Let A be a closed linear operator in a Banach space E. In the study of the nth-order abstract Cauchy problem $u^{\left(n\right)}\left(t\right)=Au\left(t\right)$, t ∈ ℝ, one is led to considering the linear Volterra equation (AVE) $u\left(t\right)=p\left(t\right)+A{ʃ}_0^{t}a(t-s)u\left(s\right)ds$, t ∈ ℝ, where $a\left(·\right)\in L_{loc}^1\left(ℝ\right)$ and p(·) is a vector-valued polynomial of the form $p\left(t\right)={\sum }_{j=0}^{n}1/(j!)x_j{t}^j$ for some elements $x_j\in E$. We describe the spectral properties of the operator A through the existence of slowly growing solutions of the (AVE). The main tool is the notion of Carleman spectrum of a vector-valued function. Moreover, an extension of a theorem...

A fixed point theorem in ordered spaces and a recently proved monotone convergence theorem are applied to derive existence and comparison results for solutions of a functional integral equation of Volterra type and a functional impulsive Cauchy problem in an ordered Banach space. A novel feature is that equations contain locally Henstock-Kurzweil integrable functions.

The aim of this work is to study the existence and uniqueness of solutions of the fractional integro-differential equations $\frac{d}{dt}[x\left(t\right)-L\left(x_t\right)]=A[x\left(t\right)-L\left(x_t\right)]+G\left(x_t\right)+\frac{1}{\Gamma \left(\alpha \right)}{\int }_{-\infty }^t{(t-s)}^{\alpha -1}\left({\int }_{-\infty }^{s}a(s-\xi )x\left(\xi \right)d\xi \right)ds+f\left(t\right)$, ($\alpha >0$) with the periodic condition $x\left(0\right)=x\left(2\pi \right)$, where $a\in L^1\left({ℝ}_{+}\right)$ . Our approach is based on the R-boundedness of linear operators $L^p$-multipliers and UMD-spaces.

We consider a nonconvex integral inclusion and we prove a Filippov type existence theorem by using an appropiate norm on the space of selections of the multifunction and a contraction principle for set-valued maps.