This paper verifies the semi-local convergence of a sixth-order iterative method for solving nonlinear systems. In the proof, the recursive relation method is used to prove the semi-local convergence of the iterative method in Banach spaces. The proof process outlined above does not necessitate the higher-order continuous differentiability of the function $L(u)$. The advantage of the iterative method is that by employing auxiliary sequences and functions, convergence can be established using only first-order Fréchet derivatives. The proposed sixth-order method reduces computational costs compared to existing sixth-order methods by requiring only one LU decomposition per iteration instead of two. Finally, this sixth-order iterative approach is utilized for solving Hammerstein equations along with real gas equation of state. The experimental results prove the effectiveness of the iterative method.