Existence and multiplicity of periodic solutions for a class of second-order ordinary differential equations
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Existence and multiplicity of periodic solutions for a class of second-order ordinary differential equations Xiaoling Han1 · Hujun Yang1 Received: 27 January 2020 / Accepted: 15 September 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract In this paper, we study the existence of positive periodic solutions for a class of nonautonomous second-order ordinary differential equations x + αx + a(t)x n − b(t)x n+1 + c(t)x n+2 = 0, where α ∈ R is a constant, n is a finite positive integer, and a(t), b(t), c(t) are continuous periodic functions. By using Mawhin’s continuation theorem, we prove the existence and multiplicity of positive periodic solutions for these equations. Keywords Second-order ordinary differential equations · Positive periodic solutions · Mawhin’s continuation theorem Mathematics Subject Classification 34B18 · 34K13 · 34C25
1 Introduction and main results In the past few years, scholars have become more and more interested in the study of differential equations in some mathematical models that arise in Biology and Physics, such as the equations x + a(t)x − b(t)x 2 + c(t)x 3 = 0,
(1.1)
Communicated by Adrian Constantin. Foundation term: This work is sponsored by the NNSF of China (No. 11561063).
B 1
Hujun Yang [email protected] Department of Mathematics, Northwest Normal University, Lanzhou 730070, People’s Republic of China
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X. Han, H. Yang
where a(t), b(t), c(t) are positive continuous periodic functions. Eq. (1.1) comes from a biomathematics model and was suggested by Cronin in [1] and Austin in [2]. Equation (1.1) description of some of the properties of an aneurysm of the circle of Willis, where x is the velocity of blood flow in the aneurysm, a(t), b(t), c(t) are coefficient functions related to aneurysm. For more equations related to the model, see [3–5]. Equation (1.1) have been studied by several authors, see [6–8]. The main tools used by these authors for obtaining their results are variational method and coincidence degree theories. At the same time, the existence of periodic solutions of nonlinear differential equations has been studied, see for instance the papers [9–17]. In this paper, our purpose is to establish the existence and multiplicity of positive periodic solutions of the non-autonomous second-order nonlinear ordinary differential equations x + αx + a(t)x n − b(t)x n+1 + c(t)x n+2 = 0,
(1.2)
where n is a positive integer, α ∈ R is a constant, and a(t), b(t), c(t) are continuous T -periodic functions on R, subject to the constraints 0 < a ≤ a(t) ≤ A, 0 < b ≤ b(t) ≤ B, 0 < c ≤ c(t) ≤ C, or −A ≤ a(t) ≤ −a < 0, −B ≤ b(t) ≤ −b < 0, −C ≤ c(t) ≤ −c < 0. We also consider the following particular case of Eq. (1.2) x + αx + a(t)x n − b(t)x n+1 = 0,
(1.3)
namely, coefficient function c(t) ≡ 0 of Eq. (1.2). We will use coincidence degree theories to prove the existence of at least two positive periodic solutions for Eq. (1.2) and at least one positive periodic solution for Eq. (1.3), under some specific assumptions on a, A, b, B, c, C, n
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