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Existence of \(S\)-asymptotically \(\omega \)-periodic solutions for abstract neutral equations. (English) Zbl 1183.34122

This work is focused on the study of the existence and uniqueness of \(S\)-asymptotically \(\omega\)-periodic mild solutions, that is, mild solutions \(u:[0,\infty) \longrightarrow X\) which are bounded continuous functions satisfying that \[ \lim_{t \rightarrow +\infty}\left[ u(t+\omega)-u(t) \right]=0, \] for the initial value problem related to the abstract neutral functional differential equations with infinite delay \[ \frac{d}{dt}\left[u(t)-f(t,u_t)\right]=Au(t)+g(t,u_t),\quad t \geq 0, \] and \[ \frac{d}{dt}D(t,u_t)=AD(t,u_t)+g(t,u_t),\quad t \geq 0, \] subject to the initial condition \[ u_0=\varphi \in \mathcal{B}, \] where \(u(t) \in X\), \(X\) is a Banach space, \(\omega>0\), the history \(u_t:(-\infty,0] \longrightarrow X\), given by \(u_t(\theta)=u(t+\theta)\), belongs to an abstract phase space \(\mathcal{B}\) which is defined axiomatically in the lines of Y. Hino, S. Murakami and T. Naito [Functional differential equations with infinite delay. Lecture Notes in Mathematics, 1473. Berlin etc.: Springer-Verlag (1991; Zbl 0732.34051)], \(D(t,\psi)=\psi(0)-f(t,\psi)\), \(f,\, g : \mathbb{R} \times \mathcal{B} \longrightarrow X\), and \(A:D(A) \subset X\longrightarrow X\) is the infinitesimal generator of a strongly continuous and uniformly exponentially stable semigroup of bounded linear operators \((T(t))_{t \geq 0}\) on \(X\).
Conditions to guarantee that an \(S\)-asymptotically \(\omega\)-periodic function is asymptotically \(\omega\)-periodic are also discussed, allowing to study the existence and uniqueness of asymptotically \(\omega\)-periodic mild solutions for these neutral systems.
The authors present some applications of the main results to a partial neutral functional differential equation with infinite delay for the heat conduction as well as a neutral system in control theory consisting of a linear distributed hereditary system with unbounded delay and a hereditary proportional-integral-differential feedback control.
Some other references which provide useful information for the development of this work are E. Hernández and H. R. Henríquez [J. Math. Anal. Appl. 221, No. 2, 452–475 (1998; Zbl 0915.35110)], for the concept of mild solution; H. R. Henríquez [Indian J. Pure Appl. Math. 27, No. 4, 357–386 (1996; Zbl 0853.34072)], for some fixed point results; and A. Pazy [Semigroups of linear operators and applications to partial differential equations. New York etc.: Springer-Verlag (1983; Zbl 0516.47023)], for the concepts and properties concerning strongly continuous semigroups.

MSC:

34K30 Functional-differential equations in abstract spaces
34K40 Neutral functional-differential equations
35R10 Partial functional-differential equations
34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
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