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 in 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 $\mathbb{R}^{n}$. Next, we prove the existence of solutions in the Sobolev space $W^{2,\infty}([0,T],\mathbb{R}^{n})$ 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.