{"title":"用不动点定理研究函数方程的渐近稳定性","authors":"Muhammad N. Islam, Jeffrey T. Neugebauer","doi":"10.12732/caa.v22i4.8","DOIUrl":null,"url":null,"abstract":"For many years, Liapunov’s direct method has been the primary technique for studying various stability also known as ‘Liapunov stability’ of functional differential equations. Recently, it has been noticed that some difficulties can arise when Liapunov’s method is applied to certain equations, and that a suitable fixed point theorem can overcome some of these difficulties. In this paper we study a particular stability which differs from the Liapunov stability. In particular, we study the existence of asymptotically stable solutions of a system of nonlinear Volterra integral equations. We employ a fixed point theorem due to Krasnosel’skii in the analysis. AMS Subject Classification: 47H10 Received: June 13, 2017 ; Accepted: October 31, 2018 ; Published: November 13, 2018. doi: 10.12732/caa.v22i4.8 Dynamic Publishers, Inc., Acad. Publishers, Ltd. http://www.acadsol.eu/caa 612 M.N. ISLAM AND J.T. NEUGEBAUER","PeriodicalId":92887,"journal":{"name":"Communications in applied analysis","volume":" ","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2018-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"ASYMPTOTIC STABILITY OF FUNCTIONAL EQUATIONS BY FIXED POINT THEOREMS\",\"authors\":\"Muhammad N. Islam, Jeffrey T. Neugebauer\",\"doi\":\"10.12732/caa.v22i4.8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For many years, Liapunov’s direct method has been the primary technique for studying various stability also known as ‘Liapunov stability’ of functional differential equations. Recently, it has been noticed that some difficulties can arise when Liapunov’s method is applied to certain equations, and that a suitable fixed point theorem can overcome some of these difficulties. In this paper we study a particular stability which differs from the Liapunov stability. In particular, we study the existence of asymptotically stable solutions of a system of nonlinear Volterra integral equations. We employ a fixed point theorem due to Krasnosel’skii in the analysis. AMS Subject Classification: 47H10 Received: June 13, 2017 ; Accepted: October 31, 2018 ; Published: November 13, 2018. doi: 10.12732/caa.v22i4.8 Dynamic Publishers, Inc., Acad. Publishers, Ltd. http://www.acadsol.eu/caa 612 M.N. ISLAM AND J.T. NEUGEBAUER\",\"PeriodicalId\":92887,\"journal\":{\"name\":\"Communications in applied analysis\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2018-11-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in applied analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.12732/caa.v22i4.8\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in applied analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.12732/caa.v22i4.8","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
ASYMPTOTIC STABILITY OF FUNCTIONAL EQUATIONS BY FIXED POINT THEOREMS
For many years, Liapunov’s direct method has been the primary technique for studying various stability also known as ‘Liapunov stability’ of functional differential equations. Recently, it has been noticed that some difficulties can arise when Liapunov’s method is applied to certain equations, and that a suitable fixed point theorem can overcome some of these difficulties. In this paper we study a particular stability which differs from the Liapunov stability. In particular, we study the existence of asymptotically stable solutions of a system of nonlinear Volterra integral equations. We employ a fixed point theorem due to Krasnosel’skii in the analysis. AMS Subject Classification: 47H10 Received: June 13, 2017 ; Accepted: October 31, 2018 ; Published: November 13, 2018. doi: 10.12732/caa.v22i4.8 Dynamic Publishers, Inc., Acad. Publishers, Ltd. http://www.acadsol.eu/caa 612 M.N. ISLAM AND J.T. NEUGEBAUER
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