Topological degree theory for multi-valued mappings of class \((S_+)_L\)
This article deals with Skrypnik-type degree theory for multivalued mappings \(T: \;\Omega \, (\subset E) \to 2^{E^*}\) of a reflexive Banach space \(E\) into subsets of its dual \(E^*\). It is assumed that: (i) \(Tx\) is bounded closed and convex for \(x \in D(T)\); (ii) \(T \cap F\) is weakly upper semicontinuous for each finite-dimensional space \(F \subset L\), \(F \cap D(T) \neq \emptyset\); (iii) if for any sequence of finite-dimensional subspaces \(F_j \subset L\) with \(L \subseteq \overline{\bigcup\limits_{j=1}^\infty F_j}\), \(h \in E^*\), \(\{x_j\}_{j=1}^\infty \subset D(T) \cap L\), \(x_j \rightharpoonup x_0\), and such that \[ \limsup_{j \to \infty} \;(f_j - h,x_j) \leq 0, \quad \lim_{j \to 0} \;(f_j - h,v) = 0 \qquad \left(v \in \bigcup_{j=1}^\infty F_j, \;\;f_j \in Tx_j\right), \] then \(x_j \to x_0 \in D(T)\) and \(h \in Tx_0\); here is \(L\) a fixed dense subspace of \(E\). Mappings with these properties form the class \((S_+)_L\); if property (iii) holds only for \(h = 0\), the corresponding mappings form the class \((S_+)_{0,L}\). The main result is the existence of the integer-valued degree \(\text{deg} \, (T,\Omega,0)\) in the case \(T \in (S_+)_{0,L}\) and \(0 \notin T(\partial\Omega \cap D(T))\) with the following properties: (1) \(\text{deg} \, (J,\Omega,0) = 1\) if \(0 \in J(\Omega)\) (\(J\) is the normalized dual mapping between \(E\) and \(E^*\)); (2) if \(\text{deg} \, (T,\Omega,0) \neq 0\), then \(0 \in Tx\) for some \(x \in \Omega \cap D(T)\); (3) if \(\Omega_1, \Omega_2 \subset \Omega\), \(\Omega = \Omega_1 \cup \Omega_2\), \(\Omega_1 \cap \Omega_2 = \emptyset\), then \(\text{deg} \, (T,\Omega,0) = \text{deg} \, (T,\Omega_1,0) +\text{deg} \, (T,\Omega_2,0)\); (4) if \(T_0\) and \(T_1\) from the class \((S_+)_{0,L}\) are homotopic (under a natural definition of homotopies in the class \((S_+)_{),L}\)), then \(\text{deg} \, (T_0,\Omega,0) = \text{deg} \, (T_1,\Omega,0)\). Two results about the calculation of this degree are presented as well. Moreover, an example with the operator \(T\) generated by the following equation with discontinuous nonlinearity \[ \sum_{i=1}^N \frac{\partial}{\partial x_i} \left(\rho^2(u) \frac{\partial u}{\partial x_i} + a_i(\left(x,u,\frac{\partial u}{\partial x}\right)\right) + f(x,u) = 0, \;\;\text{a.e.} \;\;x \in \Omega, \] \[ u(x) = 0, \;\;\;x \in \partial\Omega, \] is considered.
scientific article; zbMATH DE number 2168140