In this article, we will solve optimization problems from the financial and economic field with constants, infinite-horizon iterative techniques and elements from fixed point theory. We will resort to Ćirić contractions in Banach space and the main result consists of the existence of a fixed point to solve an important class of infinite-horizon iterative schemes for optimization problems with state constraints for which the value function is merely lower semi-continuous. The developed tools allowed us to solve the stationary infinite-horizon optimization problems, especially for the maximization of the utility of households. We present some fixed point results that are fundamental for the development of our contributions: Notably, existence, monotonicity, attainability and results in the Ćirić contribution. We show the convergence in norm with probability for an iterative procedure defined for our problem under the stated assumptions. By using the Ćirić operator and the Reich-Rus type $\psi F$-contraction, we prove the existence of the results of the optional cost function of an infinite horizon problem in a complete metric space. For a particular case, we realize a numerical simulation in C++. The conclusions are that the convergence, the existence and the uniqueness results of an optimal cost function of an infinite horizon problem in a Banach space can be treated by resorting to the Ćirić operator.