. In this paper, we consider a variational discretization combined with fully discrete splitting positive definite mixed finite element approximation of parabolic optimal control problems. For the state and co-state, Raviart-Thomas mixed finite element spaces and backward Euler scheme are used for space and time discretization, respectively. The variational discretization technique is used for the approximation of the control variable. We derive a priori error estimates for the control, state, and co-state. A numerical example is presented to demonstrate the theoretical results.
{"title":"Variational discretization combined with fully discrete splitting positive definite mixed finite elements for parabolic optimal control problems","authors":"","doi":"10.23952/jnfa.2023.11","DOIUrl":"https://doi.org/10.23952/jnfa.2023.11","url":null,"abstract":". In this paper, we consider a variational discretization combined with fully discrete splitting positive definite mixed finite element approximation of parabolic optimal control problems. For the state and co-state, Raviart-Thomas mixed finite element spaces and backward Euler scheme are used for space and time discretization, respectively. The variational discretization technique is used for the approximation of the control variable. We derive a priori error estimates for the control, state, and co-state. A numerical example is presented to demonstrate the theoretical results.","PeriodicalId":44514,"journal":{"name":"Journal of Nonlinear Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136002663","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 study efficiency of numerical simulation for integrable and nonintegrable discrete Non-linear Schr ¨ odinger equations (NLSE). We first discretize the NLSE into two classical spatial models, nonintegrable direct discrete model and integrable Ablowitz-Ladik model. By some simple transformations and Doubarx transformation, we obtain two integrable models from Ablowitz-Ladik model. Then, five different kinds of schemes can be applied to simulate four models in bright and dark cases for comparing the performance in preserving the conserved quantities’ approximations of NLSE. The numerical experiments indicate that Gauss symplectic method is more efficient than nonsymplectic schemes and splitting schemes when simulating the same model. Both intergrable models and nonintergrable model have their own advantages in preserving the conserved quanti-ties’ approximations. For the three integrable models, Ablowitz-Ladik Model and the model which has a general symplectic structure have similar simulation effects, and the model owing a cononical symplectic structure has low efficiency because the complicated Doubarx transformations make the model difficult to solve. Moreover, symplectic scheme and symmetric scheme have overwhelming superiorities over nonsymplectic schemes in preserving the invariants of Hamiltonian system.
{"title":"Comparative numerical study of integrable and nonintegrable discrete models of nonlinear Schrödinger equations","authors":"","doi":"10.23952/jnfa.2023.29","DOIUrl":"https://doi.org/10.23952/jnfa.2023.29","url":null,"abstract":". In this paper, we study efficiency of numerical simulation for integrable and nonintegrable discrete Non-linear Schr ¨ odinger equations (NLSE). We first discretize the NLSE into two classical spatial models, nonintegrable direct discrete model and integrable Ablowitz-Ladik model. By some simple transformations and Doubarx transformation, we obtain two integrable models from Ablowitz-Ladik model. Then, five different kinds of schemes can be applied to simulate four models in bright and dark cases for comparing the performance in preserving the conserved quantities’ approximations of NLSE. The numerical experiments indicate that Gauss symplectic method is more efficient than nonsymplectic schemes and splitting schemes when simulating the same model. Both intergrable models and nonintergrable model have their own advantages in preserving the conserved quanti-ties’ approximations. For the three integrable models, Ablowitz-Ladik Model and the model which has a general symplectic structure have similar simulation effects, and the model owing a cononical symplectic structure has low efficiency because the complicated Doubarx transformations make the model difficult to solve. Moreover, symplectic scheme and symmetric scheme have overwhelming superiorities over nonsymplectic schemes in preserving the invariants of Hamiltonian system.","PeriodicalId":44514,"journal":{"name":"Journal of Nonlinear Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136004409","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 use the dual variable to propose a two-step inertial adaptive iterative algorithm for solving the split common fixed point problem of directed operators in real Hilbert spaces. Under suitable conditions, we obtain the weak convergence of the proposed algorithm and give applications in the split feasibility problem. A numerical experiment is given to illustrate the efficiency of the proposed iterative algorithm.
