Positive solutions of nonlinear third-order -point BVP for an increasing homeomorphism and homomorphism with sign-changing nonlinearity

In this paper, several existence theorems of positive solutions are established for a nonlinear m -point boundary value problem for the following third-order differential equations ( ϕ ( u ″ ) ) ′ + a ( t ) f ( t , u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , ϕ ( u ″ ( 0 ) ) = ∑ i = 1 m − 2 a i ϕ ( u ″ ( ξ i ) ) , u ′ ( 0 ) = 0 , u ( 1 ) = ∑ i = 1 m − 2 b i u ( ξ i ) , where ϕ : R ⟶ R is an increasing homeomorphism and homomorphism and ϕ ( 0 ) = 0 . The nonlinear term f may change sign. As an application, an example to demonstrate our results is given.