Lipschitz continuous solutions to the Cauchy problem for 1-D first order quasi-linear hyperbolic systems are considered. Based on the methods of approximation and integral equations, the author gives two definitions o...Lipschitz continuous solutions to the Cauchy problem for 1-D first order quasi-linear hyperbolic systems are considered. Based on the methods of approximation and integral equations, the author gives two definitions of Lipschitz solutions to the Cauchy problem and proves the existence and uniqueness of solutions.展开更多
A cautious projection BFGS method is proposed for solving nonconvex unconstrained optimization problems.The global convergence of this method as well as a stronger general convergence result can be proven without a gr...A cautious projection BFGS method is proposed for solving nonconvex unconstrained optimization problems.The global convergence of this method as well as a stronger general convergence result can be proven without a gradient Lipschitz continuity assumption,which is more in line with the actual problems than the existing modified BFGS methods and the traditional BFGS method.Under some additional conditions,the method presented has a superlinear convergence rate,which can be regarded as an extension and supplement of BFGS-type methods with the projection technique.Finally,the effectiveness and application prospects of the proposed method are verified by numerical experiments.展开更多
When all the involved data in indefinite quadratic programs change simultaneously, we show the locally Lipschtiz continuity of the KKT set of the quadratic programming problem firstly, then we establish the locally Li...When all the involved data in indefinite quadratic programs change simultaneously, we show the locally Lipschtiz continuity of the KKT set of the quadratic programming problem firstly, then we establish the locally Lipschtiz continuity of the KKT solution set. Finally, the similar conclusion for the corresponding optimal value function is obtained.展开更多
In this note we prove that the corner cutting procedure preserves continuityproperties,i.e.,a sequence of polygons obtained in this way belongs to the Lipschitz classof the same constant and exponent.As a consequence ...In this note we prove that the corner cutting procedure preserves continuityproperties,i.e.,a sequence of polygons obtained in this way belongs to the Lipschitz classof the same constant and exponent.As a consequence this also holds for all functions orcurves obtained as the limit of this procedure, such as the Bernstein polynomials,Bezierand spline parametric curves,etc.展开更多
The present paper finds out that the geometric entity which characterizes the best Lipschitz constants for the Bezier nets and Bernstein polynomials over a simplex sigma is an angle Phi determined by sigma, and proves...The present paper finds out that the geometric entity which characterizes the best Lipschitz constants for the Bezier nets and Bernstein polynomials over a simplex sigma is an angle Phi determined by sigma, and proves that (1) if f(x) is Lipschitz continuous over sigma, i.e., f(x) is an element of Lip(A)(alpha,sigma), then both the n-th Bezier net <(f)over cap (n)> and the n-th Bernstein polynomial B-n(f;x) corresponding to f(x) belong to Lip(B)(alpha,sigma) , where B = Asec(alpha)Phi; and (2) if n-th Bezier net <(f)over cap (n)> is an element of Lip(A)(alpha,sigma), then the elevation Bezier net <E(f)over cap (n)> and the corresponding Bernstein polynomial. B-n(f,;x) also belong to Lip(A)(alpha,sigma). Furthermore, the constant B = Asec(alpha)Phi, in case (1) is best in some sense.展开更多
Many solutions of variational inequalities have been proposed,among which the subgradient extragradient method has obvious advantages.Two different algorithms are given for solving variational inequality problem in th...Many solutions of variational inequalities have been proposed,among which the subgradient extragradient method has obvious advantages.Two different algorithms are given for solving variational inequality problem in this paper.The problem we study is defined in a real Hilbert space and has L-Lipschitz and pseudomonotone condition.Two new algorithms adopt inertial technology and non-monotonic step size rule,and their convergence can still be proved when the value of L is not given in advance.Finally,some numerical results are designed to demonstrate the computational efficiency of our two new algorithms.展开更多
In this paper,we propose a modified two-subgradient extragradient algorithm(MTSEGA)for solving monotone and Lipschitz continuous variational inequalities with the feasible set being a level set of a smooth convex func...In this paper,we propose a modified two-subgradient extragradient algorithm(MTSEGA)for solving monotone and Lipschitz continuous variational inequalities with the feasible set being a level set of a smooth convex function in Hilbert space.The advantage of MTSEGA is that all the projections are computed onto a half-space per iteration.Moreover,MTSEGA only needs one computation of the underlying mapping per iteration.Under the same assumptions with the known algorithm,we show that the sequence generated by this algorithm is weakly convergent to a solution of the concerned problem.展开更多
