Periodic solutions for a higher order p-Laplacian neutral functional differential equation with a deviating argument

Periodic solutions for a higher order p-Laplacian neutral functional differential equation with a deviating argument

Nonlinear Analysis 71 (2009) 3906–3913 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Pe...

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Nonlinear Analysis 71 (2009) 3906–3913

Contents lists available at ScienceDirect

Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

Periodic solutions for a higher order p-Laplacian neutral functional differential equation with a deviating argumentI Kai Wang ∗ , Yanling Zhu School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, PR China

article

abstract

info

Article history: Received 16 August 2008 Accepted 13 February 2009

In this paper, a higher order p-Laplacian neutral functional differential equation with a deviating argument:

 (m) ϕp [x(t ) − c (t )x(t − σ )](n) + f (x(t ))x0 (t ) + g (t , x(t − τ (t ))) = e(t )

MSC: 34C25 34B15

has been studied by applying Mawhin’s continuation degree theorem. Some new criteria to guarantee the existence of periodic solutions are obtained. It is interesting that the power of the variable x in function g is allowed to be greater than p − 1. © 2009 Elsevier Ltd. All rights reserved.

Keywords: p-Laplacian Periodic solutions Continuation degree theorem Deviating argument

1. Introduction In this paper, we consider the following higher order neutral type p-Laplacian functional differential equation(for short P-L FDE)

 (m) ϕp [x(t ) − c (t ) x(t − σ )](n) + f (x(t )) x0 (t ) + g (t , x(t − τ (t ))) = e(t ),

(1.1)

where p > 1, ϕp (x) = |x| x for x 6= 0 and ϕp (0) = 0; g : R → R is a continuous periodic function with g (t + T , ·) ≡ g (t , ·), f ∈ C (R, R) and c , τ , e : R → R are all continuous periodic functions with c (t + T ) ≡ c (t ), τ (t + T ) ≡ τ (t ), e(t + T ) ≡ e(t ), T is a positive constant; n, m are positive integers. In the last several years, the existence of periodic solutions for functional differential equations(for short FDEs) has been studied extensively, see Refs. [1–17], especially, the P-L FDE which arises from fluid mechanical and nonlinear elastic mechanical phenomena has received more and more attention, see Refs. [5–17] for more details. Zhu and Lu’s [15] firstly introduces a kind of P-L FDE with difference operator as follows p−2

2

0 ϕp (x(t ) + c x(t − σ ))0 + g (t , x(t − τ (t ))) = e(t ). After that the existence of periodic solutions to the P-L FDE with difference operator has been studied further by [16,17]. Among plenty of results on the existence of periodic solutions for various types of FDEs, the assumptions on the function g are as follows (A0 ) x g (t , x) > 0 or x g (t , x) < 0 for |x| > d or (A1 ) x[g (t , x) − e(t )] > 0 for |x| > d or (A2 ) x[g (t , x) − e(t )] > 0 for |x| > d and I This research was supported by the NSF of educational Bureau of China (No. 207047) and the NSF of Educational Bureau of Anhui Province (No. KJ2009B103Z, KJ2008B235, 2008sk215) and the Key Young Foundation of Anhui University of Finance and Economics (No. ACKYQ0811ZD). ∗ Corresponding author. E-mail addresses: [email protected], [email protected] (K. Wang).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.02.060

K. Wang, Y. Zhu / Nonlinear Analysis 71 (2009) 3906–3913

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(A3 ) |g (t , x)| < M or |g (t , u)−g (t , v)| < L|u−v| or limx→+∞ supt ∈[0,T ] |g (t , x)/xp−1 | < r or one side bounded conditions, and the power of g respect to x is no more than p − 1. Recently, Tang’s [18] and Liu’s [19] have studied the existence of periodic solutions for the following two kinds of p-Laplacian ordinary differential equations, respectively.

