Solvability of infinite systems of second-order differential equations with boundary conditions in ℓp

Abstract

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 innite system of second-order differential equations

with p, q ∈ C([0, T ], ℝ) and the boundary conditions  _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.

Mathematics Subject Classication (2010): 47H10, 34A34, 34B27, 35E05.