A new convergence theorem for the Secant method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation. It is assumed that the divided differenc...A new convergence theorem for the Secant method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation. It is assumed that the divided difference of order one of the nonlinear operator is Lipschitz continuous. The convergence conditions differ from some existing ones and are easily satisfied. The results of the paper are justified by numerical examples that cannot be handled by earlier works.展开更多
This paper investigates the absolute exponential stability of generalized neural networks with a general class of partially Lipschitz continuous and monotone increasing activation functions. The main obtained result i...This paper investigates the absolute exponential stability of generalized neural networks with a general class of partially Lipschitz continuous and monotone increasing activation functions. The main obtained result is that if the interconnection matrix T of the neural system satisfies that - T is an H matrix with nonnegative diagonal elements, then the neural system is absolutely exponentially stable(AEST). The Hopfield network, Cellular neural network and Bidirectional associative memory network are special cases of the network model considered in this paper. So this work gives some improvements to the previous ones.展开更多
The sensitivity analysis for a class of generalized set-valued quasi-variational inclusion problems is investigated in the setting of Banach spaces. By using the resolvent operator technique, without assuming the diff...The sensitivity analysis for a class of generalized set-valued quasi-variational inclusion problems is investigated in the setting of Banach spaces. By using the resolvent operator technique, without assuming the differentiability and monotonicity of the given data, equivalence of these problems to the class of generalized resolvent equations is established.展开更多
In this paper, by using the auxiliary technique of variational inequalities, the existence and iterative algorithm; of solutions for a class of generalized mixed quasi-variational inequalities are studied. Our results...In this paper, by using the auxiliary technique of variational inequalities, the existence and iterative algorithm; of solutions for a class of generalized mixed quasi-variational inequalities are studied. Our results answer the open problems mentioned by Noor, improve and generalize some recent known results.展开更多
Under the assumption that the nonlinear operator has Lipschitz continuous divided differences for the first order,we obtain an estimate of the radius of the convergence ball for the two-step secant method.Moreover,we ...Under the assumption that the nonlinear operator has Lipschitz continuous divided differences for the first order,we obtain an estimate of the radius of the convergence ball for the two-step secant method.Moreover,we also provide an error estimate that matches the convergence order of the two-step secant method.At last,we give an application of the proposed theorem.展开更多
This paper first proves the following equations△u-m￣2u+f(x,u)=0, x(R￣n,n≥3 m>0 existence of decaying positive entire solution, then emphatically, proves this solution'suniqueness.
A modified Gauss-type Proximal Point Algorithm (modified GG-PPA) is presented in this paper for solving the generalized equations like 0 ∈T(x), where T is a set-valued mapping acts between two different Bana...A modified Gauss-type Proximal Point Algorithm (modified GG-PPA) is presented in this paper for solving the generalized equations like 0 ∈T(x), where T is a set-valued mapping acts between two different Banach spaces X and Y. By considering some necessary assumptions, we show the existence of any sequence generated by the modified GG-PPA and prove the semi-local and local convergence results by using metrically regular mapping. In addition, we give a numerical example to justify the result of semi-local convergence.展开更多
