比较定理

Rties of the system are researched by using the comparison theorem of impulsive ordinary differential equations and using the solutions of dominating inpulsive ordinary differential equation to control and estimate the solutions of the system.

利用脉冲常微分方程的比较定理以及利用相应的脉冲常微分方程的解控制和估计所讨论模型的解，研究了系统模型的解的渐近性质。

Based on Floquet theory and the comparison theorem of impulsive differential equation, the analysis of global asymptotic stability of the "infectionfree" periodic solution was given.

运用Floquet乘子理论和脉冲微分方程比较定理，证明了该周期解的全局渐近稳定性，并获得了系统一致持续生存的条件。

In this paper, the nonexistence of the closed orbit which circles the equilibrium is proved by using the Dulac function, then all the orbits of the system are bounded by the Filippov transformation and comparability theorem.

本文利用Dulac函数法，首先判定了不存在绕平衡点的闭轨线，然后利用Filippov变换和比较定理，证明了系统所有轨线的有界性，进而得到了平衡点是全局渐近稳定的。

By integral in mean and comparison theorem, boundednesses of consumer population and resources, as well as a sufficient condition for population's weak persistent in mean are given when the environment is with a small capacity.

利用积分均值法和比较定理，给出消费者种群和资源量的有界性，以及环境容量较小情形下种群弱平均持续生存的充分条件。

Chapter 4 discusses the stability of a class of nonlinear impulsive partial differential equations. The main idea is translating the stability of the considered nonlinear impulsive partial differential equation into that of the corresponding linear impulsive ordinary differential equation via Gronwall-Bellman inequality with impulse on the basis of comparison theorem.

第四章基于比较定理，主要利用含脉冲的Gronwall-Bellman不等式初步讨论了脉冲偏微分系统的稳定性，将一类非线性脉冲偏微分方程的稳定性化归为线性脉冲常微分方程的稳定性，为将脉冲常微分方程稳定性的有关结论推广到脉冲偏微分方程提供了理论依据并奠定了基础。

In chapter two, under non-Lipschitz condition, the existence and uniqueness of the solution of the second kind of BSDE is researched, based on it, the stability of the solution is proved; In chapter three, under non-Lipschitz condition, the comparison theorem of the solution of the second kind of BSDE is proved and using the monotone iterative technique , the existence of minimal and maximal solution is constructively proved; in chapter four, on the base of above results, we get some results of the second kind of BSDE which partly decouple with SDE, which include that the solution of the BSDE is continuous in the initial value of SDE and the application to optimal control and dynamic programming. At the end of this section, the character of the corresponding utility function has been discussed, e.g monotonicity, concavity and risk aversion; in chapter 5, for the first land of BSDE ,using the monotone iterative technique , the existence of minimal and maximal solution is proved and other characters and applications to utility function are studied.

首先，第二章在非Lipschitz条件下，研究了第二类方程的解的存在唯一性问题，在此基础上，又证明了解的稳定性；第三章在非Lipschitz条件下，证明了第二类BSDE解的比较定理，并在此基础上，利用单调迭代的方法，构造性证明了最大、最小解的存在性；第四章在以上的一些理论基础之上，得到了相应的与第二类倒向随机微分方程耦合的正倒向随机微分方程系统的一些结果，主要包括倒向随机微分方程的解关于正向随机微分方程的初值是具有连续性的，得到了最优控制和动态规划的一些结果，在这一章的最后还讨论了相应的效用函数的性质，如，效用函数的单调性、凹性以及风险规避性等；第五章，针对第一类倒向随机微分方程，运用单调迭代方法，证明了最大和最小解的存在性，并研究了解的其它性质及在效用函数上的应用。

By means of the comparison principle and the method of upper and lower solution,the corresponding sufficient conditions under which the models have a solution or no solution are obtained. If other parameters are given,the controllable parameters such as the population of release and the times of impulsive harvest are estimated respectively.

基于比较定理和上下解方法，分别获得了模型有解或无解的充分条件，估计了相关可控参数，即脉冲投放模型中的投放量以及脉冲收获模型中的脉冲收获次数。

We first use the method similar with that in EL.karoui[1] to get the comparison theorem of K in the case of one reflecting barrier,At last,we use the idea of penalization to prove one existence result of the solution for the multi-dimensional RBSDE where the coefficient is continuous and has the linear growth,this also help us to give the comparison theorem of K~+ and K~- in the case of two reflecting barriers.

最后，用惩罚方法证明了当多维双边界反射倒向随机微分方程的生成元满足连续和线形增长条件时解的存在性，并同时得到了双边界情形下，K~+和K~-的多维比较定理。

The boundedness of system is proved by the comparative theme of impulsive differential equation, and the local stability of trivial or semi-trivial positive periodic solution (prey annihilation periodic solution) is tested.

利用脉冲微分方程的比较定理，证明了系统的有界性，讨论了平凡周期解与半平凡周期解的局部稳定性，进而利用重合度理论证明了系统周期正解的存在性。

By applying the comparative theorem and constructing a suitable Lyapunov functional, some sufficient conditions are obtained for guaranteeing that the system is permanent and global attractivity.

利用比较定理结合构造Lyapunov泛函的方法，得到了保证系统永久持续生存及任意正解全局吸引的充分条件。

According to the clear water experiment, aeration performance of the new equipment is good with high total oxygen transfer coefficient and oxygen utilization ratio.

曝气设备的动力效率在叶轮转速为120rpm～150rpm时取得最大值，此时氧利用率和充氧能力也具有较高值。

The environmental stability of that world - including its crushing pressures and icy darkness - means that some of its most famous inhabitants have survived for eons as evolutionary throwbacks, their bodies undergoing little change.

稳定的海底环境─包括能把人压扁的压力和冰冷的黑暗─意谓海底某些最知名的栖居生物已以演化返祖的样态活了万世，形体几无变化。