In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operatorAu（x） = -△△u（x） ＋ V（x）u（x）,for all x ∈ R^n, in the Hilbert space H = L2（R^n,H1） with the operator po...In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operatorAu（x） = -△△u（x） ＋ V（x）u（x）,for all x ∈ R^n, in the Hilbert space H = L2（R^n,H1） with the operator potential V（x） ∈ C^1 （R^n, L （H1） ）, where L （H1 ） is the space of all bounded linear operators on the Hilbert space H1, while AAu is the biharmonic differential operator and△u=-∑i,j=1^n 1/√detg δ/δxi[√detgg-1（x）δu/δxj]is the Laplace-Beltrami differential operator in R^n. Here g（x） = （gij（x）） is the Riemannian matrix, while g^-1 （x） is the inverse of the matrix g（x）. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation Au = - △△u ＋ V（x） u （x） = f（x） in the Hilbert space H where f（x） ∈ H as an application of the separation approach.展开更多
For a physically possible deformation field of a continuum, the deformation gradient function F can be decomposed into direct sum of a symmetric tensor S and on orthogonal tensor R, which is called S-R decomposition t...For a physically possible deformation field of a continuum, the deformation gradient function F can be decomposed into direct sum of a symmetric tensor S and on orthogonal tensor R, which is called S-R decomposition theorem. In this paper, the S-R decomposition unique existence theorem is proved, by employing matrix and tensor method. Also, a brief proof of its objectivity is given.展开更多
An existence and uniqueness of solutions for boundary value problemsare established, by employing a priori estimates for all solutions and the Schau-der fixed point theorem. Some of these results extend earlier result...An existence and uniqueness of solutions for boundary value problemsare established, by employing a priori estimates for all solutions and the Schau-der fixed point theorem. Some of these results extend earlier results due to Usmani (1979), Yang (1988), and Pei (1997).展开更多
The paper is devoted to the proof of the uniqueness theorem for solution of the equation for the non-local ionization source in a glow discharge and a hollow cathode in general 3D geometry. The theorem is applied to w...The paper is devoted to the proof of the uniqueness theorem for solution of the equation for the non-local ionization source in a glow discharge and a hollow cathode in general 3D geometry. The theorem is applied to wide class of electric field configurations, and to the walls of discharge volume, which have a property of incomplete absorption of the electrons. Cathode is regarded as interior singular source, which is placed arbitrarily close to the wall. The existence of solution is considered also. During the proof of the theorem many of useful structure formulae are obtained. Elements of the proof structure, which have arisen, are found to have physical sense. It makes clear physical construction of non-local electron avalanche, which builds a source of ionization in glow discharge at low pressures. Last has decisive significance to understand the hollow cathode discharge configuration and the hollow cathode effect.展开更多
We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by vi...We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder＇s inequality, a suitable singular Cronwall＇s inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.展开更多
By fixed point theorem of a mixed monotone operators, we study Lidstone boundary value problems to nonlinear singular 2mth-order differential and difference equations, and provide sufficient conditions for the existen...By fixed point theorem of a mixed monotone operators, we study Lidstone boundary value problems to nonlinear singular 2mth-order differential and difference equations, and provide sufficient conditions for the existence and uniqueness of positive solution to Lidstone boundary value problem for 2mth-order ordinary differential equations and 2mth-order difference equations. The nonlinear term in the differential and difference equation may be singular.展开更多
In this paper, the existence and uniqueness of almost periodic solutions for some infinite delay integral equations are discussed. By using Krasnoselskii fixed point theorem,some new results are obtained.
This paper is concerned with a class of uncertain backward stochastic differential equations （UBSDEs） driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian c...This paper is concerned with a class of uncertain backward stochastic differential equations （UBSDEs） driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian coefficients. Such equations can be useful in mod- elling hybrid systems, where the phenomena are simultaneously subjected to two kinds of un- certainties： randomness and uncertainty. The solutions of UBSDEs are the uncertain stochastic processes. Thus, the existence and uniqueness of solutions to UBSDEs with Lipschitzian coeffi- cients are proved.展开更多
With the we of the homeomorphism theory and fixed point theory, the existence and uniqueness of solutions to boundary value problems are investigated. Two basic theorems are obtained without the boundness condition, w...With the we of the homeomorphism theory and fixed point theory, the existence and uniqueness of solutions to boundary value problems are investigated. Two basic theorems are obtained without the boundness condition, which generalizes results of Brown. When our results are applied to the existence and uniqueness of periodic solutions for nonlinear perturbed conservative systems (Newtonian equations of motion), the existence and uniqueness of the solution are obtained. The results in this note seem less restrictive than those of the former papers we have seen. Meanwhile, as far as we know, it seems that applying the homeomorphism theory to the research of this kind of problem is new.展开更多
In this paper, a mini max theorem was showed mega which the paper proves a new existent and unique result on solution of the boundary value problem for the nonlinear wave equation by using the mini max theorem.
In this paper, we study the existence and uniqueness of positive solution for 2mth-order nonlinear differential equation with boundary conditions, by using the fixed point theorems on compression and expansion of cones.
