Pub Date : 2019-06-04DOI: 10.37622/adsa/14.1.2019.67-81
Y. Raffoul
In this research, our aim is to use a new variation of parameters formula to analyze the behavior of the purely nonlinear functional delay differential equation that arise from population models x′(t) = g(x(t))− g(x(t− L)). Our approach will be based on the use of fixed point theory, by constructing suitable mapping on appropriate spaces. AMS Subject Classifications: 39A10,34A97.
{"title":"Nonlinear Functional Delay Differential Equations Arising from Population Models","authors":"Y. Raffoul","doi":"10.37622/adsa/14.1.2019.67-81","DOIUrl":"https://doi.org/10.37622/adsa/14.1.2019.67-81","url":null,"abstract":"In this research, our aim is to use a new variation of parameters formula to analyze the behavior of the purely nonlinear functional delay differential equation that arise from population models x′(t) = g(x(t))− g(x(t− L)). Our approach will be based on the use of fixed point theory, by constructing suitable mapping on appropriate spaces. AMS Subject Classifications: 39A10,34A97.","PeriodicalId":36469,"journal":{"name":"Advances in Dynamical Systems and Applications","volume":"52 2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78430114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-02DOI: 10.37622/adsa/14.1.2019.29-47
Y. Arioua
The aim of this work is to study the initial value problem of a coupled system of nonlinear fractional differential equations with Katugampola derivative. Some new existence and uniqueness results of solutions for the given problems are obtained by using the Banach contraction principle, Schauder’s and nonlinear alternative Leray–Schauder fixed point theorems. Several examples are presented to illustrate the usefulness of our main results. AMS Subject Classifications: 34A08, 34A12.
{"title":"Initial Value Problem for a Coupled System of Katugampola-Type Fractional Differential Equations","authors":"Y. Arioua","doi":"10.37622/adsa/14.1.2019.29-47","DOIUrl":"https://doi.org/10.37622/adsa/14.1.2019.29-47","url":null,"abstract":"The aim of this work is to study the initial value problem of a coupled system of nonlinear fractional differential equations with Katugampola derivative. Some new existence and uniqueness results of solutions for the given problems are obtained by using the Banach contraction principle, Schauder’s and nonlinear alternative Leray–Schauder fixed point theorems. Several examples are presented to illustrate the usefulness of our main results. AMS Subject Classifications: 34A08, 34A12.","PeriodicalId":36469,"journal":{"name":"Advances in Dynamical Systems and Applications","volume":"124 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76899762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-01DOI: 10.37622/adsa/14.1.2019.1-27
G. Anastassiou
Here we present the necessary background on nabla time scales approach. Then we give general related time scales nabla Iyengar type inequalities for all basic norms. We finish with applications to specific time scales like R, Z and qZ, q > 1. AMS Subject Classifications: 26D15, 39A12, 93C70.
{"title":"Nabla Time Scales Iyengar-Type Inequalities","authors":"G. Anastassiou","doi":"10.37622/adsa/14.1.2019.1-27","DOIUrl":"https://doi.org/10.37622/adsa/14.1.2019.1-27","url":null,"abstract":"Here we present the necessary background on nabla time scales approach. Then we give general related time scales nabla Iyengar type inequalities for all basic norms. We finish with applications to specific time scales like R, Z and qZ, q > 1. AMS Subject Classifications: 26D15, 39A12, 93C70.","PeriodicalId":36469,"journal":{"name":"Advances in Dynamical Systems and Applications","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86230352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-05-22DOI: 10.37622/adsa/14.1.2019.49-65
C. Liaw, J. Kelly, J. Osborn
The $X_m$ exceptional orthogonal polynomials (XOP) form a complete set of eigenpolynomials to a differential equation. Despite being complete, the XOP set does not contain polynomials of every degree. Thereby, the XOP escape the Bochner classification theorem. In literature two ways to obtain XOP have been presented. When m=1, Gram-Schmidt orthogonalization of a so-called "flag" was used. For general m, the Darboux transform was applied. Here, we present a possible flag for the X_m exceptional Laguerre polynomials. We can write more about this. We only want to make specific picks when we also derive determinantal representations. There is a large degree of freedom in doing so. Further, we derive determinantal representations of the X_2 exceptional Laguerre polynomials involving certain adjusted moments of the exceptional weights. We find a recursion formula for these adjusted moments. The particular canonical flag we pick keeps both the determinantal representation and the moment recursion manageable.
{"title":"Moment Representations of Type I X2 Exceptional Laguerre Polynomials","authors":"C. Liaw, J. Kelly, J. Osborn","doi":"10.37622/adsa/14.1.2019.49-65","DOIUrl":"https://doi.org/10.37622/adsa/14.1.2019.49-65","url":null,"abstract":"The $X_m$ exceptional orthogonal polynomials (XOP) form a complete set of eigenpolynomials to a differential equation. Despite being complete, the XOP set does not contain polynomials of every degree. Thereby, the XOP escape the Bochner classification theorem. \u0000In literature two ways to obtain XOP have been presented. When m=1, Gram-Schmidt orthogonalization of a so-called \"flag\" was used. For general m, the Darboux transform was applied. \u0000Here, we present a possible flag for the X_m exceptional Laguerre polynomials. We can write more about this. We only want to make specific picks when we also derive determinantal representations. There is a large degree of freedom in doing so. Further, we derive determinantal representations of the X_2 exceptional Laguerre polynomials involving certain adjusted moments of the exceptional weights. We find a recursion formula for these adjusted moments. The particular canonical flag we pick keeps both the determinantal representation and the moment recursion manageable.","PeriodicalId":36469,"journal":{"name":"Advances in Dynamical Systems and Applications","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87541908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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