Extremal solutions of classes of multivalued differential equations (Q2714294)

The authors generalize and improve results on the existence of solutions for the following problems: NEWLINE\begin{itemize}NEWLINE\item[(1)] Existence of generalized solutions in \(W^{2,1}(I,\mathbb{R}^n)\) to the multivalued boundary value problem: NEWLINE\[NEWLINE\begin{cases} \ddot{u} (t)\in \operatorname{ext}F(t,u(t),\dot{u}(t)),&\text{a.e. on } I,\\ u(0)=x_0,u(\eta)=u(T), \end{cases} \tag{\(Q^e_1\)}NEWLINE\]NEWLINEwith \(F:I\times \mathbb{R}^n\times \mathbb{R}^n\rightarrow P_{ck}\) and \(\operatorname{ext}F(.,u(.),\dot{u}(.))\) are the extremal points of \(F(.,u(.),\dot{u}(.)),\;T>0,\;\eta\in (0,T)\). NEWLINE\item[(2)] Existence of generalized solutions in \(W^{2,1}(I,\mathbb{R}^n)\) to the following single-valued boundary value problem with multivalued moving constraints: NEWLINE\[NEWLINE\begin{cases} \ddot{u} (t)=f(t,\dot{u}(t),x(t)),&\text{a.e. on }I,\\ u(0)=x_0,u(\eta)=u(T),\\ x(t)\in K(t,u(t),\dot{u}(t)),&\text{a.e. on } I, \end{cases} \tag{\(Q_2\)}NEWLINE\]NEWLINEwith \(f:I\times \mathbb{R}^n\times \mathbb{R}^n\times \mathbb{R}^n\rightarrow \mathbb{R}^n\) and \(K:I\times \mathbb{R}^n\times \mathbb{R}^n\rightarrow \mathbb{P}_k(R^m)\). NEWLINE\item[(3)] Existence of mild solutions of the evolution equation with multivalued perturbation: NEWLINE\[NEWLINE\begin{cases} \dot{x}(t)\in -A(x(t))+\operatorname{ext}G(t,x),&\text{a.e. on }I,\\ u(0)=u_0\in \overline{D(A)}, \end{cases} \tag{\(Q^e_3\)}NEWLINE\]NEWLINEwhere \(A:E\rightarrow 2^E\) is a multivalued \(m\)-accretive operator, \(E\) is a real Banach space, \(D(A)\) its domain and \(G:I\times E\rightarrow P_{cwk}(E)\). NEWLINE\end{itemize}NEWLINEIn theorem 2 and theorem 3 the authors prove under same conditions that \((Q^e_1)\) and \((Q_2)\) have solutions in \(W^{2,1}(I,\mathbb{R}^n)\) and \(W^{2,1}(I,\mathbb{R}^n)\times W^{2,1}(I,\mathbb{R}^n)\) respectively, with \(I=[0,T], T>0\) and with the initial condition \(u(0)=x_0, u(\eta)=u(T),\eta\in (0,T)\). In theorem 4 the authors prove the existence of solutions to differential equations of second order with multivalued moving constraints \((Q_2)\) and the theorem 11 is an existence theorem on mild solutions to \((Q^e_3)\). As a consequence of theorem 11, the authors prove the existence of strong solutions to the evolution equation NEWLINE\[NEWLINE\begin{cases} \dot{u}(t)\in -\partial\varphi(u(t))+\operatorname{ext}G(t,u(t)),&\text{a.e. on }I,\\ u(0)=u_0\in \overline{D(A)}, \end{cases} \tag{\(Q^e_4\)}NEWLINE\]NEWLINEwhere \(\varphi\) is a proper convex function from \(E\) to \(R\cup\{\infty\}\) and \(\partial\varphi\) is the subdifferential of the function \(\varphi\).