In this work, we study the following problem. , where ?is the fractional Laplacian and Ω?is a bounded domain in RN?with Lipschitz boundary. g: R→R?is an increasing locally Lipschitz continuous funct...In this work, we study the following problem. , where ?is the fractional Laplacian and Ω?is a bounded domain in RN?with Lipschitz boundary. g: R→R?is an increasing locally Lipschitz continuous function. and f∈Lm(Ω), . We use Stampacchia’s theorem to study existence of the solution u, and we prove the uniqueness of u by contradiction.展开更多
In this article, the existence and uniqueness of positive solution for a class of nonlinear fractional differential equations is proved by constructing the upper and lower control functions of the nonlinear term witho...In this article, the existence and uniqueness of positive solution for a class of nonlinear fractional differential equations is proved by constructing the upper and lower control functions of the nonlinear term without any monotone requirement. Our main method to the problem is the method of upper and lower solutions and Schauder fixed point theorem. Finally, we give an example to illuminate our results.展开更多
The problems of optimal control (OCPs) related to PDEs are a very active area of research. These problems deal with the processes of mechanical engineering, heat aeronautics, physics, hydro and gas dynamics, the physi...The problems of optimal control (OCPs) related to PDEs are a very active area of research. These problems deal with the processes of mechanical engineering, heat aeronautics, physics, hydro and gas dynamics, the physics of plasma and other real life problems. In this paper, we deal with a class of the constrained OCP for parabolic systems. It is converted to new unconstrained OCP by adding a penalty function to the cost functional. The existence solution of the considering system of parabolic optimal control problem (POCP) is introduced. In this way, the uniqueness theorem for the solving POCP is introduced. Therefore, a theorem for the sufficient differentiability conditions has been proved.展开更多
This article is concerned with the existence and uniqueness of positive radial solutions for a class of quasilinear elliptic system. With some reasonable assumptions on the nonlinear source functions and their coeffic...This article is concerned with the existence and uniqueness of positive radial solutions for a class of quasilinear elliptic system. With some reasonable assumptions on the nonlinear source functions and their coefficients, the existence and the upper and lower bounds of the positive solutions will be provided by using the fixed point theorem and the maximum principle for the quasilinear elliptic system.展开更多
In the paper, we consider the existence and uniqueness results for Caputo fractional differential equations with integral boundary value condition. The sufficient conditions of existence and uniqueness are obtained by...In the paper, we consider the existence and uniqueness results for Caputo fractional differential equations with integral boundary value condition. The sufficient conditions of existence and uniqueness are obtained by applying the contraction map-ping principle, Krasnoselskii＇s fixed point theorem and Leray-Schauder degree the-ory, which party improves and extends the associated results of fractional differentialequations. Four examples illustrating our main results are included.展开更多
文摘In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operatorAu（x） = -△△u（x） ＋ V（x）u（x）,for all x ∈ R^n, in the Hilbert space H = L2（R^n,H1） with the operator potential V（x） ∈ C^1 （R^n, L （H1） ）, where L （H1 ） is the space of all bounded linear operators on the Hilbert space H1, while AAu is the biharmonic differential operator and△u=-∑i,j=1^n 1/√detg δ/δxi[√detgg-1（x）δu/δxj]is the Laplace-Beltrami differential operator in R^n. Here g（x） = （gij（x）） is the Riemannian matrix, while g^-1 （x） is the inverse of the matrix g（x）. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation Au = - △△u ＋ V（x） u （x） = f（x） in the Hilbert space H where f（x） ∈ H as an application of the separation approach.
文摘For a physically possible deformation field of a continuum, the deformation gradient function F can be decomposed into direct sum of a symmetric tensor S and on orthogonal tensor R, which is called S-R decomposition theorem. In this paper, the S-R decomposition unique existence theorem is proved, by employing matrix and tensor method. Also, a brief proof of its objectivity is given.
文摘An existence and uniqueness of solutions for boundary value problemsare established, by employing a priori estimates for all solutions and the Schau-der fixed point theorem. Some of these results extend earlier results due to Usmani (1979), Yang (1988), and Pei (1997).
文摘The paper is devoted to the proof of the uniqueness theorem for solution of the equation for the non-local ionization source in a glow discharge and a hollow cathode in general 3D geometry. The theorem is applied to wide class of electric field configurations, and to the walls of discharge volume, which have a property of incomplete absorption of the electrons. Cathode is regarded as interior singular source, which is placed arbitrarily close to the wall. The existence of solution is considered also. During the proof of the theorem many of useful structure formulae are obtained. Elements of the proof structure, which have arisen, are found to have physical sense. It makes clear physical construction of non-local electron avalanche, which builds a source of ionization in glow discharge at low pressures. Last has decisive significance to understand the hollow cathode discharge configuration and the hollow cathode effect.
基金supported by Grant In Aid research fund of Virginia Military Instittue, USA
文摘We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder＇s inequality, a suitable singular Cronwall＇s inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.
