Remotely Almost Periodic Solutions of Ordinary Differential Equations

Abstract

In this paper, we analyze the existence and uniqueness of remotely almost periodic solutions for systems of ordinary differential equations. The existence and uniqueness of remotely almost periodic solutions are achieved by using the results about the exponential dichotomy and the Bi-almost remotely almost periodicity of the homogeneous part of the corresponding systems of ordinary differential equations under our consideration.

Remotely $c$-almost periodic type functions in ${\mathbb R}^{n}$

“…Keeping in mind these observations, it follows that the problem of existence or non-existence of mean value of remotely almost periodic functions is still unsolved; we want also to emphasize that our structural results established in the first two sections of [33] and the part of the third section of [33] before subsection 3.1 of this paper, where it has directly been assumed that a remotely almost periodic function has a mean value, remain completely true by assuming additionally that any considered remotely almost periodic function has a mean value. Now we will prove the existence of a bounded, uniformly continuous slowly oscillating function f : [0, ∞) → c 0 which does not have mean value, which clearly marks that the calculations given in [46,Proposition 2.4] are not true: Example 3.8.…”

“…An application in mathematical biology. The application is closely related with our recent investigation of remotely almost periodic solutions of ordinary differential equations [33] (some results established in this research article, like [33,Theorem 3], can be reformulated for remotely c-almost periodic functions but we will skip all related details for simplicity).…”

“…Further on, we provide several arguments showing that the result of [46,Proposition 2.4] is not correct and provide an example of a vector-valued slowly oscillating function without mean value. In the final section of paper, we provide certain applications to the integro-differential equations; especially, in Subsection 4.1, we continue our recent analysis from [33], a joint work with C. Maulén, S. Castillo and M. Pinto, by analyzing slowly oscillating solutions for the Richard-Chapman ordinary differential equation with an external perturbation, which plays an important role in mathematical biology.…”

“…Keeping in mind Lemma 4.3, the fact that the space of real-valued slowly oscillating functions is closed under pointwise products and sums, as well as the fact that for each positive slowly oscillating function f : R → (0, ∞) and for each real number r > 0 the function f r : R → (0, ∞) is also slowly oscillating, we can repeat verbatim the argumentation contained in the proof of [33,Theorem 6] in order to see that the following result holds true: Theorem 4.4. Suppose that the hypotheses (H1)-(H3) hold.…”

“…We need the following auxiliary lemma, which generalizes [40, Lemma 3.5] (see also [33,Lemma 4]): Lemma 4.3. Suppose that α > 0, the functions a : R → [α, ∞) and f : R → R are slowly oscillating.…”