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Let $${\Bbb T}$$ be a time scale such that 0, T ∈ $${\Bbb T}$$ . By means of the Schauder fixed point theorem and analysis method, we establish some existence criteria for positive solutions of the m–point boundary value problem on time scales where a ∈ C ld ((0, T), [0,∞)), f ∈ C ld ([0,∞) × [0,∞), [0,∞)), β, γ ∈ [0,∞), ξ i ∈ (0, ρ( T)), b, a i ∈ (0,∞) (for i = 1, . . . , m− 2) are given constants satisfying some suitable hypotheses. We show that this problem has at least one positive solution for sufficiently small b > 0 and no solution for sufficiently large b. Our results are new even for the corresponding differential equation ( $${\Bbb T}$$ = ℝ) and difference equation ( $${\Bbb T}$$ = ℤ). [ABSTRACT FROM AUTHOR]