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

time scales

{ (−1)nyδ²ⁿ(t) = g(t)f (t, y(t)), t ∈ [a, b],

y^δ2i(a) − β_i+1 yδ²ⁱ⁺¹(a) = α_i+1 yδ²ⁱ(ν),

_γi+1yδ²ⁱ(ν) = yδ²ⁱ (b), 0 ≤ i ≤ n − 1,

where ν ∈ (a, b), n ≥ 1, βi > 0, 1 < γi <b−a+βi/ν−a+βi, 0 ≤ αi <b−γiν+(γi−1)(a−βi)/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 2n-order differential equations by the use of the property of green’s function, variable transformation and the fixed point index theorem.