On topological degree to some class of multivalued mappings and its applications (Q2735874)

The main statements of topological degree theory are established for the special class of multivalued mappings \(A\) from a reflexive Banach space \(X\) to the totality of all subsets of is dual \(X^*\). For the definition of topological degree lower and upper support function were associated with the mapping \(A\). By means of this functions the special class of mappings \(A\) is described, which allows to formulate sufficient conditions that weakly convergent sequences from \(X\) be strongly convergent ones in the topology of \(X\). In terms of the lower support function for mappings \(A\) the semibounded variation and pseudomonotonicity analogues are formulated. If the graph of \(A\) is closed in \(X\times X^*\) with respect to the strong convergence on \(X\) and weak one on \(X^*\) the mapping \(A\) is called demiclosed. NEWLINENEWLINENEWLINEFor the class of demiclosed mapping \(A\) the some analogues of finite-dimension approximation is formulated. By means of this approximation the topological degree of the mapping \(A\) is studied. For this topological degree the basic properties are proved. This definition of topological degree is transfered to the class of pseudomonotone mappings.