To prove the existence of the family of exponential attractors, we first define a family of compact, invariant absorbing sets <em>B<sub>k</sub></em>. Then we prove that the solution semigroup h...To prove the existence of the family of exponential attractors, we first define a family of compact, invariant absorbing sets <em>B<sub>k</sub></em>. Then we prove that the solution semigroup has Lipschitz property and discrete squeezing property. Finally, we obtain a family of exponential attractors and its estimation of dimension by combining them with previous theories. Next, we obtain Kirchhoff-type random equation by adding product white noise to the right-hand side of the equation. To study the existence of random attractors, firstly we transform the equation by using Ornstein-Uhlenbeck process. Then we obtain a family of bounded random absorbing sets via estimating the solution of the random differential equation. Finally, we prove the asymptotic compactness of semigroup of the stochastic dynamic system;thereby we obtain a family of random attractors.展开更多
In this paper, we study the inertial manifolds for a class of asymmetrically coupled generalized Higher-order Kirchhoff equations. Under appropriate assumptions, we firstly exist Hadamard’s graph transformation metho...In this paper, we study the inertial manifolds for a class of asymmetrically coupled generalized Higher-order Kirchhoff equations. Under appropriate assumptions, we firstly exist Hadamard’s graph transformation method to structure a graph norm of a Lipschitz continuous function, then we prove the existence of a family of inertial manifolds by showing that the spectral gap condition is true.展开更多
It is shown that an arbitrary function from D Rn to Rm will become C0,a-continuous in almost every x∈ D after restriction to a certain subset with limit pointx. For n 〉 m differentiability can be obtained. Example...It is shown that an arbitrary function from D Rn to Rm will become C0,a-continuous in almost every x∈ D after restriction to a certain subset with limit pointx. For n 〉 m differentiability can be obtained. Examples show the Ho1der exponent a=min{1,n/m}is optimal.展开更多
In this paper, we consider the global error bound for the generalized complementarity problem (GCP) with analytic functions. Based on the new technique, we establish computable global error bound under milder conditio...In this paper, we consider the global error bound for the generalized complementarity problem (GCP) with analytic functions. Based on the new technique, we establish computable global error bound under milder conditions, which refines the previously known results.展开更多
Let f be a twice continuously differentiable self-mapping of a unit disk satisfying Poisson differential inequality |△f(z)| ≤ B · |Df(z)|^(2) for some B > 0 and f(0) = 0. In this note, we show that f does no...Let f be a twice continuously differentiable self-mapping of a unit disk satisfying Poisson differential inequality |△f(z)| ≤ B · |Df(z)|^(2) for some B > 0 and f(0) = 0. In this note, we show that f does not always satisfy the Schwarz-Pick type inequality (1-|z|^(2))/(1-|f(z)|^(2))≤ C(B),where C(B) is a constant depending only on B. Moreover, a more general Schwarz-Pick type inequality for mapping that satisfies general Poisson differential inequality is established under certain conditions.展开更多
In order to solve variational inequality problems of pseudomonotonicity and Lipschitz continuity in Hilbert spaces, an inertial subgradient extragradient algorithm is proposed by virtue of non-monotone stepsizes. More...In order to solve variational inequality problems of pseudomonotonicity and Lipschitz continuity in Hilbert spaces, an inertial subgradient extragradient algorithm is proposed by virtue of non-monotone stepsizes. Moreover, weak convergence and R-linear convergence analyses of the algorithm are constructed under appropriate assumptions. Finally, the efficiency of the proposed algorithm is demonstrated through numerical implementations.展开更多
文摘Lipschitz continuous solutions to the Cauchy problem for 1-D first order quasi-linear hyperbolic systems are considered. Based on the methods of approximation and integral equations, the author gives two definitions of Lipschitz solutions to the Cauchy problem and proves the existence and uniqueness of solutions.
基金supported by the Guangxi Science and Technology base and Talent Project(AD22080047)the National Natural Science Foundation of Guangxi Province(2023GXNFSBA 026063)+1 种基金the Innovation Funds of Chinese University(2021BCF03001)the special foundation for Guangxi Ba Gui Scholars.
文摘A cautious projection BFGS method is proposed for solving nonconvex unconstrained optimization problems.The global convergence of this method as well as a stronger general convergence result can be proven without a gradient Lipschitz continuity assumption,which is more in line with the actual problems than the existing modified BFGS methods and the traditional BFGS method.Under some additional conditions,the method presented has a superlinear convergence rate,which can be regarded as an extension and supplement of BFGS-type methods with the projection technique.Finally,the effectiveness and application prospects of the proposed method are verified by numerical experiments.
