Positive Solutions of Positive Linear Equations

Abstract

Let B be a real vector lattice and a Banach space under a semimonotonic norm.Suppose T is a linear operator on B which is positive and eventually compact, y is a positive vector, and A is a positive real.It is shown that (XI-TY1y is positive if, and only if, y is annihilated by the absolute value of any generalized eigenvector of T* associated with a strictly positive eigenvalue not less than /.A strictly positive eigenvalue is a positive eigenvalue having an associated positive eigenvector.For the case of B=L" this yields the result that (A/-T)~ly±i0 if, and only if, y is almost everywhere zero on a certain set which depends on X but is otherwise fixed.In some fields of applied mathematics (e.g., radiative transfer, neutron transport) there occur conditional equations of the form (1) lv = Tx + y, in which the parameter A, the known element y, and the linear operator T are all positive, in the respective appropriate senses, and one wishes, for physical reasons, to conclude existence of a positive solution, x.The theorem given below has an obvious application to such problems.Its statement and proof are the primary purpose of this note.Before stating the result, we describe the setting within which (1) is considered.The terminology and notation used is that of Day [1].Let B be a real Banach space, and denote by A a closed cone in B such that B is a vector lattice under the partial order induced by K. We further suppose that the norm on B has, relative to the order induced by K, the property termed semimonotonic by Krasnosel'skil [2]; that is, there exists a positive real constant M such that the situation O^x^y implies ||x||^A/||j||.This property is the only connection which we require between the order and the norm.For zeB we denote z++z~=(zV0) -(zaO) by \z\, with a similar notation for the conjugate space B*.For z*eB* and zeB, we will generally denote z*(z) by (z, z*).The linear operator T is defined on B, is eventually compact (Tn is compact for some positive integer «),

Locations

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