Multiple positive periodic solutions for an n-species competition predator-prey system on time scales

In this paper, we consider an n-species competition predator-prey system on time scales with Holling-type II functional response and multiple exploited (or harvesting) terms, which contains n?1 competing preys and one predator. By using the continuation theorem based on Gaines and Mawhin’s coincidence degree theory, easily verifiable criteria are established for global existence of multiple positive periodic solutions to the above system. In addition, our results generalize the corresponding results of Zhang and Hou (Nonlinear Anal. (RWA) 11:1560–1571, 2010), Fan and Wang (J. Math. Anal. Appl. 262:179–190, 2001), Ding and Lu (Appl. Math. Model. 33:2748–2756, 2009).     

Existence and continuation of periodic solutions of autonomous Newtonian systems

In this article, we study the existence and the continuation of periodic solutions of autonomous Newtonian systems. To prove the results we apply the infinite-dimensional version of the degree for SO(2)-equivariant gradient operators defined by the third author in Nonlinear Anal. Theory Methods Appl. 23(1) (1994) 83-102 and developed in Topol. Meth. Nonlinear Anal. 9(2) (1997) 383-417. Using the results due to Rabier [Symmetries, Topological degree and a Theorem of Z.Q. Wang, J. Math. 24(3) (1994) 1087-1115] and Wang [Symmetries and calculation of the degree, Chinese Ann. Math. 10 (1989) 520-536] we show that the Leray-Schauder degree is not applicable in the proofs of our theorems, because it vanishes.     

中立型Lotka-Volterra系统正周期解的存在性

利用迭合度理论的连续定理，讨论了一类中立型系统的正周期解的存在性．得到了正周期解存在的一些充分条件．     

具有时滞的半比率依赖型捕食者-食饵系统的周期性

A delayed semi-ratio-dependent predator-prey system in a periodic environment is investigated in this paper. By using a continuation theorem based on Gaines and Mawhin‘s coincidence degree,the global existence of positive periodic solution is studied. A set of easily verifiable sufficient conditions are obtained.     

Existence of positive periodic solution of n-dimensional Lotka-Volterra system with delays

With the help of a continuation theorem based on coincidence degree, easily verifiable criteria are established for the existence of positive periodic solutions of n-D Lotka-Volterra system with delays. And sufficient conditions are obtained for the persistence of the delay system. Finally, computer simulations are presented to illustrate the conclusions.     

A nonlinear diffusion system coupled via nonlinear boundary flux

In this paper, we deal with the global existence and nonexistence of solutions to a nonlinear diffusion system coupled via nonlinear boundary flux. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions. The critical curve of Fujita type is conjectured with the aid of some new results, which extend the recent results of Wang et al. [Nonlinear Anal. 71 (2009) 2134-2140] and Li et al. [J. Math. Anal. Appl. 340 (2008) 876-883] to more general equations.     

Multiple periodic solutions in generalized Gause-type predator–prey systems with non-monotonic numerical responses

By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of multiple positive periodic solutions in periodic Gause-type predator–prey systems with non-monotonic numerical responses and time delays. As corollaries, some applications are listed. In particular, our results improve and supplement those obtained by Chen [Y. Chen, Multiple periodic solutions of delayed predator–prey systems with type IV functional responses, Nonlinear Anal. Real World Appl. 5 (2004) 45–53].     

Existence of positive periodic solutions for a nonautonomous neutral delay n-species competitive model with impulses

In this paper, by using some analysis techniques and a new existence theorem, which is different from Gaines and Mawhin’s continuation theorem and abstract continuation theory for k-set contraction, we obtain some sufficient and realistic conditions for the existence of positive periodic solution of a general neutral delay n-species competitive model with impulses. As an application, we also examine some special cases which have been studied extensively in the literature. Some known results are improved and generalized.     

