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Solutions of nonlinear equations in cones and positive linear operators
Author(s) -
Webb J. R. L.
Publication year - 2010
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdq037
Subject(s) - fixed point index , mathematical proof , mathematics , fixed point theorem , nonlinear system , cone (formal languages) , invariant (physics) , operator (biology) , fixed point , linear map , index (typography) , mathematical analysis , pure mathematics , computer science , algorithm , geometry , physics , mathematical physics , biochemistry , chemistry , quantum mechanics , repressor , world wide web , transcription factor , gene , boundary value problem
We introduce a modification of the concept of a cone invariant linear operator being u 0 ‐positive, a concept due to Krasnosel'skiĭ. We show how this definition allows us to prove results closely related to some results for positive operators in a classic text of Krasnosel'skiĭ. We also show how it can be used, in conjunction with the theory of a fixed‐point index, to give some short proofs of existence and nonexistence of positive solutions for nonlinear maps. We prove a new fixed‐point index result and show that it can be applied in cases where the previous theory is not applicable.