Iterates of semidifferentiable monotone homogeneous maps
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چکیده
We study the nonlinear eigenproblem for monotone additively homogeneous maps f : Rn → Rn, and more generally, for monotone multiplicatively homogeneous maps on cones. When f is convex, monotone and additively homogeneous, we describe the nonlinear eigenspace of f using its subdifferential (or its tangent affine maps) [AG03]. This yields in particular an uniqueness result for the eigenvector of f (up to an additive constant). We also characterize the orbit lengths of f . These results generalize the maxplus spectral theorem, together with results obtained in the ergodic control literature. When f : Rn → Rn is convex, f is necessarily semidifferentiable and its semiderivative can be expressed in terms of its subdifferential. Our previous particular uniqueness result [AG03] is thus a special case of the following [AGN03]: if C is a normal cone in a Banach space and f : C → C is a monotone multiplicatively homogeneous map, then f has a unique eigenvector v (up to a multiplicative constant) provided f is semidifferentiable at v, that its semiderivative f 0 v has a unique eigenvector (up to the addition of a multiple of v) and that f 0 v satisfies a Fredholm type condition. This result extends earlier results of Nussbaum in the differentiable case, and it yields
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تاریخ انتشار 2003