat smoother presentation.
On sets of directions determined by subsets of ${\Bbb R}^d$
Peano's Existence Theorem revisited
The existence and unicity of numerical solution of initial value problems by Walsh polynomials approach
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