Global solutions of nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, by using the generalization of Darbo’s fixed point theorem, we establish the existence of global solutions of an initial value problem for a class of second-order impulsive integro-differential equations of mixed type in a real Banach space. Our results generalize and improve on the results of Guo et al. [F. Guo, L.S. Liu, Y.H. Wu, P. Siew, Global solutions of initial value problems for nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces, Nonlinear Anal. 61 (2005) 1363–1382] in the sense that the conditions for existence of global solution in our theorem is simpler and less strict. To demonstrate the application of the theorem, we give the global solutions of two mixed boundary value problems for two classes of fourth order impulsive integro-differential equations.     

Some new extensions of Edelstein-Suzuki-type fixed point theorem to G-metric and G-cone metric spaces

In this paper, we prove some fixed point theorems for generalized contractions in the setting of G-metric spaces. Our results extend a result of Edelstein [M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc., 37 (1962), 74–79] and a result of Suzuki [T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009), 5313–5317]. We prove, also, a fixed point theorem in the setting of G-cone metric spaces.     

时标上一类非自治捕食系统的周期解(英文)

本文在时标上运用迭合度理论中的Gaines和Mawhin连续性定理研究了一类非自治捕食系统的周期解的存在性,得到了此模型周期解存在的充分条件,该方法可将证明连续和离散微分方程的周期解的存在性统一起来.     

脉冲时滞微分方程的周期性和稳定性研究

利用Gaines和Mawhin的重合度理论,给出了一个周期环境下具有分布时滞的脉冲微分方程的正周期解的存在性的判定。与此同时,通过引入分片连续Lyapunov函数的方法,建立了脉冲效应下时滞微分方程的零解稳定性的充分条件。     

POSITIVE PERIODIC SOLUTIONS OF A CLASS OF DISCRETE TIME SYSTEM WITH DELAYS AND A FEEDBACK CONTROL

With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, sufficient conditions are established for the existence of periodic solutions of a class of discrete differential system with delays and feedback control.     

Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications

The purpose of this paper is by using the hybrid iterative method to prove some strong convergence theorems for approximating a common element of the set of solutions to a system of generalized mixed equilibrium problems and the set of common fixed points for two countable families of closed and asymptotically relatively nonexpansive mappings in Banach space. The results presented in the paper improve and extend the corresponding results of Su et al. [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890-3906], Li and Su [H.Y. Li, Y.F. Su, Strong convergence theorems by a new hybrid for equilibrium problems and variational inequality problems, Nonlinear Anal. 72 (2) (2010) 847-855], Chang et al. [S.S. Chang, H.W. Joseph Lee, Chi Kin Chan, A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications, Nonlinear Anal. TMA 73 (2010) 2260-2270], Kang et al. [J. Kang, Y. Su, X. Zhang, Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications, Nonlinear Anal. HS 4 (4) (2010) 755-765], Matsushita and Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theory 134 (2005) 257-266], Tan et al. [J.F. Tan, S.S. Chang, M. Liu, J.I. Liu, Strong convergence theorems of a hybrid projection algorithm for a family of quasi-?-asymptotically nonexpansive mappings, Opuscula Math. 30 (3) (2010) 341-348], Takahashia and Zembayashi [W. Takahashi, K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70 (2009) 45-57] and Wattanawitoon and Kumam [K. Wattanawitoon, P. Kumam, Strong convergence theorems by a new hybrid projection algorithm for fixed point problem and equilibrium problems of two relatively quasi-nonexpansive mappings, Nonlinear Anal. Hybrid Systems 3 (2009) 11-20] and others.     

