The paper deals with the multivalued boundary value problem $x^{\text{'}}\in A(t,x)x+F(t,x)$ for a.a. $t\in [a,b]$, $Mx\left(a\right)+Nx\left(b\right)=0$, in a separable, reflexive Banach space $E$. The nonlinearity $F$ is weakly upper semicontinuous in $x$. We prove the existence of global solutions in the Sobolev space $W^{1,p}([a,b],E)$ with $1<p<\infty$ endowed with the weak topology. We consider the case of multiple solutions of the associated homogeneous linearized problem. An example completes the discussion.

In this paper we prove two existence theorems for abstract boundary value problems controlled by semilinear evolution inclusions in which the nonlinear part is a lower Scorza-Dragoni multifunction. Then, by using these results, we obtain the existence of periodic mild solutions.

We consider a class of evolution differential inclusions defining the so-called stop operator arising in elastoplasticity, ferromagnetism, and phase transitions. These differential inclusions depend on a constraint which is represented by a convex set that is called the characteristic set. For $\mathrm{BV}$ (bounded variation) data we compare different notions of $\mathrm{BV}$ solutions and study how the continuity properties of the solution operators are related to the characteristic set. In the finite-dimensional case...