Nonimprovable, in a certain sense, sufficient conditions for the unique solvability of the boundary value problem $$u^{\text{'}}\left(t\right)=\ell \left(u\right)\left(t\right)+q\left(t\right),u\left(a\right)+\lambda u\left(b\right)=c$$ are established, where $\ell :C\left(\right[a,b];R)\to L\left(\right[a,b];R)$ is a linear bounded operator, $q\in L\left(\right[a,b];R)$, $\lambda \in R_{+}$, and $c\in R$. The question on the dimension of the solution space of the homogeneous problem $$u^{\text{'}}\left(t\right)=\ell \left(u\right)\left(t\right),u\left(a\right)+\lambda u\left(b\right)=0$$ is discussed as well.

The existence of positive periodic solutions for a kind of Rayleigh equation with a deviating argument $$x^{\text{'}\text{'}}\left(t\right)+f\left(x^{\text{'}}\left(t\right)\right)+g(t,x(t-\tau \left(t\right)))=p\left(t\right)$$ is studied. Using the coincidence degree theory, some sufficient conditions on the existence of positive periodic solutions are obtained.

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.

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,...

We consider a neutral type operator differential inclusion and apply the topological degree theory for condensing multivalued maps to justify the question of existence of its periodic solution. By using the averaging method, we apply the abstract result to an inclusion with a small parameter. As example, we consider a delay control system with the distributed control.

We propose and analyze a nonlinear mathematical model of hematopoiesis, describing the dynamics of stem cell population subject to impulsive perturbations. This is a system of two age-structured partial differential equations with impulses. By integrating these equations over the age, we obtain a system of two nonlinear impulsive differential equations with several discrete delays. This system describes the evolution of the total hematopoietic stem cell populations with impulses. We first examine...

This paper concerns with the existence of the solutions of a second order impulsive delay differential equation with a piecewise constant argument. Moreover, oscillation, nonoscillation and periodicity of the solutions are investigated.