Several Fixed Point Theorems, Iterative Schemes And Their Appliction

The fixed point theory takes a great important space on math and its applications. The development of fixed point problem brought great convenience to solve the problems of the equation and physics. Fixed point problems have close relationship with optimization problems, maximum and minimum problems, complementarity problems, variational inequality problems and equilibrium problems.In this paper,we studied the existence of the fixed point on some mappings, such as contraction mappings and expansive mappings in some spaces, got the iterative algorithm of the fixed point and studied the applications to differential equations and the equilibrium problems. This paper is divided into three parts:In the first chapter, we showed the introductions and some preliminaries. In the second chapter, in metric space, we discussed the existence of fixed points with promoted expansive mapping and the contraction mapping application to differential equations with two ordinary point; in topological linear space, we used the fixed point existence theorem to solve the equilibrium problem with upper and lower bounded and the equilibrium problems with lower and upper bounds and restricted conditions; in cone metric spaces, we studied the existence of fixed points with a general contraction mapping; in fuzzy metric spaces, discussed the existence of the fixed points theorem and the fixed point can be found by PK algorithm. In the third chapter, at first, we studied the fixed point of asymptotically k-strict pseudo-contraction mapping, proved weak and strong convergence of its iterative sequence, then showed weak and strong convergence theorems of the intersection between the solution of the equilibrium problem and fixed point problem under modified Ishikawa algorithms.In some way, the conclusions above have not only improved the conclusions some scholars have already got, but also we showed some examples and applications about relative conclusions.