Some Random Operators And Its Applications On Random Equations

In this paper, the existence and uniqueness of random fixed points for a classof random monotone operators with concavity and convexity are researched bymeans of the fixed point theory of nonlinear operator combined with the measuretheory, and the corresponding random fixed point theorems are also obtained. Asapplications, we utilize the results obtained in this paper to discuss the existenceand uniqueness of solutions for a initial value problem of random differentialequations and nonlinear random integral equations. The new results extend andimprove some findings in the relevant literatures. The overall structure of thispaper is as follows:In Chapter1, the historical background and current situation of the discussedproblems are introduced briefly. Meanwhile, the main works of this paper arestated in detail.In Chapter2, we study a class of random fixed point theorems of randommonotone operators with concavity and convexity involving random increasingoperators and random mixed monotone operators. Some corresponding randomfixed point theorems are also obtained by using the properties of cone and partialorder technique.In Chapter3, we consider the following several classes of random sum operator equations: the sum of increasing operators and increasing operators; thesum of increasing operators and decreasing operators; the sum of increasingoperators and mixed monotone operators. We get the existence and uniqueness ofsolutions for random operator equations by using random fixed point theorems ofrandom increasing operators and mixed monotone operators.In Chapter4, we apply the main results obtained in this paper to a initialvalue problem of first order random differential equations and nonlinear randomintegral equations with mixed monotonicity. What’s more, we obtain the existenceand uniqueness of solutions to our problems.