Multiple Positive Solutions of Third-Order BVP with Advanced Arguments and Stieltjes Integral Conditions

Jian Chang,Ya-Hong Zhao,Jian-Ping Sun

doi:10.1155/2016/3805804

Abstract

We consider the following third-order boundary value problem with advanced arguments and Stieltjes integral boundary conditions:u′′′t+ft,uαt=0, t∈0,1, u0=γuη1+λ1uandu′′0=0, u1=βuη2+λ2u, where0<η1<η2<1,0≤γ,β≤1,α:[0,1]→[0,1]is continuous,α(t)≥tfort∈[0,1], andα(t)≤η2fort∈[η1,η2]. Under some suitable conditions, by applying a fixed point theorem due to Avery and Peterson, we obtain the existence of multiple positive solutions to the above problem. An example is also included to illustrate the main results obtained.

Highlights

Third-order differential equations arise from a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross-section, a three-layer beam, electromagnetic waves or gravity driven flows, and so on [1].Recently, third-order boundary value problems (BVPs for short) have received much attention from many authors; see [2–20] and the references therein
We consider the following third-order boundary value problem with advanced arguments and Stieltjes integral boundary conditions: u󸀠󸀠󸀠(t)+f(t, u(α(t))) = 0, t ∈ (0, 1), u(0) = γu(η1)+λ1[u] and u󸀠󸀠(0) = 0, u(1) = βu(η2)+λ2[u], where 0 < η1 < η2 < 1, 0 ≤ γ, β ≤ 1, α : [0, 1] → [0, 1] is continuous, α(t) ≥ t for t ∈ [0, 1], and α(t) ≤ η2 for t ∈ [η1, η2]
By applying a fixed point theorem due to Avery and Peterson, we obtain the existence of multiple positive solutions to the above problem

Summary

Introduction

Third-order differential equations arise from a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross-section, a three-layer beam, electromagnetic waves or gravity driven flows, and so on [1].Recently, third-order boundary value problems (BVPs for short) have received much attention from many authors; see [2–20] and the references therein. We consider the following third-order boundary value problem with advanced arguments and Stieltjes integral boundary conditions: u󸀠󸀠󸀠(t)+f(t, u(α(t))) = 0, t ∈ (0, 1), u(0) = γu(η1)+λ1[u] and u󸀠󸀠(0) = 0, u(1) = βu(η2)+λ2[u], where 0 < η1 < η2 < 1, 0 ≤ γ, β ≤ 1, α : [0, 1] → [0, 1] is continuous, α(t) ≥ t for t ∈ [0, 1], and α(t) ≤ η2 for t ∈ [η1, η2].

Results

Conclusion