Homoclinic solutions for a class of neutral Duffing differential systems
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RESEARCH
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Homoclinic solutions for a class of neutral Duffing differential systems Wenbin Chen* * Correspondence: [email protected] School of Mathematics Science and Computer, Wu Yi University, Wu Yishan, 354300, China
Abstract By using an extension of Mawhin’s continuation theorem and some analysis methods, the existence of a set with 2kT-periodic for a n-dimensional neutral Duffing differential systems, (u(t) – Cu(t – τ )) + β (t)x (t) + g(u(t – γ (t))) = p(t), is studied. Some new results on the existence of homoclinic solutions is obtained as a limit of a certain subsequence of the above set. Meanwhile, C = [cij ]n×n is a constant symmetrical matrix and β (t) is allowed to change sign. Keywords: homoclinic solution; continuation theorem; periodic solution
1 Introduction The aim of this paper is to consider a kind of neutral Duffing differential systems as follows:
u(t) – Cu(t – τ )
+ β(t)x (t) + g u t – γ (t) = p(t),
(.)
where β ∈ C (R, R) with β(t + T) ≡ β(t), g ∈ C(Rn , Rn ), p ∈ C(R, Rn ), and γ (t) is a continuous T-periodic function with γ (t) ≥ ; T > and τ are given constants; C = [cij ]n×n is a constant symmetrical matrix and β(t) is allowed to change sign. As is well known, a solution u(t) of Eq. (.) is called homoclinic (to O) if u(t) → and u (t) → as |t| → +∞. In addition, if u = , then u is called a nontrivial homoclinic solution. Under the condition of C = O, system (.) transforms into a classic second-order Duffing equation u (t) + β(t)x (t) + g t, u t – γ (t) = p(t),
(.)
which has been studied by Li et al. [] and some new results on the existence and uniqueness of periodic solutions for (.) are obtained. Very recently, by using Mawhin’s continuation theorem, Du [] studied the following neutral differential equations:
u(t) – Cu(t – τ )
+
d ∇F u(t) + ∇G u(t) = e(t), dt
(.)
where F ∈ C (Rn , R); G ∈ C (Rn , R); e ∈ C(R, Rn ); C = diag(c , c , . . . , cn ), ci (i = , , . . . , n) and τ are given constants, obtaining the existence of homoclinic solutions for (.). ©2014 Chen; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Chen Advances in Difference Equations 2014, 2014:121 http://www.advancesindifferenceequations.com/content/2014/1/121
Page 2 of 13
In this paper, like in the work of Rabinowitz in [], Izydorek and Janczewska in [] and Tan and Xiao in [], the existence of a homoclinic solution for (.) is obtained as a limit of a certain sequence of kT-periodic solutions for the following equation:
u(t) – Cu(t – τ ) + β(t)u (t) + g u t – γ (t) = pk (t),
(.)
where k ∈ N , pk : R → Rn is a kT-periodic function such that p(t), pk (t) = p(kT – ε ) +
p(–kT)–p(kT–ε ) (t ε
t ∈ [–kT, kT – ε ), – kT + ε ), t ∈ [kT – ε , kT],
(.)
ε ∈ (,
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