The uniformly ultimate boundedness of discontinuous systems with time-delay in the sense of Filippov solutions is discussed.Based on the Lyapunov-Krasovskii functional,the Lyapunov theorem for the globally strongly un...The uniformly ultimate boundedness of discontinuous systems with time-delay in the sense of Filippov solutions is discussed.Based on the Lyapunov-Krasovskii functional,the Lyapunov theorem for the globally strongly uniformly ultimate boundedness of retarded discontinuous systems is presented.Furthermore,the result is applied to a class of mechanical systems with a retarded discontinuous friction item.展开更多
A new type criterion of globally uniformly ultimate boundedness for discrete-time nonlinear systems is introduced. In classical Lyapunov theory about globally uniformly ultimate boundedness, Lyapunov function is assum...A new type criterion of globally uniformly ultimate boundedness for discrete-time nonlinear systems is introduced. In classical Lyapunov theory about globally uniformly ultimate boundedness, Lyapunov function is assumed to be positive definite and its difference at the every latter moment and the former moment is negative definite. In this paper the condition of difference of Lyapunov function is relaxed. Under the relaxed condition, the result of this paper can be considered as the extension of the classical Lyapunov theory about uniformly ultimate boundedness.展开更多
We consider a system of neutral equations with unbounded delay, and derive conditions on Liapunov functionals to ensure that the solutions are uniformly bounded and uniformly ultimately bounded.
Stability, boundedness and persistence are three important aspects for an ecological model. In this paper, a further analysis of a class of anaerobic digestion ecological models is performed. Based on the Liupunov Met...Stability, boundedness and persistence are three important aspects for an ecological model. In this paper, a further analysis of a class of anaerobic digestion ecological models is performed. Based on the Liupunov Method, the local stability of all equilibria in the system is got. According to the vector fields described by the system, the proof of the boundedness of the solution on the anaerobic digestion processes is completed in three steps. The method proposed in the discussion on the boundedness can be generalized to the similar problems. Results in this paper give information on how to run the ecological system well by adjusting the system parameters.展开更多
We show that the lateral regularizations of the generator of any uniformly bounded set-valued composition Nemytskij operator acting in the spaces of functions of bounded variation in the sense of Riesz, with nonempty ...We show that the lateral regularizations of the generator of any uniformly bounded set-valued composition Nemytskij operator acting in the spaces of functions of bounded variation in the sense of Riesz, with nonempty bounded closed and convex values, are an affine function.展开更多
Evaluation for the performance of learning algorithm has been the main thread of theoretical research of machine learning. The performance of the regularized regression algorithm based on independent and identically d...Evaluation for the performance of learning algorithm has been the main thread of theoretical research of machine learning. The performance of the regularized regression algorithm based on independent and identically distributed（i.i.d.） samples has been researched by a large number of references. In the present paper we provide the convergence rates for the performance of regularized regression based on the inputs of p-order Markov chains.展开更多
We stress a basic criterion that shows in a simple way how a sequence of real-valued functions can converge uniformly when it is more or less evident that the sequence converges uniformly away from a finite number of ...We stress a basic criterion that shows in a simple way how a sequence of real-valued functions can converge uniformly when it is more or less evident that the sequence converges uniformly away from a finite number of points of the closure of its domain. For functions of a real variable, unlike in most classical textbooks our criterion avoids the search of extrema (by differential calculus) of their general term.展开更多
