AbstractThe existence and uniqueness of T-periodic solutions for the following boundary value problems with p-Laplacian: (ϕp(x′))′+f(t,x′)+g(t,x)=e(t),x(0)=x(T),x′(0)=x′(T) are investigated, where ϕp(u)=∣u∣p−2u with p>1 and f,g,e are continuous and are T-periodic in t with f(t,0)=0. Using coincidence degree theory, some existence and uniqueness results are presented

Kim, Yong-In

Lo, Kueiming

Yang, Xiaojing

Elsevier - Publisher Connector 

Applied Mathematics Letters 25 (2012) 586–589
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Applied Mathematics Letters
journal homepage: www.elsevier.com/locate/aml
Periodic solutions for a generalized p-Laplacian equation
Xiaojing Yang a, Yong-In Kim b,∗, Kueiming Lo c
a Department of Mathematics, Tsinghua University, Beijing 100084, China
b Department of Mathematics, University of Ulsan, Ulsan, 680-749, Republic of Korea
c School of Software, Tsinghua University, Key Laboratory for Information System Security, Ministry of Education of China, Beijing 100084, China
a r t i c l e i n f o
Article history:
Received 10 August 2010
Received in revised form 6 July 2011
Accepted 27 September 2011
Keywords:
p-Laplacian
Degree theory
Periodic solution
a b s t r a c t
The existence and uniqueness of T -periodic solutions for the following boundary value
problems with p-Laplacian:
(φp(x′))′ + f (t, x′)+ g(t, x) = e(t), x(0) = x(T ), x′(0) = x′(T )
are investigated, where φp(u) =| u |p−2 u with p > 1 and f , g, e are continuous and are
T -periodic in t with f (t, 0) = 0. Using coincidence degree theory, some existence and
uniqueness results are presented.
© 2011 Elsevier Ltd. All rights reserved.
1. Introduction
We consider the solvability of the following periodic boundary value problem:
(φp(x′))′ + f (t, x′)+ g(t, x) = e(t) (1)
x(0) = x(T ), x′(0) = x′(T ), (2)
where φp(u) = |u|p−2uwith p > 1 and f , g and e are continuous functions and are T -periodic in t with f (t, 0) = 0.
When p = 2, Eq. (1) reduces to the following second-order forced Rayleigh equation:
x′′ + f (t, x′)+ g(t, x) = e(t). (3)
The existence and uniqueness of periodic solutions of (1) and (3) have been an important research focus for the study
of dynamic behaviors of nonlinear second-order differential equations. See, for example, the research papers [1–9] and the
references therein. Recently, Wang [7] has obtained the following results:
Theorem A. Consider the following p-Laplacian equation:
(φp(x′))′ + Cx′ + g(t, x) = e(t) (4)
where C is a constant and
 T
0 e(t)dt = 0. Assume that there exists d ≥ 0 such that:
(H1) [g(t, u1)− g(t, u2)](u1 − u2) < 0 ∀ u1, u2, u1 ≠ u2, and t ∈ R.
(H2) xg(t, x) < 0∀ |x| > d, and t ∈ R.
Then (4) has a unique T-periodic solution.
∗ Corresponding author.
E-mail addresses: yangxj@mail.tsinghua.edu.cn (X. Yang), yikim@mail.ulsan.ac.kr, yikim@mail.ulsan.ac.kr (Y.-I. Kim), gluo@tsinghua.edu.cn (K. Lo).
0893-9659/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.aml.2011.09.064
X. Yang et al. / Applied Mathematics Letters 25 (2012) 586–589 587
Theorem B. Assume that g(t, x) = g(x) and (H1) holds. Then (4) has a unique T-periodic solution if and only if 0 ∈ g(R).
Remark 1. For the case of p = 2, related results were obtained by Li and Huang [2] and Wang and Shao [6].
In this work, we discuss the existence and uniqueness of T -periodic solutions of (1)–(2) under some general conditions.
The main result of this work is the following:
Theorem 1. In the problem (1)–(2), assume that:
(A1) There exists d∗ > 0 such that
x(g(t, x)− e(t)) < 0 ∀t ∈ R and |x| ≥ d∗.
(A2) There exist nonnegative constants m1, m2 and α ≤ p− 1 such that
|f (t, u)| ≤ m1|u|α +m2 ∀t, u ∈ R and f (t, 0) = 0 ∀ t ∈ R.
Then the problem (1)–(2) has at least one solution. Moreover, if p ≥ 2 and (H1) holds, then the problem (1)–(2) has a unique
solution.
2. Lemmas
We first introduce somewell-known results for p-Laplacian-like operators, which will be used in the proof of Theorem 1.
Let X = C1T be the space of all C1-functions which are T -periodic, i.e.,
X = C1T = {x(t) ∈ C1(R, R) : x(0) = x(T ), x′(0) = x′(T )}.
The norm of a function x ∈ C1T is defined by
‖x‖ := ‖x‖∞ + ‖x′‖∞,
where ‖x‖∞ := maxt∈R |x(t)| and ‖x′‖∞ := maxt∈R |x′(t)|.
Lemma 1 (Theorem 3.1 [5]). Consider the following problem:
(φp(u′))′ = h(t, u, u′), u(0) = u(T ), u′(0) = u′(T ), (5)
where φp(u) = |u|p−2u with p > 1 and h is a Caratheodory function and is T -periodic in t. Let Ω = {x ∈ C1T : ‖x‖ < r} for
some r > 0. Suppose that the following conditions hold:
(i) For each λ ∈ (0, 1), the problem
(φp(u′))′ = λh(t, u, u′), u ∈ C1T (6)
has no solution on ∂Ω .
(ii) The function H(a) defined by
H(a) := 1
T
∫ T
0
h(t, a, 0)dt
satisfies H(−r)H(r) < 0.
Then the problem (5) has at least one solution inΩ .
Lemma 2 (Generalized Bellman’s Inequality). Consider the following inequality:
|y(t)| ≤ C +M
∫ t
0
|y(s)|βds (7)
where C, M, β are nonnegative constants and t > 0. If β ≤ 1, then for t ∈ (0, T0], we have |y(t)| ≤ D, where
D =

