In this paper, the boundedness and the stability of solutions for a class of fourth order nonlinear differential equations are studied by using the method of Liapunov function. The sufficient conditions which guarante...In this paper, the boundedness and the stability of solutions for a class of fourth order nonlinear differential equations are studied by using the method of Liapunov function. The sufficient conditions which guarantee the boundedness and stability of solutions are preasented.展开更多
Let C be a nonempty bounded closed convex subset of a Banach space X, and T : C → C be uniformly L-Lipschitzian with L ≥ 1 and asymptotically pseudocontractive with a sequence {kn}(?)[1, ∞), limn→∞ kn = 1. Fix u ...Let C be a nonempty bounded closed convex subset of a Banach space X, and T : C → C be uniformly L-Lipschitzian with L ≥ 1 and asymptotically pseudocontractive with a sequence {kn}(?)[1, ∞), limn→∞ kn = 1. Fix u ∈ C. For each n ≥ 1, xn is a unique fixed point of the contraction Sn(x) = (1 - (tn)/(Lkn))u + (tn)/(Lkn)Tnx(?)x ∈ C, where {tn}(?)[0,1). Under suitable conditions, the strong convergence of the sequence{xn}to a fixed point of T is characterized.展开更多
The generalized summation integral type operators with Beta basis functions are widely studied. At present, the investigations for the properties of these operators are only limited to the functions of bounded variati...The generalized summation integral type operators with Beta basis functions are widely studied. At present, the investigations for the properties of these operators are only limited to the functions of bounded variation. Some authors studied the rate of point-wise rate of convergence, asymptotic formula of Voronovskaja type, and some direct results about these type of operators. The present paper considers the direct, inverse and equivalence theorems of modified summation integral type operators in the Lp spaces.展开更多
The dynamics anMysis of recurrent neural networks （RNNs） is a first and necessary step for any practical applications of them. In the present paper, the easily verified theorem is found to ascertain the asymptotical...The dynamics anMysis of recurrent neural networks （RNNs） is a first and necessary step for any practical applications of them. In the present paper, the easily verified theorem is found to ascertain the asymptotical stability for generic RNN model with projection mapping under the critical condition that a discriminant matrix defined by the networks is semi-positive definite. The results given here not only improve deeply upon the existing relevant critical as well as non-critical dynamics conclusions in literature, but also can be used in the practical application of RNNs directly.展开更多
We study the global dynamics of a nonlocal population model with age structure in a bounded domain. We mainly concern with the case where the birth rate decreases as the mature population size become large. The analys...We study the global dynamics of a nonlocal population model with age structure in a bounded domain. We mainly concern with the case where the birth rate decreases as the mature population size become large. The analysis is rather subtle and it is inadequate to apply the powerful theory of monotone dynamical systems. By using the method of super-sub solutions, combined with the careful analysis of the kernel function in the nonlocal term, we prove nonexistence, existence and uniqueness of positive steady states of the model.Moreover, due to the mature individuals do not diffuse, the solution semiflow to the model is not compact. To overcome the difficulty of non-compactness in describing the global asymptotic stability of the unique positive steady state, we first establish an appropriate comparison principle. With the help of the comparison principle,we can employ the theory of dissipative systems to obtain the global asymptotic stability of the unique positive steady state. The main results are illustrated with the nonlocal Nicholson's blowflies equation and the nonlocal Mackey-Glass equation.展开更多
基金The Applied Foundation of the Education Department of Yunnan Province(0012226)
文摘In this paper, the boundedness and the stability of solutions for a class of fourth order nonlinear differential equations are studied by using the method of Liapunov function. The sufficient conditions which guarantee the boundedness and stability of solutions are preasented.
基金The Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China, and The Dawn Program Fund in Shanghai.
文摘Let C be a nonempty bounded closed convex subset of a Banach space X, and T : C → C be uniformly L-Lipschitzian with L ≥ 1 and asymptotically pseudocontractive with a sequence {kn}(?)[1, ∞), limn→∞ kn = 1. Fix u ∈ C. For each n ≥ 1, xn is a unique fixed point of the contraction Sn(x) = (1 - (tn)/(Lkn))u + (tn)/(Lkn)Tnx(?)x ∈ C, where {tn}(?)[0,1). Under suitable conditions, the strong convergence of the sequence{xn}to a fixed point of T is characterized.
基金the National Natural Science Foundation of China (No.10571040)
文摘The generalized summation integral type operators with Beta basis functions are widely studied. At present, the investigations for the properties of these operators are only limited to the functions of bounded variation. Some authors studied the rate of point-wise rate of convergence, asymptotic formula of Voronovskaja type, and some direct results about these type of operators. The present paper considers the direct, inverse and equivalence theorems of modified summation integral type operators in the Lp spaces.
基金supported by the National Nature Science Foundation of China under Grant Nos.11101327,11471006,and 11171270the National Basic Research Program of China(973 Program)under Grant No.2013C13329406the Fundamental Research Funds for the Central Universities under Grant Nos.xjj20100087 and 2011jdhz30
文摘The dynamics anMysis of recurrent neural networks （RNNs） is a first and necessary step for any practical applications of them. In the present paper, the easily verified theorem is found to ascertain the asymptotical stability for generic RNN model with projection mapping under the critical condition that a discriminant matrix defined by the networks is semi-positive definite. The results given here not only improve deeply upon the existing relevant critical as well as non-critical dynamics conclusions in literature, but also can be used in the practical application of RNNs directly.
基金supported by National Natural Science Foundation of China(Grant Nos.11031002 and 11371107)the Research Fund for the Doctoral Program of Higher Education of China(Grant No.20124410110001)
文摘We study the global dynamics of a nonlocal population model with age structure in a bounded domain. We mainly concern with the case where the birth rate decreases as the mature population size become large. The analysis is rather subtle and it is inadequate to apply the powerful theory of monotone dynamical systems. By using the method of super-sub solutions, combined with the careful analysis of the kernel function in the nonlocal term, we prove nonexistence, existence and uniqueness of positive steady states of the model.Moreover, due to the mature individuals do not diffuse, the solution semiflow to the model is not compact. To overcome the difficulty of non-compactness in describing the global asymptotic stability of the unique positive steady state, we first establish an appropriate comparison principle. With the help of the comparison principle,we can employ the theory of dissipative systems to obtain the global asymptotic stability of the unique positive steady state. The main results are illustrated with the nonlocal Nicholson's blowflies equation and the nonlocal Mackey-Glass equation.
