Sturm–Liouville Differential Inclusions with Set-Valued Reaction Term Depending on a Parameter

In this paper we study the controllability for a Cauchy problem governed by a nonlinear differential inclusion driven by a Sturm-Liouville type operator. In particular, the considered second order differential inclusion involves a set-valued reaction term depending on a parameter. The key tool in the proof of the controllability result we provide is a multivalued version of the theorem recently proved by Haddad-Yarou, here established for an initial conditions problem monitored by a nonlinear second order differential inclusion presenting the sum of two multimaps on the right-hand side. We thereby deduce the existence of a local admissible pair for the considered control problem, that is the existence of a couple of functions consisting of a control, which is a measurable function, and the correspondent trajectory, which is an absolutely continuous function with absolutely continuous derivative. Secondly, under appropriate assumptions on the involved multimaps, we obtain an increased regularity for the solutions produced by our existence result. This regularity is the same of that recently tested by Bonanno, Iannizzotto and Marras for a different type of problem, which however involves the Sturm-Liouville operator

Existence Results for Implicit Nonlinear Second-Order Differential Inclusions

In this paper, we consider a Cauchy problem driven by an implicit nonlinear second-order differential inclusion presenting the sum of two real-valued multimaps, one taking convex values and the other assuming closed values, on the right-hand side. We first obtain, on the basis of a selection theorem proved by Kim, Prikry and Yannelis and on an existence result proved by Cubiotti and Yao (Adv Differ Equ 214:1- 10, 2016), an existence theorem for an initial value problem governed by a non implicit second-order differential inclusion involving two multimaps whose values are subsets of R-n. Next, we prove the existence of solutions in the Sobolev space W-2,W-infinity([0, T],R-n) for the considered implicit problem. A fundamental tool employed to achieve our goal is a profound result of B. Ricceri on inductively open functions. Moreover, we derive from the aforementioned results two corollaries that examine the viable cases. An application to Sturm-Liouville differential inclusions is also discussed. Lastly, we focus on a Cauchy problem monitored by a second-order differential inclusion having as nonlinearity on the second-order derivative a trigonometric map