Structure of fixed point sets of condensing maps in \(B_0\) spaces with applications to differential equations in unbounded domain (Q2769132)

This article deals with continuous mappings in topological vector space whose topology is determined by a countably family \(Q\) of seminorms \(q\); these mappings are defined on the set \({\mathcal B}=\cap_{q\in Q}B_q\), \(B_q =\{u:(u)<1\}\) and assumed to be condensing with respect to the family \(\{\gamma: q\in Q\}\) of ``measures of noncompactness'' defined by \(\gamma_q(A)= \inf\{ \delta> 0\): there exists a finite set \(S\subset E\) such that \(A\subseteq S+\delta B_q\}\). More exactly, a continuous map \(F:\overline \Omega\to E\) \((\Omega \subset E\) an open set) is \((Q,{\mathcal R}_Q)\)-condensing iff \(F(\overline\Omega) \in{\mathcal R}_Q\) and \(\forall_{q\in Q}\) \(\forall_{A \subseteq \overline\Omega} \gamma_q(F(A)) <\gamma_q(A)\); analogously, such a map \(F\) is called a \((Q,{\mathcal R}_Q)\)-set-contraction iff \(F(\overline\Omega) \in{\mathcal R}_Q\) and \(\forall_{q\in Q} \exists_{k_q<1} \forall_{A\subseteq \overline\Omega}\) \(\gamma_q\) \((F(A)) <\gamma_q (A)\); here \({\mathcal R}_Q\) is a family of ``regular'' sets, distinguished in a special way (for example, the set of bounded sets in \(E)\). For both classes, the author develops a degree theory that is analogous to the standard degree theory for vector fields with condensing mappings and \(k\)-set-contractions. In the end of the article, some results about the structure of the fixed point set for mappings from these classes are presented; the latter are similar to known results by W. V. Petryshyn. Some applications to boundary value problems in unbounded domains are also given.