Sufficient condition for existence of solutions for higher-order resonance boundary value problem with one-dimensional p-Laplacian

(1.2) where p > 1 is a constant; φp : R → R,φp(u) = |u|u; f : [0, 1]×R → R is a continuous function and 1 ≤ i ≤ n−1 is a fixed integer, e(t) ∈ L[0, 1], αj(1 ≤ j ≤ m− 2) ∈ R, η, ξ, ξj ∈ (0, 1), j = 1, · · · ,m− 2, 0 < ξ1 < · · · < ξm−2 < 1. We notice that the operator φp(u) = |u|u is called the (one-dimensional) p-Laplacian and it appears in many contexts. For example, it is used extensively in non-Newtonian fluids, in some reaction-diffusion problems, in flow through porous media, in nonlinear elasticity, glaceology and petroleum extraction. The boundary value problem (1.1), (1.2) is said to be at resonance in the sense that the associate homogeneous problem (φp(x (t))) = 0, 0 ≤ t ≤ 1