Solutions of fuzzy differential equations provide a noteworthy example of time-dependent fuzzy sets The purpose of this paper is to introduce functions of a suitable Lyapunov-like type and to show the existence and ...Solutions of fuzzy differential equations provide a noteworthy example of time-dependent fuzzy sets The purpose of this paper is to introduce functions of a suitable Lyapunov-like type and to show the existence and uniqueness theorem for the Cauchy problem of fuzzy differential equations under non-Lipschitz conditions The comparison principles and the existence and uniqueness theorems of this paper generalize many well-known results up to now展开更多
The aim of this paper is to develop some basic theories of stochastic functional differential equations (SFDEs) under the local Lipschitz condition in continuous functions space C. Firstly, we establish a global exist...The aim of this paper is to develop some basic theories of stochastic functional differential equations (SFDEs) under the local Lipschitz condition in continuous functions space C. Firstly, we establish a global existence-uniqueness lemma for the SFDEs under the global Lipschitz condition in C without the linear growth condition. Then, under the local Lipschitz condition in C, we show that the non-continuable solution of SFDEs still exists if the drift coefficient and diffusion coefficient are square-integrable with respect to t when the state variable equals zero. And the solution of the considered equation must either explode at the end of the maximum existing interval or exist globally. Furthermore, some more general sufficient conditions for the global existence-uniqueness are obtained. Our conditions obtained in this paper are much weaker than some existing results. For example, we need neither the linear growth condition nor the continuous condition on the time t. Two examples are provided to show the effectiveness of the theoretical results.展开更多
The authors consider the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems and obtain a necessary and sufficient condition to guarantee the existence and uniqueness of global wea...The authors consider the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems and obtain a necessary and sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution.展开更多
Most known results on existence, uniqueness and stability for solutions of the polynomial-like iterative equation ∑ni=1λifi（x） = F（x） were obtained in the case of λ1 ≠0. In this paper, we construct Co decreasi...Most known results on existence, uniqueness and stability for solutions of the polynomial-like iterative equation ∑ni=1λifi（x） = F（x） were obtained in the case of λ1 ≠0. In this paper, we construct Co decreasing solutions of the iterative equation in the case that A1 can vanish to answer the Leading Coeffi- cient Problem. Moreover, we also give the necessary and sufficiently condition for uniqueness of solutions.展开更多
In this paper the concept of first boundary condition （i）（i = 0, 1, 2,…, n） is proposed based on [1], the existence of two times spline interpolant under first boundary condition is proved using constructivity me...In this paper the concept of first boundary condition （i）（i = 0, 1, 2,…, n） is proposed based on [1], the existence of two times spline interpolant under first boundary condition is proved using constructivity method and the uniqueness of the two times spline interpolant under first boundary condition（n） is proved too.展开更多
Techniques of Burton-Keane,developed earlier for independent percolation on Zd,are adapted to the setting of locally dependent percolation on Zd for d 2.The main result of this paper is a uniqueness theorem,that there...Techniques of Burton-Keane,developed earlier for independent percolation on Zd,are adapted to the setting of locally dependent percolation on Zd for d 2.The main result of this paper is a uniqueness theorem,that there exists almost surely a unique infinite out-cluster in locally dependent percolation on Zd,under the finite energy condition.展开更多
文摘Solutions of fuzzy differential equations provide a noteworthy example of time-dependent fuzzy sets The purpose of this paper is to introduce functions of a suitable Lyapunov-like type and to show the existence and uniqueness theorem for the Cauchy problem of fuzzy differential equations under non-Lipschitz conditions The comparison principles and the existence and uniqueness theorems of this paper generalize many well-known results up to now
基金supported by National Natural Science Foundation of China (Grant Nos.11271270, 11201320 and 11101298)Youth Foundation of Sichuan University (Grant No. 2011SCU11111)
文摘The aim of this paper is to develop some basic theories of stochastic functional differential equations (SFDEs) under the local Lipschitz condition in continuous functions space C. Firstly, we establish a global existence-uniqueness lemma for the SFDEs under the global Lipschitz condition in C without the linear growth condition. Then, under the local Lipschitz condition in C, we show that the non-continuable solution of SFDEs still exists if the drift coefficient and diffusion coefficient are square-integrable with respect to t when the state variable equals zero. And the solution of the considered equation must either explode at the end of the maximum existing interval or exist globally. Furthermore, some more general sufficient conditions for the global existence-uniqueness are obtained. Our conditions obtained in this paper are much weaker than some existing results. For example, we need neither the linear growth condition nor the continuous condition on the time t. Two examples are provided to show the effectiveness of the theoretical results.
基金Project supported by the Special Funds for Major State Basic Research Projects of China
文摘The authors consider the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems and obtain a necessary and sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution.
基金supported by National Natural Science Foundation of China(Grant No.11201184)the Natural Science Foundation of Chongqing Normal University(Grant No. 12XLB019)
文摘Most known results on existence, uniqueness and stability for solutions of the polynomial-like iterative equation ∑ni=1λifi（x） = F（x） were obtained in the case of λ1 ≠0. In this paper, we construct Co decreasing solutions of the iterative equation in the case that A1 can vanish to answer the Leading Coeffi- cient Problem. Moreover, we also give the necessary and sufficiently condition for uniqueness of solutions.
文摘In this paper the concept of first boundary condition （i）（i = 0, 1, 2,…, n） is proposed based on [1], the existence of two times spline interpolant under first boundary condition is proved using constructivity method and the uniqueness of the two times spline interpolant under first boundary condition（n） is proved too.
基金the National Natural Science Foundation of China (No.10531070)
文摘Techniques of Burton-Keane,developed earlier for independent percolation on Zd,are adapted to the setting of locally dependent percolation on Zd for d 2.The main result of this paper is a uniqueness theorem,that there exists almost surely a unique infinite out-cluster in locally dependent percolation on Zd,under the finite energy condition.
