#### Periodic solutions to a generalized Liénard neutral functional differential system with p-Laplacian

Journal of Inequalities and Applications
p-Laplacian
Qing Yang 0 1 2
Bo Du 0 1 2
0 Jiangsu 223300 , P.R. China
1 Huaiyin Normal University , Huaiyin
2 Department of Mathematics
By means of the generalized Mawhin's continuation theorem, we present some sufficient conditions which guarantee the existence of at least one T -periodic solution for a generalized Liénard neutral functional differential system with p-Laplacian. n
periodic solutions; neutral equations; generalized Mawhin's continuation
1 Introduction
tial system:
ϕp(x) = |x|p–x =
A : CT → CT ,
continuous bounded inverse A– satisfying
A–f (t) =
if |c| < , ∀f ∈ CT ,
j≥ c–jf (t + jτ ), if |c| > , ∀f ∈ CT .
j= ji= c(t – (i – )τ )f (t – jτ ), c < , ∀f ∈ CT ,
∞
j∞= ji+= c(t+iτ) f (t + jτ + τ ), σ > , ∀f ∈ CT .
⎧
A–f (t) dt ≤ ⎨⎩ σ––c TT||ff((tt))||ddtt,, cσ><,,∀∀ff ∈∈CCTT.,
2 Main lemmas
|x| = max x(t) ,
≤t≤T
x(t) =
U is a complex such that
UBU– = Eλ = diag(J, J, . . . , Jn)
is a Jordan’s normal matrix, where
A : CT → CT ,
Furthermore, we suppose that γ (t) ∈ C(R, R) with γ (t) < , ∀t ∈ R. It is obvious that the
function t – γ (t) has a unique inverse denoted by μ(t).
Lemma . ([]) Suppose that the matrix U and the operator A are defined by (.) and
(.), respectively, and for all i = , , . . . , l, |λi| = . Then A has its inverse A– : CT → CT
with the following properties:
() A– ≤ |U–||U|σ, σ = li= jn=i jk= |–λi|k .
() For all f ∈ CT , T |[A–f ](s)|p ds ≤ |U–|p|U|pσ T |f (s)|p ds, p ∈ [, +∞), where
p ∈ [, +∞)
and q > is a constant with /p + /q = .
R
() A–f ∈ CT , [A–f ] (t) = [A–f ](t), for all f ∈ CT , t ∈ .
Definition . ([]) Let X and Z be two Banach spaces with norms · X , · Z,
respectively. A continuous operator
M : X ∩ dom M → Z
M : X ∩ dom M → Z
Gl(a) =
Gl(a) = .
Lemma . ([]) The function Gl has the following properties:
() For any fixed l ∈ Z, there must be a unique a˜ = a˜ (l) such that the equation
3 Main results
A : Z → Z,
R
(Ax)(t) = x(t) – Bx(t – τ ), t ∈ ,
M : dom M ∩ X → Z,
(Mx)(t) = ϕp (Ax)
then (Nλx)(t) = λF. By (.)-(.), Eq. (.) is equivalent to the operator equation Nx = Mx,
where N = N . Then we have
Im M = z ∈ Z :
P : X → Ker M,
Q : Z → Z/ Im M,
"
λ F r, x(r) – (QF)(r) dr ds , t ∈ [, T ],
"
λ F r, x(r) – (QF)(r) dr ds, t ∈ [, T ],
Hence, R(x, λ) is equicontinuous on ¯ × [, ]. By using the Arzelà-Ascoli theorem, we
have R(x, λ) is completely continuous on ¯ × [, ].
Secondly, we show that Nλ is M-compact in four steps, i.e., the conditions of
Definition . are all satisfied.
Step . By Q = Q, we have Q(I – Q)Nλ( ¯ ) = θ , so (I – Q)Nλ( ¯ ) ⊂ Ker Q = Im M, here θ is
an n-dimension zero vector. On the other hand, ∀z ∈ Im M. Clearly, Qz = θ , so z = z – Qz =
(I – Q)z, then z ∈ (I – Q)Z. So, we have
(I – Q)Nλ( ¯ ) ⊂ Im M ⊂ (I – Q)Z.
Step . We show that QNλx = θ , λ ∈ (, ) ⇔ QNx = θ , ∀x ∈ . Because QNλx =
T λF dr = θ , we get T F dr = θ , i.e., QNx = θ . The inverse is true.
Step . When λ = , from the above proof, we have ax = θ . So, we get R(·, ) = θ . ∀x ∈
λ = {x ∈ ¯ : Mx = Nλx}, we have (ϕp[(Ax) ]) = λF and QF = θ . In this case, when ax =
ϕp[(Ax) ()], we have
"
λ F r, x(r) – (QF)(r) dr ds
ϕp (Ax) (r)
s
(Ax) (s) ds
t
t
t
= A–
= A–
= A–
(Ax) (s) ds
= (I – P)x (t).
s
s
"
λ F r, x(r) – (QF)(r) dr ds
ϕp (Ax) (r)
Step . ∀x ∈ ¯ , we have
= #ϕp#!
