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Existence and uniqueness of periodic solutions for a kind of duffing type p-Laplacian equation
Nonlinear Analysis: Real World Applications 9 (2008) 985 – 989 www.elsevier.com/locate/nonrwa
Existence and uniqueness of periodic solutions for a kind of duffing type p-Laplacian equation Fuxing Zhanga,∗ , Ya Lib a Department of Mathematics, Shaoyang University, Shaoyang, Hunan 422000, PR China b Editorial Department of Journal of Hunan University, Changsha 410082, PR China
Received 28 November 2006; accepted 19 January 2007
Abstract By using Manasevich–Mawhin continuation theorem and some analysis skill, we obtain some sufficient conditions for the existence and uniqueness of periodic solutions for Duffing type p-Laplacian differential equation. 䉷 2007 Elsevier Ltd. All rights reserved. MSC: 34K15; 34C25 Keywords: p-Laplacian; Periodic solutions; Duffing equation; Manasevich–Mawhin continuation theorem
1. Introduction In the last several years, the existence of periodic solutions for second-order differential equation has received a lot of attention. For example, in [2–5,7–9], the Duffing equation, Rayleigh equation and Liénard type equation. However, so far as we know, fewer papers discuss the existence and uniqueness of periodic solutions for Duffing type p-Laplacian differential equation. In this paper, we deal with the existence and uniqueness of T -periodic solutions of the Duffing type p-Laplacian differential equation of the form: (p (x (t))) + Cx (t) + g(t, x(t)) = e(t),
(1)
where p > 1 and p : R → R is given by p (s) = |s|p−2 s for s = 0 and p (0) = 0, C is a constant, g is a continuous function defined on R 2 and is periodic about t with g(t, ·) = g(t + T , ·), t ∈ R. e is a continuous periodic function T defined on R with period T , 0 e(t) dt = 0 and T > 0. Because the p-Laplacian operator p : p (x) = |x|p−2 x is nonlinear, the continuation theorem of Mawhin [1] is not available, which leads to the difficulty for solving the problem equation (1). By using Manasevich–Mawhin continuation theorem and some analysis skill, we establish some sufficient conditions for the existence and uniqueness of T -periodic solutions of Eq. (1). The results of this paper are new and they complement previously known results. ∗ Corresponding author.
E-mail address: [email protected] (F. Zhang). 1468-1218/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2007.01.013
986
F. Zhang, Y. Li / Nonlinear Analysis: Real World Applications 9 (2008) 985 – 989
For convenience, define CT1 : ={x ∈ C 1 (R) : x is T -periodic}, which is a Banach space endowed with the norm . defined by x = |x|∞ + |x|∞ , for all x, and |x|∞ = max |x(t)|, t∈[0,T ]
|x |∞ = max |x (t)|. t∈[0,T ]
For the T-periodic boundary value problem (p (x (t))) = f(t, x, x ),
(2)
where f is a continuous function and T -periodic in the first variable we have the following result. Lemma 1 (Manasevich–Mawhin [6]). Let B be the open in CT1 of center 0 and radius r. Assume that the following conditions hold: (i) For each ∈ (0, 1) the problem (p (x )) = f (t, x, x ) has no solution on the board of B. T (ii) The continuous function F defined on R by F (a) = 1/T 0 f (t, a, 0) dt is such that F (−r)F (r) < 0. Then (2) has ¯ at least one solution in B. Lemma 2. Suppose that the following condition hold: (A1 ) (g(t, u1 ) − g(t, u2 ))(u1 − u2 ) < 0, for i = 1, 2, ui ∈ R, ∀t ∈ R and u1 = u2 . Then Eq. (1) has at most one T-periodic solution. Proof. Suppose that x1 (t) and x2 (t) are two T-periodic solutions of Eq. (1). Then, we obtain (p (x1 (t)) − p (x2 (t))) + C(x1 (t) − x2 (t)) + (g1 (t, x1 (t)) − g1 (t, x2 (t))) = 0.
