In this work we adapt Bourgain’s ideas in [?] to a coupled system and we prove the L2 concentration of blow-up solutions for two-coupled nonlinear Schrödinger equations at critical dimension. Mathematics subject classification (2010): 35A01, 35Q55.
{"title":"L^2-concentration for a coupled nonlinear Schrödinger system","authors":"X. Carvajal, P. Gamboa","doi":"10.7153/DEA-2019-11-04","DOIUrl":"https://doi.org/10.7153/DEA-2019-11-04","url":null,"abstract":"In this work we adapt Bourgain’s ideas in [?] to a coupled system and we prove the L2 concentration of blow-up solutions for two-coupled nonlinear Schrödinger equations at critical dimension. Mathematics subject classification (2010): 35A01, 35Q55.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"135 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130447","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}
. In this manuscript, we examine the Hyer’s-Ulam stability and exact controllability results for impulsive evolution system on time scales. This manuscript has two segments: the fi rst segment of the work is concerned with the Hyer’s-Ulam type’s stability analysis and the other segment is to exact controllability results. We used the Banach fi xed point theorem, evolution operator theory and nonlinear functional analysis to establish these results. At last, we have presented some theoretical and numerical examples to outcome the utilization of these developed analytical results. scales by de fi ning the Picard operators and it is proved the proposed results are more general than some existing works.
{"title":"Stability and controllability results of evolution system with impulsive condition on time scales","authors":"Vipin Kumar, Muslim Malik","doi":"10.7153/dea-2019-11-27","DOIUrl":"https://doi.org/10.7153/dea-2019-11-27","url":null,"abstract":". In this manuscript, we examine the Hyer’s-Ulam stability and exact controllability results for impulsive evolution system on time scales. This manuscript has two segments: the fi rst segment of the work is concerned with the Hyer’s-Ulam type’s stability analysis and the other segment is to exact controllability results. We used the Banach fi xed point theorem, evolution operator theory and nonlinear functional analysis to establish these results. At last, we have presented some theoretical and numerical examples to outcome the utilization of these developed analytical results. scales by de fi ning the Picard operators and it is proved the proposed results are more general than some existing works.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130494","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}
The stability for delay differential equations with non-instantaneous impulses is studied using Lyapunov like functions and the Razumikhin technique. In these differential equations we have impulses, which start abruptly at some points and their action continue on given finite intervals. Sufficient conditions are given and they use comparison results for nonlinear scalar non-instantaneous impulsive equation without any delay. Examples are given to illustrate our stability properties and the influence of non-instantaneous impulses on the behavior of the solution.
{"title":"Razumikhin method to delay differential equations with non-instantaneous impulses","authors":"R. Agarwal, S. Hristova, D. Regan","doi":"10.7153/dea-2019-11-05","DOIUrl":"https://doi.org/10.7153/dea-2019-11-05","url":null,"abstract":"The stability for delay differential equations with non-instantaneous impulses is studied using Lyapunov like functions and the Razumikhin technique. In these differential equations we have impulses, which start abruptly at some points and their action continue on given finite intervals. Sufficient conditions are given and they use comparison results for nonlinear scalar non-instantaneous impulsive equation without any delay. Examples are given to illustrate our stability properties and the influence of non-instantaneous impulses on the behavior of the solution.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130458","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}
. An existence result for positive solutions to a fourth order differential inclusion with boundary values is given. This is accomplished by using a fi xed point theorem on cones for multivalued maps, L 1 selections and a generalization of the Ascoli theorem. The inclusion allows the function and its fi rst three derivatives to be on the right-hand side. The proof involves a Green’s function and a positive eigenvalue of a particular operator. An example is provided.
{"title":"Positive solutions for a fourth order differential inclusion based on the Euler-Bernoulli equation for a Cantilever beam","authors":"John S. Spraker","doi":"10.7153/dea-2019-11-26","DOIUrl":"https://doi.org/10.7153/dea-2019-11-26","url":null,"abstract":". An existence result for positive solutions to a fourth order differential inclusion with boundary values is given. This is accomplished by using a fi xed point theorem on cones for multivalued maps, L 1 selections and a generalization of the Ascoli theorem. The inclusion allows the function and its fi rst three derivatives to be on the right-hand side. The proof involves a Green’s function and a positive eigenvalue of a particular operator. An example is provided.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130481","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}
In this paper we prove some coupled and mixed coupled hybrid fixed point theorems involving different algebraic combinations of three operators and coupled operators in a partially ordered Banach algebra by an application of a coupled hybrid fixed point principle for partially condensing coupled mappings developed in Dhage [J. Fixed Point Theory Appl. 19 (2017), 2541–2575]. Our approach is based on the partial Kuratowskii measure of noncompactness with maximum property and is somewhat different from the approach of coupled hybrid fixed point theorems presented in Dhage [J. Fixed Point Theory Appl. 19 (2017), 3231–3264]. We apply our newly developed abstract mixed coupled hybrid fixed point theorems along with algorithms to a couple of nonlinear first and second order coupled quadratically perturbed hybrid differential equations with the periodic boundary conditions for proving the existence and approximation theorems under certain mixed hybrid conditions from algebra, analysis and topology. The abstract existence and approximation results of the coupled quadratic periodic boundary value problems of first and second order ordinary differential equations are also illustrated by presenting a few numerical examples. We claim that the results of this paper are new to the literature on nonlinear analysis applications. Mathematics subject classification (2010): 47H07, 47H10, 34A12, 34A45.
