Summary: The purpose of this article is to prove some existence theorems for solutions of a certain system of condensing operator equations in Banach spaces. Moreover, the main results are applied to prove the existence theorems of solutions for two kinds of systems of nonlinear integral equations. To get the expected results, some results about the measure of noncompactness are obtained in Banach space \(BC([0,+\infty))\) consisting of all real bounded and continuous functions defined on \([0,+\infty)\). The results of this article improve and extend the previous classical results given in the literature.