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
This article studies the existence and uniqueness of the mild solution of a family of control systems with a delay that are governed by the nonlinear fractional evolution differential equations in Banach spaces.Moreov...This article studies the existence and uniqueness of the mild solution of a family of control systems with a delay that are governed by the nonlinear fractional evolution differential equations in Banach spaces.Moreover,we establish the controllability of the considered system.To do so,first,we investigate the approximate controllability of the corresponding linear system.Subsequently,we prove the nonlinear system is approximately controllable if the corresponding linear system is approximately controllable.To reach the conclusions,the theory of resolvent operators,the Banach contraction mapping principle,and fixed point theorems are used.While concluding,some examples are given to demonstrate the efficacy of the proposed results.展开更多
The linear equilibrium theory of thermoelasticity with microtemperatures for microstretch solids is considered.The basic internal and external boundary value problems(BVPs)are formulated and uniqueness theorems are gi...The linear equilibrium theory of thermoelasticity with microtemperatures for microstretch solids is considered.The basic internal and external boundary value problems(BVPs)are formulated and uniqueness theorems are given.The single-layer and double-layer thermoelastic potentials are constructed and their basic properties are established.The integral representation of general solutions is obtained.The existence of regular solutions of the BVPs is proved by means of the potential method(boundary integral method)and the theory of singular integral equations.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This article studies the existence and uniqueness of the mild solution of a family of control systems with a delay that are governed by the nonlinear fractional evolution differential equations in Banach spaces.Moreover,we establish the controllability of the considered system.To do so,first,we investigate the approximate controllability of the corresponding linear system.Subsequently,we prove the nonlinear system is approximately controllable if the corresponding linear system is approximately controllable.To reach the conclusions,the theory of resolvent operators,the Banach contraction mapping principle,and fixed point theorems are used.While concluding,some examples are given to demonstrate the efficacy of the proposed results.
文摘The linear equilibrium theory of thermoelasticity with microtemperatures for microstretch solids is considered.The basic internal and external boundary value problems(BVPs)are formulated and uniqueness theorems are given.The single-layer and double-layer thermoelastic potentials are constructed and their basic properties are established.The integral representation of general solutions is obtained.The existence of regular solutions of the BVPs is proved by means of the potential method(boundary integral method)and the theory of singular integral equations.
