R E S E A R C H
Open Access
Multiple periodic solutions for a class of
second-order neutral functional differential
equations
Zhiguo Luo
1and Jingli Xie
2**Correspondence: [email protected] 2College of Mathematics and Statistics, Jishou University, Jishou, Hunan 416000, P.R. China Full list of author information is available at the end of the article
Abstract
In this paper we consider a class of second-order neutral functional differential equations. Under certain conditions, we establish the existence of multiple periodic solutions by means ofZ2group index theory and variational methods. The main result is also illustrated with an example.
Keywords: neutral functional differential equations;Z2group index theory; periodic solution
1 Introduction
In this paper we consider a class of second-order neutral functional differential equations described by
⎧ ⎨ ⎩
(p(t)u(t–sτ))–q(t)u(t–sτ) +λf(t,u(t),u(t–τ), . . . ,u(t– sτ)) = ,
u() –u(kτ) =u() –u(kτ) = , (.)
wherep∈C([,τ],R+),q∈C([,τ],R+) and they areτ-periodic.f ∈C(R(s+),R),λ∈R, andτ> .kandsare given positive integers withk>s. A functionu∈C(R,R) is a solution of system (.) if the functionusatisfies (.).
The necessity to study delay differential equations is due to the fact that these equations are useful mathematical tools in modeling many real processes and phenomena studied in economics, biology, electronics, optimal control, mechanics, medicine, etc. [, ].
In recent years many researchers have focused on the existence of periodic solutions of delay differential equations; see, for example, [–]. Several available approaches to tackle the existence of periodic solutions for delay differential equations include the dual Lyapunov method, the Fourier analysis method, fixed point theory, and the coincidence degree theory [–]. Recently, some researchers have studied the existence of periodic solutions for delay differential equations via variational methods [–].
Whenp(t) =q(t) = ands= , system (.) reduces to the following equation:
⎧ ⎨ ⎩
u(t–τ) –u(t–τ) +λf(t,u(t),u(t–τ),u(t– τ)) = ,
u() –u(kτ) =u() –u(kτ) = .
In [], Shu and Xu obtained the following result.
Theorem A Assume that the following conditions are satisfied.
(H) ∂f(t,u∂t,u,u) = .
(H) There exists a functionF(t,u,u)∈C(R,R)such that
∂F(t,u,u) ∂u
+∂F(t,u,u) ∂u
=f(t,u,u,u).
(H) F(t,u,u)isτ-periodic int.
(H) FsatisfiesF(t, –u, –u) =F(t,u,u)andf(t, –u, –u, –u) = –f(t,u,u,u). (H) F(t,u,u) = if and only if(u,u) = ,∀t∈[,τ].
(H) lim|u|→F(t,|u|u,u)= ,where|u|= (|u|+|u|)
,t∈[,τ].
(H) There exists a constantα> such that when|u|+|u|>α, F(t,u,u) < ,t∈ [,τ].
Moreover,if there exists an integer m> such thatλsatisfies
λ>m
(π+kτ)
kτ , (.)
then the system
⎧ ⎨ ⎩
u(t–τ) –u(t–τ) +λf(t,u(t),u(t–τ),u(t– τ)) = ,
u() –u(kτ) =u() –u(kτ) = , (.)
possesses at leastm non-zero solutions with the periodkτ.
Remark . Based on our analysis, (.) should be replaced by
λ>m
(π+kτ)
kτ .
Compared to system (.), the neutral functional differential system (.) admits four control parametersp,q,λ,s. We aim to derive conditions in terms of these four control parameters for the existence and multiplicity of periodic solutions of a class of second-order neutral functional differential equation (.).
Our approach is based on theZgroup index theory and some techniques of mathe-matical analysis. We remark that theZ group index theory and the variational method have also been employed to prove the existence of multiple periodic solutions of mixed type differential equations in []. When (.) reduces to the special case, see system (.), we obtain a more accurate result (see Remark .). Moreover, our result generalizes the existence result obtained in [] as the equation considered in [] is a special case of our system (.) withp(t) =q(t) =s= .
To prove our main result, we first make the following assumptions.
(H) There exists a functionF(t,u,u, . . . ,us+)∈C(Rs+,R)such that
∂F(t,u,u, . . . ,us+)
∂us+
+∂F(t,u, . . . ,us+,us+) ∂us+
+· · ·+∂F(t,us+,us+, . . . ,us+) ∂us+
=f(t,u,u, . . . ,us+).
