This paper deals with the system of functional-differential equations $$\frac{dx\left(t\right)}{dt}=p\left(x\right)\left(t\right)+q\left(t\right),$$ where $p:C_{\omega }\left(R^n\right)\to L_{\omega }\left(R^n\right)$ is a linear bounded operator, $q\in L_{\omega }\left(R^n\right)$, $\omega >0$ and $C_{\omega }\left(R^n\right)$ and $L_{\omega }\left(R^n\right)$ are spaces of $n$-dimensional $\omega$-periodic vector functions with continuous and integrable on $[0,\omega ]$ components, respectively. Conditions which guarantee the existence of a unique $\omega$-periodic solution and continuous dependence of that solution on the right hand side of the system considered are established.

We investigate the existence of solutions on a compact interval to second order boundary value problems for a class of functional differential inclusions in Banach spaces. We rely on a fixed point theorem for condensing maps due to Martelli.

Consider boundary value problems for a functional differential equation $$\left\{\begin{array}{cc}{x}^{\left(n\right)}\left(t\right)=\left(T^{+}x\right)\left(t\right)-\left(T^{-}x\right)\left(t\right)+f\left(t\right),\hfill & t\in [a,b],\hfill \\ lx=c,\hfill \end{array}\right.$$ where $T^{+},T^{-}:𝐂[a,b]\to 𝐋[a,b]$ are positive linear operators; $l:{\mathrm{𝐀𝐂}}^{n-1}[a,b]\to {ℝ}^n$ is a linear bounded vector-functional, $f\in 𝐋[a,b]$, $c\in {ℝ}^n$, $n\ge 2$. Let the solvability set be the set of all points $({𝒯}^{+},{𝒯}^{-})\in {ℝ}_2^{+}$ such that for all operators $T^{+}$, $T^{-}$ with $\parallel T^{\pm }{\parallel }_{𝐂\to 𝐋}={𝒯}^{\pm }$ the problems have a unique solution for every $f$ and $c$. A method of finding the solvability sets are proposed. Some new properties of these sets are obtained in various cases. We continue the investigations of the solvability sets started in R. Hakl,...

Consider the homogeneous equation $$u^{\text{'}}\left(t\right)=\ell \left(u\right)\left(t\right)\text{for}\text{a.e.}t\in [a,b]$$ where $\ell :C\left(\right[a,b];ℝ)\to L\left(\right[a,b];ℝ)$ is a linear bounded operator. The efficient conditions guaranteeing that the solution set to the equation considered is one-dimensional, generated by a positive monotone function, are established. The results obtained are applied to get new efficient conditions sufficient for the solvability of a class of boundary value problems for first order linear functional differential equations.

We study the existence of one-signed periodic solutions of the equations $$\begin{array}{ccc}& x^{\text{'}\text{'}}\left(t\right)-a^2\left(t\right)x\left(t\right)+\mu f(t,x\left(t\right),x^{\text{'}}\left(t\right))=0,\hfill & \hfill x^{\text{'}\text{'}}\left(t\right)+a^2\left(t\right)x\left(t\right)=\mu f(t,x\left(t\right),x^{\text{'}}\left(t\right)),\end{array}$$ where $\mu >0$, $a:(-\infty ,+\infty )\to (0,\infty )$ is continuous and 1-periodic, $f$ is a continuous and 1-periodic in the first variable and may take values of different signs. The Krasnosielski fixed point theorem on cone is used.