In this paper, we study the existence, uniqueness and exponential stability of the square-mean almost automorphic solution for stochastic evolution equation with impulses on time scales. For this purpose, we introduce the concept of equipotentially square-mean almost automorphic sequence and square-mean almost automorphic functions with impulses on time scales. At the end, a numerical example is given to illustrate the effectiveness of the obtained theoretical results.
{"title":"Square mean almost automorphic solution of stochastic evolution equations with impulses on time scales","authors":"Soniya Dhama, Syed Abbas","doi":"10.7153/DEA-2018-10-30","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-30","url":null,"abstract":"In this paper, we study the existence, uniqueness and exponential stability of the square-mean almost automorphic solution for stochastic evolution equation with impulses on time scales. For this purpose, we introduce the concept of equipotentially square-mean almost automorphic sequence and square-mean almost automorphic functions with impulses on time scales. At the end, a numerical example is given to illustrate the effectiveness of the obtained theoretical results.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"12 1","pages":"449-469"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86854813","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 article is devoted to study the fractional Neumann elliptic problem ⎧⎪⎨ ⎪⎩ ε2s(−Δ)su+u = up in Ω, ∂νu = 0 on ∂Ω, u > 0 in Ω, where Ω is a smooth bounded domain of RN , N > 2s , 0 < s s0 < 1 , 1 < p < (N +2s)/(N− 2s) , ε > 0 and ν is the outer normal to ∂Ω . We show that there exists at least one nonconstant solution uε to this problem provided ε is small. Moreover, we prove that uε ∈ L∞(Ω) by using Moser-Nash iteration.
{"title":"Existence of positive solutions for nonlinear fractional Neumann elliptic equations","authors":"Haoqi Ni, Aliang Xia, Xiongjun Zheng","doi":"10.7153/DEA-2018-10-07","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-07","url":null,"abstract":"This article is devoted to study the fractional Neumann elliptic problem ⎧⎪⎨ ⎪⎩ ε2s(−Δ)su+u = up in Ω, ∂νu = 0 on ∂Ω, u > 0 in Ω, where Ω is a smooth bounded domain of RN , N > 2s , 0 < s s0 < 1 , 1 < p < (N +2s)/(N− 2s) , ε > 0 and ν is the outer normal to ∂Ω . We show that there exists at least one nonconstant solution uε to this problem provided ε is small. Moreover, we prove that uε ∈ L∞(Ω) by using Moser-Nash iteration.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"26 1","pages":"115-129"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89380038","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 note is concerned with the problem of existence and uniqueness of solutions for a fourth order boundary value problem that models the deflection of a hinged plate of nonconstant thickness.
本文讨论了一类四阶边值问题解的存在唯一性问题，该问题模拟了非定厚铰接板的挠度。
{"title":"On a hinged plate equation of nonconstant thickness","authors":"C. Danet","doi":"10.7153/DEA-2018-10-16","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-16","url":null,"abstract":"This note is concerned with the problem of existence and uniqueness of solutions for a fourth order boundary value problem that models the deflection of a hinged plate of nonconstant thickness.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"177 1","pages":"235-238"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86225567","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 work we study a third-order three-point boundary-value problem (BVP). We derive sucient conditions that guarantee the positivity of the solution of the corresponding linear BVP Then, based on the classi- cal Guo-Krasnosel'skii's fixed point theorem, we obtain positive solutions to the nonlinear BVP. Additional hypotheses guarantee the uniqueness of the solution.
{"title":"Existence and uniqueness of monotone positive solutions for a third-order three-point boundary value problem","authors":"A. Palamides, N. Stavrakakis","doi":"10.7153/DEA-2018-10-18","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-18","url":null,"abstract":"In this work we study a third-order three-point boundary-value problem (BVP). We derive sucient conditions that guarantee the positivity of the solution of the corresponding linear BVP Then, based on the classi- cal Guo-Krasnosel'skii's fixed point theorem, we obtain positive solutions to the nonlinear BVP. Additional hypotheses guarantee the uniqueness of the solution.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"33 1","pages":"251-260"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88620767","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 differential equation with cubic nonlinearity x′′ = −ax + bx3 is considered together with the boundary conditions x(−1) = x(1) = 0 . In the autonomous case, b = const > 0 , the exact number of solutions for the boundary value problem is given. For nonautonomous case, where b = β(t) is a step-wise function, the existence of additional solutions is detected. The reasons for such behaviour are revealed. The example considered in this paper is supplemented by a number of visualizations.
