Let T ∈ ℕ be an integer with T > 1 , 𝕋 : = { 1 , … , T } , 𝕋 ^ : = { 0 , 1 , … , T + 1 } . We consider boundary value problems of nonlinear second-order difference equations of the form Δ 2 u ( t − 1 ) + λ a ( t ) f ( u ( t ) ) = 0 , t ∈ 𝕋 , u ( 0 ) = u ( T + 1 ) = 0 , where a : 𝕋 → ℝ + , f ∈ C ( [ 0 , ∞ ) , [ 0 , ∞ ) ) and, f ( s ) > 0 for s > 0 , and f 0 = f ∞ = 0 , f 0 = lim ⁡ s → 0 + f ( s ) / s , f ∞ = lim ⁡ s → + ∞ f ( s ) / s . We investigate the global structure of positive solutions by using the Rabinowitz's global bifurcation theorem.