Some applications of degree theory to bifurcation problems (Q1081744)

Let X,Y be locally convex Hausdorff spaces and D,K\(\subseteq X\) subset sets with \(D_ K=D\cap \neq \emptyset\). Let \({\mathcal C}_ K(D)\) be some family of nonempty subsets of \(D_ K\). For \(\Omega\in {\mathcal C}_ K(D)\) by \({\mathcal B}_ K(\Omega,Y)\) we denote the class of multivalued mappings F from \(\Omega\) into Y for which one can define the topological degree of F on \(\Omega\) at the origin with respect to K, denoted by \(\deg_ K(F,\Omega,0)\), which satisfies the existence and the homotopy axioms. The main purpose of the paper is to show that under some necessary additional conditions on the mappings \(F\in {\mathcal B}_ K(\Omega,Y)\) we have \(\deg_ K(F,\Omega,0)=\{0\}\) provided \(0\not\in F(\partial_ K\Omega)\). This result is applied to consider the existence of bifurcation solutions of the equation \[ F(\lambda,u)=0,\quad (\lambda,u)\in \Lambda \times D_ K, \] where \(\Lambda\) is an open subset of some metric space, D,K are subsets of a Banach space X with \(0\in int D,D_ K=D\cap K\neq \emptyset\) and F is a mapping from \(\Lambda \times D_ K\) into another Banach space Y. As a special case we consider the mapping F in the form \[ F(\lambda,u)=u-M(u)-N(\lambda,u)- H(\lambda,u),\quad (\lambda,u)\in \Lambda \times D_ K, \] where \(\Lambda\) is an open subset of some normed space with a partial ordering \(\prec\), D is a subset of a Banach space X with \(0\in int D\) and K is a closed convex cone in X, M,N(\(\lambda\),.) are linear continuous mappings from \(D_ K\) into X and H(\(\lambda\),.) is a nonlinear continuous mapping with \(H(\lambda,0)=0\), \(\| H(\lambda,u)\| =o(\| u\|)\) as \(\| u\| \to 0\). Let \(\lambda_ 0\in \Lambda\) be the smallest eigenvalue of the pair (M,N) with respect to K which isolated from the right-hand side. Then \((\lambda_ 0,0)\) is a bifurcation solution of the equation \(F(\lambda,u)=0\). The obtained result is applied to investigate bifurcation solutions of elliptic partial differential equations of second order.