{"title":"On boundary value problem for equations with cubic nonlinearity and step-wise coefficient","authors":"A. Kirichuka, F. Sadyrbaev","doi":"10.7153/dea-2018-10-29","DOIUrl":"https://doi.org/10.7153/dea-2018-10-29","url":null,"abstract":"The differential equation with cubic nonlinearity x′′ = −ax + bx3 is considered together with the boundary conditions x(−1) = x(1) = 0 . In the autonomous case, b = const > 0 , the exact number of solutions for the boundary value problem is given. For nonautonomous case, where b = β(t) is a step-wise function, the existence of additional solutions is detected. The reasons for such behaviour are revealed. The example considered in this paper is supplemented by a number of visualizations.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"91 1 1","pages":"433-447"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86466915","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 definition of normal and extremal solutions of the scalar Riccati equation with complex coefficients is given. Some properties of normal and extremal solutions to Riccati equation are studied. On the basis of the obtained, some theorems which describe the asymptotic behavior of solutions of the system of two linear first order ordinary differential equations are proved (in particular a minimality theorem of a solution of the system of two linear first order ordinary differential equations is proved).
{"title":"Properties of solutions of the scalar Riccati equation with complex coefficients and some their applications","authors":"G. Grigorian","doi":"10.7153/DEA-2018-10-20","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-20","url":null,"abstract":"The definition of normal and extremal solutions of the scalar Riccati equation with complex coefficients is given. Some properties of normal and extremal solutions to Riccati equation are studied. On the basis of the obtained, some theorems which describe the asymptotic behavior of solutions of the system of two linear first order ordinary differential equations are proved (in particular a minimality theorem of a solution of the system of two linear first order ordinary differential equations is proved).","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"60 1","pages":"277-298"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90455730","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 work we provide conditions for the existence of solutions to nonlinear SturmLiouville problems of the form (p(t)x′(t))′ +q(t)x(t)+λx(t) = f (x(t)) subject to non-local boundary conditions ax(0)+bx′(0) = η1(x) and cx(1)+dx′(1) = η2(x). Our approach will be topological, utilizing Schaefer’s fixed point theorem and the LyapunovSchmidt procedure.
{"title":"Existence theory for nonlinear Sturm-Liouville problems with non-local boundary conditions","authors":"D. Maroncelli, Jesús F. Rodríguez","doi":"10.7153/DEA-2018-10-09","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-09","url":null,"abstract":"In this work we provide conditions for the existence of solutions to nonlinear SturmLiouville problems of the form (p(t)x′(t))′ +q(t)x(t)+λx(t) = f (x(t)) subject to non-local boundary conditions ax(0)+bx′(0) = η1(x) and cx(1)+dx′(1) = η2(x). Our approach will be topological, utilizing Schaefer’s fixed point theorem and the LyapunovSchmidt procedure.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"11 1","pages":"147-161"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88614180","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 is essentially a survey on grand and small Lebesgue spaces, which are rearrangement-invariant Banach function spaces of interest not only from the point of view of Function Spaces theory, but also from the point of view of their applications: the corresponding Sobolev spaces are of interest, for instance, in the theory of PDEs. We discuss results of existence, uniqueness and regularity of certain Dirichlet problems, where the knowledge of these spaces plays a central role. The novelty of this paper relies in an unified treatment containing a number of equivalent quasinorms, all written making explicit the dependence of |Ω| , in the discussion of the sharpness of Hölder’s inequality, and in the connection of the results in PDEs with some existing literature. 1. Grand and small Lebesgue spaces: a short overview 1.1. The original motivation Let Ω ⊂ Rn be a bounded domain and f : Ω →Rn , f = ( f 1, ..., f n) be a mapping of Sobolev class W 1,n loc (Ω,R n) . Let us denote by Df (x) : Rn → Rn the differential and by J(x, f ) = detD f (x) the Jacobian of f . After the elementary remark that by Hölder’s inequality the Jacobian J(x, f ) is in Lloc(Ω) , the first fundamental result on the integrability of the Jacobian was due to Müller ([135]): f ∈W (Ω,R), J(x, f ) 0 a.e. ⇒ J(x, f ) ∈ L logLloc(Ω). Mathematics subject classification (2010): 46E30, 35J65.