(3)
Since x1 (t) and x2 (t) are T-periodic, multiplying x1 (t) − x2 (t) and (3) and then integrating it from 0 to T, we get T 0 − (p (x1 (t)) − p (x2 (t)))(x1 (t) − x2 (t)) dt 0 T (p (x1 (t)) − p (x2 (t))) (x1 (t) − x2 (t)) dt = 0 T (g(t, x1 (t)) − g(t, x2 (t)))(x1 (t) − x2 (t)) dt = − 0.
0
(4)
In view of (A1 ) and (4), we have x1 (t) ≡ x2 (t)
for all t ∈ R.
Therefore, Eq. (1) has at most one T-periodic solution. The proof of Lemma 2 is now complete. 2. Main results By using Lemmas 1 and 2, we obtain our main results: Theorem 1. Let (A1 ) hold. Suppose there exist positive constants K and M such that: (A2 ) xg(t, x) < 0 for |x| > 0 and t ∈ R. (A3 ) 22−p MT p < 1, g(t, x) − M|x|p−1 − K, for x 0 and t ∈ R. Then Eq. (1) has a unique T-periodic solution.
F. Zhang, Y. Li / Nonlinear Analysis: Real World Applications 9 (2008) 985 – 989
987
Proof. Consider the homotopic equation (1) as following: (p (x (t))) + Cx (t) + g(t, x(t)) = e(t),
∈ (0, 1).
(5)
By Lemma 2, together with (A1 ), it is easy to see that Eq. (1) has at most one T-periodic solution. Thus, to prove Theorem 1, it suffices to show that Eq. (1) has at least one T-periodic solution. To do this, we shall apply Lemma 1. Firstly, we will claim that the set of all possible T-periodic solutions of Eq. (5) are bounded in CT1 . Let x(t) ∈ CT1 be an arbitrary solution of Eq. (5) with period T. By integrating two sides of Eq. (5) over [0, T ], and noticing that x (0) = x (T ), we have T g(t, x(t)) dt = 0. (6) 0
As x(0) = x(T ), there exists t0 ∈ [0, T ] such that x (t0 ) = 0, while p (0) = 0 we see
|p (x (t))| = |
t t0
(p (x (s))) ds|
T
|g(t, x(t))| dt.
(7)
0
From (6), there exists a ∈ [0, T ] such that: g(, x()) = 0. In view of (A2 ), we obtain x() = 0. Then, we have |x(t)| = |x() +
t
x (s) ds|
t
|x (s)| ds,
t ∈ [, + T ]
and |x(t)| = |x(t − T )| = |x() −
t−T
x (s) ds|
t−T
|x (s)| ds,
t ∈ [, + T ].
Combining the above two inequalities, we obtain t 1 1 |x|∞ = max |x(t)| = max |x(t)| max |x (s)| ds + |x (s)| ds T |x |∞ . t∈[0,T ] 2 t∈[,+T ] t∈[,+T ] 2 t−T
(8)
Denote E1 = {t : t ∈ [0, T ], x(t) < 0},
E2 = {t : t ∈ [0, T ], x(t)0}.
Then, in view of (6), (A2 ) and (A3 ), we can get |g(t, x(t))| dt = g(t, x(t)) dt = − E1
E1 T 0
which yields T |g(t, x(t))| dt 0
E2
E2
(M|x(t)|p−1 + K) dt
p−1
(M|x(t)|p−1 + K) dt MT |x|∞ + KT ,
E1
g(t, x(t)) dt
|g(t, x(t))| dt +
p−1
E2
|g(t, x(t))| dt 2MT |x|∞ + 2KT .