{"title":"Coupled and mixed coupled hybrid fixed point principles in a partially ordered Banach algebra and PBVPs of nonlinear coupled quadratic differential equations","authors":"B. Dhage","doi":"10.7153/dea-2019-11-01","DOIUrl":"https://doi.org/10.7153/dea-2019-11-01","url":null,"abstract":"In this paper we prove some coupled and mixed coupled hybrid fixed point theorems involving different algebraic combinations of three operators and coupled operators in a partially ordered Banach algebra by an application of a coupled hybrid fixed point principle for partially condensing coupled mappings developed in Dhage [J. Fixed Point Theory Appl. 19 (2017), 2541–2575]. Our approach is based on the partial Kuratowskii measure of noncompactness with maximum property and is somewhat different from the approach of coupled hybrid fixed point theorems presented in Dhage [J. Fixed Point Theory Appl. 19 (2017), 3231–3264]. We apply our newly developed abstract mixed coupled hybrid fixed point theorems along with algorithms to a couple of nonlinear first and second order coupled quadratically perturbed hybrid differential equations with the periodic boundary conditions for proving the existence and approximation theorems under certain mixed hybrid conditions from algebra, analysis and topology. The abstract existence and approximation results of the coupled quadratic periodic boundary value problems of first and second order ordinary differential equations are also illustrated by presenting a few numerical examples. We claim that the results of this paper are new to the literature on nonlinear analysis applications. Mathematics subject classification (2010): 47H07, 47H10, 34A12, 34A45.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130524","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}
In this paper, we consider the large-time behavior of solutions of a reaction diffusion system arising from a nuclear reactor model with the Robin boundary conditions, which consists of two real-valued unknown functions. In particular, we show that global solutions of this system are uniformly bounded in a suitable norm with respect to time. Since this system has no variational structure, we cannot apply the standard methods relying on the Lyapunov functional in order to obtain a priori estimates of global solutions. To cope with this difficulty, we make use of the weighted $L^1$ norm characterized by the first eigenfunction of Laplacian with the Robin boundary condition.
{"title":"Bounds for global solutions of a reaction diffusion system with the Robin boundary conditions","authors":"K. Kita, M. Otani","doi":"10.7153/dea-2019-11-09","DOIUrl":"https://doi.org/10.7153/dea-2019-11-09","url":null,"abstract":"In this paper, we consider the large-time behavior of solutions of a reaction diffusion system arising from a nuclear reactor model with the Robin boundary conditions, which consists of two real-valued unknown functions. In particular, we show that global solutions of this system are uniformly bounded in a suitable norm with respect to time. Since this system has no variational structure, we cannot apply the standard methods relying on the Lyapunov functional in order to obtain a priori estimates of global solutions. To cope with this difficulty, we make use of the weighted $L^1$ norm characterized by the first eigenfunction of Laplacian with the Robin boundary condition.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46233244","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}
In this paper, we propose a generalized Gronwall inequality through the fractional integral with respect to another function. The Cauchy-type problem for a nonlinear differential equation involving the $psi$-Hilfer fractional derivative and the existence and uniqueness of solutions are discussed. Finally, through generalized Gronwall inequality, we prove the continuous dependence of data on the Cauchy-type problem.
{"title":"A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator","authors":"J. Sousa, E. C. Oliveira","doi":"10.7153/DEA-2019-11-02","DOIUrl":"https://doi.org/10.7153/DEA-2019-11-02","url":null,"abstract":"In this paper, we propose a generalized Gronwall inequality through the fractional integral with respect to another function. The Cauchy-type problem for a nonlinear differential equation involving the $psi$-Hilfer fractional derivative and the existence and uniqueness of solutions are discussed. Finally, through generalized Gronwall inequality, we prove the continuous dependence of data on the Cauchy-type problem.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2017-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45489229","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}
{"title":"Stability of nonautonomous impulsive evolution system on time scale","authors":"A. Zada, Y. Arafat, Syed Omar Shah","doi":"10.7153/dea-2021-13-20","DOIUrl":"https://doi.org/10.7153/dea-2021-13-20","url":null,"abstract":"","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130715","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}
. In this article, we propose a general facultative mutualistic model with harvesting and investigate an associated optimal control problem. The suf fi cient and the necessary conditions for the existence of the optimal control are studied. Numerical simulations are carried out to show the ef fi ciency of the proposed control.
{"title":"Optimal control to a facultative mutualistic model with harvesting","authors":"Liangcheng Wang, Min Wang","doi":"10.7153/DEA-2020-12-02","DOIUrl":"https://doi.org/10.7153/DEA-2020-12-02","url":null,"abstract":". In this article, we propose a general facultative mutualistic model with harvesting and investigate an associated optimal control problem. The suf fi cient and the necessary conditions for the existence of the optimal control are studied. Numerical simulations are carried out to show the ef fi ciency of the proposed control.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130088","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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