Existence of solutions for second-order differential inclusions associated with highly non-semicontinuous multifunctions
Applicable Nonlinear Analysis, Volume 1, Issue 1, June 2024, Pages: 20–43
PAOLO CUBIOTTI
Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Viale F. Stagno d’Alcontres 31, 98166 Messina, Italy
JEN-CHIH YAO
Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taichung, 404328, Taiwan
Abstract
In this paper we deal with the existence of generalized solutions for the Cauchy problem associated with a second-order differential inclusion, both in explicit and in implicit form. We firstly prove an existence result for an inclusion of the type \( u' \in F(t, u, u') \) where \( F : [0, T] \times \mathbb{R}^n \times \mathbb{R}^n \to 2^{\mathbb{R}^n} \) is a given closed-valued multifunction. The main peculiarity of this latter result is as follows: our assumptions do not imply any kind of semicontinuity for the multifunction \( F(t, \cdot, \cdot) \). That is, a multifunction \(F\) can satisfy all the assumptions and, at the same time, for every \(t \in [0,T]\) the multifunction \(F(t, \cdot , \cdot) \)can be neither upper nor lower semicontinuous even at each point \( (x, z) \in \mathbb{R}^n \times \mathbb{R}^n \). A viable version of this result is also proved. Furtherly, as an application, an analogous result is proved for an inclusion of the type \( u'' \in Q(t, u, u') + S(t, u, u'), \) where \( Q : [0, T] \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n \) has convex values, and \(S : [0, T] \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n\) has closed values. Again, our assumptions do not imply any kind of semicontinuity for the multifunctions \( Q(t, \cdot , \cdot) \) and \( S(t, \cdot , \cdot) \). Then we consider an application to the implicit differential inclusion \( n \psi(u'') \in F(t, u, u') + G(t, u, u') \), where \(F\) is convex-valued and \(G\) is closed-valued. As regards the function \(\psi\), we only assume that it is continuous and locally nonconstant. Finally, we present a further application to the Cauchy problem associated with a Sturm-Liouville type differential inclusion.
Cite this Article as
Paolo Cubiotti and Jen-Chih Yao, Existence of solutions for second-order differential inclusions associated with highly non-semicontinuous multifunctions, Applicable Nonlinear Analysis, 1 (2024), 20–43