基金Supported by the National Natural Science Foundation of China(10571141,70971109,71371152)supported by the Talents Fund of Xi’an Polytechnic University(BS1320)the Mathematics Discipline Development Fund of Xi’an Ploytechnic University(107090701)
文摘When all the involved data in indefinite quadratic programs change simultaneously, we show the locally Lipschtiz continuity of the KKT set of the quadratic programming problem firstly, then we establish the locally Lipschtiz continuity of the KKT solution set. Finally, the similar conclusion for the corresponding optimal value function is obtained.
文摘In this note we prove that the corner cutting procedure preserves continuityproperties,i.e.,a sequence of polygons obtained in this way belongs to the Lipschitz classof the same constant and exponent.As a consequence this also holds for all functions orcurves obtained as the limit of this procedure, such as the Bernstein polynomials,Bezierand spline parametric curves,etc.
文摘The present paper finds out that the geometric entity which characterizes the best Lipschitz constants for the Bezier nets and Bernstein polynomials over a simplex sigma is an angle Phi determined by sigma, and proves that (1) if f(x) is Lipschitz continuous over sigma, i.e., f(x) is an element of Lip(A)(alpha,sigma), then both the n-th Bezier net <(f)over cap (n)> and the n-th Bernstein polynomial B-n(f;x) corresponding to f(x) belong to Lip(B)(alpha,sigma) , where B = Asec(alpha)Phi; and (2) if n-th Bezier net <(f)over cap (n)> is an element of Lip(A)(alpha,sigma), then the elevation Bezier net <E(f)over cap (n)> and the corresponding Bernstein polynomial. B-n(f,;x) also belong to Lip(A)(alpha,sigma). Furthermore, the constant B = Asec(alpha)Phi, in case (1) is best in some sense.
文摘Many solutions of variational inequalities have been proposed,among which the subgradient extragradient method has obvious advantages.Two different algorithms are given for solving variational inequality problem in this paper.The problem we study is defined in a real Hilbert space and has L-Lipschitz and pseudomonotone condition.Two new algorithms adopt inertial technology and non-monotonic step size rule,and their convergence can still be proved when the value of L is not given in advance.Finally,some numerical results are designed to demonstrate the computational efficiency of our two new algorithms.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1187105911801455)+1 种基金Sichuan Science and Technology Program(Grant No.2019YFG0299)General Cultivation Program of China West Normal University(Grant No.20A024)。
文摘In this paper,we propose a modified two-subgradient extragradient algorithm(MTSEGA)for solving monotone and Lipschitz continuous variational inequalities with the feasible set being a level set of a smooth convex function in Hilbert space.The advantage of MTSEGA is that all the projections are computed onto a half-space per iteration.Moreover,MTSEGA only needs one computation of the underlying mapping per iteration.Under the same assumptions with the known algorithm,we show that the sequence generated by this algorithm is weakly convergent to a solution of the concerned problem.
基金Supported by the National Natural Science Foundation of China (10871178)the Natural Science Foundation of Zhejiang Province of China (Y606154)Foundation of the Education Department of Zhejiang Province of China (20071362)
文摘A new convergence theorem for the Secant method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation. It is assumed that the divided difference of order one of the nonlinear operator is Lipschitz continuous. The convergence conditions differ from some existing ones and are easily satisfied. The results of the paper are justified by numerical examples that cannot be handled by earlier works.
文摘This paper investigates the absolute exponential stability of generalized neural networks with a general class of partially Lipschitz continuous and monotone increasing activation functions. The main obtained result is that if the interconnection matrix T of the neural system satisfies that - T is an H matrix with nonnegative diagonal elements, then the neural system is absolutely exponentially stable(AEST). The Hopfield network, Cellular neural network and Bidirectional associative memory network are special cases of the network model considered in this paper. So this work gives some improvements to the previous ones.
基金Project supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education, China (No.0705)the Dawn Program Fund of Shanghai of China (No.BL200404)Shanghai Leading Academic Discipline Project (No.T0401)
文摘The sensitivity analysis for a class of generalized set-valued quasi-variational inclusion problems is investigated in the setting of Banach spaces. By using the resolvent operator technique, without assuming the differentiability and monotonicity of the given data, equivalence of these problems to the class of generalized resolvent equations is established.
