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Poisson equations associated with a homogeneous and monotone function: necessary and sufficient conditions for a solution in a weakly convex case. (English) Zbl 1215.47052
Let \(f\) be a monotone and homogeneous self-map of the \(n\)-dimensional positive cone. Let \(\mathcal{F}\) be a family of functions constructed by multiplying the components of \(f\) by positive numbers. Let \(M^f\) be a communication matrix, i.e., the matrix whose \((i,j)\) entry indicates whether or not the \(i\)-th component of \(f\) is an unbounded function of the \(j\)-th coordinate. Assuming that the function \(f\) satisfies a weak form of convexity, the authors obtain two results concerning the eigenvalues for family \(\mathcal{F}\). First, the structure of matrix \(M^f\) which is necessary and sufficient for the existence of an eigenvalue of each function in the family \(\mathcal{F}\) is determined. Second, a necessary and sufficient criterion is given in terms of the recession function of the function \(f\).

47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
Full Text: DOI
[1] Gaubert, S.; Gunawardena, J., The perron – frobenius theorem for homogeneous, monotone functions, Trans. amer. math. soc., 356, 12, 4931-4950, (2004) · Zbl 1067.47064
[2] Akian, M.; Gaubert, S.; Lemmens, B.; Nussbaum, R., Iteration of order preserving subhomogeneous maps on a cone, Math. proc. Cambridge philos. soc., 140, 157-176, (2006) · Zbl 1101.37032
[3] Nussbaum, R.D., Hilberts projective metric and iterated nonlinear maps, Mem. amer. math. soc., 75, 391, (1988)
[4] Nussbaum, R.D., Iterated nonlinear maps and hilberts projective metric, Mem. amer. math. soc., 79, 401, (1989)
[5] Lemmens, B.; Scheutzow, M., On the dynamics of sup-norm nonexpansive maps, Ergodic theory dynam. systems, 25, 3, 861-871, (2005) · Zbl 1114.47044
[6] Gunawardena, J., From MAX-plus algebra to nonexpansive maps: A nonlinear theory for discrete event systems, Theoret. comput. sci., 293, 141-167, (2003) · Zbl 1036.93045
[7] Kolokoltsov, V.N., Nonexpansive maps and option pricing theory, Kybernetika, 34, 713-724, (1998) · Zbl 1274.91420
[8] Akian, M.; Gaubert, S., Spectral theorem for convex monotone homogeneous maps, and ergodic control, Nonlinear anal., 52, 2, 637-679, (2003) · Zbl 1030.47048
[9] Cavazos-Cadena, R.; Hernández-Hernández, D., Necessary and sufficient conditions for a solution to the risk-sensitive Poisson equation on a finite state space, Systems control lett., 58, 254-258, (2009) · Zbl 1159.93030
[10] Dellacherie, C., (), 52-104
[11] Zijm, W.H.M., Generalized eigenvectors and sets of nonnegative matrices, Linear algebra appl., 59, 91-113, (1984) · Zbl 0548.15015
[12] Kolokoltsov, V.N., On linear, additive, and homogeneous operators in idempotent analysis, (), 87-101 · Zbl 0925.47016
[13] Crandall, G.; Tartar, L., Some relations between nonexpansive and order preserving mappings, Proc. amer. math. soc., 78, 3, 385-390, (1980) · Zbl 0449.47059
[14] Oshime, Y., An extension of morishimas nonlinear perron – frobenius theorem, J. math. Kyoto univ., 23, 803-830, (1983) · Zbl 0548.34034
[15] Minc, H., Nonnegative matrices, (1988), Wiley New York · Zbl 0638.15008
[16] Seneta, E., Non-negative matrices and Markov chains, (1980), Springer New York · Zbl 1099.60004
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