In this paper, we extend the third order $ p $-Laplacian boundary value problem researched by S. Iyase and O. Imaga in [11] to the fractional differential equation. Firstly, we construct a mild Banach space and establish an appropriate compactness criterion. Then applying the Schauder's fixed point theorem, we obtain a sufficient condition for existence of at least one solution to the fractional differential equation with $ p $-Laplacian operator on an infinite interval. As an application, an example is given to illustrate our main result.
本文将S. Iyase和O. Imaga在[11]中研究的三阶$ p $-拉普拉斯边值问题推广到分数阶微分方程。首先构造了一个温和的Banach空间，并建立了一个适当的紧性准则。然后应用Schauder不动点定理，得到了具有p $-拉普拉斯算子的分数阶微分方程在无限区间上存在至少一个解的充分条件。作为应用，给出了一个例子来说明我们的主要结果。
{"title":"SOLVABILITY OF A FRACTIONAL BOUNDARY VALUE PROBLEM WITH <i>P</i>-LAPLACIAN OPERATOR ON AN INFINITE INTERVAL","authors":"Xingfang Feng, Yucheng Li","doi":"10.11948/20220329","DOIUrl":"https://doi.org/10.11948/20220329","url":null,"abstract":"In this paper, we extend the third order $ p $-Laplacian boundary value problem researched by S. Iyase and O. Imaga in [<xref ref-type=\"bibr\" rid=\"b11\">11</xref>] to the fractional differential equation. Firstly, we construct a mild Banach space and establish an appropriate compactness criterion. Then applying the Schauder's fixed point theorem, we obtain a sufficient condition for existence of at least one solution to the fractional differential equation with $ p $-Laplacian operator on an infinite interval. As an application, an example is given to illustrate our main result.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"2014 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we are concerned with a class of Schrödinger-Poisson systems in $mathbb{R}^3$ with zero mass and periodic potential. Under some 3-superlinear assumptions on the nonlinearity, one nontrivial generalized solution is obtained by a combination of variational methods and perturbation method.
{"title":"NONTRIVIAL GENERALIZED SOLUTION OF SCHRÖDINGER-POISSON SYSTEM IN <inline-formula><tex-math id=\"M1\">$mathbb{R}^3$</tex-math></inline-formula> WITH ZERO MASS AND PERIODIC POTENTIAL","authors":"Anran Li, Chongqing Wei, Leiga Zhao","doi":"10.11948/20230122","DOIUrl":"https://doi.org/10.11948/20230122","url":null,"abstract":"In this paper, we are concerned with a class of Schrödinger-Poisson systems in $mathbb{R}^3$ with zero mass and periodic potential. Under some 3-superlinear assumptions on the nonlinearity, one nontrivial generalized solution is obtained by a combination of variational methods and perturbation method.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"89 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We are concerned with the nonlinear Schrödinger equation
$ begin{equation*}-Delta u+lambda u=g(u)text{ in }mathbb{R}^{N}text{, }lambda inmathbb{R}, end{equation*} $ with prescribed begin{document}$L^{2}$end{document}-norm begin{document}$int_{mathbb{R}^{N}}u^{2}dx=rho ^{2}$end{document}. Under general assumptions about the nonlinearity which allows at least mass critical growth, we prove the existence of a ground state solution to the problem via a clear constrained minimization method.
We are concerned with the nonlinear Schrödinger equation $ begin{equation*}-Delta u+lambda u=g(u)text{ in }mathbb{R}^{N}text{, }lambda inmathbb{R}, end{equation*} $ with prescribed begin{document}$L^{2}$end{document}-norm begin{document}$int_{mathbb{R}^{N}}u^{2}dx=rho ^{2}$end{document}. Under general assumptions about the nonlinearity which allows at least mass critical growth, we prove the existence of a ground state solution to the problem via a clear constrained minimization method.