{"title":"Two-step inertial adaptive iterative algorithm for solving the split common fixed point problem of directed operators","authors":"","doi":"10.23952/jnfa.2023.20","DOIUrl":"https://doi.org/10.23952/jnfa.2023.20","url":null,"abstract":". In this paper, we use the dual variable to propose a two-step inertial adaptive iterative algorithm for solving the split common fixed point problem of directed operators in real Hilbert spaces. Under suitable conditions, we obtain the weak convergence of the proposed algorithm and give applications in the split feasibility problem. A numerical experiment is given to illustrate the efficiency of the proposed iterative algorithm.","PeriodicalId":44514,"journal":{"name":"Journal of Nonlinear Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136004421","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 study, we investigate the existence of at least one weak solution for a nonlinear Steklov boundary-value problem involving weighted p ( · ) -Laplacian. Our technical approach is based on variational methods. In addition, an example to illustrate our results is given
{"title":"Existence of one weak solution for a Steklov problem involving the weighted $p(cdot)$-Laplacian","authors":"","doi":"10.23952/jnfa.2023.8","DOIUrl":"https://doi.org/10.23952/jnfa.2023.8","url":null,"abstract":". In this study, we investigate the existence of at least one weak solution for a nonlinear Steklov boundary-value problem involving weighted p ( · ) -Laplacian. Our technical approach is based on variational methods. In addition, an example to illustrate our results is given","PeriodicalId":44514,"journal":{"name":"Journal of Nonlinear Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136004426","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":"Strong convergence analysis for solving quasi-monotone variational inequalities and fixed point problems in reflexive Banach spaces","authors":"","doi":"10.23952/jnfa.2023.30","DOIUrl":"https://doi.org/10.23952/jnfa.2023.30","url":null,"abstract":"","PeriodicalId":44514,"journal":{"name":"Journal of Nonlinear Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136002661","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}
. This paper investigates the linear convergence of a projection algorithm for solving the split equality mixed equilibrium problem (SEMEP). We introduce the notion of bounded linear regularity property for the SE-MEP and construct several sufficient conditions to prove its linear convergence. Furthermore, the result of the linear convergence of the SEMEP is applied to split equality equilibrium problems, split equality convex minimization problems, split equality mixed variational inequality problems, and split equality variational inequality problems. Finally, numerical results are provided to verify the effectiveness of our proposed algorithm
{"title":"Linear convergence of an iterative algorithm for solving the split equality mixed equilibrium problem","authors":"","doi":"10.23952/jnfa.2023.14","DOIUrl":"https://doi.org/10.23952/jnfa.2023.14","url":null,"abstract":". This paper investigates the linear convergence of a projection algorithm for solving the split equality mixed equilibrium problem (SEMEP). We introduce the notion of bounded linear regularity property for the SE-MEP and construct several sufficient conditions to prove its linear convergence. Furthermore, the result of the linear convergence of the SEMEP is applied to split equality equilibrium problems, split equality convex minimization problems, split equality mixed variational inequality problems, and split equality variational inequality problems. Finally, numerical results are provided to verify the effectiveness of our proposed algorithm","PeriodicalId":44514,"journal":{"name":"Journal of Nonlinear Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136002675","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":"Superclose analysis of $H^1$-Galerkin mixed finite element methods combined with two-grid scheme for semilinear parabolic equations","authors":"","doi":"10.23952/jnfa.2023.3","DOIUrl":"https://doi.org/10.23952/jnfa.2023.3","url":null,"abstract":"","PeriodicalId":44514,"journal":{"name":"Journal of Nonlinear Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136004434","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":"Existence and multiplicity of solutions for a class of fractional Hamiltonian systems with separated variables","authors":"","doi":"10.23952/jnfa.2023.25","DOIUrl":"https://doi.org/10.23952/jnfa.2023.25","url":null,"abstract":"","PeriodicalId":44514,"journal":{"name":"Journal of Nonlinear Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136002655","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":"Exact null controllability for semilinear differential equations with nonlocal conditions in Hilbert spaces","authors":"","doi":"10.23952/jnfa.2023.21","DOIUrl":"https://doi.org/10.23952/jnfa.2023.21","url":null,"abstract":"","PeriodicalId":44514,"journal":{"name":"Journal of Nonlinear Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136002659","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":"Some common coupled fixed point results for the mappings with a new contractive condition in a Menger PbM-metric space","authors":"","doi":"10.23952/jnfa.2023.9","DOIUrl":"https://doi.org/10.23952/jnfa.2023.9","url":null,"abstract":"","PeriodicalId":44514,"journal":{"name":"Journal of Nonlinear Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136002681","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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