Abstract
Suppose that E is a real Banach space with a cone K and suppose that the homogeneous additive operator A that is positive on K is focusing, i.e.,ak⊂k uop for certain u0∈ K and ρ ⩾ 1. Then, as is well known, the operator A uniformly reduces the oscillation (osc) between the elements of K. In this paper we show that only the focusing operators have this property.
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P. P. Zabreiko, M. A. Krasnosel'skii, and Yu. V. Pokornyi, “On a certain class of positive linear operators,” Funktional'. Analiz i Ego Prilozhen.,5, No. 4, 9–17 (1971).
M. G. Krein and M. A. Rutman, “Linear operators that leave a cone in a Banach space invariant,” Usp. Mat. Nauk,2, No. 1(23), 3–95 (1948).
M. A. Krasnosel'skii, Positive Solutions of Operator Equations, Noordhoff, Groningen (1964).
Yu. V. Pokornyi, “Oscillations of a positive operator,” Trudy Mat. Fak. Voronezh Gos. Univ., No. 3, 92–102 (1971).
E. Hopf, “An inequality for positive linear integral operators,” J. Math. and Mech.,12, No. 5, 683–692 (1963).
Yu. V. Pokornyi, “On certain estimates of the Green's function of a multipoint boundaryvalue problem,” Mat. Zametki,4, No. 5, 533–540 (1968).
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Translated from Matematicheskii Zametki, Vol. 20, No. 5, pp. 753–760, November, 1976.
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Pokornyi, Y.V., Smitskikh, S.V. Inversion of the oscillatory property of focusing operators. Mathematical Notes of the Academy of Sciences of the USSR 20, 980–984 (1976). https://doi.org/10.1007/BF01146924
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DOI: https://doi.org/10.1007/BF01146924