{"title":"REMARKS ON NORMALIZED GROUND STATES OF SCHRÖDINGER EQUATION WITH AT LEAST MASS CRITICAL NONLINEARITY","authors":"Yanyan Liu, Leiga Zhao","doi":"10.11948/20230139","DOIUrl":"https://doi.org/10.11948/20230139","url":null,"abstract":"We are concerned with the nonlinear Schrödinger equation <p class=\"disp_formula\">$ begin{equation*}-Delta u+lambda u=g(u)text{ in }mathbb{R}^{N}text{, }lambda inmathbb{R}, end{equation*} $ with prescribed <inline-formula><tex-math id=\"M1\">begin{document}$L^{2}$end{document}</tex-math></inline-formula>-norm <inline-formula><tex-math id=\"M2\">begin{document}$int_{mathbb{R}^{N}}u^{2}dx=rho ^{2}$end{document}</tex-math></inline-formula>. Under general assumptions about the nonlinearity which allows at least mass critical growth, we prove the existence of a ground state solution to the problem via a clear constrained minimization method.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a multiscale finite element scheme on a graded mesh for solving a singularly perturbed convection-diffusion problem efficiently. Twin boundary layers phenomena are shown in the one-dimensional model, and an adaptively graded mesh is applied to probe the twin boundary jumps. We evoke an updated multiscale strategy through the multiscale basis functions in a linear Lagrange style. Detailed mapping behaviors are investigated on fine as well as on coarse scales, thus incorporating information at the micro-scale into the macroscopic data. High-order stability theorems in an energy norm of multiscale errors are addressed. Our approach can achieve a parameter-uniform superconvergence with limited computational costs on the coarse graded mesh. Numerical results support the high-order convergence theorem and validate the advantages over other prevalent methods in the literature, especially for the singular perturbation with very small parameters. The proposed method is twin boundary layers resolving as well as parameter uniform superconvergent.
{"title":"PARAMETER-UNIFORM SUPERCONVERGENCE OF MULTISCALE COMPUTATION FOR SINGULAR PERTURBATION EXHIBITING TWIN BOUNDARY LAYERS","authors":"Shan Jiang, Xiao Ding, Meiling Sun","doi":"10.11948/20230020","DOIUrl":"https://doi.org/10.11948/20230020","url":null,"abstract":"We propose a multiscale finite element scheme on a graded mesh for solving a singularly perturbed convection-diffusion problem efficiently. Twin boundary layers phenomena are shown in the one-dimensional model, and an adaptively graded mesh is applied to probe the twin boundary jumps. We evoke an updated multiscale strategy through the multiscale basis functions in a linear Lagrange style. Detailed mapping behaviors are investigated on fine as well as on coarse scales, thus incorporating information at the micro-scale into the macroscopic data. High-order stability theorems in an energy norm of multiscale errors are addressed. Our approach can achieve a parameter-uniform superconvergence with limited computational costs on the coarse graded mesh. Numerical results support the high-order convergence theorem and validate the advantages over other prevalent methods in the literature, especially for the singular perturbation with very small parameters. The proposed method is twin boundary layers resolving as well as parameter uniform superconvergent.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pradip Ramesh Patle, Moosa Gabeleh, Manuel De La Sen
In this article, a class of cyclic (noncyclic) condensing operators is defined on a Banach space using the notion of measure of noncompactness and $ C $-class functions. For these newly defined condensing operators, best proximity point (pair) results are manifested. Then the obtained main results are applied to demonstrate the existence of optimum solutions of a system of fractional differential equations involving $ psi $-Hilfer fractional derivatives.
