本文的主要目的是寻找半线性方程\[ L u+N(u)=h \]在Hilbert空间$H$中解的存在性条件，其中$L$是线性自伴算子，$N$是非线性映射，$h\in H$ .在这个项目中，我们集中在$0$的情况下，属于运营商$L$这是以前没有研究在这个一般设置的本质谱。在第二章中，我们还假设$0$是$L$的本征谱的下确界。我们将$(S_+)$类稠定映射的度理论应用于方程左边给出的算子。我们假设非线性部分$N$是拟单调的，并且满足次线性增长条件。此外，由于$0$可以有非平凡的本征空间，我们利用所谓的衰退功能连接到$L$和$N$，它允许控制非线性部分的行为的核心$L$。在第三章中，我们允许$L$的本征谱低于零。我们的方法是基于观察到算子$L$的某些扰动在$L$的谱的非负部分所对应的$H$的子空间上是极大单调的，并且我们假设算子$N$也满足一定的单调性条件.其次，我们探讨了极大单调算子的满射性质，以证明该类扰动方程解的存在性。最后利用退化泛函和$N$上的a增长条件证明了方程的可解性。给出了该方法在定常半线性薛定谔方程可解性中的应用。
英文摘要：
The main goal of this dissertation is to find conditions which will guarantee the existence of solutions in the Hilbert space $H$ of semilinear equation \[ L u+N(u)=h \] where $L$ is a linear and self-adjoint operator, $N$ a non-linear mapping and $h\in H$. In this project we concentrate on the case when $0$ belongs to the essential spectrum of operator $L$ which was not previously studied in this general setting.
In chapter 2 we additionally assume that $0$ is the infimum of the essential spectrum of $L$. We apply the degree theory for densely defined mappings of class $(S_+)$ to the operator given by the left hand side of the equation. We assume that non-linear part $N$ is quasi-monotone and satisfies sublinear growth condition. Moreover, since $0$ can have non-trivial eigenspace, we make use of the so called recession functional connected with $L$ and $N$ which allows to control the behaviour of non-linear part on the kernel of $L$.
In chapter 3 we allow the essential spectrum of $L$ to lay below zero. Our method is based on the observation that certain perturbation of operator $L$ is maximal monotone on the subspace of $H$ corresponding to non-negative part of the spectrum of $L$ and we assume that operator $N$ satisfies certain monotonicity condition as well. Next we explore the surjectivity properties of maximal monotone operators to show the existence of solutions to the class of perturbed equations. Finally with the help of recession functional and a growth conditions on $N$ we prove the solvability of the equation. The application to the solvability of stationary semilinear Schrödinger equation is given.
Trotter-Kato approximations of semilinear stochastic evolution equations in Hilbert spaces
受T. E. Govindan在[5，8，9]中的结果，本文研究了一个更一般的半线性随机发展方程。本文所考虑的方程与前一个方程的不同之处在于，它对随机积分中的非线性函数做了一些改变，这也取决于当时随机过程的概率分布。首先，本文考虑了这类方程的温和解的存在唯一性。进一步，对该问题的弱解引入Trotter-Kato逼近系统，得到了诱导概率测度和零级逼近的弱收敛性.然后我们考虑了这类方程的参数依赖性的经典极限定理。最后，给出了一个随机偏微分方程的例子来说明我们的结果。
Weyl Pseudo Almost Periodic Type Solutions to Semilinear Stochastic Evolution Equations Driven by Fractional Brownian Motion
Virginia Military Institute (VMI)（弗吉尼亚军事学院）；Faculty of Technical Sciences, University of Novi Sad（诺维萨德大学技术科学学院）；Department for Mathematics and Informatics, Faculty of Civil Engineering, Ss. Cyril and Methodius University in Skopje（斯科普里圣西里尔和圣美多德大学土木工程学院数学与信息系）
