Positive solutions of semipositone higher-order differential equations on time scales

In this paper, we are concerned with the following 2 n th-order differential equations on time scales { ( − 1 ) n y Δ 2 n ( t ) = g ( t ) f ( t , y ( t ) ) , t ∈ [ a , b ] , y Δ 2 i ( a ) − β i + 1 y Δ 2 i + 1 ( a ) = α i + 1 y Δ 2 i ( ν ) , γ i + 1 y Δ 2 i ( ν ) = y Δ 2 i ( b ) , 0 ≤ i ≤ n − 1 , where ν ∈ ( a , b ) , n ≥ 1 , β i > 0 , 1 γ i b − a + β i ν − a + β i , 0 ≤ α i b − γ i ν + ( γ i − 1 ) ( a − β i ) b − ν , i = 1 , 2 , … , n . The functions g : [ a , b ] → [ 0 , + ∞ ) and f : [ a , b ] × [ 0 , + ∞ ) → ( − ∞ , + ∞ ) are continuous, or g is singular at t = a and/or t = b . We obtain some properties and sharp estimates of the corresponding Green’s function and investigate the existence of positive solutions of the semipositone problems for 2 n -order differential equations by the use of the property of Green’s function, variable transformation and the fixed point index theorem.