= #ϕp#
(Ax)() + AA–
" " $$
λ F r, x(r) – (QF)(r) dr ds
"
λ F r, x(r) – (QF)(r) dr ds
(H) xigi(xi) > , ∀xi ∈ R, |xi| > D, for each i = , , . . . , n,
(H) |gi(u) – gi(u)| ≤ l|u – u|, u, u ∈ R, for each i = , , . . . , n,
(H) |fi(xi)| ≥ σ , xi ∈ R, for each i = , , . . . , n.
σ > √l max γ (t)
t∈[,T]
σ > √l max γ (t) ,
t∈[,T]
for < q < or
U– |U|σ√nTfR < for q = ,
ϕp (Ax)
Integrating both sides of (.) over [, T ], we have
g x t – γ (t) dt = θ ,
xi(ηi) ≤ D, for each i = , , . . . , n.
|xi| ≤ D +
xi(s) ds, for each i = , , . . . , n.
By (.), we have
x (s) ds
|x| ≤
By (.), we have
By assumption (H), we have
ϕp (Ax)
dt =
fi xi(t) xi(t)
e(t) dt .
and integrating them over [, T ], we have
ϕp (Ax)
x (t) f x(t) x (t) dt
From (.) and (.), we have
fi xi(t)
fi xi(t) xi(t)
e(t) dt.
Then by (.),
x (s) ds ≤ R.
By (.), we have
|x| ≤
n(D + R) := R.
e(t) dt
ϕp (Ax)
≤ fR x (t) + gR + |e|,
y ≤ fR x (t) + gR + |e|
by (.), we get
y(t) ≤
nT y ≤
nTfR x (t) +
nTgR +
nT |e|
(Ax) (t) ≤
nTfR x (t) +
nTgR +
nT |e|
By (.) and Lemma ., we have
x (t) =
A–Ax (t) ≤ U– |U|σ (Ax) (t)
nTfR x (t) +
nTgR +
nT |e|
nTfR x (t) +
nTgR +
nT |e|
nTfR x (t)
nTfR |x (t)|
pendent on p only, such that
( + x)p < + ( + p)x,
∀x ∈ , h(p) .
√
Now, we consider ( nTfR |x (t)| +
nTgR +
nT |e|)q–. In the formal case, we get
x (t) <
Case .. If
nTfR x (t) +
nTgR +
nT |e|
nT |e|
nTfR h
nTfR x (t)
nTfR x (t)
:= M.
nTfR x (t)
nTfR x (t)
nT |e|) $
nTgR +
√
nT |e| ( nTfR )q– x (t)
x (t) ≤
U– |U|σ(√nTfR )q– x (t)
√
nTgR + T |e| ( nTfR )q– x (t)
nTfR < , we know that there exists a constant M > such
From (.) and (.), we have
x (t) ≤ M.
x (t) ≤ M.
When < q < , there must be a constant M > such that
Hence, from (.), (.), (.) and (.), we have
∩ Ker M, we
H(x, μ) = aμ – ( – μ)
T
= aμ + ( – μ)g(a),
–g(a) + e(t) dt
then we have
deg{JQN ,
∩ Ker M, } = deg H(·, ),
= deg H(·, ),
= deg{I,
∩ Ker M, } = .
∩ Ker M,
∩ Ker M,
Applying Lemma ., we complete the proof.
Remark Assumption (H) guarantees that condition (A) of Lemma . is satisfied.
Furthermore, using assumptions (H)-(H), we can easily estimate prior boud of the solution
to Eq. (.).
As an application, we consider the following example:
x(t) =
xx((tt)) ∈ R,
g(x) =
B =
e(t) = (sin t, cos t) , τ = γ = π , p = ., T = π , f (x) = ( + sin x, + cos x).
Obviously, λ = = ±, λ = – = ±,
e(t) dt =
for|x| > D > ,
xg(x) =
for |x| > D > ,
gi(u) – gi(u) ≤
|u – u|,
u, u ∈ R, for each i = , ,
f(x) = | + sin x| ≥ ,
f(x) = | + cos x| ≥ ,
U =
U– =
such that
UBU– =
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The first author QY gave an example for verifying the paper’s results. The corresponding author BD gave the proof for all
the theorems. QY and BD read and approved the final manuscript.
Acknowledgements
This work was supported by NSF of Jiangsu education office (11KJB110002), Postdoctoral Fundation of Jiangsu
(1102096C), Postdoctoral Fundation of China (2012M511296) and Jiangsu province fund (BK2011407).
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