In this paper, governed by the fundamental solutions we introduce the Green’s function of the second-order differential equations in general form with respect to boundary conditions and deal with the solvability of the infinite system of second-order differential equations

with p, q ∈ C([0, T ], ℝ) and the boundary conditions u_i(0) = u_i(T ) = 0. We remark that the subjected system has not been previously considered and this investigation complements several results in the literature. Using the ideas of Hausdorff measure of noncompactness and Meir-Keleer condensing operator we seek the sufficient conditions to justify the existence of solutions for the aforementioned system in Banach sequence space ℓ_p (1 ≤ p < ∞). Finally, an example is given to ascertain the efficiency of the results.