存在性定理

Under suit upper and lower solution, iteration sequences were constructed, and existence and unique of solutions of linear boundary value problems on second order nonlinear Hammerstein type integro-differential-difference equation were obtained by means of applying the Arzela-Ascoli theorem Lebesque control convergence theorem and disproof method.

在上下解存在的条件下，构造迭代序列，由Arzea-Ascoli定理和Lebesque控制收敛定理得到了二阶非线性Hammerstein型积分微分差分方程的线性边值问题的解的存在性。再利用反证法获得了解的唯一性。

The contents are the following:In chapter two, the existence and multiplicity results for the following equation of p-Laplacian type are obtained.For the elliptic quasilinear hemivariational inequality involving the p-Laplacian operator,in order to use the mountain pass theorem proving the existence result, the authors usually need to use the uniform convexity of the Sobolev space to prove the energy function satisfies the PS condition. But for the p-Laplacian type equation mentioned above, this method is no use. To overcome this difficulty, the potential function is assumed to be convex, then I prove the existence result and by using the extension of the Ricceri theorem, the multiplicity result for the problem is obtained.

在第二章我们首先考虑关于以下p-Laplacian型(p-Laplacian type)方程非平凡解及多解的存在性对于带有p-Laplacian算子的椭圆拟线性半边分不等式问题，为应用非光滑的山路引理证明解的存在性，在证明方程所对应的能量泛函满足非光滑的PS条件时，需利用Sobolev空间的一致凸性，但是对于具有更一般形式的算子的p-Laplacian型方程，不具备上述性质，在文中为克服这一困难，本人对位势泛函做了一致凸的假设，从而证明了解的存在性，并应用推广的Ricceri定理，证明了方程三个解的存在性。

As a application of the above results, some minimax inequalities and saddle point theorems can be obtained. In addition, fixed point theorems and section theorems, maximal element theorems and the existence theorems of equilibrium for abstract economies can also be discussed.

作为应用，可以得到L-凸空间中极大极小不等式和鞍点定理；进一步，还可以探讨不动点定理和截口定理，极大元定理和抽象经济的平衡存在性等问题。

According to the existence results of general equilibrium problems and vector equilibrium problems have been studied more and more. Inspired and motivated by these research results, this paper is devoted to study systematically a class of equilibrium problems, which is unify and extension of a large number of known equilibrium problems and variational inequalities problems. The research is carried on from three aspects.Firstly, in finitely continuous topological spaces, we introduce four new types of the system of generalized vector quasi-equilibrium problems, and we derive some existence results of a solution for the system of generalized vector quasi-equilibrium problems via the maximal element theorems in product finitely continuous topological spaces.Secondly, in complete metric spaces, we provide the Ekeland variational principle to equilibrium problems with set-valued maps. And via the Ekeland variational principle, existence results for vector equilibrium problem with set-valued maps and the system of vector equilibrium problem with set-valued maps.

针对一般的均衡问题和向量均衡问题解的存在性，已有许多研究成果，受这些成果的启发，本文主要从理论上较为系统地研究了一类均衡问题，它统一和推广了许多已有的均衡问题和变分不等式问题，研究分有三个方面；首先，在有限连续拓扑空间中，我们提出了四类广义向量拟均衡系，并借助于有限连续拓扑空间中的极大元定理讨论了这四类均衡系问题的解的存在性问题，然后，在完备度量空间中，我们给出了关于集值均衡问题的Ekeland变分原理，并利用Ekeland变分原理分别讨论了集值向量均衡问题和集值向量均衡系问题的解的存在性。

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。

A complete novel assumption is presented. Then, we prove the existence of two nontrivial solutions for a class of second order superlinear difference systems with the help of Minimax principle and Linking theorem in critical point theory,.6. Two applications in physics are given. Firstly, a coupled differential system coming from second harmonic waves is discussed.

利用Ricceri建立的三临界点定理，证明了一类含参数二阶差分方程在某些新的条件下至少存在三个解；随后，利用临界点理论中的极小极大原理和环绕定理，讨论了一类二阶超线性差分方程组多重解的存在性。

The existence of solutions for two-point boundary value problem subject to nonlinear second order differential equation with mixed boundary conditions is studied by using topological transversality theorem, and some known results are improved.

现利用拓扑横截定理，考虑了二阶常微分方程两点边值问题在组合边界条件下的解的存在性，对二阶常微分方程两点边值问题所对应的辅助问题作先验界估计，并利用拓扑横截定理，得到了边值问题解的存在性，推广了一些已有结果。

The existence and uniqueness of the limit cycle are proved by applying Poincare-Bendixson theorem and Zhang Zhifen's Uniqueness theorem.

运用Poincare-Bendixson环域定理，证明了极限环的存在性。这用张芷芬惟一性定理，注明了极限环的惟一性。

This chapter proposes three conceptions, i.e., Kernelled quasidiferential, star-kernel and star-diferential, and establishes their operational properties. A sufficient theorem and a sufficent and necessity theorem for a quasi-kernel being a kernelled quasidiferential are proven. Both the existence of star-kernel for a quasidiferentiable function and the existence of star-differential for a direnction-ally diferentiable function are established.

在这一章里，首先给出核拟微分，星核与星微分的定义及其它们的运算性质；然后证明了拟核微分的一个充分条件定理及一个充要条件定理；最后讨(来源：A27BC论文网www.abclunwen.com)论拟可微函数星核的存在性及方向可微函数星微分的存在性以及Penot-微分与上下导数之间的关系。

Thischapter proposes three concepts,i.e.,kernelled quasidifferential,star-kerneland star-differential,and establishes their operational properties.A sufficiencytheorem and a sufficiency and necessity theorem for a quasi-kernel being a ker-nelled quasidifferential are proven.Both the existence of star-kernel for a qua-sidifferentiable function and the existence of star-differential for a directionallydifferentiable function are established.The relationships between sub-and su-per derivatives and Penot differentials are discussed as well.

这一章我们引入了三个概念，即核拟微分，星核与星微分并建立了它们的运算性质；证明了拟核为核拟微分的一个充分条件定理及一个充分与必要条件定理；建立了拟可微函数星核的存在性及方向可微函数星微分的存在性并讨论了Penot微分与上、下导数的关系。

With Death guitarist Schuldiner adopting vocal duties, the band made a major impact on the scene.

随着死亡的吉他手Schuldiner接受主唱的职务，乐队在现实中树立了重要的影响。

But he could still end up breakfasting on Swiss-government issue muesli because all six are accused of nicking around 45 million pounds they should have paid to FIFA.

不过他最后仍有可能沦为瑞士政府&议事餐桌&上的一道早餐，因为这所有六个人都被指控把本应支付给国际足联的大约4500万英镑骗了个精光。