For large and sparse saddle point problems, Zhu studied a class of generalized local Hermitian and skew-Hermitian splitting iteration methods for non-Hermitian saddle point problem [M.-Z. Zhu, Appl. Math. Comput. 218 ...For large and sparse saddle point problems, Zhu studied a class of generalized local Hermitian and skew-Hermitian splitting iteration methods for non-Hermitian saddle point problem [M.-Z. Zhu, Appl. Math. Comput. 218 （2012） 8816-8824 ]. In this paper, we further investigate the generalized local Hermitian and skew-Hermitian splitting （GLHSS） iteration methods for solving non-Hermitian generalized saddle point problems. With different choices of the parameter matrices, we derive conditions for guaranteeing the con- vergence of these iterative methods. Numerical experiments are presented to illustrate the effectiveness of our GLHSS iteration methods as well as the preconditioners.展开更多
Recently,some authors(Shen and Shi,2016)studied the generalized shiftsplitting(GSS)iteration method for singular saddle point problem with nonsymmetric positive definite(1,1)-block and symmetric positive semidefinite(...Recently,some authors(Shen and Shi,2016)studied the generalized shiftsplitting(GSS)iteration method for singular saddle point problem with nonsymmetric positive definite(1,1)-block and symmetric positive semidefinite(2,2)-block.In this paper,we further apply the GSS iteration method to solve singular saddle point problem with nonsymmetric positive semidefinite(1,1)-block and symmetric positive semidefinite(2,2)-block,prove the semi-convergence of the GSS iteration method and analyze the spectral properties of the corresponding preconditioned matrix.Numerical experiment is given to indicate that the GSS iteration method with appropriate iteration parameters is effective and competitive for practical use.展开更多
Two results about the multiplicity of nontrivial periodic bouncing solutions for sublinear damped vibration systems-x=g(t)x+f(t,x) are obtained via the Generalized Nonsmooth Saddle Point Theorem and a technique establ...Two results about the multiplicity of nontrivial periodic bouncing solutions for sublinear damped vibration systems-x=g(t)x+f(t,x) are obtained via the Generalized Nonsmooth Saddle Point Theorem and a technique established by Wu Xian and Wang Shaomin.Both of them imply the condition "f≥0" required in some previous papers can be weakened,furthermore,one of them also implies the condition about ■F(t,x)/■t required in some previous papers,such as "|■F(t,x)/■t|=σ_(0)F(t,x)" and "|■F(t,x)/■t|≤C(1+F(t,x))", is unnecessary,where F(t,x):=∫_(0)~xf(t,x)ds,and σ_(0),C are positive constants.展开更多
In this paper, the theorem of the alternative based on separation functions in ordered locally convex topological vector spaces has been established by using the concept on set valued mappings. The optimality con...In this paper, the theorem of the alternative based on separation functions in ordered locally convex topological vector spaces has been established by using the concept on set valued mappings. The optimality conditions in ref. for D convex function have been generalized to ordered locally convex topological vector space and the similarly optimality conditions for D subconvexlike functions, such as the necessary and sufficient conditions of nondominated solutions, the generalized saddle point theorems and the lagrange duality theorems, have been obtained.展开更多
基金We would like to express our sincere gratitude to the anonymous referees whose constructive comments have the presentation of this paper greatly improved. The work was supported by the National Natural Science Foundation （No.11171371 and No.11101195）.
文摘For large and sparse saddle point problems, Zhu studied a class of generalized local Hermitian and skew-Hermitian splitting iteration methods for non-Hermitian saddle point problem [M.-Z. Zhu, Appl. Math. Comput. 218 （2012） 8816-8824 ]. In this paper, we further investigate the generalized local Hermitian and skew-Hermitian splitting （GLHSS） iteration methods for solving non-Hermitian generalized saddle point problems. With different choices of the parameter matrices, we derive conditions for guaranteeing the con- vergence of these iterative methods. Numerical experiments are presented to illustrate the effectiveness of our GLHSS iteration methods as well as the preconditioners.
基金Supported by Guangxi Science and Technology Department Specific Research Project of Guangxi for Research Bases and Talents(Grant No.GHIKE-AD23023001)Natural Science Foundation of Guangxi Minzu University(Grant No.2021KJQD01)Xiangsi Lake Young Scholars Innovation Team of Guangxi University for Nationalities(Grant No.2021RSCXSHQN05)。
文摘Recently,some authors(Shen and Shi,2016)studied the generalized shiftsplitting(GSS)iteration method for singular saddle point problem with nonsymmetric positive definite(1,1)-block and symmetric positive semidefinite(2,2)-block.In this paper,we further apply the GSS iteration method to solve singular saddle point problem with nonsymmetric positive semidefinite(1,1)-block and symmetric positive semidefinite(2,2)-block,prove the semi-convergence of the GSS iteration method and analyze the spectral properties of the corresponding preconditioned matrix.Numerical experiment is given to indicate that the GSS iteration method with appropriate iteration parameters is effective and competitive for practical use.
基金Supported by the National Natural Science Foundation of China (Grant No. 12171355)Elite Scholar Program in Tianjin University,P. R. China。
文摘Two results about the multiplicity of nontrivial periodic bouncing solutions for sublinear damped vibration systems-x=g(t)x+f(t,x) are obtained via the Generalized Nonsmooth Saddle Point Theorem and a technique established by Wu Xian and Wang Shaomin.Both of them imply the condition "f≥0" required in some previous papers can be weakened,furthermore,one of them also implies the condition about ■F(t,x)/■t required in some previous papers,such as "|■F(t,x)/■t|=σ_(0)F(t,x)" and "|■F(t,x)/■t|≤C(1+F(t,x))", is unnecessary,where F(t,x):=∫_(0)~xf(t,x)ds,and σ_(0),C are positive constants.
文摘In this paper, the theorem of the alternative based on separation functions in ordered locally convex topological vector spaces has been established by using the concept on set valued mappings. The optimality conditions in ref. for D convex function have been generalized to ordered locally convex topological vector space and the similarly optimality conditions for D subconvexlike functions, such as the necessary and sufficient conditions of nondominated solutions, the generalized saddle point theorems and the lagrange duality theorems, have been obtained.
