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H. Pühl, W. Schirotzek, M. Weber (1998)
On matrices satisfying a maximum principle with respect to a coneLinear Algebra and its Applications, 277
G. Stoyan (1986)
On maximum principles for monotone matricesLinear Algebra and its Applications, 78
G. Stoyan (1982)
On a Maximum Principle for Matrices, and on Conservation of Monotonicity. With Applications to Discretization MethodsZamm-zeitschrift Fur Angewandte Mathematik Und Mechanik, 62
C. Aliprantis, Kim Border (1994)
Infinite dimensional analysis
B. Beauzamy (1985)
Introduction to Banach spaces and their geometry
(1978)
Special topics of geometry of cones in normed spaces
C. Türke, M. Weber (1995)
On a maximum principle for inverse monotone matricesLinear Algebra and its Applications, 218
B.Z. Vulikh (1977)
Introduction to the theory of cones in normed spaces
The equation $$Ax = y$$ for a positive linear continuous operator is considered in an ordered normed space $$\left( {X,K,\left\| \cdot \right\|} \right)$$ , where the cone is assumed to be closed and having a nonempty interior. Then the dual cone $$K'$$ of K possesses a base $${\mathcal{F}}$$ . Generalizing the well known maximum principle for positive matrices an operator A is said to satisfy the maximum principle, if for any $$x \in K,\;x \ne 0$$ there exists a positive linear continuous functional $$f \in {\mathcal{F}}$$ which is both, maximal on the element Ax, i.e. $$f\left( {Ax} \right) = \sup _{g \in {\mathcal{F}}} g\left( {Ax} \right)$$ , and positive on the element x, i.e. $$f\left( x \right) >0$$ . This property is studied and characterized both analytically by some extreme point condition and geometrically by means of the behaviour under A of the faces of the cone K. It is shown that the conditions which have been obtained for finite dimensional spaces in earlier relevant papers are special cases of conditions presented in this paper. The maximum pinciple is proved for simple operators in the spaces $$C\left[ {0,1} \right]$$ and c.
Positivity – Springer Journals
Published: Oct 25, 2004
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