Eigenvalues for a Class of Homogeneous Cone Maps Arising from Max-plus Operators
نویسندگان
چکیده
We study the nonlinear eigenvalue problem f(x) = λx for a class of maps f : K → K which are homogeneous of degree one and order-preserving, where K ⊆ X is a closed convex cone in a Banach space X. Solutions are obtained, in part, using a theory of the “cone spectral radius” which we develop. Principal technical tools are the generalized measure of noncompactness and related degree-theoretic techniques. We apply our results to a class of problems max t∈J(s) a(s, t)x(t) = λx(s) arising from so-called “max-plus operators,” where we seek a nonnegative eigenfunction x ∈ C[0, μ] and eigenvalue λ. Here J(s) = [α(s), β(s)] ⊆ [0, μ] for s ∈ [0, μ], with a, α, and β given functions, and the function a nonnegative.
منابع مشابه
A Class of compact operators on homogeneous spaces
Let $varpi$ be a representation of the homogeneous space $G/H$, where $G$ be a locally compact group and $H$ be a compact subgroup of $G$. For an admissible wavelet $zeta$ for $varpi$ and $psi in L^p(G/H), 1leq p <infty$, we determine a class of bounded compact operators which are related to continuous wavelet transforms on homogeneous spaces and they are called localization operators.
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