(9)
988
F. Zhang, Y. Li / Nonlinear Analysis: Real World Applications 9 (2008) 985 – 989
Together with (7), (8), and (9), we have p−1 |x |∞
t
= max {|p (x (t))|} = max (p (x (s))) ds
t∈[0,T ] t∈[0,T ] t0 T p−1 p−1 |g(t, x(t))| dt 2MT |x|∞ + 2KT 22−p MT p |x |∞ + 2KT .
(10)
0
Since 22−p MT p < 1, we can get some positive constant M1 such that for all t ∈ R, |x (t)|M1 , in view of (8), which implies that for all t ∈ R, |x(t)| 21 T M 1 < M2 .
(11)
Hence taking r = 1/2T M 1 + M1 + 1, we have that the set of all possible T-periodic solutions of Eq. (5) is a subset of B. T On the other hand, it is clear that, in our case F (a) = −1/T 0 g(t, a) dt. From (A2 ) it follows that F (−r)F (r) < 0. In consequence we can apply Manasevich–Mawhin continuation theorem to deduce that Eq. (5) has at least one solution ¯ This completes the proof. in B. Theorem 2. Let (A1 ) hold. Suppose there exist positive constants K, and M such that: (A2 ) xg(t, x) < 0 for |x| > 0 and t ∈ R. (A3 ) 22−p MT p < 1, g(t, x) M|x|p−1 + K, for x 0 and t ∈ R. Then Eq. (1) has a unique T-periodic solution. Remark. To prove the existence of at least one T-periodic solution of Eq. (1) we need only (A2 ), (A3 ) in Theorem 1, respectively, (A2 ), (A3 ) in Theorem 2. 3. An example As an application, let us consider the following equation: (p x (t)) + 100x (t) − g(t, x) = cos t, (12) √ where p = 2, g(t, x)=−1/(100 +cos2 t)|x(t)|p−2 x(t) for all t ∈ R, x > 0, and g(t, x)=−x 33 (t) for all t ∈ R, x 0. We can easily check the conditions (A1 ).(A3 ) hold. By Theorem 1, Eq. (12) has a unique 2-periodic solution. √ Remark 1. Since p = 2, one can easily to see that all the results in [1–9] and the references therein are cannot be applicable to (12) to obtain the existence and uniqueness of 2-periodic solutions. This implies that the results of this paper are essentially new. Acknowledgments The author is grateful to the referee for his or her suggestions on the first draft of the paper. References [1] R.E. Gaines, J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Mathematics, vol. 568, Springer, Berlin, New York, 1977. [2] X. Huang, Z. Xiang, On the existence of 2-periodic solution for delay Duffing equation x (t) + g(t, x(t − )) = p(t), Chin. Sci. Bull. 39 (3) (1994) 201–203. [3] Y. Li, Periodic solutions of the Liénard equation with deviating arguments, J. Math. Res. Exp. 18 (4) (1998) 565–570 (in Chinese).
F. Zhang, Y. Li / Nonlinear Analysis: Real World Applications 9 (2008) 985 – 989
989
[4] S. Lu, W. Ge, Some new results on the existence of periodic solutions to a kind of Rayleigh equation with a deviating argument, Nonlinear Anal. TMA 56 (2004) 501–514. [5] S. Lu, Z. Gui, On the existence of periodic solutions to p-Laplacian Rayleigh differential equation with a delay, J. Math. Anal. Appl. 325 (2007) 685–702. [6] R. Manásevich, J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. Differential Equations 145 (1998) 367–393. [7] G.Q. Wang, S.S. Cheng, A priori bounds for periodic solutions of a delay Rayleigh equation, Appl. Math. Lett. 12 (1999) 41–44. [8] G.Q. Wang, J. Yan, On existence of periodic solutions of the Rayleigh equation of retarded type, Int. J. Math. Math. Sci. 23 (1) (2000) 65–68. [9] M. Zong, H. Liang, Periodic solutions for Rayleigh type p-Laplacian equation with deviating arguments, Appl. Math. Lett. 206 (2007) 43–47.
