arXiv subjects
Publications and source records attributed to Philippe Cieutat.
We give an extension of Bochner's criterion for the almost periodic functions. By using our main result, we extend two results of A. Haraux. The first is a generalization of Bochner's criterion which is useful for periodic dynamical systems. The second is a characterization of periodic functions in term of Bochner's criterion.
We study the superposition operators (also called Nemytskii operators) between spaces of almost periodic (respectively almost automorphic) functions in the sense of Stepanov. We state new results on the superposition, notably we give a necessary and sufficient condition for that these operators are well-defined and continuous.
We disclose an interesting connection between the gradient flow of a $\mathcal{C}^2$-smooth function $ψ$ and evanescent orbits of the second order gradient system defined by the square-norm of $\nablaψ$, under adequate convexity assumption. As a consequence, we obtain the following surprising result for two $\mathcal{C}^2$, convex and bounded from below functions $ψ_1$, $ψ_2$: if $||\nablaψ_1||=||\nablaψ_2||$, then $ψ_1=ψ_2 + k$, for some $k\in \mathbb{R}$.
We give a new proof of a characterization of the closeness of the range of a continuous linear operator and of the closeness of the sum of two closed vector subspaces of a Banach space. Then we state sufficient conditions for the closeness of the sum of two closed subspaces of the Banach space of bounded functions and apply this result on various pseudo almost periodic spaces and pseudo almost automorphic spaces.