$ (\omega,\mathbb{T}) $-periodic solutions of impulsive evolution equations

Abstract

<p style='text-indent:20px;'>In this paper, we study <inline-formula><tex-math id="M2">\begin{document}$ (\omega,\mathbb{T}) $\end{document}</tex-math></inline-formula>-periodic impulsive evolution equations via the operator semigroups theory in Banach spaces <inline-formula><tex-math id="M3">\begin{document}$ X $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{T}: X\rightarrow X $\end{document}</tex-math></inline-formula> is a linear isomorphism. Existence and uniqueness of <inline-formula><tex-math id="M5">\begin{document}$ (\omega,\mathbb{T}) $\end{document}</tex-math></inline-formula>-periodic solutions results for linear and semilinear problems are obtained by Fredholm alternative theorem and fixed point theorems, which extend the related results for periodic impulsive differential equations.</p>

Doss ρ-Almost Periodic Type Functions in Rn

“…By Proposition 4(iv), the set BCD (α k );c (R : X) equipped with the metric d(•, •) := • − • ∞ is a complete metric space. Suppose now that a mapping F : Λ × X → Y satisfies the estimate (15). We say that a continuous function u : R → X is a mild solution of the semilinear Cauchy inclusion…”

“…A further generalization of the concept c-almost periodicity presents the concept ρ-almost periodicity, which has recently been introduced and analyzed in [14]; here, ρ denotes a general binary relation acting on a corresponding pivot space (see also M. Fečkan et. al [15] for the first steps made in the investigation of general classes of ρ-almost periodic type functions; the main assumption used in [15] is that ρ = T is a linear isomorphism). The Stepanov and Weyl classes of multi-dimensional ρ-almost periodic functions have recently been studied in [16]; it is also worth noting that multi-dimensional (S, D, B)-asymptotically (ω, ρ)-periodic type functions, multi-dimensional quasi-asymptotically ρ-almost periodic type functions and multi-dimensional ρ-slowly oscillating type functions have recently been analyzed in [17].…”

$(ω,ρ)$-Periodic solutions of abstract integro-differential impulsive equations on Banach space

“…In [1], the authors studied the existence and uniqueness of (ω, c)-periodic solutions for semilinear evolution equations u ′ = Au + f (t, u) in complex Banach spaces. The notion of (ω, c)-periodicity was generalized in [12], by M. Fečkan, K. Liu and J. Wang, who considered (ω, T)periodic solutions for this class of semilinear evolution equations, where T is linear isomorphism on a Banach space X. For some other generalizations of this concept see [13].…”

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

“…On the other hand, the notion of (ω, c)-periodicity and various generalizations of this concept have recently been introduced and analyzed by E. Alvarez, A. Gómez, M. Pinto [1], E. Alvarez, S. Castillo, M. Pinto [2]- [3] and M. Fečkan, K. Liu, J.-R. Wang [22]. In our joint research article [29] with M. T. Khalladi, A. Rahmani, M. Pinto and D. Velinov, we have investigated c-almost periodic type functions and their applications (the notion of c-almost periodicity, depending only on the parameter c, is substantially different from the notion of (ω, c)-periodicity and the recently analyzed notion of (ω, c)-almost periodicity; see the forthcoming research monograph [33] for more details about the subject).…”