35 research outputs found
Peano's Existence Theorem revisited
We present new proofs to four versions of Peano's Existence Theorem for
ordinary differential equations and systems. We hope to have gained readability
with respect to other usual proofs. We also intend to highlight some ideas due
to Peano which are still being used today but in specialized contexts: it
appears that the lower and upper solutions method has one of its oldest roots
in Peano's paper of 1886
Existence of minimal and maximal solutions to first--order differential equations with state--dependent deviated arguments
We prove some new results on existence of solutions to first--order ordinary
differential equations with deviating arguments. Delay differential equations
are included in our general framework, which even allows deviations to depend
on the unknown solutions. Our existence results lean on new definitions of
lower and upper solutions introduced in this paper, and we show with an example
that similar results with the classical definitions are false. We also
introduce an example showing that the problems considered need not have the
least (or the greatest) solution between given lower and upper solutions, but
we can prove that they do have minimal and maximal solutions in the usual
set--theoretic sense. Sufficient conditions for the existence of lower and
upper solutions, with some examples of application, are provided too
Mean value integral inequalities
Let have zero derivative in a dense subset of
. What else we need to conclude that is constant in ? We
prove a result in this direction using some new Mean Value Theorems for
integrals which are the real core of this paper. These Mean Value Theorems are
proven easily and concisely using Lebesgue integration, but we also provide
alternative and elementary proofs to some of them which keep inside the scope
of the Riemann integral
A roller coaster approach to integration and Peano’s existence theorem
This is a didactic proposal on how to introduce the Newton integral in just three or four sessions in elementary courses. Our motivation for this paper were Talvila’s work on the continuous primitive integral and Koliha’s general approach to the Newton integral. We introduce it independently of any other integration theory, so some basic results require somewhat nonstandard proofs. As an instance, showing that continuous functions on compact intervals are Newton integrable (or, equivalently, that they have primitives) cannot lean on indefinite Riemann integrals. Remarkably, there is a very old proof (without integrals) of a more general result, and it is precisely that of Peano’s existence theorem for continuous nonlinear ODEs, published in 1886. Some elements in Peano’s original proof lack rigor, and that is why his proof has been criticized and revised several times. However, modern proofs are based on integration and do not use Peano’s original ideas. In this note we provide an updated correct version of Peano’s original proof, which obviously contains the proof that continuous functions have primitives, and it is also worthy of remark because it does not use the Ascoli-Arzelà theorem, uniform continuity, or any integration theoryS
Fourier series for nonperiodic functions
We introduce a small change in the definition of the Fourier series so that
we can guarantee the coincidence with the given function at the endpoints of
the interval even if the function does not assume the same value at the
endpoints. This definition of the Fourier series also wipes out the Gibbs
phenomenom at the endpoints of the interval and proves useful in the resolution
of antiperiodic boundary value problems with linear partial differential
equations