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) ;

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