Boltzmann equation is an equation which is related to the three variables of x, v, t. In this paper, we mainly study the space-uniform Boltzmann equation which unknown function F is not related to the position variabl...Boltzmann equation is an equation which is related to the three variables of x, v, t. In this paper, we mainly study the space-uniform Boltzmann equation which unknown function F is not related to the position variable x. We mainly use the contraction mapping theorem to find the existence of the solution, so our mainly work is to prove the self-mapping, i.e. to prove its uniformly bounded, and then to prove the contraction mapping. There we can get the range of ||B(θ)||L1(L∞), next we can figure out the range of M and T from the conditions what we know. Finally, from these conditions, we can find the existence of the solution.展开更多
In this paper,we first obtain a unified integral representation on the analytic varieties of the general bounded domain in Stein manifolds(the two types bounded domains in[3]are regarded as its special cases).Secondly...In this paper,we first obtain a unified integral representation on the analytic varieties of the general bounded domain in Stein manifolds(the two types bounded domains in[3]are regarded as its special cases).Secondly we get the integral formulas of the solution of∂-equation.And we use a new and unique method to give a uniform estimate of the solution of∂-equation,which is different from Henkin's method.展开更多
It is shown that the famous Banach-Steinhaus theorem can be generalized to some families of nonlinear functionals defined on some topological groups and topological vector space, e.g. the F-spaced lβ(0 <β < 1)...It is shown that the famous Banach-Steinhaus theorem can be generalized to some families of nonlinear functionals defined on some topological groups and topological vector space, e.g. the F-spaced lβ(0 <β < 1) and S[a, b].展开更多
This note deals with the functional differential equations with infinite delay x’=f（t, x,）, （1）where x∈R<sup>n</sup>, f: [0,∞) ×C<sub>9</sub>→R<sup>n</sup>, C<sub>...This note deals with the functional differential equations with infinite delay x’=f（t, x,）, （1）where x∈R<sup>n</sup>, f: [0,∞) ×C<sub>9</sub>→R<sup>n</sup>, C<sub>g</sub> is the phase space of （1）, which is defined as follows:C=∮(（-∞, 0], R<sup>n</sup>） is the set of all the continuous vector-functions mapping (-∞, 0] toR<sup>n</sup>. The function g:（-∞, 0]→ [1, ∞） is continuous and nonincreasing with g （0）=1 andg（-∞）=∞.展开更多
This paper presents a discussion of differential equations with two terms generated by delays. They are as follows x′(t) = a(t)x(t - p(t)) - b(t)x(t - r(t)).Sufficient conditions for its solution with uniform bound a...This paper presents a discussion of differential equations with two terms generated by delays. They are as follows x′(t) = a(t)x(t - p(t)) - b(t)x(t - r(t)).Sufficient conditions for its solution with uniform bound and for its zero solution with uniform stability are established.展开更多
Using the energy estimate and Gagliardo-Nirenberg-type inequalities,the existence and uniform boundedness of the global solutions to a strongly coupled reaction-diffusion system are proved. This system is a generaliza...Using the energy estimate and Gagliardo-Nirenberg-type inequalities,the existence and uniform boundedness of the global solutions to a strongly coupled reaction-diffusion system are proved. This system is a generalization of the two-species Lotka-Volterra predator-prey model with self and cross-diffusion. Suffcient condition for the global asymptotic stability of the positive equilibrium point of the model is given by constructing Lyapunov function.展开更多
基金supported by the National Natural Science Foundation of China (No. 60874006)
文摘The uniformly ultimate boundedness of discontinuous systems with time-delay in the sense of Filippov solutions is discussed.Based on the Lyapunov-Krasovskii functional,the Lyapunov theorem for the globally strongly uniformly ultimate boundedness of retarded discontinuous systems is presented.Furthermore,the result is applied to a class of mechanical systems with a retarded discontinuous friction item.
文摘A new type criterion of globally uniformly ultimate boundedness for discrete-time nonlinear systems is introduced. In classical Lyapunov theory about globally uniformly ultimate boundedness, Lyapunov function is assumed to be positive definite and its difference at the every latter moment and the former moment is negative definite. In this paper the condition of difference of Lyapunov function is relaxed. Under the relaxed condition, the result of this paper can be considered as the extension of the classical Lyapunov theory about uniformly ultimate boundedness.
文摘We consider a system of neutral equations with unbounded delay, and derive conditions on Liapunov functionals to ensure that the solutions are uniformly bounded and uniformly ultimately bounded.