CeMT0 , if β = 1,
(C1−β +MT0(1− β))
1
1−β , if β < 1.
Proof of Theorem 1. Let h(t, x, x′) = e(t)− f (t, x′)− g(t, x). Then (6) reduces to
(φp(x′))′ + λf (t, x′)+ λ[g(t, x)− e(t)] = 0, λ ∈ (0, 1). (8)
We first show that the set of all possible T -periodic solutions of (8) is bounded. Let x(t) be an arbitrary T -periodic solution
of (8). Assume that
x(t1) = max
t∈[0,T ]
x(t), x(t2) = min
t∈[0,T ] x(t), t1, t2 ∈ [0, T ].
Then we have
x′(t1) = x′(t2) = 0, x′′(t1) ≤ 0, x′′(t2) ≥ 0.
588 X. Yang et al. / Applied Mathematics Letters 25 (2012) 586–589
Setting t = t1 and t = t2 in (8) and using f (t, 0) = 0 and (φp(x′(t)))′ = (p− 1)|x′(t)|p−2x′′(t), we then have
g(t1, x(t1))− e(t1) ≥ 0 and g(t2, x(t2))− e(t2) ≤ 0.
Then from the assumption (A1), we must have x(t1) < d∗ and x(t2) > −d∗. Hence by the periodicity of x(t), we get
−d∗ < x(t2) ≤ x(t) ≤ x(t1) < d∗ ∀ t ∈ R.
This implies that x(t) is bounded and ‖x‖∞ := maxt∈R |x(t)| < d∗.
Next we show that x′(t) is bounded. Let y(t) = φp(x′(t)). Then x′ = φq(y), where q = pp−1 > 1, and it follows from (8)
that
y′(t) = −λf (t, φq(y(t)))− λg(t, x(t))+ λe(t).
Let x(t1) = maxt∈[0,T ] x(t). Then x′(t1) = 0 and hence y(t1) = 0. Thus for any t ∈ [0, T ] such that 0 ≤ t1 ≤ t , by the
assumption (A2), we can write
|y(t)| =
y(t1)+ ∫ t
t1
y′(s)ds
 = ∫ t
t1
|y′(s)|ds ≤
∫ t
0
|y′(s)|ds
=
∫ t
0
λf (s, φq(y(s)))+ λ[g(s, x(s))− e(s)] ds
≤
∫ t
0
|f (s, φq(y(s)))|ds+
∫ T
0
|g(t, x(t))− e(t)|dt
≤ T max
t∈[0,T ],|x|≤d∗
|g(t, x)− e(t)| + Tm2 +m1
∫ t
0
|y(s)|(q−1)αds. (9)
Let D1 = T maxt∈[0,T ],|x|≤d∗ |g(t, x)− e(t)| + Tm2 and β = (q− 1)α = αp−1 . Then from (9) we get
|y(t)| ≤ D1 +m1
∫ t
0
|y(s)|βds.
Note that 0 ≤ β ≤ 1 since 0 ≤ α ≤ p − 1 from the assumption (A2). Now from the generalized Bellman’s inequality (7),
we obtain |y(t)| ≤ D2 for all t ∈ [t1, T ], where
D2 =

D1em1T , if α = p− 1,
D
p−1−α
p−1
1 +
m1T (p− 1− α)
p− 1
 p−1
p−1−α
, if α < p− 1.
If 0 ≤ t ≤ t1, we have 0 ≤ t1 ≤ t + T ≤ 2T and from the T -periodicity of y(t) and the assumption (A2), we obtain
|y(t)| = |y(t + T )| =
y(t1)+ ∫ t+T
t1
y′(s)ds
 = ∫ t+T
t1
y′(s)ds

