Nondegeneracy and Uniqueness of Periodic Solution for a Neutral Differential Equation
- PDF / 374,967 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 66 Downloads / 154 Views
Nondegeneracy and Uniqueness of Periodic Solution for a Neutral Differential Equation Zhibo Cheng1,2 Received: 11 June 2020 / Accepted: 20 September 2020 © Springer Nature Switzerland AG 2020
Abstract We analyze the nondegeneracy of second-order linear neutral differential equation (x(t) − cx(t − τ )) = a(t)x(t), where c is a constant. By applications of the nondegeneracy of this linear neutral equation and an extension of Mawhin’s continuation theorem, we obtain existence and uniqueness of periodic solution for the prescribed second-order neutral differential equations. At last, we give two examples to show the applications of the theorems. Keywords Nondegeneracy · Uniqueness · Periodic solution · Neutral equation Mathematics Subject Classification 34C25
1 Introduction In this paper, we consider the nondegeneracy of the following second-order linear neutral differential equation (x(t) − cx(t − τ )) = a(t)x(t),
(1.1)
Research is supported by NSFC Project (No. 11501170), Postdoctoral fund in China (2016M590886), Young backbone teachers of colleges and universities in Henan Province (2017GGJS057) and Fundamental Research Funds for the Universities of Henan Provience (NSFRF170302).
B
Zhibo Cheng [email protected]
1
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China
2
Department of Mathematics, Sichuan University, Chengdu 610064, China 0123456789().: V,-vol
92
Page 2 of 16
Z. Cheng
where c is a constant and |c| = 1, τ is a constant and 0 ≤ τ < ω, a ∈ L p (R) is an ω-periodic function, here p is a constant and 1 ≤ p < ∞. The nondegeneracy of (1.1) is understood in the same sense as a matrix. It is known that a matrix A is referred to be nondegenerate if the linear equation Ax = 0 has only the trivial solution x ≡ 0. In the same sense, (1.1) is referred to be nondegenerate if (1.1) has only a trivial solution x(t) ≡ 0 in 2, p
x ∈ Wω2, p (R) := {x ∈ Wloc (R) : x(t + ω) ≡ x(t), ∀ t ∈ R}, n, p
where Wloc (R) := {x| D i x ∈ L p (R), 1 = 0, 1, 2, . . . , n} is a Sobolev space, D i x denotes the i-th order weak derivative of function x. Moreover, note that when c = 0 the neutral operator x(t) → x(t) − cx(t − τ ) reduces to the identity operator x → x and the (1.1) is of the linear differential equation form x (t) = a(t)x(t).
(1.2)
The study of nondegeneracy of linear differential equations began with the paper of Lasota and Opial. The authors [8] in 1964 introduced the concept “nondegenerate” to the linear differential equation (1.2), where a ∈ L 1 (R) is an ω-periodic function. The 2 2 authors proved that (1.2) ω is nondegenerate in xω ∈ Cω := {x ∈ C (R), x(t + ω) ≡ x(t), ∀ t ∈ R)} if 0 a(t)dt ≤ 0 and ω 0 |a(t)|dt ≤ 16. Afterwards, Ortega and Zhang [15] in 2005 improved the results of [8], the authors proved that (1.2) is 2, p nondegenerate in x ∈ Wω (R) if a ∈ L p (R) is an ω-periodic function, and a− p :=
0
ω
1 |a− (t)| p dt
p
Data Loading...
