We study the relation between the solutions set to a perturbed semilinear differential inclusion with nonconvex and non-Lipschitz right-hand side in a Banach space and the solutions set to the relaxed problem corresponding to the original one. We find the conditions under which the set of solutions for the relaxed problem coincides with the intersection of closures (in the space of continuous functions) of sets of δ-solutions to the original problem.

We consider a second order regular differential operator whose coefficients are nonselfadjoint bounded operators acting in a Hilbert space. An estimate for the resolvent and a bound for the spectrum are established. An operator is said to be stable if its spectrum lies in the right half-plane. By the obtained bounds, stability and instability conditions are established.

The aim of this paper is to give a complete proof of the formula for the resolvent of a nonautonomous linear delay functional differential equations given in the book of Hale and Verduyn Lunel [9] under the assumption alone of the continuity of the right-hand side with respect to the time,when the notion of solution is a differentiable function at each point, which satisfies the equation at each point, and when the initial value is a continuous function.

In this paper we study a linear integral equation $x\left(t\right)=a\left(t\right)-{\int }_0^{t}C(t,s)x\left(s\right)\mathrm{d}s$, its resolvent equation $R(t,s)=C(t,s)-{\int }_s^{t}C(t,u)R(u,s)\mathrm{d}u$, the variation of parameters formula $x\left(t\right)=a\left(t\right)-{\int }_0^{t}R(t,s)a\left(s\right)\mathrm{d}s$, and a perturbed equation. The kernel, $C(t,s)$, satisfies classical smoothness and sign conditions assumed in many real-world problems. We study the effects of perturbations of $C$ and also the limit sets of the resolvent. These results lead us to the study of nonlinear perturbations.

The paper deals with the boundary value problem $$u^{\text{'}\text{'}}+ku=g\left(u\right)+e\left(t\right),u\left(0\right)=u\left(2\pi \right),u^{\text{'}}\left(0\right)=u^{\text{'}}\left(2\pi \right),$$ where $k\in ℝ$, $gI\mapsto ℝ$ is continuous, $e\in 𝕃J$ and ${lim}_{x\to 0+}{\int }_x^{1}g\left(s\right)\text{d}s=\infty .$ In particular, the existence and multiplicity results are obtained by using the method of lower and upper functions which are constructed as solutions of related auxiliary linear problems.

This paper deals with the asymptotic behavior as $t\to \infty$ of solutions $u$ to the forced Preisach oscillator equation $\ddot{w}\left(t\right)+u\left(t\right)=\psi \left(t\right)$, $w=u+𝒫\left[u\right]$, where $𝒫$ is a Preisach hysteresis operator, $\psi \in L^{\infty }(0,\infty )$ is a given function and $t\ge 0$ is the time variable. We establish an explicit asymptotic relation between the Preisach measure and the function $\psi$ (or, in a more physical terminology, a balance condition between the hysteresis dissipation and the external forcing) which guarantees that every solution remains bounded for all times. Examples show...

We study the vibration of lumped parameter systems whose constituents are described through novel constitutive relations, namely implicit relations between the forces acting on the system and appropriate kinematical variables such as the displacement and velocity of the constituent. In the classical approach constitutive expressions are provided for the force in terms of appropriate kinematical variables, which when substituted into the balance of linear momentum leads to a single governing ordinary...