Minimal complexity of semi-bounded components in bifurcation theory
In this interesting article, the authors use topological degree theory to count the number of solutions of nonlinear eigenvalue problems: Let \(E\) be a real Banach space, let \(\mathcal{L}:\mathbb{R}\to L(E)\) be a continuous map into the space of bounded linear operators on \(E\) and \(\mathcal{N}:\mathbb{R}\times E\to E\) continuous and compact. Put \(\mathcal{F}(\lambda,x):=\mathcal{L}(\lambda)x+\mathcal{N}(\lambda,x)\). Assume that \(\mathcal{L}(\lambda)-I\) is compact for each \(\lambda\) and that \(\lim_{x\to0}\sup_{\lambda\in K}\frac{\| \mathcal{N}(\lambda,x)\| }{\| x\| }=0\) whenever \(K\subset\mathbb{R}\) is compact. The spectrum of \(\mathcal{L}\) is defined to be the set \(\Sigma\) of all \(\lambda\in\mathbb{R}\) such that \(\mathcal{L}(\lambda)\) is not an isomorphism. Call \(\mathcal{S}\) the set of nontrivial solutions to \(\mathcal{F}(\lambda,x)=0\), i.e., \((\lambda,x)\in\mathcal{S}\) if \(\lambda\in \Sigma\) and \(x=0\) or \(F(\lambda,x)=0\) and \(x\not=0\). The main goal of the present article is the analysis of the structure of the compact components of \(\mathcal{S}\cap(J\times E)\) whenever \(J\) is a closed unbounded interval. If \(\mathcal{A}\) is a locally finite nonempty family of disjoint nonempty open intervals such that \(A\cap\Sigma=\emptyset\) whenever \(A\in\mathcal{A}\), then it is easy to see that there are \(r\leq s\in\overline{\mathbb{Z}}:=\mathbb{Z}\cup\{-\infty\}\cup\{\infty\}\) and a map \(J:[r,s]\to\mathcal{A}\) such that \(J(i-1)\) lies on the left hand side of \(J(i)\) whenever \(r+1\leq i\leq s\), where we have written \([r,s]:=\{n\in\overline{\mathbb{Z}}\mid r\leq n\leq s\}\). Call \(I(i):=[\sup J_{i-1},\inf J(i)]\). It is obvious that \(\lambda\mapsto\text{ind}(E,\mathcal{L}(\lambda),0)\) is constant on \(J(i)\) and takes values only in \(\{-1,1\}\). Call this value \(a_i\) and let \(P(i)=\frac12(a_i-a_{i-1})\). Denote by \(\mathcal{S}_{\mathcal{A}}\) the set of all \((\lambda,x)\) such that \((\mathcal{F}(\lambda,x)=0\) and \(x\not=0\)) or (\(\lambda\notin\bigcup\mathcal{A}\) and \(x=0\)). Let \(\mathcal{C}\) be a bounded component of \(\mathcal{S}_{\mathcal{A}}\cap(J\times E)\) and denote by \(\mathcal{B}\) the (finite!) set of \(i\in[r+1,s]\) such that \(\mathcal{C}\) meets \(I(i)\times\{0\}\). Choose \(\lambda^*\in J\setminus\bigcup_{i\in\mathcal{B}}I(i)\). If \(J\) is bounded from below, denote by \(J^*\) the set of \(i\in\mathcal{B}\) such that \(\lambda^*\) lies on the left hand side of \(I(i)\) and let \(\operatorname {sign}J^*=1\). Similarly, if \(J\) is bounded from above, denote by \(J^*\) the set of \(i\in\mathcal{B}\) such that \(\lambda^*\) lies on the right hand side of \(I(i)\) and let \(\operatorname {sign}J^*=-1\) with the appropriate modification if \(J=\mathbb{R}\). The main result then states: If \(\Omega\) is an open isolating neighbourhood of \(\mathcal{C}\) in \(J\times E\), then there is a \(\rho_0>0\) such that \(\deg(E,\mathcal{F}(\lambda^*,\cdot),\Omega_{\lambda^*}\setminus\overline{B} (0;\rho))=2\operatorname {sign}J^*\sum_{i\in J^*}P(i)\), where \(\Omega_\lambda=\{x\in E\mid (\lambda,x)\in\Omega\}\). This result can then be used to estimate the cardinality of the \(\lambda\)-slices of \(\mathcal{C}\).
scientific article; zbMATH DE number 2105446
scientific article