The first main purpose of this project is to study the operators concerning the heat equations and the wave equation such operator are the Laplace, the ultra hyperbolic the Diamond operators and the mixed operator. In doing research for such operators, we are succeeded in obtaining the interesting solution that all solutions cover all old area of solution before and all such solution are beautiful uniqueness. The second purpose of this project is to construct new iterative methods for approximating a fixed point and common fixed points of nonlinear mappings. In this part, we introduce a new three-step iteration with errors for nonexpansive nonself-mappings in a uniformly convex Banach space. Weak and strong convergence theorems of the new three-step iteration under certain control conditions are established. We also modify Noor iterations for non-Lipshitzian mappings in Banach spaces and prove weak and strong convergence theorems of the modified Noor iterations under some control conditions. For finding a common fixed point of a finite family of nonexpansive mappings, we introduce a new interactive method for them and prove weak and strong convergence theorems under some suitable control conditions. Moreover, we introduce new methods for finding a common element of a fixed point set of nonlinear mappings and the set of solutions of equilibrium problems. Our results improve and extend story results in this area