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 Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{n}$$\end{document}. Next, we prove the existence of solutions in the Sobolev space W2,∞([0,T],Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{2,\infty }([0,T],{\mathbb {R}}^{n})$$\end{document} 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.