带有扩散、脉冲和时滞的非自治捕食系统的正周期解

考虑了一类食饵在斑块环境中扩散具有脉冲和时滞的捕食系统，通过灵活地运用Gaines和Mawhin的连续拓扑度定理，获得了一系列易验证的正周期解存在的充分条件．     

Multiple solutions of nonlinear elliptic equations for oscillation problems

We discussed oscillating equations with Neumann boundary value in [Nonlinear Anal. 54 (2003) 431-443] and [J. Math. Anal. Appl. 298 (2004) 14-32] and prove the existence of infinitely many nonconstant solutions. However, it seems difficult to find infinitely many disjoint order intervals for oscillating equations with Dirichlet boundary value. To get rid of this difficulty, in this paper, we build up a mountain pass theorem in half-order intervals and use it to study oscillating problems with Dirichlet boundary value in which we only have the existence of super-solutions (or sub-solutions) and obtain new results on the exactly infinitely many solutions.     

Weighted pseudo-almost periodic solutions of a class of abstract differential equations

For abstract linear functional differential equations with a weighted pseudo-almost periodic forcing term, we prove that the existence of a bounded solution on R⁺ implies the existence of a weighted pseudo-almost periodic solution. Our results extend the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations. To illustrate the results, we consider the Lotka-Volterra model with diffusion.     

On generalized operator quasi-equilibrium problems

In this paper, we will introduce the generalized operator equilibrium problem and generalized operator quasi-equilibrium problem which generalize the operator equilibrium problem due to Kazmi and Raouf [K.R. Kazmi, A. Raouf, A class of operator equilibrium problems, J. Math. Anal. Appl. 308 (2005) 554-564] into multi-valued and quasi-equilibrium problems. Using a Fan-Browder type fixed point theorem in [S. Park, Foundations of the KKM theory via coincidences of composites of upper semicontinuous maps, J. Korean Math. Soc. 31 (1994) 493-519] and an existence theorem of equilibrium for 1-person game in [X.-P. Ding, W.K. Kim, K.-K. Tan, Equilibria of non-compact generalized games with L^∗-majorized preferences, J. Math. Anal. Appl. 164 (1992) 508-517] as basic tools, we prove new existence theorems on generalized operator equilibrium problem and generalized operator quasi-equilibrium problem which includes operator equilibrium problems.     

Equilibria of noncompact abstract economies in general almost convex strategy spaces

We first extend the concept of almost convex condition and establish a fixed point theorem for correspondences with convex values only on an almost convex subset for their ranges. This generalizes both results of Himmelberg [C.J. Himmelberg, Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 (1972) 205–207] and the result of Jafari and Sehgal [F. Jafari, V.M. Sehgal, An extension to a theorem of Himmelberg, J. Math. Anal. Appl. 327 (2007) 298–301]. Furthermore, applying it, we have existence theorems for equilibria of noncompact abstract economies in general almost convex strategy spaces. Our theorems generalize the corresponding results of Zhou [Jianxin Zhou, On the existence of equilibrium for abstract economies, J. Math. Anal. Appl. 193 (1995) 839–858], Tan and Wu [K.-K. Tan, Z. Wu, A note on abstract economies with upper semicontinuous correspondence, Appl. Math. Lett. 11 (5) (1998) 21–22] in several ways. In particular, we answer the question raised by Zhou in the reference cited above in the affirmative with weaker hypotheses.     

Existence theorem and algorithm for a general implicit variational inequality in Banach space

By using the generalized f-projection operator, the existence theorem of solutions for the general implicit variational inequality GIVI(T-ξ,K) is proved without assuming the monotonicity of operators in reflexive and smooth Banach space. An iterative algorithm for approximating solution of the general implicit variational inequality is suggested also, and the convergence for this iterative scheme is shown. These theorems extend the corresponding results of Wu and Huang [K.Q. Wu, N.J. Huang, Comput. Math. Appl. 54 (2007) 399–406], Wu and Huang [K.Q. Wu, N.J. Huang, Bull. Austral. Math. Soc. 73 (2006) 307–317], Zeng and Yao [L.C. Zeng, J.C. Yao, J. Optimiz. Theory Appl. 132 (2) (2007) 321–337] and Li [J. Li, J. Math. Anal. Appl. 306 (2005) 55–71].