Periodic solutions of differential inclusions with retards (Q5929989)

The authors study the existence of periodic solutions to differential inclusions with retarded arguments of the form NEWLINE\[NEWLINE \dot{x}(t)\in F(t,x(t-\tau_1),\ldots,x(t-\tau_m)) \tag{1} NEWLINE\]NEWLINE for \(t\in [0,T]\) a.e., \(x(t)=x(t+T)\) for every \(t\in [-\tau,0],\)where \(F:[0,T]\times \mathbb{R}^{mn}\multimap \mathbb{R}^{n}\) is a multivalued map, \(T>0, \tau_{1},\ldots,\tau_{m}>0, \tau=\max\{\tau_{1},\ldots,\tau_{m}\}.\) NEWLINENEWLINENEWLINEIf \(V:\mathbb{R}^{n}\to \mathbb{R}\) is a locally Lipschitzian function, then the symbol \(\partial V(x)\) is used for the generalized gradient of \(V\) at \(x\in \mathbb{R}^{n}.\) The lower inner product of nonempty compact subsets of \(\mathbb{R}^{n}\) is denoted by\(\langle A,B\rangle^{-} =\inf\{\langle a,b\rangle:a\in A, b\in B\)\}. NEWLINENEWLINENEWLINEA locally Lipschitzian function \(V:\mathbb{R}^{n}\to \mathbb{R}\) is called a nonsingular potential, provided there exists a nonzero radius \(r_{0}>0\) such that \(\langle\partial V(x),\partial V(x)\rangle^{-}>0\) for every \( \|x\|\geq r_{0}.\) NEWLINENEWLINENEWLINELet \(F\) be a \(\mu\)-integrable bounded multimap with compact values and \(M=\max\{\|\mu\|_{1},T\}.\) A nonsingular potential \(V:\mathbb{R}^{n}\to \mathbb{R}\) is called a guiding potential for \(F,\) if there is \(r_{0}>0\) such that for every \((t,x,x_{1},\ldots,x_{m})\in (0,T) \times \mathbb{R}^{n}\times \mathbb{R}^{mn},\) with \(\|x\|\geq r_{0}\) and \(x_{i}\in B^{n}(x,M)\) for \(i=1,\ldots,m\) we have \(\langle\partial V(x),F(t,x_{1},\ldots,x_{m})\rangle^{-}\geq 0.\) The number \(\text{Ind}(V)=\text{Deg}(\partial V,B^{n}(r))\), where \(r\geq r_{0}\), is called the index of the nonsingular potential \(V.\) NEWLINENEWLINENEWLINEThe main result of the paper is the theorem:NEWLINENEWLINENEWLINELet \(F\) be a \(\mu\)-integrably bounded upper semicontinuous multimap with compact convex values and \(V:\mathbb{R}^{n}\to \mathbb{R}\) be a guiding potential for \(F\) with Ind\((V)\neq 0.\) Then the periodic problem (1) has a solution.