Existence of a Periodic Solution for Some Partial Functional Differential Equations with Infinite Delay

Abstract

This paper deals with the existence of periodic solutions for some partial functional differential equations with infinite delay. We suppose that the linear part is nondensely defined and satisfies the Hille᎐Yosida condition. In the nonlinear case we give several criteria to ensure the existence of a periodic solution. In the nonhomogeneous linear case, we prove the existence of a periodic solution under the existence of a bounded solution. ᮊ 2001 Academic Press t 1 Ž . dt ¢ x s g B B, 0

Variation of constants formula and almost periodic solutions for some partial functional differential equations with infinite delay

“…In order to illustrate the previous results, we consider the following Lotka-Volterra model with diffusion, which has been studied in periodic case in [8] ⎧ This lemma implies that (H 0 ) is satisfied. Moreover, D(A) = ψ ∈ C [0, π]; R : ψ(0) = ψ(π) = 0 .…”

“…In [8], the authors have chosen the following space B = C γ , γ > 0, where C γ = φ ∈ C (−∞, 0]; X : lim θ→−∞ e γ θ φ(θ) exists in X with the following norm φ γ = sup −∞<θ 0 e γ θ φ(θ) , for φ ∈ C γ . Lemma 6.2.…”

“…The problem of periodic and almost periodic solutions for differential equations has been studied extensively in the literature, for the reader, we refer to [4,8,11,[15][16][17][18]20], and the references therein. We recall a classical theorem for the existence of almost periodic solutions for differential equations in finite-dimensional space.…”

Ultimate boundedness and periodicity for some partial functional differential equations with infinite delay

“…There is an extensive literature related to this topics, for instance, we refer to [6][7][8][9][11][12][13]16,17,[20][21][22]. Fixed point theorems is a powerful tool to investigate this problem.…”

Global Attractor for Some Partial Functional Differential Equations with Infinite Delay

“…With regard to the infinite delay case, the situation is di¤erent, since properties of solutions depend on the choice of the phase space. The fundamental theory related to functional di¤erential equations with infinite delay can be found in [25] and we refer the reader to works in [1,2,3,6,10,11,12,13,14,17,24] and the references therein, in which the results are about some quantitative and qualitative aspects of study.…”