0 ϕp (x(t ))0 + C x0 (t ) + g (x(t )) = e(t ) and

0 ϕp (x(t ))0 + f (x(t )) x0 (t ) + g (x(t )) = e(t ), under the assumption that: (A1 ) x[g (x) − e(t )] < 0 for |x| > d. As far as we know, few result on the existence of periodic solutions for FDEs has been established only under the sign condition (A0 )–(A2 ) without (A3 ), or the power of function g respect to x is more than p − 1, especially for the P-L FDE. Stimulated by the above reason, in this paper we will establish some sufficient results for guaranteeing the existence of periodic solutions to Eq. (1.1) without assumption (A3 ). Furthermore, the power of variable x in function g is allowed to be greater than p − 1. One can easily see that papers [1–20] are the special cases of Eq. (1.1), so our results is essentially new and improve the results in previous literatures. 2. Lemmas In order to apply Mawhin’s continuation degree theorem to study the existence of periodic solutions for Eq. (1.1), we set CT = {φ ∈ C (R, R) : φ(t + T ) ≡ φ(t )} with the norm |φ|∞ = maxt ∈[0,T ] |φ(t )|; X = {x = (x1 (t ), x2 (t )) ∈ C (R, R2 ) : x(t + T ) ≡ x(t )} with the norm |x|∞ = max{|x1 |∞ , |x2 |∞ }; Y = {x = (x1 (t ), x2 (t )) ∈ C 1 (R, R2 ) : x(t + T ) ≡ x(t )} with the norm kxk = max{|x|∞ , |x0 |∞ }. Clearly, X and Y are both Banach spaces. Define difference operator A as follows A : X → X,

(Ax)(t ) = x(t ) − c (t ) x(t − σ )

and rewrite Eq. (1.1) in the form:

(Ax1 )(n) (t ) = ϕq (x2 (t )), (2.1) (m) x2 (t ) = −f (x1 (t )) x01 (t ) − g (t , x1 (t − τ (t ))) + e(t ), where 1/p + 1/q = 1. Obviously, the existence of periodic solutions to Eq. (1.1) is equivalent to the existence of periodic 

solutions to Eqs. (2.1). Thus, the problem of finding a T -periodic solution for Eq. (1.1) reduces to finding one for Eqs. (2.1). Let     (Ax1 )(n) x L : Dom(L) ⊂ Y → X , L 1 = (m) x2 x2 , where Dom(L) = {x ∈ C n+m (R, R2 ) : x(t + T ) ≡ x(t )}. And set N : X → X,

  N

x1 x2

 =

ϕq (x2 (t )) −f (x1 (t )) x1 (t ) − g (t , x1 (t − τ (t ))) + e(t )

 (2.2)

0

.

One can easily see that Eqs. (2.1) can be converted to the abstract equation Lx = Nx. Moreover, we see from the definition of L that



Ker L = R2 ,

Im L =

T

Z



y∈Y : 0

y1 (s) ds = y2 (s)



  0 0

,

thus L is a Fredholm operator with index zero. Set project operators P and Q in the following form, respectively.

  P

x1 x2

Q : Y → Im Q ⊂ R2 ,

Q

P : X → Ker L,

 =

 (Ax1 )(0) x2 (0) .

and

  y1 y2

=

1 T

T

Z 0



y1 (s) ds. y2 (s)



1 2 Denote the inverse of L|KerP ∩D(L) by L− P . Obviously, Ker L = Im Q = R and



−1

LP

  y1 y2

(t ) =



A−1 Gy1 (t ) (Gy2 )(t )



 (2.3) ,

where

Z t n −1 X 1 1 (Ax1 )(i) (0)t i + (t − s)n−1 y1 (s)ds, i ! ( n − 1 )! 0 i =1 Z t m −1 X 1 (i) 1 (Gy2 )(t ) = x2 (0)t i + (t − s)m−1 y2 (s)ds, i ! ( m − 1 )! 0 i=1 (Gy1 )(t ) =

3908

K. Wang, Y. Zhu / Nonlinear Analysis 71 (2009) 3906–3913

where (Ax1 )(i) (0), i = 1, 2, . . . , n − 1 are decided by the following equation



1  c1  c  2

E1 Z = B,

where E1 =   .. .

