Positive solutions of a nonlinear algebraic system with sign-changing coefficient matrix
Abstract Existence of positive solutions for the nonlinear algebraic system x = λ G F ( x ) $x=\lambda GF ( x ) $ has been extensively studied when the n × n $n\times n$ coefficient matrix G is positive or nonnegative. However, to the best of our knowledge, few results have been obtained when the co...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2020-11-01
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| Series: | Advances in Difference Equations |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13662-020-03090-1 |
| _version_ | 1818219199802114048 |
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| author | Yanping Jia Ying Gao Wenying Feng Guang Zhang |
| author_facet | Yanping Jia Ying Gao Wenying Feng Guang Zhang |
| author_sort | Yanping Jia |
| collection | DOAJ |
| description | Abstract Existence of positive solutions for the nonlinear algebraic system x = λ G F ( x ) $x=\lambda GF ( x ) $ has been extensively studied when the n × n $n\times n$ coefficient matrix G is positive or nonnegative. However, to the best of our knowledge, few results have been obtained when the coefficient matrix changes sign. In this case, some commonly applied analysis methods such as the cone theory, the Krein–Rutman theorem, the monotone iterative techniques, and so on cannot be directly applied. In this note, we prove the existence of positive solutions for the above nonlinear algebraic system with sign-changing coefficient matrix taking the advantages of the classical Brouwer fixed point theorem combined with a decomposition condition on the coefficient matrix. We provide an example in solving a second-order difference equation with periodic boundary conditions to illustrate the applications of the results. |
| first_indexed | 2024-12-12T07:35:52Z |
| format | Article |
| id | doaj.art-0172e492372346bdbb04494272394fb2 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-12T07:35:52Z |
| publishDate | 2020-11-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-0172e492372346bdbb04494272394fb22022-12-22T00:32:55ZengSpringerOpenAdvances in Difference Equations1687-18472020-11-01202011610.1186/s13662-020-03090-1Positive solutions of a nonlinear algebraic system with sign-changing coefficient matrixYanping Jia0Ying Gao1Wenying Feng2Guang Zhang3School of Mathematics and Statistics, Shanxi Datong UniversitySchool of Mathematics and Statistics, Shanxi Datong UniversityDepartment of Mathematics, Trent UniversitySchool of Science, Tianjin University of CommerceAbstract Existence of positive solutions for the nonlinear algebraic system x = λ G F ( x ) $x=\lambda GF ( x ) $ has been extensively studied when the n × n $n\times n$ coefficient matrix G is positive or nonnegative. However, to the best of our knowledge, few results have been obtained when the coefficient matrix changes sign. In this case, some commonly applied analysis methods such as the cone theory, the Krein–Rutman theorem, the monotone iterative techniques, and so on cannot be directly applied. In this note, we prove the existence of positive solutions for the above nonlinear algebraic system with sign-changing coefficient matrix taking the advantages of the classical Brouwer fixed point theorem combined with a decomposition condition on the coefficient matrix. We provide an example in solving a second-order difference equation with periodic boundary conditions to illustrate the applications of the results.http://link.springer.com/article/10.1186/s13662-020-03090-1Fixed point theoremNonlinear algebraic systemPositive solutionDifference equation |
| spellingShingle | Yanping Jia Ying Gao Wenying Feng Guang Zhang Positive solutions of a nonlinear algebraic system with sign-changing coefficient matrix Advances in Difference Equations Fixed point theorem Nonlinear algebraic system Positive solution Difference equation |
| title | Positive solutions of a nonlinear algebraic system with sign-changing coefficient matrix |
| title_full | Positive solutions of a nonlinear algebraic system with sign-changing coefficient matrix |
| title_fullStr | Positive solutions of a nonlinear algebraic system with sign-changing coefficient matrix |
| title_full_unstemmed | Positive solutions of a nonlinear algebraic system with sign-changing coefficient matrix |
| title_short | Positive solutions of a nonlinear algebraic system with sign-changing coefficient matrix |
| title_sort | positive solutions of a nonlinear algebraic system with sign changing coefficient matrix |
| topic | Fixed point theorem Nonlinear algebraic system Positive solution Difference equation |
| url | http://link.springer.com/article/10.1186/s13662-020-03090-1 |
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