{"title":"A monotone iterative method for second order nonlinear problems with boundary conditions driven by maximal monotone multivalued operators","authors":"D. A. Béhi","doi":"10.22436/jnsa.017.01.01","DOIUrl":null,"url":null,"abstract":"In this paper, we study the following second order differential equation: −( Φ ( u (cid:48) ( t ))) (cid:48) + φ p ( u ( t )) = εf ( t , u ( t )) a.e. on Ω = [ 0, T ] under nonlinear multivalued boundary value conditions which incorporate as special cases the classicals boundary value conditions of type Dirichlet, Neumann, and Sturm-Liouville. Using monotone iterative method coupled with lower and upper solutions method, multifunction analysis, theory of monotone operators, and theory of topological degree, we show existence of solution and extremal solutions when the lower and upper solutions are well ordered or not. Since the boundary value conditions do not include the periodic one, we show that our method stay true for the periodic problem.","PeriodicalId":48799,"journal":{"name":"Journal of Nonlinear Sciences and Applications","volume":"252 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Nonlinear Sciences and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.22436/jnsa.017.01.01","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we study the following second order differential equation: −( Φ ( u (cid:48) ( t ))) (cid:48) + φ p ( u ( t )) = εf ( t , u ( t )) a.e. on Ω = [ 0, T ] under nonlinear multivalued boundary value conditions which incorporate as special cases the classicals boundary value conditions of type Dirichlet, Neumann, and Sturm-Liouville. Using monotone iterative method coupled with lower and upper solutions method, multifunction analysis, theory of monotone operators, and theory of topological degree, we show existence of solution and extremal solutions when the lower and upper solutions are well ordered or not. Since the boundary value conditions do not include the periodic one, we show that our method stay true for the periodic problem.
本文研究以下二阶微分方程：-( Φ ( u (cid:48) ( t )) (cid:48) + φ p ( u ( t )) = εf ( t , u ( t )) a.e. on Ω = [ 0, T ] 在非线性多值边界值条件下，这些条件作为特例包含了迪里夏特、诺伊曼和斯特姆-柳维尔类型的经典边界值条件。利用单调迭代法与上下解法、多元函数分析、单调算子理论和拓扑度理论相结合，我们证明了在上下解有序与否的情况下，解和极值解的存在性。由于边界值条件不包括周期性条件，我们证明了我们的方法对于周期性问题是正确的。
期刊介绍：
The Journal of Nonlinear Science and Applications (JNSA) (print: ISSN 2008-1898 online: ISSN 2008-1901) is an international journal which provides very fast publication of original research papers in the fields of nonlinear analysis. Journal of Nonlinear Science and Applications is a journal that aims to unite and stimulate mathematical research community. It publishes original research papers and survey articles on all areas of nonlinear analysis and theoretical applied nonlinear analysis. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics. Manuscripts are invited from academicians, research students, and scientists for publication consideration. Papers are accepted for editorial consideration through online submission with the understanding that they have not been published, submitted or accepted for publication elsewhere.
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