0 1 c1

0 0 1

··· ··· ···

cn−4 cn−3

cn−5 cn−4

··· ···

.. .

 c

n −3

cn−2

.. .

Z = (Ax1 )(n−1) (0), . . . , (Ax1 )00 (0), (Ax1 )0 (0)

>

(i)



0 0 0

0 0  0

.. .

..   . 0

1 c1

1 (n−1)×(n−1)

, B = (b1 , b2 , . . . , bn−1 )> , bi = − i!1T

RT 0

(T − s)i y1 (s)ds and cj =

Tj

(j+1)!

,j =

1, 2, . . . , n − 2. And x2 (0), i = 1, 2, . . . , m − 1 are decided by the equation



0 1 c1

0 0 1

··· ··· ···

cm−4 cm−3

cm−5 cm−4

··· ···

1  c1  c  2

E2 W = F ,

where E2 =   ..

.. .

 . c

m−3

cm−2

0 0 0

.. .

.. .

1 c1



0 0  0

..   . 0 1 (m−1)×(m−1)

> RT (m−1) W = x2 (0), . . . , x002 (0), x02 (0) , F = (d1 , d2 , . . . , dm−1 )> , di = − i!1T 0 (T − s)i y2 (s)ds and cj = 1, 2, . . . , m − 2. From (2.2) and (2.3), we know that N is L-compact on Ω , where Ω is an arbitrary open bounded subset of X . 

Tj

(j+1)!

,j =

Lemma 2.1 (See [4]). If |c (t )| 6= 1 then operator A has continuous inverse A−1 on X , which satisfies

(1)

−1

A

f (t ) =



 j −1 ∞ Y X    f ( t ) + c (t − iσ )f (t − jσ )  

for |c |∞ < 1, f ∈ X ,

j +1 ∞ Y X  f (t + (j + 1)σ ) f (t + σ )   − −   c (t + σ ) c ( t + iσ )

for |c |L > 1, f ∈ X ,

j=1 i=0

j=1 i=1

 |f | ∞   1 − |c |∞ −1 (2) |A f | 6 |f |∞   |c |L − 1 (3)

T

Z

−1  A f (t ) dt 6 0

for |c |∞ < 1, f ∈ X , for |c |L > 1, f ∈ X .

      

T

Z

1 1 − | c |∞

Z

1

|c |L − 1

|f (t )|dt

for |c |∞ < 1, f ∈ X , where |c |L := min |c (t )|.

0 T

|f (t )|dt

for |c |L > 1, f ∈ X .

t ∈[0,T ]

0

Lemma 2.2 (See [21]). Let X and Y be Banach spaces, L : Dom(L) ⊂ X → Y be a Fredholm operator with index zero, Ω ⊂ Y be an open bounded set and N : Ω → X be L-compact on Ω . If the following conditions hold: [1] Lx 6= λNx, for x ∈ ∂ Ω ∩ Dom(L), λ ∈ (0, 1); [2] Nx 6∈ Im L, for x ∈ ∂ Ω ∩ Ker L; [3] deg{JQN , Ω ∩ Ker L, 0} 6= 0, where J : Im Q → Ker L is an isomorphism, then equation Lx = Nx has a solution on T Ω Dom(L). 3. Main results Theorem 3.1. Assume that there exist positive constants d and L such that (H1 ) x(g (t , x) − e(t )) > 0 for |x| > d, (H2 ) g (t , x) > 0 or g (t , x) < 0 for (t , x) ∈ ([0, T ], R) , |f (x)| (H3 ) lim|x|→+∞ |x|p−2 6 L for x ∈ R, and LT m+(p−1)n−1 < 2m+(p−1)n−1 (1 − |c |∞ )p−1 LT

m+(p−1)n−1

m+(p−1)n−1

<2

(|c |L − 1)

p−1

Then Eq. (1.1) has at least one periodic solution.

for |c |∞ < 1. for |c |L > 1.

