## 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
Full Text:

### References:

  Aizicovici, S.; McKibben, M., Existence results for a class of abstract nonlocal Cauchy problems, Nonlinear anal., 39, 649-668, (2000) · Zbl 0954.34055  Aizicovici, S.; Lee, H., Nonlinear nonlocal Cauchy problems in Banach spaces, Appl. math. lett., 18, 401-407, (2005) · Zbl 1084.34002  Bochner, S., A new approach to almost-periodicity, Proc. natl. acad. sci. USA, 48, 2039-2043, (1962) · Zbl 0112.31401  Chen, G., Control and stabilization for the wave equation in a bounded domain, SIAM J. control, 17, 66-81, (1979) · Zbl 0402.93016  Desch, W.; Grimmer, R.; Schappacher, W., Some considerations for linear integro-differential equations, J. math. anal. appl., 104, 219-234, (1984) · Zbl 0595.45027  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  H.S. Ding, T.J. Xiao, J. Liang, Positive almost automorphic solutions of some delay integral equations, submitted for publication  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  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  Grimmer, R., Resolvent operators for integral equations in a Banach space, Trans. amer. math. soc., 273, 333-349, (1982) · Zbl 0493.45015  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)  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  Liang, J.; van Casteren, J.; Xiao, T.J., Nonlocal Cauchy problems for semilinear evolution equations, Nonlinear anal., 50, 173-189, (2002) · Zbl 1009.34052  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  Liang, J.; Xiao, T.J., Semilinear integrodifferential equations with nonlocal initial conditions, Comput. math. appl., 47, 863-875, (2004) · Zbl 1068.45014  Lin, Y.; Liu, J.H., Semilinear integrodifferential equations with nonlocal Cauchy problem, Nonlinear anal., 26, 1023-1033, (1996) · Zbl 0916.45014  Liu, J.H.; Ezzinbi, K., Non-autonomous integrodifferential equations with nonlocal conditions, J. integral equations appl., 15, 79-93, (2003) · Zbl 1044.45002  Miller, R.K., An integrodifferential equation for rigid heat conuctors with memory, J. math. anal. appl., 66, 313-332, (1978) · Zbl 0391.45012  N’Guérékata, G.M., Almost automorphic functions and almost periodic functions in abstract spaces, (2001), Kluwer Academic/Plenum Publishers New York · Zbl 1001.43001  N’Guérékata, G.M., Topics in almost automorphy, (2005), Springer-Verlag New York · Zbl 1073.43004  Xiao, T.J.; Liang, J., Existence of classical solutions to nonautonomous nonlocal parabolic problems, Nonlinear anal., 63, e225-e232, (2005) · Zbl 1159.35383
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.