本文不仅从函数空间理论的角度，而且从其应用的角度对大小勒贝格空间进行了综述，这些空间是重排不变的Banach函数空间，其相应的Sobolev空间在偏微分方程理论中具有重要意义。讨论了一类Dirichlet问题的存在性、唯一性和正则性的结果，其中这些空间的知识起着中心作用。本文的新颖之处在于，它采用了统一的处理方法，其中包含了若干等价的拟规范，这些拟规范都明确了|Ω|的依赖性，它讨论了Hölder不等式的尖锐性，并将偏微分方程的结果与一些现有文献联系起来。1. 勒贝格空间的大小:简要概述设Ω∧Rn为有界域，f: Ω→Rn, f = (f1，…)， f n)是Sobolev类w1,n loc (Ω，R n)的映射。我们用Df (x)表示Rn→Rn微分用J(x, f) = detD f (x)表示f的雅可比矩阵。在通过Hölder不等式证明雅可比矩阵J(x, f)在Lloc(Ω)中之后，关于雅可比矩阵可积性的第一个基本结果是由m([135])得出的:f∈W (Ω，R)， J(x, f) 0 a.e.⇒J(x, f)∈lloglloc (Ω)。数学学科分类(2010):46E30, 35J65。
{"title":"On grand and small Lebesgue and Sobolev spaces and some applications to PDE's","authors":"A. Fiorenza, M. R. Formica, Amiran Gogatishvili","doi":"10.7153/DEA-2018-10-03","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-03","url":null,"abstract":"This paper is essentially a survey on grand and small Lebesgue spaces, which are rearrangement-invariant Banach function spaces of interest not only from the point of view of Function Spaces theory, but also from the point of view of their applications: the corresponding Sobolev spaces are of interest, for instance, in the theory of PDEs. We discuss results of existence, uniqueness and regularity of certain Dirichlet problems, where the knowledge of these spaces plays a central role. The novelty of this paper relies in an unified treatment containing a number of equivalent quasinorms, all written making explicit the dependence of |Ω| , in the discussion of the sharpness of Hölder’s inequality, and in the connection of the results in PDEs with some existing literature. 1. Grand and small Lebesgue spaces: a short overview 1.1. The original motivation Let Ω ⊂ Rn be a bounded domain and f : Ω →Rn , f = ( f 1, ..., f n) be a mapping of Sobolev class W 1,n loc (Ω,R n) . Let us denote by Df (x) : Rn → Rn the differential and by J(x, f ) = detD f (x) the Jacobian of f . After the elementary remark that by Hölder’s inequality the Jacobian J(x, f ) is in Lloc(Ω) , the first fundamental result on the integrability of the Jacobian was due to Müller ([135]): f ∈W (Ω,R), J(x, f ) 0 a.e. ⇒ J(x, f ) ∈ L logLloc(Ω). Mathematics subject classification (2010): 46E30, 35J65.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"27 1","pages":"21-46"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78327821","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}
We establish new Lyapunov-type inequalities for linear Hadamard fractional differential equations with pointwise boundary conditions. Furthermore, we employ the contraction mapping principle to obtain the criterion of the existence of a unique solution for a nonlinear fractional Hadamard type boundary value problem.
{"title":"On linear and nonlinear fractional Hadamard boundary value problems","authors":"Sougata Dhar","doi":"10.7153/DEA-2018-10-23","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-23","url":null,"abstract":"We establish new Lyapunov-type inequalities for linear Hadamard fractional differential equations with pointwise boundary conditions. Furthermore, we employ the contraction mapping principle to obtain the criterion of the existence of a unique solution for a nonlinear fractional Hadamard type boundary value problem.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"10 1","pages":"329-339"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78908933","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}
We are interested in the system of conservation laws modeling the pressureless magnetogasdynamics. Firstly, we solve the Riemann problem and obtain five kinds of structures consisting of combinations of shocks, rarefaction waves and contact discontinuities. Secondly, we study the vanishing magnetic field limits of the Riemann solutions to the pressureless magnetogasdynamics and show that the density and velocity in the Riemann solutions to the pressureless magnetogasdynamics converge to the Riemann solutions to the pressureless gas dynamics. The formation processes of delta-shocks and vacuum states are discussed in details.
{"title":"Vanishing magnetic field limits of solutions to the pressureless magnetogasdynamics","authors":"Hongjun Cheng, Zhongshun Sun","doi":"10.7153/dea-2018-10-10","DOIUrl":"https://doi.org/10.7153/dea-2018-10-10","url":null,"abstract":"We are interested in the system of conservation laws modeling the pressureless magnetogasdynamics. Firstly, we solve the Riemann problem and obtain five kinds of structures consisting of combinations of shocks, rarefaction waves and contact discontinuities. Secondly, we study the vanishing magnetic field limits of the Riemann solutions to the pressureless magnetogasdynamics and show that the density and velocity in the Riemann solutions to the pressureless magnetogasdynamics converge to the Riemann solutions to the pressureless gas dynamics. The formation processes of delta-shocks and vacuum states are discussed in details.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"396 1","pages":"163-182"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74945722","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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