基金supported by Scientific Research Fund of Heilongjiang Provincial Education Department (11544032)the National Natural Science Foundation of China (10571021, 10701020)
文摘By fixed point theorem of a mixed monotone operators, we study Lidstone boundary value problems to nonlinear singular 2mth-order differential and difference equations, and provide sufficient conditions for the existence and uniqueness of positive solution to Lidstone boundary value problem for 2mth-order ordinary differential equations and 2mth-order difference equations. The nonlinear term in the differential and difference equation may be singular.
基金supported by the National Natural Science Foundation of China（11371027） the Projects of Outstanding Young Talents of Universities in Anhui Province（gxyq2018116）+2 种基金 the Teaching Groups in Anhui Province（2016jxtd080,2015jxtd048） the NSF of Educational Bureau of Anhui Province（KJ2017A702,KJ2017A704） the NSF of Bozhou University（BZSZKYXM201302,BSKY201539）
文摘In this paper, the existence and uniqueness of almost periodic solutions for some infinite delay integral equations are discussed. By using Krasnoselskii fixed point theorem,some new results are obtained.
基金Supported by National Natural Science Foundation of China(71171003,71210107026)Anhui Natural Science Foundation(10040606003)Anhui Natural Science Foundation of Universities(KJ2012B019,KJ2013B023)
文摘This paper is concerned with a class of uncertain backward stochastic differential equations （UBSDEs） driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian coefficients. Such equations can be useful in mod- elling hybrid systems, where the phenomena are simultaneously subjected to two kinds of un- certainties： randomness and uncertainty. The solutions of UBSDEs are the uncertain stochastic processes. Thus, the existence and uniqueness of solutions to UBSDEs with Lipschitzian coeffi- cients are proved.
文摘With the we of the homeomorphism theory and fixed point theory, the existence and uniqueness of solutions to boundary value problems are investigated. Two basic theorems are obtained without the boundness condition, which generalizes results of Brown. When our results are applied to the existence and uniqueness of periodic solutions for nonlinear perturbed conservative systems (Newtonian equations of motion), the existence and uniqueness of the solution are obtained. The results in this note seem less restrictive than those of the former papers we have seen. Meanwhile, as far as we know, it seems that applying the homeomorphism theory to the research of this kind of problem is new.
基金the Natural Science Foundation of Southern Yangtze University China(0371)
文摘In this paper, a mini max theorem was showed mega which the paper proves a new existent and unique result on solution of the boundary value problem for the nonlinear wave equation by using the mini max theorem.
文摘In this paper, we study the existence and uniqueness of positive solution for 2mth-order nonlinear differential equation with boundary conditions, by using the fixed point theorems on compression and expansion of cones.
文摘In this work, we study the following problem. , where ?is the fractional Laplacian and Ω?is a bounded domain in RN?with Lipschitz boundary. g: R→R?is an increasing locally Lipschitz continuous function. and f∈Lm(Ω), . We use Stampacchia’s theorem to study existence of the solution u, and we prove the uniqueness of u by contradiction.
基金supported by Science and Technology Project of Chongqing Municipal Education Committee (kJ110501) of ChinaNatural Science Foundation Project of CQ CSTC (cstc2012jjA20016) of ChinaNational Natural Science Foundation of China (11101298)
文摘In this article, the existence and uniqueness of positive solution for a class of nonlinear fractional differential equations is proved by constructing the upper and lower control functions of the nonlinear term without any monotone requirement. Our main method to the problem is the method of upper and lower solutions and Schauder fixed point theorem. Finally, we give an example to illuminate our results.
文摘The problems of optimal control (OCPs) related to PDEs are a very active area of research. These problems deal with the processes of mechanical engineering, heat aeronautics, physics, hydro and gas dynamics, the physics of plasma and other real life problems. In this paper, we deal with a class of the constrained OCP for parabolic systems. It is converted to new unconstrained OCP by adding a penalty function to the cost functional. The existence solution of the considering system of parabolic optimal control problem (POCP) is introduced. In this way, the uniqueness theorem for the solving POCP is introduced. Therefore, a theorem for the sufficient differentiability conditions has been proved.
基金This work was supported by the National Natural Science Foundation of China （No. 1117 -1092 and 11471164） the Graduate Students Education and Innovation of Jiangsu Province （No. KYZZ_0209） and the Natural Science Foundation of Educational Department of Jiangsu Province （No. 08KJB110005）.
文摘This article is concerned with the existence and uniqueness of positive radial solutions for a class of quasilinear elliptic system. With some reasonable assumptions on the nonlinear source functions and their coefficients, the existence and the upper and lower bounds of the positive solutions will be provided by using the fixed point theorem and the maximum principle for the quasilinear elliptic system.
文摘In the paper, we consider the existence and uniqueness results for Caputo fractional differential equations with integral boundary value condition. The sufficient conditions of existence and uniqueness are obtained by applying the contraction map-ping principle, Krasnoselskii＇s fixed point theorem and Leray-Schauder degree the-ory, which party improves and extends the associated results of fractional differentialequations. Four examples illustrating our main results are included.
