This article is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. A family of weak regularizing operators is introduced. If the spectrum of A is contain...This article is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. A family of weak regularizing operators is introduced. If the spectrum of A is contained in a sector of right-half complex plane and its resolvent is polynomially bounded, the weak regularization for such ill-posed Cauchy problem can be shown by using the quasi-reversibilky method and regularized semigroups. Finally, an example is given.展开更多
Some characterizations of preregular operators between two Banach lattices are presented. Then several sufficient conditions for preregular operators being regular are given, and some related results are also obtained.
Given general quasi-differential expressions , each of order n with complex coefficients and their formal adjoint are on the interval [a,b) respectively, we give a characterization of all regularly solvable operators ...Given general quasi-differential expressions , each of order n with complex coefficients and their formal adjoint are on the interval [a,b) respectively, we give a characterization of all regularly solvable operators and their adjoints generated by a general ordinary quasi-differential expression in the direct sum Hilbert spaces . The domains of these operators are described in terms of boundary conditions involving -solutions of the equations and their adjoint on the intervals [a<sub>p</sub>,b<sub>p</sub>). This characterization is an extension of those obtained in the case of one interval with one and two singular end-points of the interval (a,b), and is a generalization of those proved in the case of self-adjoint and J-self-adjoint differential operators as a special case, where J denotes complex conjugation.展开更多
In this paper, we have considered the general ordinary quasi-differential operators generated by a general quasi-differential expression τ<sub>p,q</sub> in L<sup>p</sup>w</sub>-spaces of...In this paper, we have considered the general ordinary quasi-differential operators generated by a general quasi-differential expression τ<sub>p,q</sub> in L<sup>p</sup>w</sub>-spaces of order n with complex coefficients and its formal adjoint τ<sup>+</sup><sub>q',p' </sub>in L<sup>p</sup>w</sub>-spaces for arbitrary p,q∈[1,∞). We have proved in the case of one singular end-point that all well-posed extensions of the minimal operator T<sub>0</sub> (τ<sub>p,q</sub>) generated by such expression τ<sub>p,q</sub> and their formal adjoint on the interval [a,b) with maximal deficiency indices have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions can be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while others are new.展开更多
In this paper we define the tensor products of spaces of exponential type vectors of closed unbounded operators in Banach spaces. Using the real method of interpolation （K-functional） we prove the interpolation theo...In this paper we define the tensor products of spaces of exponential type vectors of closed unbounded operators in Banach spaces. Using the real method of interpolation （K-functional） we prove the interpolation theorems that permit to characterize of tensor products of spaces of exponential type vectors, We show an application of abstract results to the theory of regular elliptic operators on bounded domains. For such operators the exponential type vectors are root vectors. Thus we describe the tensor products of root vectors of regular elliptic operators on bounded domains.展开更多
Non-self-adjoint quasi-differential expression M and its formal adjoint M＋may generate nonsymmetric ordinary differential operators. Although minimal operators T0, T＋0 generated by M, M＋are not symmetric, they form...Non-self-adjoint quasi-differential expression M and its formal adjoint M＋may generate nonsymmetric ordinary differential operators. Although minimal operators T0, T＋0 generated by M, M＋are not symmetric, they form an adjoint pair. In this paper, author studies regularly solvable operators with respect to the adjoint pair T0, T＋0 in two kinds of conditions and give their geometry description in the corresponding ways.展开更多
In this paper,we study the continuous dependence of eigenvalue of Sturm-Liouville differential operators on the boundary condition by using of implicit function theorem.The work not only provides a new and elementary ...In this paper,we study the continuous dependence of eigenvalue of Sturm-Liouville differential operators on the boundary condition by using of implicit function theorem.The work not only provides a new and elementary proof of the above results,but also explicitly presents the expressions for derivatives of the n-th eigenvalue with respect to given parameters.Further-more,we obtain the new results of the position and number of the generated double eigenvalues under the real coupled boundary condition.展开更多
The regularization of ill-posed problems has become a useful tool in studying initial value problems that do not adhere to certain desired properties such as continuous dependence of solutions on initial data.Because ...The regularization of ill-posed problems has become a useful tool in studying initial value problems that do not adhere to certain desired properties such as continuous dependence of solutions on initial data.Because direct computation of the solution becomes difficult in this situation,many authors have alternatively approximated the solution by the solution of a closely defined well posed problem.In this paper,we demonstrate this process of regularization for nonautonomous ill-posed problems including the backward heat equation with a time-dependent diffusion coefficient.In the process,we provide two different approximate well posed models and numerically compare convergence rates of their solutions to a known solution of the original ill-posed problem.展开更多
This paper deals with a discrete-time dynamical system generated by a modified susceptible-infected-recovered-dead model(SIRD model;nonlinear operator)in threedimensional simplex.We introduce a novel approach that inc...This paper deals with a discrete-time dynamical system generated by a modified susceptible-infected-recovered-dead model(SIRD model;nonlinear operator)in threedimensional simplex.We introduce a novel approach that incorporates the SIRD model with the quadratic stochastic operator(QSO)that allows for real-time forecasting.The basic reproductive number Ro is obtained.We describe the set of fixed points of the operator and demonstrate that all fixed points are non-hyperbolic.Further,we study the asymptotical behavior of the trajectories of this system and show that SIRD operators havea regularity property.展开更多
基金This project was supported by TRAPOYT, the Key Project of Chinese Ministry of Education(104126) the NNSF of China(10371046)
文摘This article is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. A family of weak regularizing operators is introduced. If the spectrum of A is contained in a sector of right-half complex plane and its resolvent is polynomially bounded, the weak regularization for such ill-posed Cauchy problem can be shown by using the quasi-reversibilky method and regularized semigroups. Finally, an example is given.