本文利用非紧性测度的概念和$ C $-类函数在Banach空间上定义了一类循环(非循环)压缩算子。对于这些新定义的凝聚算子，得到了最佳接近点(对)结果。然后应用所得的主要结果，证明了一类包含$ psi $-Hilfer分数阶导数的分数阶微分方程组最优解的存在性。
{"title":"ON BEST PROXIMITY POINT APPROACH TO SOLVABILITY OF A SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS","authors":"Pradip Ramesh Patle, Moosa Gabeleh, Manuel De La Sen","doi":"10.11948/20230007","DOIUrl":"https://doi.org/10.11948/20230007","url":null,"abstract":"In this article, a class of cyclic (noncyclic) condensing operators is defined on a Banach space using the notion of measure of noncompactness and $ C $-class functions. For these newly defined condensing operators, best proximity point (pair) results are manifested. Then the obtained main results are applied to demonstrate the existence of optimum solutions of a system of fractional differential equations involving $ psi $-Hilfer fractional derivatives.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Eduardo A. Alarcon, Marcos R. Batista, Alysson Cunha, Jesus C. Da Mota, Ronaldo A. Santos
This study proved that the Cauchy problem for a one-dimensional reaction-diffusion-convection system is locally and globally well-posed in $ mathtt{H}^2(mathbb{R})$. The system modeled a gasless combustion front through a multi-layer porous medium when the fuel concentration in each layer was a known function. Combustion has critical practical porous media applications, such as in in-situ combustion processes in oil reservoirs and several other areas. Earlier studies considered physical parameters (e.g., porosity, thermal conductivity, heat capacity, and initial fuel concentration) constant. Here, we consider a more realistic model where these parameters are functions of the spatial variable rather than constants. Furthermore, in previous studies, we did not consider the continuity of the solution regarding the initial data and parameters, unlike the current study. This proof uses a novel approach to combustion problems in porous media. We follow the abstract semigroups theory of operators in the Hilbert space and the well-known Kato's theory for a well-posed associated initial value problem.
{"title":"APPLICATION OF THE SEMIGROUP THEORY TO A COMBUSTION PROBLEM IN A MULTI-LAYER POROUS MEDIUM","authors":"Eduardo A. Alarcon, Marcos R. Batista, Alysson Cunha, Jesus C. Da Mota, Ronaldo A. Santos","doi":"10.11948/20220333","DOIUrl":"https://doi.org/10.11948/20220333","url":null,"abstract":"This study proved that the Cauchy problem for a one-dimensional reaction-diffusion-convection system is locally and globally well-posed in $ mathtt{H}^2(mathbb{R})$. The system modeled a gasless combustion front through a multi-layer porous medium when the fuel concentration in each layer was a known function. Combustion has critical practical porous media applications, such as in in-situ combustion processes in oil reservoirs and several other areas. Earlier studies considered physical parameters (e.g., porosity, thermal conductivity, heat capacity, and initial fuel concentration) constant. Here, we consider a more realistic model where these parameters are functions of the spatial variable rather than constants. Furthermore, in previous studies, we did not consider the continuity of the solution regarding the initial data and parameters, unlike the current study. This proof uses a novel approach to combustion problems in porous media. We follow the abstract semigroups theory of operators in the Hilbert space and the well-known Kato's theory for a well-posed associated initial value problem.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we consider some q-curl-curl equations with lack of compactness. Our analysis is developed in the abstract setting of exterior domains. We first recall a decomposition of $ {rm curl} $-free space based on $ L^{r} $-Helmholtz-Weyl decomposition in exterior domains. Then by reducing the original system into a div-curl system and a $ p $-Laplacian equation with Neumann boundary condition, we obtain the solvability of solutions for the q-curl-curl equation.
本文考虑了一些缺乏紧性的q-旋-旋方程。我们的分析是在外部域的抽象设置中进行的。我们首先回顾了基于外域$ L^{r} $-Helmholtz-Weyl分解的$ {rm curl} $自由空间的分解。然后通过将原系统简化为一个div-旋度系统和一个具有Neumann边界条件的$ p $- laplace方程，得到了q-旋度方程解的可解性。
{"title":"SOLVABILITY OF QUASILINEAR MAXWELL EQUATIONS IN EXTERIOR DOMAINS","authors":"Zifei Shen, Shuijin Zhang","doi":"10.11948/20230121","DOIUrl":"https://doi.org/10.11948/20230121","url":null,"abstract":"In this paper we consider some q-curl-curl equations with lack of compactness. Our analysis is developed in the abstract setting of exterior domains. We first recall a decomposition of $ {rm curl} $-free space based on $ L^{r} $-Helmholtz-Weyl decomposition in exterior domains. Then by reducing the original system into a div-curl system and a $ p $-Laplacian equation with Neumann boundary condition, we obtain the solvability of solutions for the q-curl-curl equation.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135106048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we develop almost-periodic tori bifurcation theory for $ 2 $-dimensional degenerate Hamiltonian vector fields. With KAM theory and singularity theory, we show that the universal unfolding of completely degenerate Hamiltonian $ N(x, y)=x^2y+y^l $ and partially degenerate Hamiltonian $ M(x, y)=x^2+y^l, $ respectively, can persist under any small almost-periodic time-dependent perturbation and some appropriate non-resonant conditions on almost-periodic frequency $ omega=(cdots, omega_i, cdots)_{iin mathbb{Z}}in mathbb{R}^mathbb{Z}. $ We extend the analysis about almost-periodic bifurcations of one-dimensional degenerate vector fields considered in [21] to $ 2 $-dimensional degenerate vector fields. Our main results (Theorem 2.1 and Theorem 2.2) imply infinite-dimensional degenerate umbilical tori or normally parabolic tori bifurcate according to a generalised umbilical catastrophe or generalised cuspoid catastrophe under any small almost-periodic perturbation. For the proof in this paper we use the overall strategy of [21], which however has to be substantially developed to deal with the equations considered here.
