## Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions.(English)Zbl 1142.45005

Consider a Banach space $$(X,\|\cdot\|)$$. – A continuous function $$f:\mathbb{R}\to X$$ is called almost automorphic if for each real sequence $$(s_n)_n$$ there exists a subsequence $$(s_{i_n})_n$$ of $$(s_n)_n$$ such that
$(\forall t\in \mathbb{R})\Bigl(\exists g(t)=\lim_n f(t+ s_{i_n})\wedge \lim_n\,g(t- s_{i_n})= f(t)\Bigr).$
Denote by $$AA(\mathbb{R},X)$$ the set of almost automorphic functions defined in $$\mathbb{R}$$ and valued in $$X$$.
Denote by $$C_0(\mathbb{R}^+,X)$$ the space of continuous functions $$h: \mathbb{R}^+\to X$$ such that $$\lim_{t\to+\infty}\, h(t)= 0$$.
A continuous function $$f: \mathbb{R}^+\to X$$ is called asymptotically almost automorphic if
$(\exists g\in AA(\mathbb{R},X))(\exists h\in C_0(\mathbb{R}^+, X))(\forall t\in \mathbb{R}^+)(f(t)= g(t)+ h(t)).$
Denote by $$AAA(\mathbb{R}^+,X)$$ the set of asymptotically almost automorphic functions defined on $$\mathbb{R}^+$$ and valued in $$X$$, this set being a Banach space with the norm $$|\cdot|$$ defined by
$(\forall g\in AA(\mathbb{R}, X))(\forall h\in C_0(\mathbb{R}^+, X))(|f|= \sup_{t\in\mathbb{R}}\,\| g(t)\|+ \sup_{t\in \mathbb{R}^+}\,\| h(t)\|).$
The authors study the existence of asymptotically almost automorphic solutions of the integro-differential equation
$u'(t)= Au(t)+ \int^t_0 B(t- s)u(s)\,ds+ f(t, u(t)),\quad t\geq 0,\tag{1}$ with nonlocal initial condition
$u(0)= u_0+ g(u)\tag{2}$
in a Banach space $$(X,\|\cdot\|)$$, where $$u_0\in X$$, $$A: X\to X$$ and $$\forall t\geq 0$$ $$B(t): X\to X$$ are densely defined closed linear operators in $$X$$ and
(3) there exists a resolvent exponentially stable operator $$R$$ of (1) so that
$(\exists M> 0)(\exists\omega> 0)(\forall t\geq 0)(\| R(t)\|\leq M e^{-\omega t}),$
$f\in AAA(\mathbb{R}^+\times X,X)\quad\text{and}\tag{4}$
$$(\exists L_f: \mathbb{R}^+\to \mathbb{R}^+)(\forall r\geq 0)(\forall(u, v)\in X\times X)(\| u\|\leq r)(\| v\|\leq r)(\forall t\in \mathbb{R}^+)(\| f(t, u)- f(t,v)\|\leq L_f(r)\| u-v\|)$$,
$g: C(\mathbb{R}^+, X)\to X\quad\text{and}\tag{5}$
$$(\exists L_g: \mathbb{R}^+\to \mathbb{R}^+)(\forall r\geq 0)(\forall(u,v)\in X\times X)(\| u\|\leq r)(\| v\|\leq r)(\| g(u)- g(v)\|\leq L_g(r)\| u-v\|)$$,
$\sup_{r> 0}\,(M^{-1}\omega r-\omega rL_g(r)- rL_f(r))> \omega(\| u_0\|+\| g(0)\|)+ \sup_{s\in\mathbb{R}}\,\| f(s,0)\|.\tag{6}$
Under the hypothesis (3)–(6) the authors prove that there exists an asymptotically almost automorphic mild solution to equations (1),(2).
Reviewer: D. M. Bors (Iaşi)

### MSC:

 45N05 Abstract integral equations, integral equations in abstract spaces 45J05 Integro-ordinary differential equations 45G10 Other nonlinear integral equations 45M05 Asymptotics of solutions to integral equations
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### References:

 [1] Aizicovici, S.; McKibben, M., Existence results for a class of abstract nonlocal Cauchy problems, Nonlinear Anal., 39, 649-668 (2000) · Zbl 0954.34055 [2] Aizicovici, S.; Lee, H., Nonlinear nonlocal Cauchy problems in Banach spaces, Appl. Math. Lett., 18, 401-407 (2005) · Zbl 1084.34002 [3] Bochner, S., A new approach to almost-periodicity, Proc. Natl. Acad. Sci. USA, 48, 2039-2043 (1962) · Zbl 0112.31401 [4] Chen, G., Control and stabilization for the wave equation in a bounded domain, SIAM J. Control, 17, 66-81 (1979) · Zbl 0402.93016 [5] Desch, W.; Grimmer, R.; Schappacher, W., Some considerations for linear integro-differential equations, J. Math. Anal. Appl., 104, 219-234 (1984) · Zbl 0595.45027 [6] Dianaga, T.; N’Guérékata, G. M., Almost automorphic solutions to some classes of partial evolution equations, Appl. Math. Lett., 20, 462-466 (2007) · Zbl 1169.35300 [7] H.S. Ding, T.J. Xiao, J. Liang, Positive almost automorphic solutions of some delay integral equations, submitted for publication; H.S. Ding, T.J. Xiao, J. Liang, Positive almost automorphic solutions of some delay integral equations, submitted for publication [8] Ezzinbi, K.; N’Guérékata, G. M., Massera type theorem for almost automorphic solutions of functional differential equations of neutral type, J. Math. Anal. Appl., 316, 707-721 (2006) · Zbl 1122.34052 [9] Goldstein, J. A.; N’Guérékata, G. M., Almost automorphic solutions of semilinear evolution equations, Proc. Amer. Math. Soc., 133, 2401-2408 (2005) · Zbl 1073.34073 [10] Grimmer, R., Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273, 333-349 (1982) · Zbl 0493.45015 [11] Hernandez, M. E.; dos Santos, J. P.C., Asymptotically almost periodic and almost periodic solutions for a class of partial integrodifferential equations, Electron. J. Differential Equations, 38, 1-8 (2006) · Zbl 1085.45001 [12] Hetzer, G.; Shen, W., Uniform persistence, coexistence, and extinction in almost periodic/nonautonomous competition diffusion systems, SIAM J. Math. Anal., 34, 204-227 (2002) · Zbl 1020.35006 [13] Liang, J.; van Casteren, J.; Xiao, T. J., Nonlocal Cauchy problems for semilinear evolution equations, Nonlinear Anal., 50, 173-189 (2002) · Zbl 1009.34052 [14] Liang, J.; Liu, J. H.; Xiao, T. J., Nonlocal Cauchy problems for nonautonomous evolution equations, Comm. Pure Appl. Anal., 5, 529-535 (2006) · Zbl 1143.34320 [15] Liang, J.; Xiao, T. J., Semilinear integrodifferential equations with nonlocal initial conditions, Comput. Math. Appl., 47, 863-875 (2004) · Zbl 1068.45014 [16] Lin, Y.; Liu, J. H., Semilinear integrodifferential equations with nonlocal Cauchy problem, Nonlinear Anal., 26, 1023-1033 (1996) · Zbl 0916.45014 [17] Liu, J. H.; Ezzinbi, K., Non-autonomous integrodifferential equations with nonlocal conditions, J. Integral Equations Appl., 15, 79-93 (2003) · Zbl 1044.45002 [18] Miller, R. K., An integrodifferential equation for rigid heat conuctors with memory, J. Math. Anal. Appl., 66, 313-332 (1978) · Zbl 0391.45012 [19] N’Guérékata, G. M., Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces (2001), Kluwer Academic/Plenum Publishers: Kluwer Academic/Plenum Publishers New York · Zbl 1001.43001 [20] N’Guérékata, G. M., Topics in Almost Automorphy (2005), Springer-Verlag: Springer-Verlag New York · Zbl 1073.43004 [21] Xiao, T. J.; Liang, J., Existence of classical solutions to nonautonomous nonlocal parabolic problems, Nonlinear Anal., 63, e225-e232 (2005) · Zbl 1159.35383
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