K. Wang, Y. Zhu / Nonlinear Analysis 71 (2009) 3906–3913

3909

Proof. Without lost of generality, we suppose |c |∞ < 1 and g (t , x) > 0. Consider the operator equation L x = λ Nx,

λ ∈ (0, 1).

Set Ω1 = {x : L x = λ N x, λ ∈ (0, 1)}. If x(t ) = (x1 (t ), x2 (t ))> ∈ Ω1 , then



(Ax1 )(n) (t ) = λ ϕq (x2 (t )), (m) x2 (t ) = −λ f (x1 (t )) x01 (t ) − λ g (t , x1 (t − τ (t ))) + λ e(t ).

(3.1)

The second equation of (3.1) and x2 (t ) = λ1−p ϕp [(Ax1 )(n) (t )] imply



  (m) ϕp (Ax1 )(n) (t ) + λp f (x1 (t )) x01 (t ) + λp g (t , x1 (t − τ (t ))) = λp e(t ).

(3.2)

Integration of both sides of Eq. (3.2) from 0 to T gives T

Z

g (t , x1 (t − τ (t ))) dt =

T

Z

e(t )dt ,

(3.3)

0

0

which together with (H1 ) yields that there exists at least one ξ ∈ [0, T ] such that

|x1 (ξ − τ (ξ ))| 6 d, i.e.,

|x1 (η)| = |x1 (ξ − τ (ξ ))| 6 d,

(3.4)

where η ∈ [0, T ] with η = ξ − τ (ξ ) ± kT , k is an integer. Thus we have

|x1 (t )| 6 d +

t

Z

|x0 (s)|ds for t ∈ [η, η + T ]

η

and

|x1 (t )| = |x(t − T )| 6 d +

η

Z

|x0 (s)|ds for t ∈ [η, η + T ],

t −T

which yield

|x1 |∞ 6 d +

1

T

Z

2

|x01 (s)|ds 6 · · · 6 d + 0

T i −1

T

Z

2i

0

(i) x1 (s) ds for i = 1, 2, . . . , n

(3.5a)

and 0

|x1 |∞ 6

T i−1

T

Z

2i

 i T (i+1) (i+1) x1 (s) ds 6 x1 2

0



for i = 1, 2, . . . , n − 1.

(3.5b)

From the first equation in Eqs. (3.1) and Lemma 2.1, we obtain that

(Ax1 )(n) (t ) = Ax(n) (t ) 6 ϕq (|x2 |∞ )

(3.6a)

x2 (ζ ) = 0.

(3.6b)

1

and

where ζ ∈ [0, T ] is a constant. Similar to (3.5) one can easily get

(i) x2



6

T j −1 2j

T

Z 0

 j T (i+j) (i+j) x2 (s) ds 6 x2 , 2



for i + j 6 1, 2, . . . , m,

(3.7)

where i, j are both nonnegative integers. In order to finish the estimation of a priori bounds, now we introduce the classical elementary inequality. For constant δ > 0, which is only dependent on k > 0, we have

(1 + x)k 6 1 + (1 + k) x for x ∈ (0, δ].

(3.8) (m−1)

On the other hand, there must be γ ∈ [0, T ] such that x2 (γ ) = 0, and for any small constant ε > 0 assumption (H3 ) yields |f (x)| 6 (L + ε)|x|p−2 , it follows from (3.3) and assumption (H2 ) that T

Z

|g (s, x1 (s − τ (s))) |ds = 0

T

Z

g (s, x1 (s − τ (s))) ds 6 0

T

Z

|e(s)| ds, 0

3910

K. Wang, Y. Zhu / Nonlinear Analysis 71 (2009) 3906–3913

which together with the integration of the second Eq. (3.1) on interval [0, T ] gives



(m−1)

