Existence of solutions for a third-order multipoint boundary value problem with p-Laplacian

In this paper, we consider multipoint boundary value problem for third-order differential equations with p-Laplacian at resonance [ ϕ p ( x ″ ( t ) ) ] ′ = f ( t , x ( t ) , x ′ ( t ) , x ″ ( t ) ) , 0 < t < 1 , x ( 0 ) = 0 , x ′ ( 1 ) = ∑ i = 1 m − 2 a i x ′ ( ξ i ) , x ″ ( 0 ) = 0 , where f : [ 0 , 1 ] × R 3 → R is continuous function; m ⩾ 3 is an integer, a i ⩾ 0 are constants satisfying ∑ i = 1 m − 2 a i = 1 and 0 < ξ 1 < ξ 2 < ⋯ < ξ m − 2 < 1 . Under various assumptions on the degree of the power with respect to the variables z in the function f(t, x, y, z), we obtain the existence of solutions for the problem.