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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:

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