Periodic solutions for the p -Laplacian neutral functional differential system
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RESEARCH
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Periodic solutions for the p-Laplacian neutral functional differential system Zhenyou Wang and Changxiu Song* * Correspondence: [email protected] School of Applied Mathematics, Guangdong University of Technology, Guangzhou, 510006, China
Abstract By using the generalized Borsuk theorem in coincidence degree theory, we prove the existence of periodic solutions for the p-Laplacian neutral functional differential system. MSC: 34C25 Keywords: p-Laplacian; periodic solutions; coincidence degree
1 Introduction In recent years, the existence of periodic solutions for the Rayleigh equation and the Liénard equation has been studied (see [–]). By using topological degree theory, some results on the existence of periodic solutions are obtained. Motivated by the works in [–], we consider the existence of periodic solutions of the following system: d d φp x(t) – Cx(t – τ ) + grad F x(t) + grad G x(t) = e(t), dt dt
(.)
where F ∈ C (Rn , R), G ∈ C (Rn , R), e ∈ C(R, Rn ) are periodic functions with period T; C = [cij ]n×n is an n × n symmetric matrix of constants, τ ∈ R is a constant. φp : Rn → Rn is given by T φp (u) = φp (u , . . . , un ) := |u |p– u , . . . , |un |p– un ,
< p < ∞.
The φp is a homeomorphism of Rn with the inverse φq . By using the theory of coincidence degree, we obtain some results to guarantee the existence of periodic solutions. Even for p = , the results in this paper are also new. In what follows, we use ·, · to denote the Euclidean inner product in Rn and | · |p to denote the lp -norm in Rn , i.e., |x|p = ( ni= |xi |p )/p . The norm in Rn×n is defined by Ap = sup|x|ρ =,x∈Rn |Ax|p . The corresponding Lp -norm in Lp ([, T], Rn ) is defined by xp =
n i=
xi (t) p dt
T
p
=
x(t) p dt
T
p
p ,
©2013 Wang and Song; 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.
Wang and Song Advances in Difference Equations 2013, 2013:367 http://www.advancesindifferenceequations.com/content/2013/1/367
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and the L∞ -norm in L∞ ([, T], Rn ) is x∞ = max xi ∞ , ≤i≤n
where xi ∞ = supt∈[,ω] |xi (t)| (i = , . . . , n). Let W = W ,p ([, T], R] be the Sobolev space. Lemma . (See []) Suppose u ∈ W and u() = u(T) = , then
T u , p πp
up ≤ where
(p–)/p
πp =
ds ( –
sp /p ) p–
=
π(p – )/p . p sin( πp )
In order to use coincidence degree theory to study the existence of T-periodic solutions for (.), we rewrite (.) in the following form:
(x(t) – Cx(t – τ )) (t) = φq (y(t)), y (t) = dtd grad F(x(t)) – grad G(x(t)) + e(t).
(.)
x(t) If z(t) = y(t) is a T-periodic solution of (.), x(t) must be a T-periodic solution of (.). Thus, the problem of finding a T-periodic solution for (.) reduces to finding one for (.). Let CT =
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