基金Supported by the National Natural Science Foundation of China (No.60372012) and NSF of Chongqing (No.0831)
文摘Stability, boundedness and persistence are three important aspects for an ecological model. In this paper, a further analysis of a class of anaerobic digestion ecological models is performed. Based on the Liupunov Method, the local stability of all equilibria in the system is got. According to the vector fields described by the system, the proof of the boundedness of the solution on the anaerobic digestion processes is completed in three steps. The method proposed in the discussion on the boundedness can be generalized to the similar problems. Results in this paper give information on how to run the ecological system well by adjusting the system parameters.
文摘We show that the lateral regularizations of the generator of any uniformly bounded set-valued composition Nemytskij operator acting in the spaces of functions of bounded variation in the sense of Riesz, with nonempty bounded closed and convex values, are an affine function.
基金Supported by the National Natural Science Foundation of China (10871226)the Natural Science Foundation of Zhejiang Province (Y6100096)
文摘Evaluation for the performance of learning algorithm has been the main thread of theoretical research of machine learning. The performance of the regularized regression algorithm based on independent and identically distributed（i.i.d.） samples has been researched by a large number of references. In the present paper we provide the convergence rates for the performance of regularized regression based on the inputs of p-order Markov chains.
文摘We stress a basic criterion that shows in a simple way how a sequence of real-valued functions can converge uniformly when it is more or less evident that the sequence converges uniformly away from a finite number of points of the closure of its domain. For functions of a real variable, unlike in most classical textbooks our criterion avoids the search of extrema (by differential calculus) of their general term.
文摘Boltzmann equation is an equation which is related to the three variables of x, v, t. In this paper, we mainly study the space-uniform Boltzmann equation which unknown function F is not related to the position variable x. We mainly use the contraction mapping theorem to find the existence of the solution, so our mainly work is to prove the self-mapping, i.e. to prove its uniformly bounded, and then to prove the contraction mapping. There we can get the range of ||B(θ)||L1(L∞), next we can figure out the range of M and T from the conditions what we know. Finally, from these conditions, we can find the existence of the solution.
文摘In this paper,we first obtain a unified integral representation on the analytic varieties of the general bounded domain in Stein manifolds(the two types bounded domains in[3]are regarded as its special cases).Secondly we get the integral formulas of the solution of∂-equation.And we use a new and unique method to give a uniform estimate of the solution of∂-equation,which is different from Henkin's method.
文摘It is shown that the famous Banach-Steinhaus theorem can be generalized to some families of nonlinear functionals defined on some topological groups and topological vector space, e.g. the F-spaced lβ(0 <β < 1) and S[a, b].
基金Project partially supported by the National Natural Science Foundation of China.
文摘This note deals with the functional differential equations with infinite delay x’=f（t, x,）, （1）where x∈R<sup>n</sup>, f: [0,∞) ×C<sub>9</sub>→R<sup>n</sup>, C<sub>g</sub> is the phase space of （1）, which is defined as follows:C=∮(（-∞, 0], R<sup>n</sup>） is the set of all the continuous vector-functions mapping (-∞, 0] toR<sup>n</sup>. The function g:（-∞, 0]→ [1, ∞） is continuous and nonincreasing with g （0）=1 andg（-∞）=∞.
文摘This paper presents a discussion of differential equations with two terms generated by delays. They are as follows x′(t) = a(t)x(t - p(t)) - b(t)x(t - r(t)).Sufficient conditions for its solution with uniform bound and for its zero solution with uniform stability are established.
基金Partially supported by the National Natural Science Foundation of China (10671158)the NSFof Gansu Province (3ZS061-A25-015)+1 种基金the Scientific Research Fund of Gansu Provincial Educatio nDepartment (0601-21)NWNU-KJCXGC-03-18, 39 Foundations
文摘Using the energy estimate and Gagliardo-Nirenberg-type inequalities,the existence and uniform boundedness of the global solutions to a strongly coupled reaction-diffusion system are proved. This system is a generalization of the two-species Lotka-Volterra predator-prey model with self and cross-diffusion. Suffcient condition for the global asymptotic stability of the positive equilibrium point of the model is given by constructing Lyapunov function.
