In the paper we consider lower semicontinuous differential inclusions with one sided Lipschitz and compact valued right hand side in a Banach space with uniformly convex dual. We examine the nonemptiness and some qualitative properties of the solution set.

The classical framework for studying the equations governing the motion of lumped parameter systems presumes one can provide expressions for the forces in terms of kinematical quantities for the individual constituents. This is not possible for a very large class of problems where one can only provide implicit relations between the forces and the kinematical quantities. In certain special cases, one can provide non-invertible expressions for a kinematical quantity in terms of the force, which then...

Modern physics theories claim that the dynamics of interfaces between the two-phase is described by the evolution equations involving the curvature and various kinematic energies. We consider the motion of spiral-shaped polygonal curves by its crystalline curvature, which deserves a mathematical model of real crystals. Exploiting the comparison principle, we show the local existence and uniqueness of the solution.

Let Φ : H → R be a C2 function on a real Hilbert space and ∑ ⊂ H x R the manifold defined by ∑ := Graph (Φ). We study the motion of a material point with unit mass, subjected to stay on Σ and which moves under the action of the gravity force (characterized by g>0), the reaction force and the friction force ($\gamma >0$ is the friction parameter). For any initial conditions at time t=0, we prove the existence of a trajectory x(.) defined on R+. We are then interested in the asymptotic behaviour of...

Let $\Phi :H\to ℝ$ be a ${𝒞}^2$ function on a real Hilbert space and $\Sigma \subset H\times ℝ$ the manifold defined by $\Sigma :=$ Graph $\left(\Phi \right)$. We study the motion of a material point with unit mass, subjected to stay on $\Sigma$ and which moves under the action of the gravity force (characterized by $g\>0$), the reaction force and the friction force ($\gamma \>0$ is the friction parameter). For any initial conditions at time $t=0$, we prove the existence of a trajectory $x(.)$ defined on ${ℝ}_{+}$. We are then interested in the asymptotic behaviour of the trajectories when $t\to +\infty$. More precisely,...

The aim of this paper is to establish an existence and uniqueness result for a class of the set functional differential equations of neutral type$$\left\{\begin{array}{c}{D}_{H}X\left(t\right)=F(t,X_t,D_H{X}_t),\hfill \\ {X|}_{[-r,0]}=\Psi ,\hfill \end{array}\right.$$ where $F:[0,b]\times {𝒞}_0\times {𝔏}_0^1\to K_c\left(E\right)$ is a given function, $K_c\left(E\right)$ is the family of all nonempty compact and convex subsets of a separable Banach space $E$, ${𝒞}_0$ denotes the space of all continuous set-valued functions $X$ from $[-r,0]$ into $K_c\left(E\right)$, ${𝔏}_0^1$ is the space of all integrally bounded set-valued functions $X:[-r,0]\to K_c\left(E\right)$, $\Psi \in {𝒞}_0$ and $D_H$ is the Hukuhara derivative. The continuous dependence of solutions on initial data and...