Abstract
This work is devoted to the study of the existence and periodicity of solutions of initial differential problems, paying special attention to the explicit computation of the period. These problems are also connected with some particular initial and boundary value problems with reflection, which allows us to prove existence of solutions of the latter using the existence of the former.
Highlights
The idea behind this paper appeared in another work of the authors [ ] where the following lemmas were proved.Definition
We study the range of H+. g(x (t)) is positive as long as x (t) is positive
Observe that x (t ) = and x(t ) = F–– (G(g(c )) + F(c )). This last equality comes from evaluating equation ( ) at t and Rolle’s theorem as we show : the other possibility would be x(t ) = F+– (G(g(c )) + F(c ))
Summary
The idea behind this paper appeared in another work of the authors [ ] where the following lemmas were proved. If we consider the problem with the f – -Laplacian f – ◦ xc (t) + f xc(t) = , xc(a) = c, xc(a) = f (c), and we assume there exist c , c ∈ R, c < c , such that a unique solution of problem ( ) exists for every c ∈ [c , c ] and (xc (b) – c )(xc (b) – c ) < , problem ( ) must have at least a solution due to the continuity of xc on c and Bolzano’s theorem For this reason we will be interested in studying the properties of problem ( ) and its solutions in this paper. We will study the existence, uniqueness and periodicity of solutions of problem ( ) and in Section we will apply these results to the case of problems with reflection
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