(H) F(t,u,u, . . . ,us+)isτ-periodic int. (H) Fsatisfies
F(t, –u, –u, . . . , –us+) =F(t,u,u, . . . ,us+)
and
f(t, –u, –u, . . . , –us+) = –f(t,u,u, . . . ,us+).
(H) F(t,u,u, . . . ,us+) = if and only if(u,u, . . . ,us+) = ,∀t∈[,τ]. (H) lim|u|→F(t,u,|u|u,..., us+) = , where|u|= (|u|+· · ·+|us+|)
,t∈[,τ].
(H) There exists a constantα > such thatF(t,u,u, . . . ,us+) < for t∈[,τ]when |u|+· · ·+|us+|>α.
Note that system (.) is equivalent to the following system:
p(t)u(t–sτ)–q(t)u(t–sτ) +λFs+t,u(t),u(t–τ), . . . ,u(t–sτ)
+· · ·+Ft,u(t–sτ),ut– (s+ )τ, . . . ,u(t– sτ)= . (.) The rest of this paper is organized as follows. In Section , we present some preliminar-ies, which will be used to prove our main results. In Section we state and prove our main results. Finally, we provide one example to illustrate the applicability of our results.
2 Some preliminaries
Let
Hkτ=u:R→R|u,u∈L[, kτ],R,u() =u(kτ),u() =u(kτ). ThenH
kτ is a separable and reflexive Banach space and the inner product
(u,v) = kτ
p(t)u(t)v(t) +q(t)u(t)v(t)dt,
induces the norm
uH kτ=
kτ
p(t)u(t)+q(t)u(t)dt
.
We introduce the following notations. Denote
Fs+(t,u,u, . . . ,us+) =
∂F(t,u,u, . . . ,us+)
Fs(t,u, . . . ,us+,us+) =
∂F(t,u, . . . ,us+,us+)
∂us+
,
· · ·
F(t,us+,us+, . . . ,us+) =
∂F(t,us+,us+, . . . ,us+)
∂us+
.
Define a functionalϕas
ϕ(u) =
kτ
p(t)u(t)+q(t)u(t)dt
–λ kτ
Ft,u(t), . . . ,u(t–sτ)dt, u∈Hkτ. (.)
Thenϕis Fréchet differentiable at anyu∈H
kτ. For anyv∈Hkτ, by a simple calculation,
we have
ϕ(u)(v) = kτ
p(t)u(t)v(t) +q(t)u(t)v(t)dt–λ kτ
Ft,u(t), . . . ,u(t–sτ)v(t) +Ft,u(t),u(t–τ), . . . ,u(t–sτ)v(t–τ)
+· · ·
+Fs+t,u(t),u(t–τ), . . . ,u(t–sτ)v(t–sτ)dt.
From (H),p(t)∈C([,τ],R+),q(t)∈C([,τ],R+), and their periodicity, we have
ϕ(u)(v) = kτ
–p(t)u(t)+q(t)u(t)v(t)dt–λ kτ
Ft,u(t), . . . ,u(t–sτ)v(t) +Ft,u(t+τ),u(t), . . . ,ut– (s– )τv(t)
+· · ·
+Fs+t,u(t+sτ),ut+ (s– )τ, . . . ,u(t)v(t)dt =
kτ
–p(t)u(t)+q(t)u(t) –λFt,u(t), . . . ,u(t–sτ) +Ft,u(t+τ),u(t), . . . ,ut– (s– )τ
+· · ·
+Fs+t,u(t+sτ),ut+ (s– )τ, . . . ,u(t)v(t)dt.
Therefore, the corresponding Euler equation of functionalϕis
p(t)u(t)–q(t)u(t) +λFt,u(t), . . . ,u(t–sτ) +Ft,u(t+τ),u(t), . . . ,ut– (s– )τ +· · ·
Note that system (.) is equivalent of system (.). Hence, critical points of the functional ϕare classical kτ-periodic solutions of system (.).
Definition . []. LetEbe a real reflexive Banach space, and
=A|A⊂E\ {}is closed, symmetric set. Defineγ :→Z+∪ {+∞}as follows:
γ(A) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
min{n∈Z+: there exists an odd continuous map
ϕ:A→Rn\ {}}; , ifA=∅;
+∞, if there is no odd continuous mapϕ:A→Rn\ {}
for anyn∈Z+.
(.)
Then we sayγ is the genus of.
Denotei(ϕ) =lima→–γ(ϕa) andi(ϕ) =lima→–∞γ(ϕa) whereϕa={u∈E|ϕ(u)≤a}.
