Fixed Point Theorems With Applications Of Several Classes Of Nonlinear Operators

In this thesis, we mainly investigate the existences of fixed points for weakly inward 1-set-contraction mappings and monotone operators with their applications. The whole thesis concludes three chapters.It is the purpose of Chapter 1 to introduce a fixed point index of weakly inward 1-set-contraction mappings. With the aid of the new index, we obtain some new fixed point theorems, nonzero fixed point theorems and multiple positive fixed points for this class of mappings. As an application of nonzero fixed point theorems, we discuss an eigenvalue problem. And our results probably supply some tools to study the existence of nontrivial solutions for the Hammerstein Integral Equationin the case of not requiring the nonlinear terms and kernel function non-negative.In Chapter 2, iterative techniques and partial order theory are used to study the existence of coupled quasi-solutions and solutions for a mixed monotone operator equation Lx = N(x, y) in complete metric space and Banach space, respectively. And then main results are applied to two point boundary value problems of an integro-differential equation.In Chapter 3, we introduce some new concepts such as random monotone operators, random Mann iteration and so on in a separable real Banach space. And the existence and uniqueness theorems of random fixed points for random monotone operators satisfying Condition (H) are proved.