Solvability of Higher Order Iterative System with Non-Homogeneous Integral Boundary Conditions

Abstract

The aim of this paper is to establish the existence of positive solutions by determining the eigenvalue intervals of the parameters μ₁, μ₂, ..., μ_m for the iterative system of nonlinear differential equations of order p

              w_i^{(p) ₍}x_{^{) + μ}i} a_i (x) f_i (w_i+1 (x) ) = 0, 1 ≤ i ≤ m, x∈ [0,1], 

                             w_m+1 (x) = w₁ (x), x ∈ [0,1],

satisfying non-homogeneous integral boundary conditions

                          w_i (0) = 0, w_i' (0) = 0, ..., w_i^(p-2) (0) = 0,

                       w_i^(r) - η_i ∫₀¹g_i(τ)w_i^(r)(τ)dτ = λ_i, 1 ≤ i ≤ m,

where r ∈ {1, 2, ..., p−2} but fixed, p ≥ 3 and η_i, λ_i ∈ (0, ∞) are parameters. The fundamental tool in this paper is an application of the Guo-Krasnosel'skii fixed point theorem to establish the existence of positive solutions of the problem for operators on a cone in a Banach space. Here the kernels play a fundamental role in defining an appropriate operator on a suitable cone.