Positive eigenvectors of wedge maps
The paper has three main purposes. The first is the study of the basic properties of a very large class of maps acting on cones, or more generally on wedges of Banach spaces. A typical result of this part is the following. Let W be a wedge, \(\Omega\subset W\) be open and bounded, and \(f: {\bar \Omega}\to W\) be an \(\alpha\)-contraction with constant p. Let \(x_ 0\in W\setminus {\bar \Omega}\) and assume that \(x-x_ 0-tf(x)\neq 0\) for every \(x\in \partial W\) and \(t\in [0,1]\). Then there exists an \(\alpha\)- contraction \(h: {\bar \Omega}\to W\) with constant \(k\leq p\) and \(h(x)=0\) for all \(x\in \partial W\), such that \(x-x_ 0-f(x)\neq h(x)\) for all \(x\in {\bar \Omega}.\) The second part uses the results of the first to obtain the existence of positive eigenvalues and corresponding eigenvectors for maps acting on wedges or cones of Banach spaces. The theorem below exemplifies the results of this part. Let W be a cone and \(\Omega\subset W\) be an open, bounded neighborhood of the origin. Let \(f: {\bar \Omega}\to W\) be \(\alpha\)-Lipschitz with constant p and assume that \(\delta \gamma >p\rho\) where \(\rho =\sup \{\| x\|:\) \(x\in \partial \Omega \}\), \(\delta =\inf \{\| f(x)\|:\) \(x\in \partial \Omega \}\) and \(\gamma\) is the quasi-normality constant of the cone. Then there exists \(x\in \partial \Omega\), \(\lambda >\delta /\rho\) such that \(f(x)=\lambda x.\) In the third part we use the results obtained in the other two to establish a theorem on eigenvectors of positively homogeneous, order preserving and noncompact maps, acting on cones. More precisely we prove that given a cone K and a positively homogeneous map \(f: K\to K\) such that \(\alpha (f)<r(f)\) and \(\limsup_{n\to +\infty}\| f^ n(u)\| /\delta^ n>0\) for some \(\| u\| =1\) and \(\delta\in (\alpha (f),r(f)]\), there exists \(\| x_ 0\| =1\) and \(\mu\in [\delta,r(f)]\) such that \(f(x_ 0)=\mu x_ 0\).
existence of positive eigenvalues and corresponding eigenvectors for maps acting on wedges or cones of Banach spaces
eigenvectors of positively homogeneous, order preserving and noncompact maps, acting on cones
scientific article; zbMATH DE number 4002896
scientific article