一般Lidstone边值问题的解的存在性

考察了含有各阶导数的一般Lidstone边值问题的存在性. 利用Leray-Schauder不动点定理、等价范数及积分方程组技巧证明了解或正解的两个存在定理. 这些定理在应用上非常方便. 换句话说, 通过考察非线性项在某个有界子集上的“高度”就能决定解的存在性.     

Positive periodic solution for a neutral delay competitive system

In this paper we study the existence of positive periodic solutions for a class of neutral type competitive system with delay. Applying the continuation theorem of coincidence degree theory, a general criteria on the existence of positive periodic solutions are established. Some the related results are generalized and improved. It is shown that the arguments delays have no effect on the existence positive periodic solution of system.     

On the integrability of the differential systems in dimension two and of the polynomial differential systems in arbitrary dimension

This is a survey on recent results providing sufficient conditions for the existence of a first integral, first for vector fields defined on real surfaces, and second for polynomial vector fields in \(R^n\) or \(C^n\) with \(n\geq 2\). We also provide an open question and some applications based on the existence of such first integrals.               

Existence of a solution for a thermoelectric model with several phase changes and a Carathéodory thermal conductivity

We present an existence result for the time domain eddy current model for electromagnetic field, coupled with a Stefan problem for temperature. We have extended the existence result proved by Bermúdez et al. (2005) [40] to the case of materials with several phase changes and thermal conductivity depending on both position and temperature.The new proof starts from a totally implicit time discretization of a truncated system. The thermal part can be rewritten as a variational inequality of the second kind. Then, a priori estimates independent of the truncation parameter are obtained for the solution of the truncated problem, using a technique that adapts the method of Boccardo and Gallouët (1989)  [28] to the case of materials with several phase changes.     

Zero Mach number limit in critical spaces for compressible Navier-Stokes equations

We are concerned with the existence and uniqueness of local or global solutions for slightly compressible viscous fluids in the whole space. In [6] and [7], we proved local and global well-posedness results for initial data in critical spaces very close to the one used by H. Fujita and T. Kato for incompressible flows (see [14]). In the present paper, we address the question of convergence to the incompressible model (for ill-prepared initial data) when the Mach number goes to zero. When the initial data are small in a critical space, we get global existence and convergence. For large initial data and a bit of additional regularity, the slightly compressible solution is shown to exist as long as the corresponding incompressible solution does. As a corollary, we get global existence (and uniqueness) for slightly compressible two-dimensional fluids.     

Generalized Nash equilibrium problems for lower semi-continuous strategy maps

We consider a class of generalized Ky Fan inequalities (quasi-variational inequalities) in which the involved multi-valued mapping is lower semi-continuous. We present a relaxed version of the generalized Nash equilibrium problem involving strategy maps, which are only lower semi-continuous. This relaxed version may have no exact Nash equilibrium, but we prove that it has an ε-Nash equilibrium for every ε > 0. Easy examples of such problems show no existence of exact solutions, but existence of ε-solutions for every ε > 0. We give positive answers to two questions (in the compact case) raised in a recent paper of Cubiotti and Yao.     

Existence of multidimensional phase transitions in a non-isothermal van der Waals fluid

In this paper, we prove the existence of multidimensional subsonic phase transitions in a non-isothermal van der Waals fluid. The argument is based on the result of the existence of travelling waves given in [S.-Y. Zhang, Existence of travelling waves in non-isothermal phase dynamics, J. Hyperbolic Differ. Equ. 4 (3) (2007) 391–400] and the result of multidimensional stability given in [S.-Y. Zhang, Stability of multidimensional subsonic phase transition in a non-isothermal van der Waals fluid, Preprint]. An iteration technique [A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 281 (1983) 1–93] is applied to achieve the result.     

Numerical solution of fuzzy Camassa-Holm equation by using Homotopy analysis methods

In this paper, a fuzzy Camassa-Holm equation is solved by using the  homotopy analysis method (HAM). The approximation solution of this equation is calculated in the form of series which its components are computed by applying a recursive relation. The existence and uniqueness of the solution and the convergence of the proposed method are proved.     

Existence of global attractors for a nonlinear wave equation

Using a new method developed in [Q. Ma, S. Wang, C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Math. J. 5 (6) (2002) 1542–1558], we prove the existence of the global attractor for a nonlinear wave equation.     

Fixed point theorems for random pseudo-contractive mappings

Very recently in Fierro et al. (2009) [6], we obtained a general principle to prove the existence of Random Fixed Point Theorems. As a consequence of this, we have been able to obtain various generalizations for pseudo-contractive mappings with rather simple proofs. In addition, while we were deriving these extensions for random operators, some deterministic results arose, which also appear to be new.     

