{"title":"一类具有齐次性的非线性差分方程组的可解性","authors":"Stevo Stević","doi":"10.1016/j.aml.2025.109595","DOIUrl":null,"url":null,"abstract":"<div><div>We show that the following nonlinear system of difference equations of interest <span><span><span><math><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow><mrow><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow></mrow></msubsup><mo>=</mo><mfrac><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>l</mi></mrow></msub><munderover><mrow><mo>∏</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>j</mi><mo>≠</mo><mi>l</mi></mrow><mrow><mi>k</mi></mrow></munderover><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mrow><mo>(</mo><mi>j</mi><mo>)</mo></mrow></mrow></msubsup></mrow><mrow><mi>f</mi><mrow><mo>(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></msubsup><mo>)</mo></mrow></mrow></mfrac><mo>,</mo><mspace></mspace><mi>n</mi><mo>∈</mo><mi>N</mi><mo>,</mo><mspace></mspace><mi>l</mi><mo>=</mo><mover><mrow><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mo>¯</mo></mover><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow><mrow><mrow><mo>(</mo><mi>j</mi><mo>)</mo></mrow></mrow></msubsup><mo>∈</mo><mi>ℂ</mi><mo>∖</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow><mo>,</mo></mrow></math></span> <span><math><mrow><mi>j</mi><mo>=</mo><mover><mrow><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mo>¯</mo></mover><mo>,</mo></mrow></math></span> and the function <span><math><mrow><mi>f</mi><mo>:</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>→</mo><mi>ℂ</mi></mrow></math></span> is homogeneous of degree <span><math><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></math></span>, is solvable in a closed form considerably extending some results in the literature.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"169 ","pages":"Article 109595"},"PeriodicalIF":2.9000,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Solvability of a class of nonlinear system of difference equations with homogeneity\",\"authors\":\"Stevo Stević\",\"doi\":\"10.1016/j.aml.2025.109595\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We show that the following nonlinear system of difference equations of interest <span><span><span><math><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow><mrow><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow></mrow></msubsup><mo>=</mo><mfrac><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>l</mi></mrow></msub><munderover><mrow><mo>∏</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>j</mi><mo>≠</mo><mi>l</mi></mrow><mrow><mi>k</mi></mrow></munderover><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mrow><mo>(</mo><mi>j</mi><mo>)</mo></mrow></mrow></msubsup></mrow><mrow><mi>f</mi><mrow><mo>(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></msubsup><mo>)</mo></mrow></mrow></mfrac><mo>,</mo><mspace></mspace><mi>n</mi><mo>∈</mo><mi>N</mi><mo>,</mo><mspace></mspace><mi>l</mi><mo>=</mo><mover><mrow><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mo>¯</mo></mover><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow><mrow><mrow><mo>(</mo><mi>j</mi><mo>)</mo></mrow></mrow></msubsup><mo>∈</mo><mi>ℂ</mi><mo>∖</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow><mo>,</mo></mrow></math></span> <span><math><mrow><mi>j</mi><mo>=</mo><mover><mrow><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mo>¯</mo></mover><mo>,</mo></mrow></math></span> and the function <span><math><mrow><mi>f</mi><mo>:</mo><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>→</mo><mi>ℂ</mi></mrow></math></span> is homogeneous of degree <span><math><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></math></span>, is solvable in a closed form considerably extending some results in the literature.</div></div>\",\"PeriodicalId\":55497,\"journal\":{\"name\":\"Applied Mathematics Letters\",\"volume\":\"169 \",\"pages\":\"Article 109595\"},\"PeriodicalIF\":2.9000,\"publicationDate\":\"2025-05-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applied Mathematics Letters\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0893965925001454\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics Letters","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0893965925001454","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Solvability of a class of nonlinear system of difference equations with homogeneity
We show that the following nonlinear system of difference equations of interest where , and the function is homogeneous of degree , is solvable in a closed form considerably extending some results in the literature.
期刊介绍:
The purpose of Applied Mathematics Letters is to provide a means of rapid publication for important but brief applied mathematical papers. The brief descriptions of any work involving a novel application or utilization of mathematics, or a development in the methodology of applied mathematics is a potential contribution for this journal. This journal''s focus is on applied mathematics topics based on differential equations and linear algebra. Priority will be given to submissions that are likely to appeal to a wide audience.