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# On a certain maximum principle for positively invertible matrices and operators in an ordered normed space

DOI link for On a certain maximum principle for positively invertible matrices and operators in an ordered normed space

On a certain maximum principle for positively invertible matrices and operators in an ordered normed space book

# On a certain maximum principle for positively invertible matrices and operators in an ordered normed space

DOI link for On a certain maximum principle for positively invertible matrices and operators in an ordered normed space

On a certain maximum principle for positively invertible matrices and operators in an ordered normed space book

## ABSTRACT

Let a mathematical model of some physical, technological, economical, biological or other problem lead to an equation

Ax = y (0.1)

with a positive or an inverse monotone linear operator A in a normed space X (or a matrix A in a finite dimensional space) ordered by some cone K . In this paper for the equation (0.1) with some operator A we understand y to be a given arbitrary element of the cone K and look for the solution x and its properties. If y € K, y 0 (i. e. if the input y is positive) then besides the question whether (the output or response) x is positive, in the case of a finite dimensional space X and a matrix A it is also of interest, whether at least one of the maximal components of x is attained at the set of indices, where the corresponding components of y are strongly positive. More exactly, let A be an invertible (n, n)-matrix satisfying A-1 > 0 considered as a linear operator in the ordered normed space (R w, R+, || • ||), where R+ denotes the cone of all vectors x = (x \ , . . . , xn)T e Wl with nonnegative components and ||jc|| == (Y?!=\ x f ) 1^ 2 the norm. Then the mentioned property can be expressed as

the system where is a nonzero influence. A similar problem arises for operators in spaces of continuous functions on some com

pact topological space. As a further example we consider the following homogeneous Dirichlet (boundary

(k is some positive constant) and satisfy the conditions c(jc) > 0, f ( x ) > 0 for x e where Q denotes the closure of Q with respect to the norm || • || in Rw. Denote by supp(/) the closure of the set {.v e Q: / ( x) > 0} and, in order to avoid the trivial case, assume supp(/ ) ^ Q,.