Lemma .([]) Let E be a real Banach space,ϕ∈C(E,R)withϕeven functional and satisfying the Palais-Smale(PS)condition.Supposeϕ() = and
(i) if there exist anm-dimensional subspaceXofEand a constantr> such that
sup
u∈X∩Br
ϕ(u) < , (.)
whereBris an open ball of radiusrinEcentered at,then we havei(ϕ)≥m; (ii) if there exists aj-dimensional subspaceVofEsuch that
inf
u∈V⊥ϕ(u) > –∞, (.)
then we havei(ϕ)≤j.
Moreover,if m≥j,thenϕpossesses at least(m–j)distinct critical points.
3 Main result
In this section, we state and prove our main result. SetP=maxt∈[,τ]p(t),Q=maxt∈[,τ]q(t).
Theorem . Assume that(H)-(H)are satisfied.If there exists an integer m> such thatλsatisfies
λ>m
(Pπ+Qkτ)
(s+ )kτ , (.)
then system(.)admits at leastm non-zero solutions with the periodkτ.
of the system (.). Therefore, the solutions of the system (.) are a set which is symmetric with respect to the origin inH
kτ. It follows directly from (.) and (H) thatϕis even in
uandϕ() = . The rest of the proof is divided into three steps.
Step: We show that the functionalϕsatisfies the assumption (ii) of Lemma .. It follows from (H) that there exists a constantM> such that
max
t∈R F
t,u(t), . . . ,u(t–sτ)≤ max (t,u,u,...,us+)∈
F(t,u, . . . ,us+)≤M, (.)
where= [,τ]×[–α,α]×[–α,α]× · · · ×[–α,α]. Combining (.) and (.), we get
ϕ(u) = u
Hkτ–λ
kτ
Ft,u(t), . . . ,u(t–sτ)dt
≥
u
Hkτ– λMkτ> –∞, (.)
which implies thatϕis bounded from below. By the condition (ii) of Lemma ., we have i(ϕ) = .
Step: We show that the functionalϕsatisfies PS condition.
For any given sequence{un} ∈Hkτ such that{ϕ(un)}is bounded andlimn→∞ϕ(un) = ,
there exists a constantCsuch that
ϕ(un)≤C, ϕ(un)(H
kτ)∗≤C, ∀n∈N,
where (Hkτ)∗is the dual space ofHkτ. Therefore, we have
u
Hkτ≤C+ λMkτ.
It follows thatunH
kτ is bounded.
SinceH
kτ is a reflexive Banach space, we can pick{un}be a weakly convergent sequence
touinHkτ and{un}converges uniformly touinC[, kτ]. So, we have
kτ
Ft,un(t), . . . ,un(t–sτ)
–Ft,u(t), . . . ,u(t–sτ)un(t) –u(t)
dt→,
kτ
Ft,un(t), . . . ,un(t–sτ)
–Ft,u(t), . . . ,u(t–sτ)
×un(t–τ) –u(t–τ)
dt→,
· · ·
kτ
Fs+t,un(t), . . . ,un(t–sτ)
–Fs+t,u(t),u(t–τ), . . . ,u(t–sτ)
×un(t–sτ) –u(t–sτ)
dt→,
un(t) –u(t)→ asn→ ∞,t∈[, kτ].
(.)
Therefore, by (.), we haveun–uHkτ →. Hence the functionalϕ satisfies the PS
Step: We show that the functionalϕsatisfies the assumption (i) of Lemma ..
Define them-dimensional linear subspace as follows:
Next we provide an example to illustrate the applicability of our result.
Example . Consider (.) withs= ,p(t) =q(t) = ( +sinπtτ ),
ft,u(t), . . . ,u(t– τ) = u(t– τ) –
+cosπt τ
u(t– τ)
×u(t) + u(t–τ) + u(t– τ) + u(t– τ) +u(t– τ)
and
F(t,u,u,u) =u+u+u–
+cosπt τ
u+u+u.
It is easy to verify thatf,Fsatisfy the assumptions of Theorem .. Therefore system (.) admits at least mnon-zero solutions with the period kτ. Noteu= is also a solution of system (.).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors made an equal contribution.
Author details
1Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, P.R. China.2College of Mathematics and Statistics, Jishou University, Jishou, Hunan 416000, P.R. China.
Acknowledgements
This work was partially supported by Hunan Provincial Natural Science Foundation of China (No: 2016JJ6122), National Natural Science Foundation of P.R. China (No: 11661037 and 11471109), and Jishou University Doctor Science Foundation (No: jsdxxcfxbskyxm201504).
Received: 9 August 2016 Accepted: 22 December 2016
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