2 x2

Z (t ) 6 2 x(2m−1) (γ ) + 6 λ

T 0



(m)

x2 (s)ds

T

Z

f (x1 (s)) x0 (s) + g (s, x1 (s − τ (s))) − e(s) ds 1

0 T

Z

T

Z

f (x1 (s)) x0 (s) ds +

6

|g (s, x1 (s − τ (s))) |ds +

1

6 (L + ε)

|e(s)| ds 0

0

0

T

Z

T

Z

|x1 (s)|p−2 |x01 (s)|ds + 2T |e|∞ 0

6 (L + ε)T |x01 |∞

 d+

1 2

T

Z

|x01 (s)|ds

p−2

+ 2T |e|∞ .

(3.9)

0

Case I. p 6 2, it follows from (3.9) that



(m−1)

2 x2

(t ) 6 22−p (L + ε)T p−1 |x01 |p∞−1 + 2T |e|∞ .

(3.10a)

Case II. p > 2.

RT

, then it follows from (3.5) that |x1 |∞ 6 (1 + 1δ )d, which together with (3.7) and (3.9) implies that If 0 |x01 (s)|ds 6 2d δ there exits positive constant M (independent of λ) such that |x2 |∞ 6 M . Otherwise, R T 2d 6 δ, it follows from (3.8) and (3.9) that 0 0

|x1 (s)|ds

 Z 1 (m−1) 2 x2 (t ) 6 (L + ε)T |x01 |∞ d + 2

2 −p

6 2

T

|x01 (s)|ds

p−2 + 2T |e|∞

0

(L + ε)T |x1 |∞ 1 + R T 0

0

!p−2 Z

2d

|x01 (s)|ds

2 d (p − 1) 1+ RT |x01 (s)|ds 0

6 22−p (L + ε)T |x01 |∞

T

p−2 |x01 (s)|ds + 2T |e|∞

0

! Z

T

|x01 (s)|ds

p−2 + 2T |e|∞

0

6 22−p (L + ε)T p−1 |x01 |p∞−1 + 23−p (L + ε) d(p − 1) T p−1 |x01 |p∞−2 + 2T |e|∞ .

(3.10b)

From (2) of Lemma 2.1 and (3.6a), one can obtain 1

|x(1n) (t )| = |A−1 Ax(1n) (t )| 6

1 − |c |∞

|Ax(1n) (t )| 6

1 1 − |c |∞

ϕq (|x2 |∞ ),

which together with (3.5b) gives 0

|x1 |∞

 n−1 T (n) 6 x1 2



6

1 1 − |c |∞

 n−1 T

2

ϕq (|x2 |∞ ).

(3.11)

Combination of (3.7), (3.10b) and (3.11) implies

|x2 |∞

 m−1 T (m−1) 6 x2 2

6 6

T



m−1

22−p (L + ε)T p−1 |x01 |p∞−1 + 2T |e|∞

 2m



(L + ε)T m+(p−1)n−1 Tm |x2 |∞ + m−1 |e|∞ for p 6 2, p − 1 (1 − |c |∞ ) 2

2m+(p−1)n−1

and

|x2 |∞

 m−1 T (m−1) 6 x2 2

6 6

T



m−1

22−p (L + ε)T p−1 |x01 |p∞−1 + 23−p (L + ε) d(p − 1) T p−2 |x01 |p∞−2 + 2T |e|∞

 2m



(L + ε)T m+(p−1)n−1 (L + ε)d T m+(p−2)n−1 (p − 1) Tm 2−q |x2 |∞ + m+(p−1)n−2 |x2 |∞ + m−1 |e|∞ for p > 2, p − 1 p − 2 (1 − |c |∞ ) 2 (1 − | c |∞ ) 2

2m+(p−1)n−1

K. Wang, Y. Zhu / Nonlinear Analysis 71 (2009) 3906–3913

3911

which together with q > 1 and assumption LT m+(p−1)n−1 < 2m+(p−1)n−1 (1 − |c |∞ )p−1 yields that there must exists positive b b . Thus case I–II gives constant M(independent of λ) such that |x2 |∞ 6 M

b } := M1 . |x2 |∞ 6 max{M , M

(3.12)

From (3.11) and (3.12), we obtain that

|x01 |∞ 6

T n−1 2n−1 (1 − |c |∞ )

T n −1

ϕq (|x2 |∞ ) 6

2n−1 (1 − |c |∞ )

ϕq (M1 ) := M2 .