{"title":"ALMOST-PERIODIC BIFURCATIONS FOR 2-DIMENSIONAL DEGENERATE HAMILTONIAN VECTOR FIELDS","authors":"Xinyu Guan, Wen Si","doi":"10.11948/20220163","DOIUrl":"https://doi.org/10.11948/20220163","url":null,"abstract":"In this paper, we develop almost-periodic tori bifurcation theory for $ 2 $-dimensional degenerate Hamiltonian vector fields. With KAM theory and singularity theory, we show that the universal unfolding of completely degenerate Hamiltonian $ N(x, y)=x^2y+y^l $ and partially degenerate Hamiltonian $ M(x, y)=x^2+y^l, $ respectively, can persist under any small almost-periodic time-dependent perturbation and some appropriate non-resonant conditions on almost-periodic frequency $ omega=(cdots, omega_i, cdots)_{iin mathbb{Z}}in mathbb{R}^mathbb{Z}. $ We extend the analysis about almost-periodic bifurcations of one-dimensional degenerate vector fields considered in [<xref ref-type=\"bibr\" rid=\"b21\">21</xref>] to $ 2 $-dimensional degenerate vector fields. Our main results (Theorem 2.1 and Theorem 2.2) imply infinite-dimensional degenerate umbilical tori or normally parabolic tori bifurcate according to a generalised umbilical catastrophe or generalised cuspoid catastrophe under any small almost-periodic perturbation. For the proof in this paper we use the overall strategy of [<xref ref-type=\"bibr\" rid=\"b21\">21</xref>], which however has to be substantially developed to deal with the equations considered here.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135106055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, building on the previous published work by Li and Wu [Appl. Math, Lett., 2015, 44, 26–29], we extend the single-step HSS (SHSS) method for saddle point problems. Based on the idea of SHSS method, the SHSS preconditioner for solving saddle point problems is introduced. We discuss the spectral properties of the preconditioned matrix in detail. By some numerical experiments, we demonstrate the effectiveness of the SHSS preconditioner.
{"title":"THE SHSS PRECONDITIONER FOR SADDLE POINT PROBLEMS","authors":"Cuixia Li, Shiliang Wu","doi":"10.11948/20220552","DOIUrl":"https://doi.org/10.11948/20220552","url":null,"abstract":"In this paper, building on the previous published work by Li and Wu [Appl. Math, Lett., 2015, 44, 26–29], we extend the single-step HSS (SHSS) method for saddle point problems. Based on the idea of SHSS method, the SHSS preconditioner for solving saddle point problems is introduced. We discuss the spectral properties of the preconditioned matrix in detail. By some numerical experiments, we demonstrate the effectiveness of the SHSS preconditioner.","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135105589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A MORE ACCURATE HALF-DISCRETE HILBERT-TYPE INEQUALITY INVOLVING ONE HIGHER-ORDER DERIVATIVE FUNCTION","authors":"J. Zhong, Bicheng Yang, Qiang Chen","doi":"10.11948/20210223","DOIUrl":"https://doi.org/10.11948/20210223","url":null,"abstract":"","PeriodicalId":48811,"journal":{"name":"Journal of Applied Analysis and Computation","volume":"34 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87959680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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