In the present work, the canonical form of the differential equations is derived from the Hamiltonian function H which is obtained for the system of the four-body problem. This canonical form is considered as the equations of motion, the equilibrium points of the restricted four-body problem are studied under the effects of radiation pressure and oblatness Lyapunov function is used to provide a method for showing that equilibrium points are stable or asymptotically stable. If the system has an equilibrium point conditionally the eigenvalues of the system contain negative real parts, the scalar potential function is positive definite, then The Lyapunov center's theorem is used to analyze the stability and periodicity of the motion of orbits about these equilibrium points of the restricted four-body problem. From this theorem, the Lyapunov function is found. Also, the stability regions are studied by using The Poincare maps, an analytical and numerical approach had been used. A cod of Mathematica is constructed to truncate these steps. The periodic orbits around the equilibrium points are investigated for the Sun-Earth-Moon system.