(3.13)

Hence 1

|x1 |∞ 6 d +

T

Z

2

|x01 (s)|ds 6 d + 0

TM2 2

:= M3 .

Then it follows from (3.13), (3.9) and (3.7) that there exists positive constant M4 (independent of λ) such that |x02 |∞ 6 M4 . Let M0 = max{M1 , M2 , M3 , M4 } + 1, Ω = {x = (x1 , x2 )> : kxk < M0 } and Ω2 = {x : x ∈ ∂ Ω ∩ Ker L}, then

  QN

x1 x2

=

1

T

Z

T

0



ϕq (x2 ) −g (t , x1 ) + e(t )



dt .

If x ∈ Ω2 , then x2 = 0, x1 = M0 or −M0 . Thus, from (H1 ) we can easily get QNx 6= 0, i.e., Nx 6∈ Im L for x ∈ Ω . So conditions [1] and [2] of Lemma 2.2 are satisfied. In order to show that condition [3] of Lemma 2.2 is also satisfied, we define an isomorphism J in the following form:

 

J : Im Q → Ker L,

J

x1 x2

 =

x2 −x 1



.

Let H (µ, x) = −µ x + (1 − µ)JQNx

for (µ, x) ∈ [0, 1] × Ω2 ,

then

 H (µ, x) =

(1 − µ)

T

 (g (t , x1 ) − e(t ))dt  , T 0 −µ x2 − (1 − µ)ϕq (x2 )

−µ x1 −

Z

it follows from condition (H1 ) that xH (µ, x) < 0. Hence deg{JQN , Ω ∩ Ker L, 0} = deg{H (0, x), Ω ∩ Ker L, 0} = deg{H (1, x), Ω ∩ Ker L, 0} = deg{I , Ω ∩ Ker L, 0} 6= 0. Thus, condition [3] of Lemma 2.2 is satisfied. So by applying Lemma 2.2, we conclude that equation Lx = Nx has a solution x(t ) = (x1 (t ), x2 (t ))> on Ω ∩ D(L). So Eq. (1.1) has a T −periodic solution x1 (t ). The proof of Theorem 3.1 is now finished.  Corollary 3.1. Assume that there exist positive constants d and L such that (H1 ) x(g (t , x) − e(t )) < 0 for |x| > d, (H2 ) g (t , x) > 0 or g (t , x) < 0 for (t , x) ∈ ([0, T ], R) , |f (x)| (H3 ) lim|x|→+∞ |x|p−2 6 L for x ∈ R, and LT m+(p−1)n−1 < 2m+(p−1)n−1 (1 − |c |∞ )p−1 , LT

m+(p−1)n−1

m+(p−1)n−1

<2

(|c |L − 1)

p−1

,

for |c |∞ < 1. for |c |L > 1.

Then Eq. (1.1) has at least one periodic solution. If m = n and c (t ) ≡ 0, then Eq. (1.1) reduces to the following P-L FDE:

ϕp x(n) (t )

(n)

+ f (x(t )) x0 (t ) + g (t , x(t − τ (t ))) = e(t ).

Similar to Theorem 3.1 and Corollary 3.1, we obtain the following results. Theorem 3.2. Assume that there exists positive constant d such that (H1 ) x(g (t , x) − e(t )) < 0 for |x| > d, (H2 ) g (t , x) > 0 or g (t , x) < 0 for (t , x) ∈ ([0, T ], R) . Then Eq. (1.1) has at least one periodic solution.

