The existence of a positive solution to a second-order delta–nabla -Laplacian BVP on a time scale

In this work we consider a second-order delta–nabla dynamic boundary value problem of the form ( ϕ p ( y Δ ( t ) ) ) ∇ = − a ( t ) f ( y ( t ) ) ,  t ∈ ( 0 , T ) T κ ∩ T κ , ϕ p ( y Δ ( 0 ) ) = 0 , y ( T ) = ψ ( y ) : = ∑ i = 1 n c i y ( ξ i ) , where T is a time scale, a : [ 0 , T ] T → [ 0 , + ∞ ) and f : [ 0 , + ∞ ) → [ 0 , + ∞ ) are continuous functions, and ψ : C ld ( [ 0 , T ] T ) → R is a given functional. We show that even if some of the c i ’s are negative, the boundary value problem may still admit a positive solution. Our results extend and generalize some recent results on this type of problem, and we illustrate this by way of a numerical example.