## 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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