Existence and uniqueness of periodic solutions for a kind of duffing type p-Laplacian equation Fuxing Zhanga,∗ , Ya Lib a Department of Mathematics, Shaoyang University, Shaoyang, Hunan 422000, PR China b Editorial Department of Journal of Hunan University, Changsha 410082, PR China
Received 28 November 2006; accepted 19 January 2007
Abstract By using Manasevich–Mawhin continuation theorem and some analysis skill, we obtain some sufficient conditions for the existence and uniqueness of periodic solutions for Duffing type p-Laplacian differential equation. 䉷 2007 Elsevier Ltd. All rights reserved. MSC: 34K15; 34C25 Keywords: p-Laplacian; Periodic solutions; Duffing equation; Manasevich–Mawhin continuation theorem
1. Introduction In the last several years, the existence of periodic solutions for second-order differential equation has received a lot of attention. For example, in [2–5,7–9], the Duffing equation, Rayleigh equation and Liénard type equation. However, so far as we know, fewer papers discuss the existence and uniqueness of periodic solutions for Duffing type p-Laplacian differential equation. In this paper, we deal with the existence and uniqueness of T -periodic solutions of the Duffing type p-Laplacian differential equation of the form: (p (x (t))) + Cx (t) + g(t, x(t)) = e(t),
(1)
where p > 1 and p : R → R is given by p (s) = |s|p−2 s for s = 0 and p (0) = 0, C is a constant, g is a continuous function defined on R 2 and is periodic about t with g(t, ·) = g(t + T , ·), t ∈ R. e is a continuous periodic function T defined on R with period T , 0 e(t) dt = 0 and T > 0. Because the p-Laplacian operator p : p (x) = |x|p−2 x is nonlinear, the continuation theorem of Mawhin [1] is not available, which leads to the difficulty for solving the problem equation (1). By using Manasevich–Mawhin continuation theorem and some analysis skill, we establish some sufficient conditions for the existence and uniqueness of T -periodic solutions of Eq. (1). The results of this paper are new and they complement previously known results. ∗ Corresponding author.
E-mail address: [email protected] (F. Zhang). 1468-1218/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2007.01.013
986
F. Zhang, Y. Li / Nonlinear Analysis: Real World Applications 9 (2008) 985 – 989
For convenience, define CT1 : ={x ∈ C 1 (R) : x is T -periodic}, which is a Banach space endowed with the norm . defined by x = |x|∞ + |x|∞ , for all x, and |x|∞ = max |x(t)|, t∈[0,T ]
|x |∞ = max |x (t)|. t∈[0,T ]
For the T-periodic boundary value problem (p (x (t))) = f(t, x, x ),
(2)
where f is a continuous function and T -periodic in the first variable we have the following result. Lemma 1 (Manasevich–Mawhin [6]). Let B be the open in CT1 of center 0 and radius r. Assume that the following conditions hold: (i) For each ∈ (0, 1) the problem (p (x )) = f (t, x, x ) has no solution on the board of B. T (ii) The continuous function F defined on R by F (a) = 1/T 0 f (t, a, 0) dt is such that F (−r)F (r) < 0. Then (2) has ¯ at least one solution in B. Lemma 2. Suppose that the following condition hold: (A1 ) (g(t, u1 ) − g(t, u2 ))(u1 − u2 ) < 0, for i = 1, 2, ui ∈ R, ∀t ∈ R and u1 = u2 . Then Eq. (1) has at most one T-periodic solution. Proof. Suppose that x1 (t) and x2 (t) are two T-periodic solutions of Eq. (1). Then, we obtain (p (x1 (t)) − p (x2 (t))) + C(x1 (t) − x2 (t)) + (g1 (t, x1 (t)) − g1 (t, x2 (t))) = 0.