基金the Natural Science Foundation of Sichuan Education Com mittee
文摘In this paper, by using the auxiliary technique of variational inequalities, the existence and iterative algorithm; of solutions for a class of generalized mixed quasi-variational inequalities are studied. Our results answer the open problems mentioned by Noor, improve and generalize some recent known results.
基金supported by National Natural Science Foundation of China(11771393,11371320,11632015)Zhejiang Natural Science Foundation(LZ14A010002,LQ18A010008)Scientific Research Fund of Zhejiang Provincial Education Department(FX2016073)
文摘Under the assumption that the nonlinear operator has Lipschitz continuous divided differences for the first order,we obtain an estimate of the radius of the convergence ball for the two-step secant method.Moreover,we also provide an error estimate that matches the convergence order of the two-step secant method.At last,we give an application of the proposed theorem.
文摘This paper first proves the following equations△u-m￣2u+f(x,u)=0, x(R￣n,n≥3 m>0 existence of decaying positive entire solution, then emphatically, proves this solution'suniqueness.
文摘A modified Gauss-type Proximal Point Algorithm (modified GG-PPA) is presented in this paper for solving the generalized equations like 0 ∈T(x), where T is a set-valued mapping acts between two different Banach spaces X and Y. By considering some necessary assumptions, we show the existence of any sequence generated by the modified GG-PPA and prove the semi-local and local convergence results by using metrically regular mapping. In addition, we give a numerical example to justify the result of semi-local convergence.
文摘To prove the existence of the family of exponential attractors, we first define a family of compact, invariant absorbing sets <em>B<sub>k</sub></em>. Then we prove that the solution semigroup has Lipschitz property and discrete squeezing property. Finally, we obtain a family of exponential attractors and its estimation of dimension by combining them with previous theories. Next, we obtain Kirchhoff-type random equation by adding product white noise to the right-hand side of the equation. To study the existence of random attractors, firstly we transform the equation by using Ornstein-Uhlenbeck process. Then we obtain a family of bounded random absorbing sets via estimating the solution of the random differential equation. Finally, we prove the asymptotic compactness of semigroup of the stochastic dynamic system;thereby we obtain a family of random attractors.
文摘In this paper, we study the inertial manifolds for a class of asymmetrically coupled generalized Higher-order Kirchhoff equations. Under appropriate assumptions, we firstly exist Hadamard’s graph transformation method to structure a graph norm of a Lipschitz continuous function, then we prove the existence of a family of inertial manifolds by showing that the spectral gap condition is true.
文摘It is shown that an arbitrary function from D Rn to Rm will become C0,a-continuous in almost every x∈ D after restriction to a certain subset with limit pointx. For n 〉 m differentiability can be obtained. Examples show the Ho1der exponent a=min{1,n/m}is optimal.
基金supported by National Natural Science Foundation of China (Nos. 11171180 and 11101303)Specialized Research Fund for the Doctoral Program of Chinese Higher Education (No. 20113705110002)Shandong Provincial Natural Science Foundation (Nos. ZR2010AL005 and ZR2011FL017)
文摘In this paper, we consider the global error bound for the generalized complementarity problem (GCP) with analytic functions. Based on the new technique, we establish computable global error bound under milder conditions, which refines the previously known results.
基金supported by NNSF of China(11701111)NNSFs of Guangdong Province (2016A030310257 and 2015A030313346)the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University when the authors worked as visiting scholars。
文摘Let f be a twice continuously differentiable self-mapping of a unit disk satisfying Poisson differential inequality |△f(z)| ≤ B · |Df(z)|^(2) for some B > 0 and f(0) = 0. In this note, we show that f does not always satisfy the Schwarz-Pick type inequality (1-|z|^(2))/(1-|f(z)|^(2))≤ C(B),where C(B) is a constant depending only on B. Moreover, a more general Schwarz-Pick type inequality for mapping that satisfies general Poisson differential inequality is established under certain conditions.
文摘In order to solve variational inequality problems of pseudomonotonicity and Lipschitz continuity in Hilbert spaces, an inertial subgradient extragradient algorithm is proposed by virtue of non-monotone stepsizes. Moreover, weak convergence and R-linear convergence analyses of the algorithm are constructed under appropriate assumptions. Finally, the efficiency of the proposed algorithm is demonstrated through numerical implementations.