(1.1*)

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K. Wang, Y. Zhu / Nonlinear Analysis 71 (2009) 3906–3913

Proof. Corresponding to (3.1), Eq. (1.1*) reduces to the following form:



(n)

x1 (t ) = λ ϕq (x2 (t )), (n) x2 (t ) = −λ f (x1 (t )) x01 (t ) − λ g (t , x1 (t − τ (t ))) + λ e(t ),

(3.1*)

(n)

which together with x2 (t ) = λ1−p ϕp (x1 (t )) yields

  (n) ϕp x(1n) (t ) = −λp f (x1 (t )) x01 (t ) − λp g (t , x1 (t − τ (t ))) + λp e(t ).

(3.14)

Now we only need to estimate the prior bounds of x ∈ Ω1 . Multiplying both sides of Eq. (3.14) with x1 (t ) and integrating them from 0 to T , we get T

Z 0

Z T  (n) (n) p ϕp x(1n) (t ) x(t ) dt x1 (t ) dt = − 0 Z T x1 (t )(g (t , x1 (t − τ (t ))) − e(t )) dt = λp 0 Z T  Z T |g (t , x1 (t − τ (t )))|dt + |e(t )|dt , 6 |x1 |∞ 0

0

it follows from (3.3) and assumption (H2 ) that T

Z 0

(n) p x1 (t ) dt 6 2 T |e|∞ |x1 |∞ ,

which together with (3.5b) yields

|x01 |∞ 6

T n −2

T

Z

2n−1

|x(1n) (t )|dt 6

T n−2+1/q

0

2n−1

Z

T

|x(1n) (t )|p dt

1/p 6

0

T n−1+1/q 2n

|e|∞ |x1 |1∞/p .

Thus there is a positive constant M10 (independent of λ) such that

|x01 |∞ < M10 .

(3.15)

Furthermore, from (3.5a) one can easily get

|x1 |∞ < d +

T 2

M10 := M20 .

(3.16)

Lastly, from the second equation of (3.1*), (3.15) and (3.16) we claim that there exist two positive constants M30 , M40 (independent of λ) such that |x02 |∞ < M30 and |x2 |∞ < M40 . The reminder of the proof is similar to Theorem 3.1, thus we omit it here.  Corollary 3.2. Assume that there exists positive constant d such that (H1 ) x(g (t , x) − e(t )) > 0 for |x| > d, (H2 ) g (t , x) > 0 or g (t , x) < 0 for (t , x) ∈ ([0, T ], R) . Then Eq. (1.1) has at least one periodic solution. As an application, we consider the following equation

ϕ3 (x(5) (t ))

(5)

+ x6 (t )x0 (t ) + (3 + sin π t ) x4 (t − 2 cos π t ) ex(t −2 cos π t ) = sin π t + 20.

(3.17)

Comparing with Corollary 3.2, we choose g (t , x) = (3 + sin π t ) x e , e(t ) = sin π t + 20, T = 2, thus 4

g (t , x) = (3 + sin π t ) x4 ex > 0,

x

for (t , x) ∈ ([0, 2], R),

x(g (t , x) − e(t )) > x(x4 ex − 21) > 0,

for (t , |x|) ∈ ([0, 2], [2, +∞)) .

Thus we know that Eq. (3.17) has at least one 2-periodic solution. Obviously, all theorems in [5–7,10–15,19–21] can not be applied to study the existence of periodic solutions for Eq. (3.17). 4. Conclusion In this paper we investigated two kinds of P-L FDEs, Eq. (1.1) and Eq. (1.1*). Some sufficient conditions are established for guaranteeing the existence of periodic solutions for them. It is obviously that in Theorems 3.1 and 3.2. and Corollaries 3.1 and 3.2. the conditions we imposed on function g are different from the previous ones, especially for Eq. (1.1*), our results (Theorem 3.2 and Corollary 3.2) are weaker than the ones in paper [1–20]. The significance is that the power of x in function g is allowed to be greater than p − 1. So our results are essentially new.

K. Wang, Y. Zhu / Nonlinear Analysis 71 (2009) 3906–3913

3913

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