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Optimal mild solutions and weighted pseudo-almost periodic classical solutions of fractional integro-differential equations. (English) Zbl 1213.34089
The aim of this paper is to study the existence of classical solutions to the following fractional integro-differential equation $$\frac{d^\alpha x(t)}{dt^\alpha}= Ax(t)+f(t,x(t),G x(t)),\tag1$$ where $\frac{d^\alpha x(t)}{dt^\alpha}$ stands for the Riemann-Liouville derivative of order $\alpha$, $0<\alpha<1$, $A$ is the infinitesimal generator of an analytic semigroup $\{Q(t)\}_{t\geq 0}$ in a Banach space $\mathbb{X},$ $f:\mathbb{R}\times \mathbb{X}_q\times \mathbb{X}_q\to \mathbb{X}$ is a suitable function, and $\mathbb{X}_q$ a Banach space. $Gx(t)$, which may be interpreted as a control of the system, is defined by $Gx(t)=\int_{t_0}^t k(t,s,x(s))\,ds$ and $k:D\times\mathbb{X}_q\to \mathbb{X}_q$, $D=\{(t,s):t_0\leq s\leq t\leq T\}$. By means of the contraction mapping, the authors prove the existence and uniqueness of a classical solution of the initial value problem associated to (1). Then, they show the existence and uniqueness of an optimal mild solution among all the solutions of (1) which are bounded over $\mathbb{R}$. Note that the notion of optimal solution was introduced by {\it G. M. N’Guérékata} in [Riv. Mat. Univ. Parma, IV. Ser. 9, 145--151 (1983; Zbl 0547.34049)]. Finally, they study sufficient conditions for the existence and uniqueness of a weighted pseudo-almost periodic classic solution. An example is also given to illustrate the abstract results.

34K30Functional-differential equations in abstract spaces
34A08Fractional differential equations
34C27Almost and pseudo-almost periodic solutions of ODE
Full Text: DOI
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