Existence of positive solutions of nonlinear -point BVP for an increasing homeomorphism and positive homomorphism on time scales

In this paper, we will consider the following multipoint boundary value problem for the following second-order dynamic equations on time scales ( ϕ ( u Δ ) ) ∇ + a ( t ) f ( t , u ( t ) ) = 0 , t ∈ ( 0 , T ) , u ( 0 ) = ∑ i = 1 m − 2 a i u ( ξ i ) , ϕ ( u Δ ( T ) ) = ∑ i = 1 m − 2 b i ϕ ( u Δ ( ξ i ) ) , where ϕ : R ⟶ R is an increasing homeomorphism and positive homomorphism and ϕ ( 0 ) = 0 . By using fixed point theorems, we obtain an existence theorem of positive solutions for the above boundary value problem, which includes and improve some related results in the relevant literature. As an application, an example to demonstrate our results is given.