This paper is devoted to the solvability of the Lyapunov equation A*U + UA = I, where A is a given nonselfadjoint differential operator of order 2m with nonlocal boundary conditions, A* is its adjoint, I is the identity operator and U is the selfadjoint operator to be found. We assume that the spectra of A* and -A are disjoint. Under this restriction we prove the existence and uniqueness of the solution of the Lyapunov equation in the class of bounded operators.

The aim of this note is 1. to show that some results (concerning the structure of the solution set of equations (18) and (21)) obtained by Czarnowski and Pruszko in [6] can be proved in a rather different way making use of a simle generalization of a theorem proved by Vidossich in [8]; and 2. to use a slight modification of the “main theorem” of Aronszajn from [1] applying methods analogous to the above mentioned idea of Vidossich to prove the fact that the solution set of the equation (24), (25)...

We propose a theoretical framework for solving a class of worst scenario problems. The existence of the worst scenario is proved through the convergence of a sequence of approximate worst scenarios. The main convergence theorem modifies and corrects the relevant results already published in literature. The theoretical framework is applied to a particular problem with an uncertain boundary value problem for a nonlinear ordinary differential equation with an uncertain coefficient.

We study the Ambrosetti–Prodi and Ambrosetti–Rabinowitz problems.We prove for the first one the existence of a continuum of solutions with shape of a reflected $C$ ($\supset$-shape). Next, we show that there is a relationship between these two problems.

The existence of optimal control for nonlinear delay systems having an implicit derivative with quadratic performance criteria is proved. The results are established by an iterative technique and using the Darbo fixed point theorem.

In this paper, we study the existence of periodic solutions to a class of functional differential system. By using Schauder's fixed point theorem, we show that the system has aperiodic solution under given conditions. Finally, four examples are given to demonstrate the validity of our main results.

By means of the Krasnoselskii fixed piont theorem, periodic solutions are found for a neutral type delay differential system of the form $$x^{\text{'}}\left(t\right)+c{x}^{\text{'}}\left(t-\tau \right)=A\left(t,x\left(t\right)\right)x\left(t\right)+f\left(t,x\left(t-r_1\left(t\right)\right),\cdots ,x\left(t-r_k\left(t\right)\right)\right).$$

We consider first order neutral functional differential equations with multiple deviating arguments of the form ${(x\left(t\right)+Bx(t-\delta ))}^{\text{'}}=g₀(t,x\left(t\right))+{\sum }_{k=1}^n{g}_k(t,x(t-{\tau }_k\left(t\right)))+p\left(t\right)$. By using coincidence degree theory, we establish some sufficient conditions on the existence and uniqueness of periodic solutions for the above equation. Moreover, two examples are given to illustrate the effectiveness of our results.

By using the coincidence degree theory of Mawhin, we study the existence of periodic solutions for $n$ th order delay differential equations with damping terms $x^{\left(n\right)}\left(t\right)=\sum _{i=1}^s{b}_i{\left[x^{\left(i\right)}\left(t\right)\right]}^{2k-1}+f\left(x(t-\tau \left(t\right))\right)+p\left(t\right)$. Some new results on the existence of periodic solutions of the investigated equation are obtained.

In this paper, we consider periodic solutions for a class of nonlinear evolution equations with non-instantaneous impulses on Banach spaces. By constructing a Poincaré operator, which is a composition of the maps and using the techniques of a priori estimate, we avoid assuming that periodic solution is bounded like in [1-4] and try to present new sufficient conditions on the existence of periodic mild solutions for such problems by utilizing semigroup theory and Leray-Schauder's fixed point theorem....