Existence of asymptotically almost periodic solutions for some second-order hyperbolic integrodifferential equations
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Existence of asymptotically almost periodic solutions for some second‑order hyperbolic integrodifferential equations Toka Diagana1 · Jamilu H. Hassan2 · Salim A. Messaoudi3 Received: 20 March 2020 / Accepted: 8 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper we study and obtain the existence of asymptotically almost periodic solutions to some classes of second-order hyperbolic integrodifferential equations of Gurtin–Pipkin type in a separable Hilbert space H. To illustrate our abstract results, the existence of asymptotically almost periodic mild solutions to the well-known Kirchoff plate equation is studied.
1 Introduction Integrodifferential equations play an important role when it comes to describing various practical problems, see, e.g., [4–6, 13, 14, 17, 21–23]. In particular, these types of equations are utilized to study practical problems in which some memory effect is taken into account, such as the heat conduction in materials with memory or the sound propagation in viscoelastic media or in homogenization problems in perforated media (Darcy’s Law), see, e.g., [1, 5, 15, 16, 18].
Communicated by Jerome A. Goldstein. * Toka Diagana [email protected] Jamilu H. Hassan [email protected] Salim A. Messaoudi [email protected] 1
Department of Mathematical Sciences, University of Alabama in Huntsville, 301 Sparkman Drive, Huntsville, AL 35899, USA
2
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O.Box 5046, Dhahran 31261, Saudi Arabia
3
Department of Mathematics, University of Sharjah, P.O.Box 27272, Sharjah, United Arab Emirates
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Let (H, ‖ ⋅ ‖H , ⟨⋅, ⋅⟩H ) be a separable Hilbert space. The main purpose of this paper consists of establishing the existence of asymptotically almost periodic mild solutions to a class of second-order hyperbolic integrodifferential equations of Gurtin–Pipkin type given by t
d2 u + A2 u − g(t − s)A2 u(s)ds = f (t, u), t > 0 ∫−∞ dt2 with initial conditions
u(−t) = u0 (t), t ≥ 0 and u� (0) = u1 ,
(1)
(2)
where A ∶ D(A) ⊂ H ↦ H is a positive self-adjoint operator which is bounded below, that is, there exists a constant 𝜔 > 0 such that
‖Au‖H ≥ 𝜔‖u‖H for all u ∈ D(A),
(3)
the function f ∶ [0, ∞) × H ↦ H is asymptotically almost periodic in the first variable uniformly in the second one, and the non-increasing differentiable relaxation (kernel) function g ∶ [0, ∞) ⟶ [0, ∞) satisfies the following assumptions, ∞
g(s)ds > 0 ; and ∫0 (A.2) there exists a positive constant 𝜉 such that g� (t) ≤ −𝜉g(t) for all t ≥ 0. (A.1) g(0) > 0 and 𝛽 ∶= 1 −
Our main task in this paper consists of showing that problem (1)–(2), under some suitable assumptions, has an asymptotically almost periodic mild solution. To achieve that, our strategy consists of transforming such a system into a first-order evolution Eq. (7) below. Under assumptions (A.1) and (A.2), it will be shown that the linear operator A appearing in Eq. (7) is the infinit
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