In this work, we present a new extension of Darbo's theorem for two different classes of altering distance functions via measure of non-compactness. Using two-variable contractions we obtain the wellknown results in this literature (see [22]). We also use these results to discuss the existence of solutions for a system of integral equations. Finally, we provide an example to confirm the results obtained. Definition 1.1. ([11]) A function µ : M E → R + is called a measure of non-compactness in E if it satisfies the following hypothesis: (BM1) The family ker µ = X ∈ M E : µ (X) = 0 ∅ and ker µ ⊂ N E ; (BM2) X ⊂ Y ⇒ µ (X) ≤ µ (Y) ;

We propose a new notion of contraction mappings for two class of functions involving measure of noncompactness in Banach space. In this regard we present some theory and results on the existence of tripled fixed points and some basic Darbo's type fixed points for a class of operators in Banach spaces. Also as an application we discuss the existence of solutions for a general system of nonlinear functional integral equations which satisfy in new certain conditions. Further we give an example to verify the effectiveness and applicability of our results.