Positive solutions of nonlinear -point boundary value problems on time scales

In this paper, by using fixed point theorems in a cone and the associated Green’s function, we study the existence of at least two and three positive solutions to the m -point boundary value problem (BVP) on time scales, u Δ ∇ ( t ) + f ( t , u ( t ) ) = 0 , t ∈ [ 0 , 1 ] ⊂ T , β u ( 0 ) − γ u Δ ( 0 ) = 0 , u ( 1 ) = ∑ i = 1 m − 2 α i u ( ε i ) , m ≥ 3 , where T is a time scale, f ∈ C l d ( [ 0 , 1 ] × [ 0 , ∞ ) , [ 0 , ∞ ) ) , β , γ ∈ [ 0 , ∞ ) , α i ∈ [ 0 , ∞ ) for i = 1 , 2 , … , m − 2 , and ε i satisfy 0 < ε 1 < ε 2 < ⋯ < ε m − 2 < ρ ( 1 ) .