≤
∫ t+T
t1
y′(s) ds ≤ ∫ t+T
0
|y′(s)|ds
=
∫ t+T
0
λf (s, φq(y(s)))+ λ[g(s, x(s))− e(s)] ds
≤
∫ t+T
0
|f (s, φq(y(s)))|ds+
∫ 2T
0
|g(t, x(t))− e(t)|dt
≤ 2T max
t∈[0,T ],|x|≤d∗
|g(t, x)− e(t)| + 2Tm2 +m1
∫ t+T
0
|y(s)|(q−1)αds.
From the above inequality, it follows that for 0 ≤ t ≤ t1, we have |y(t)| = |y(t+T )| ≤ 2D1+m1
 t+T
0 |y(s)|βds. This implies
that for 0 ≤ t ≤ t1, we have |y(t)| ≤ D3, where
D3 =

2D1em1T , if α = p− 1,
(2D1)
p−1−α
p−1 + 2m1T (p− 1− α)
p− 1
 p−1
p−1−α
, if α < p− 1.
Since D2 ≤ D3, the above inequalities imply that ‖y‖∞ := maxt∈[0,T ] |y(t)| ≤ D3, which again implies that ‖x′‖∞ ≤ Dq−13 =
D
1
p−1
3 . We have therefore ‖x‖ := ‖x‖∞ + ‖x′‖∞ < d∗ + D
1
p−1
3 =: M . Thus we have shown that the set of all T -periodic
X. Yang et al. / Applied Mathematics Letters 25 (2012) 586–589 589
solutions x(t) of (8) is bounded, i.e., ‖x(t)‖ < M . Now setΩ = {x ∈ X : ‖x‖ = ‖x‖∞ + ‖x′‖∞ < M}. Then Eq. (8) has no
solution on ∂Ω for λ ∈ (0, 1). Moreover, for x = ±M ∈ R, x ∈ ∂Ω and sinceM > d∗ and f (t, 0) = 0, using the assumption
(A1), we see that H(−M)H(M) < 0. Now Lemma 1 implies that problem (1)–(2) has a solution inΩ .
Now to prove uniqueness, assume that p ≥ 2 and (H1) holds. Let x1(t) and x2(t) be any two solutions of (1)–(2), and let
y1(t) = φp(x′1(t)) and y2(t) = φp(x′2(t)). Also, let u(t) = x1(t)− x2(t) and v(t) = y1(t)− y2(t). Since x′ = φ−1p (y) = φq(y),
it follows from (1) that
u′(t) = φq(y1(t))− φq(y2(t)),
v′(t) = [f (t, φq(y2(t)))− f (t, φq(y1(t)))] + [g(t, x2(t))− g(t, x1(t))]. (10)
We will show that u(t) ≤ 0∀ t ∈ [0, T ]. Suppose there exists a t0 ∈ [0, T ) such that u(t0) = maxt∈[0,T ] u(t) =
x1(t0) − x2(t0) > 0. Then u′(t0) = φq(y1(t0)) − φq(y2(t0)) = 0, which implies that y1(t0) = y2(t0) and u′′(t0) ≤ 0.
Note that since p ≥ 2, we have q = pp−1 ≤ 2. Define |u|q−2 = 1 for u = 0 and q = 2, and |u|q−2 = +∞ for u = 0 and q < 2.
Then it follows from (H1), (10) and y1(t0) = y2(t0) that
u′′(t0) = (q− 1)|y1(t0)|q−2y′1(t0)− (q− 1)|y2(t0)|q−2y′2(t0)
= (q− 1)|y1(t0)|q−2[y′1(t0)− y′2(t0)] = (q− 1)|y1(t0)|q−2v′(t0)
= (q− 1)|y1(t0)|q−2[f (t0, φq(y2(t0)))− f (t0, φq(y1(t0)))]
+ (q− 1)|y1(t0)|q−2[g(t0, x2(t0))− g(t0, x1(t0))]
= (q− 1)|y1(t0)|q−2[g(t0, x2(t0))− g(t0, x1(t0))]
> 0(or = +∞),
which is a contradiction. Hence maxt∈[0,T ] u(t) ≤ 0. Similarly, exchanging the role of x1 and x2, we can show that
maxt∈[0,T ] u(t) ≥ 0. This implies that u(t) ≡ 0. Therefore, the problem (1)–(2) has at most one solution. The proof of
Theorem 1 is now complete. 
Remark 2. Note that in this work the term Cx′ in Theorem A has been replaced by a more general function f (t, x′) in
Theorem 1. Thus the method used in the proof of existence and uniqueness of T -periodic solutions of (4) can no longer
be applied in the proof of Theorem 1 and the conditions on g and e must be replaced by our condition (A1). Therefore the
result and themethod used in our work can be regarded as a generalization and improvement of those in [7]. For 1 < p < 2,
we could not show the uniqueness when (H1) is satisfied. This is left as an open question for further discussion.
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Periodic solutions for a generalized p-Laplacian equation 

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Periodic solutions for a generalized p-Laplacian equation

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