文摘Some characterizations of preregular operators between two Banach lattices are presented. Then several sufficient conditions for preregular operators being regular are given, and some related results are also obtained.
文摘Given general quasi-differential expressions , each of order n with complex coefficients and their formal adjoint are on the interval [a,b) respectively, we give a characterization of all regularly solvable operators and their adjoints generated by a general ordinary quasi-differential expression in the direct sum Hilbert spaces . The domains of these operators are described in terms of boundary conditions involving -solutions of the equations and their adjoint on the intervals [a<sub>p</sub>,b<sub>p</sub>). This characterization is an extension of those obtained in the case of one interval with one and two singular end-points of the interval (a,b), and is a generalization of those proved in the case of self-adjoint and J-self-adjoint differential operators as a special case, where J denotes complex conjugation.
文摘In this paper, we have considered the general ordinary quasi-differential operators generated by a general quasi-differential expression τ<sub>p,q</sub> in L<sup>p</sup>w</sub>-spaces of order n with complex coefficients and its formal adjoint τ<sup>+</sup><sub>q',p' </sub>in L<sup>p</sup>w</sub>-spaces for arbitrary p,q∈[1,∞). We have proved in the case of one singular end-point that all well-posed extensions of the minimal operator T<sub>0</sub> (τ<sub>p,q</sub>) generated by such expression τ<sub>p,q</sub> and their formal adjoint on the interval [a,b) with maximal deficiency indices have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions can be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while others are new.
文摘In this paper we define the tensor products of spaces of exponential type vectors of closed unbounded operators in Banach spaces. Using the real method of interpolation （K-functional） we prove the interpolation theorems that permit to characterize of tensor products of spaces of exponential type vectors, We show an application of abstract results to the theory of regular elliptic operators on bounded domains. For such operators the exponential type vectors are root vectors. Thus we describe the tensor products of root vectors of regular elliptic operators on bounded domains.
文摘Non-self-adjoint quasi-differential expression M and its formal adjoint M＋may generate nonsymmetric ordinary differential operators. Although minimal operators T0, T＋0 generated by M, M＋are not symmetric, they form an adjoint pair. In this paper, author studies regularly solvable operators with respect to the adjoint pair T0, T＋0 in two kinds of conditions and give their geometry description in the corresponding ways.
基金supported in part by the National Natural Science Foundation of China(Grant No.11571212).
文摘In this paper,we study the continuous dependence of eigenvalue of Sturm-Liouville differential operators on the boundary condition by using of implicit function theorem.The work not only provides a new and elementary proof of the above results,but also explicitly presents the expressions for derivatives of the n-th eigenvalue with respect to given parameters.Further-more,we obtain the new results of the position and number of the generated double eigenvalues under the real coupled boundary condition.
文摘The regularization of ill-posed problems has become a useful tool in studying initial value problems that do not adhere to certain desired properties such as continuous dependence of solutions on initial data.Because direct computation of the solution becomes difficult in this situation,many authors have alternatively approximated the solution by the solution of a closely defined well posed problem.In this paper,we demonstrate this process of regularization for nonautonomous ill-posed problems including the backward heat equation with a time-dependent diffusion coefficient.In the process,we provide two different approximate well posed models and numerically compare convergence rates of their solutions to a known solution of the original ill-posed problem.
文摘This paper deals with a discrete-time dynamical system generated by a modified susceptible-infected-recovered-dead model(SIRD model;nonlinear operator)in threedimensional simplex.We introduce a novel approach that incorporates the SIRD model with the quadratic stochastic operator(QSO)that allows for real-time forecasting.The basic reproductive number Ro is obtained.We describe the set of fixed points of the operator and demonstrate that all fixed points are non-hyperbolic.Further,we study the asymptotical behavior of the trajectories of this system and show that SIRD operators havea regularity property.