一类带奇异项的双调和方程的多解性

利用纤维方法及亏格理论对一类带奇异项的双调和方程进行了研究,证明了方程在两种不同情形下解的存在性及多解性.     

Periodic solutions for nonautonomous discrete-time neural networks

In this paper, we theoretically prove the existence of periodic solutions for a nonautonomous discrete-time neural networks by using the topological degree theory. Sufficient conditions are also obtained for the existence of an asymptotically stable periodic solution. As a special case, we obtain the existence of a fixed point to the corresponding autonomous discrete-time neural networks which corrects the error in [W.R. Zhao, W. Lin, R.S. Liu, J. Ruan, Asymptotical stability in discrete-time neural networks, IEEE Trans. Circuits Syst. I 49 (2002) 1516–1520]. Numerical simulations are given at the end of the paper.     

Global solutions to the damped MHD system

The global existence of solutions to a damped MHD system in Besov spaces is obtained. This research is motivated by the open question [Wu et al., J. Nonlinear Sci. 25 (2015), 157–192, Remark 5.1] on the possibility of extension to the existence of global small solutions from a potential Banach space to a Besov space. The present result positively confirms the open question. What is more, the initial data are not required to be uniformly small.     

Existence of Optimal Controls for Semilinear Parabolic Equations without Cesari-Type Conditions

   Abstract. Optimal control problems governed by semilinear parabolic partial differential equations are considered. No Cesari-type conditions are assumed. By proving the existence theorem and the Pontryagin maximum principle of optimal ``state-control" pairs for the corresponding relaxed problems, an existence theorem of optimal pairs for the original problem is established.     

一类带有趋化性扩散的细菌模型一致有界解的整体存在性

考虑二维空间中一类带有趋化性扩散的生物模型的一致有界解的整体存在性.利用细致的能量估计、不同希尔伯特空间(包括V_2(Q_(t,t+1))、W_(p,p)~(1,2)(Q_(t,t+1))、L_(p,q)(q_(t,t+1))的先验估计以及一致Gronwall不等式,证明了一类带有出生率和死亡率项的生物模型的一致有界解的整体存在性.     

非线性奇异边值问题的正解

该文利用锥上的不动点定理, 在较弱的条件下,讨论了非线性Sturm-Liouville方程奇异边值问题正解的存在性, 并获得了当特征值λ在某一范围内取值时, 边值问题至少存在一个正解的结论.作者的结果包含推广并改进了许多已知的结果。     

The existence of minimal large positive solutions for a class of nonlinear cooperative systems

In this paper we establish the existence of the minimal large positive solution for a general class of nonlinear cooperative systems including the simplest prototype of García-Melián et al. (2016). Precisely, based on the existence of a large positive supersolution, we can infer the existence of the minimal large positive solution. Moreover, we also give some sufficient easily computable conditions for the existence of a large positive supersolution. Our results generalize, very substantially, some of the findings of García-Melián et al. (2016) adopting a rather novel methodology.     

一类广义Liénard方程的概周期解

研究广义 Liénard方程: x'=φ(y)-F(x), y'=-g(x)+p(t), 利用 Amerio的结果证明方程的解部分变元的最终有界性意味着概周期解的存在性, 推广了Cieutat^[1]的主要结果.     

Existence of directed BIBDs with block size 7 and related perfect 5-deletion-correcting codes of length 7

Wang and Yin have established the existence of a directed BIBD with block size 7 and index 1 (a DBIBD(v, 7, 1)) for all v ≡ 1, 7 (mod 21) except for v = 22 and possibly for 68 other cases. In this paper, we reduce the number of possible exceptions to 4, namely v = 274, 358, 400, 526. Correspondingly, for all such v, our results establish the existence of a T(2, 7, v)-code or equivalently a perfect 5-deletion-correcting code with words of length 7 over an alphabet of size v, where all the coordinates must be different. In the process, we also reduce the possible exceptions for (v, 7, 2)-BIBDs to 2 cases, v = 274 and 358 (in addition to the non-existent (22, 7, 2)-BIBD).     

Random dynamics and limiting behaviors for 3D globally modified Navier–Stokes equations driven by colored noise

This paper is mainly concerned with the long-term random dynamics for the nonautonomous 3D globally modified Navier–Stokes equations with nonlinear colored noise. We first prove the existence of random attractors of the nonautonomous random dynamical system generated by the solution operators of such equations. Then we establish the existence of invariant measures supported on the random attractors of the underlying system. Random Liouville-type theorem is also derived for such invariant measures. Moreover, we further investigate the limiting relationship of invariant measures between the above equations and the corresponding limiting equations when the noise intensity approaches to zero. In addition, we show the invariant measures of such equations with additive white noise can be approximated by those of the corresponding equations with additive colored noise as the correlation time of the colored noise goes to zero.