Аннотация:The operator equations of the maximum principle are constructed in nonlinear optimal control problems in the form of fixed point problems in the control space. The equivalence of operator equations to the condition of the maximum principle is shown. The constructed operator forms of the maximum principle make it possible to apply and modify the well-known apparatus of the theory and methods of fixed points to search for extreme controls. The control operators under consideration define new iterative algorithms for finding extreme controls. The proposed iterative algorithms of fixed points of the maximum principle have the property of nonlocality of successive control approximations and the absence of a parametric procedure for improving the approximation at each iteration, which is characteristic of the well-known standard gradient type methods.