Abstract
We study the existence and multiplicity of positive periodic solutions to the nonlinear differential equation:u5(t)+ku4(t)-βu3-ξu″(t)+αu'(t)+ωu(t)=λh(t)f(u), in 0≤t≤1, ui(0)=ui(1), i=0,1,2,3,4, wherek,α,ω,λ>0, β,ξ∈R,h∈C(R,R)is a 1-periodic function. The proof is based on the Krasnoselskii fixed point theorem.
Highlights
Fifth-order boundary value problems (BVPs) are known to arise in the mathematical modeling of viscoelastic flow and other branches of mathematical, physical, and engineering sciences; see [1,2,3,4]
We study the existence and multiplicity of positive periodic solutions to the nonlinear differential equation: u(5)(t) + ku(4)(t)− βu(3) − ξu(t) + αu(t) + ωu(t) = λh(t)f(u), in 0 ≤ t ≤ 1, ui(0) = ui(1), i = 0, 1, 2, 3, 4, where k, α, ω, λ > 0, β, ξ ∈ R, h ∈ C(R, R) is a 1-periodic function
Multiplicity, and nonexistence of solutions to the fifth-order BVPs are noticed by many authors
Summary
Fifth-order boundary value problems (BVPs) are known to arise in the mathematical modeling of viscoelastic flow and other branches of mathematical, physical, and engineering sciences; see [1,2,3,4]. Agarwal and Odda [5, 6] gave some theorems which list conditions for the existence and uniqueness of solutions to the fifth-order BVPs by the topological methods; Gamel and Lv [7, 8] focused their attention on the study of numerical solution to the fifth-order BVPs. On the other hand, all kinds of topological methods, such as the method of upper and lower solutions, degree theory, and some fixed point theorems in cones, have been widely applied to study the singular and regular periodic boundary value problems; see [9,10,11,12,13,14,15,16,17,18,19]. This paper is organized as follows: in Section 2, some preliminaries are given; in Section 3, we give the main results; in Section 4, we give an example to illustrate our main results
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