Existence of Solutions of Periodic Boundary Value Problems for Impulsive Functional Duffing Equations at Nonresonance Ca
- PDF / 143,040 Bytes
- 17 Pages / 600.05 x 792 pts Page_size
- 11 Downloads / 217 Views
Research Article Existence of Solutions of Periodic Boundary Value Problems for Impulsive Functional Duffing Equations at Nonresonance Case Xingyuan Liu1 and Yuji Liu2 1 2
Department of Mathematics, Shaoyang University, Shaoyang, Hunan 422000, China Department of Mathematics, Guangdong University of Business Studies, Guangzhou, Guangdong 510320, China
Correspondence should be addressed to Xingyuan Liu, [email protected] Received 19 June 2008; Accepted 27 August 2008 Recommended by Zhitao Zhang This paper deals with the existence of solutions of the periodic boundary value problem of the impulsive Duffing equations: x t αx t βxt ft, xt, xα1 t, . . . , xαn t, a.e. t ∈ 0, T , Δxtk Ik xtk , x tk , k 1, . . . , m, Δx tk Jk xtk , x tk , k 1, . . . , m, xi 0 xi T , i 0, 1. Sufficient conditions are established for the existence of at least one solution of above-mentioned boundary value problem. Our method is based upon Schaeffer’s fixed-point theorem. Examples are presented to illustrate the efficiency of the obtained results. Copyright q 2008 X. Liu and Y. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction In recent years, many authors studied the solvability of the periodic boundary value problems PBVPs for short for second-order ordinary or functional differential equations with or without impulse effects; see 1–24 and the references therein. For example, consider the following PBVP: x t f t, xt , t ∈ 0, 2π, 1.1 x0 x2π, x 0 x 2π, the well-known result is that if f satisfies the nonresonance condition −N 12 ≤ fu t, u ≤ − − N 2 ,
1.2
where N is a nonnegative integer and is a positive constant, then PBVP1.1 has a unique solution; see 1.
2
Boundary Value Problems For PBVP of the Duffing equation x t cx t g t, xt et, x0 x2π,
t ∈ 0, 2π,
1.3
x 0 x 2π,
in 16, the authors proved the following results. Theorem 1.1. Suppose g is a L2 -Caratheodory function, and there are a ≤ A, r < 0 < R such that gt, x ≥ A,
x ≥ R, t ∈ 0, 2π,
gt, x ≤ a,
x ≤ r, t ∈ 0, 2π,
1.4
and further there is r ∈ L0, 2π with ||r||∞ < 1 C2 such that lim
|x| → ∞
gt, x < rt, x
t ∈ 0, 2π.
1.5
Then, PBVP1.3 has at least one solution for each e ∈ L2 0, 2π with a ≤ 1/2π
2π 0
esds ≤ A.
In 17, Nieto and Rodr´ıguez-Lopez studied the following PBVP: ´ x t ax t bxt cx t dx t σt, x0 xT ,
t ∈ 0, T ,
x 0 x T .
1.6
They gave Green’s function to express the unique solution for the correspondence secondorder functional differential equation with periodic boundary conditions and the functional dependence given by the piecewise constant function. Using upper and lower solution methods, they presented sufficient conditions to
Data Loading...
