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, ..., μm for the iterative system of nonlinear differential equations of order p                 wi(p) (x) + μi ai (x) fi (wi+1 (x) ) = 0, 1 ≤ i ≤ m, x∈ [0,1],                                 wm+1 (x) = w1 (x), x ∈ [0,1],   satisfying non-homogeneous integral boundary conditions                             wi (0) = 0, wi' (0) = 0, ..., wi(p-2) (0) = 0,                          wi(r) - ηi ∫01gi(τ)wi(r)(τ)dτ = λi, 1 ≤ i ≤ m,   where r ∈ {1, 2, ..., p−2} but fixed, p ≥ 3 and ηi, λi ∈ (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.