The existence of three positive solutions of -point boundary value problems for some dynamic equations on time scales

In this paper, we define a new operator which improves and generalizes a p -Laplacian operator for some p > 1 . By using this operator, we consider the existence of triple positive solutions of m -point boundary value problems for some dynamic equations on time scales [ φ ( p ( t ) u Δ ( t ) ) ] ∇ + a ( t ) f ( u ( t ) ) = 0 , t ∈ [ 0 , T ] T κ ∩ T k , u ( 0 ) − B 0 ( ∑ i = 1 m − 2 α i u Δ ( ξ i ) ) = 0 , u Δ ( T ) = 0 , where φ : R → R is an increasing homeomorphism and positive homomorphism and φ ( 0 ) = 0 . We show the existence of at least three positive solutions with suitable growth conditions imposed on the nonlinear term by using a new fixed-point theorem.