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This article deals with the following second-order multi-point boundary value problem x′′(t) = f(t,x(t),x′(t))+e(t),t ∈(0,1),x′(0)=(α_ix′(ξ_i) from i=1 to m,x(1)=(β_jx(ηj) from j=1 to n .Under the resonance conditions α_i from i=1 to m =1,β_j from j=1 to