(3)
Since x1 (t) and x2 (t) are T-periodic, multiplying x1 (t) − x2 (t) and (3) and then integrating it from 0 to T, we get T 0 − (p (x1 (t)) − p (x2 (t)))(x1 (t) − x2 (t)) dt 0 T (p (x1 (t)) − p (x2 (t))) (x1 (t) − x2 (t)) dt = 0 T (g(t, x1 (t)) − g(t, x2 (t)))(x1 (t) − x2 (t)) dt = − 0.
0
(4)
In view of (A1 ) and (4), we have x1 (t) ≡ x2 (t)
for all t ∈ R.
Therefore, Eq. (1) has at most one T-periodic solution. The proof of Lemma 2 is now complete. 2. Main results By using Lemmas 1 and 2, we obtain our main results: Theorem 1. Let (A1 ) hold. Suppose there exist positive constants K and M such that: (A2 ) xg(t, x) < 0 for |x| > 0 and t ∈ R. (A3 ) 22−p MT p < 1, g(t, x) − M|x|p−1 − K, for x 0 and t ∈ R. Then Eq. (1) has a unique T-periodic solution.
F. Zhang, Y. Li / Nonlinear Analysis: Real World Applications 9 (2008) 985 – 989
987
Proof. Consider the homotopic equation (1) as following: (p (x (t))) + Cx (t) + g(t, x(t)) = e(t),
∈ (0, 1).
(5)
By Lemma 2, together with (A1 ), it is easy to see that Eq. (1) has at most one T-periodic solution. Thus, to prove Theorem 1, it suffices to show that Eq. (1) has at least one T-periodic solution. To do this, we shall apply Lemma 1. Firstly, we will claim that the set of all possible T-periodic solutions of Eq. (5) are bounded in CT1 . Let x(t) ∈ CT1 be an arbitrary solution of Eq. (5) with period T. By integrating two sides of Eq. (5) over [0, T ], and noticing that x (0) = x (T ), we have T g(t, x(t)) dt = 0. (6) 0
As x(0) = x(T ), there exists t0 ∈ [0, T ] such that x (t0 ) = 0, while p (0) = 0 we see
|p (x (t))| = |
t t0
(p (x (s))) ds|
T
|g(t, x(t))| dt.
(7)
0
From (6), there exists a ∈ [0, T ] such that: g(, x()) = 0. In view of (A2 ), we obtain x() = 0. Then, we have |x(t)| = |x() +
t
x (s) ds|
t
|x (s)| ds,
t ∈ [, + T ]
and |x(t)| = |x(t − T )| = |x() −
t−T
x (s) ds|
t−T
|x (s)| ds,
t ∈ [, + T ].
Combining the above two inequalities, we obtain t 1 1 |x|∞ = max |x(t)| = max |x(t)| max |x (s)| ds + |x (s)| ds T |x |∞ . t∈[0,T ] 2 t∈[,+T ] t∈[,+T ] 2 t−T
(8)
Denote E1 = {t : t ∈ [0, T ], x(t) < 0},
E2 = {t : t ∈ [0, T ], x(t)0}.
Then, in view of (6), (A2 ) and (A3 ), we can get |g(t, x(t))| dt = g(t, x(t)) dt = − E1
E1 T 0
which yields T |g(t, x(t))| dt 0
E2
E2
(M|x(t)|p−1 + K) dt
p−1
(M|x(t)|p−1 + K) dt MT |x|∞ + KT ,
E1
g(t, x(t)) dt
|g(t, x(t))| dt +
p−1
E2
|g(t, x(t))| dt 2MT |x|∞ + 2KT .
(9)
988
F. Zhang, Y. Li / Nonlinear Analysis: Real World Applications 9 (2008) 985 – 989
Together with (7), (8), and (9), we have p−1 |x |∞
t
= max {|p (x (t))|} = max (p (x (s))) ds
t∈[0,T ] t∈[0,T ] t0 T p−1 p−1 |g(t, x(t))| dt 2MT |x|∞ + 2KT 22−p MT p |x |∞ + 2KT .
(10)
0
Since 22−p MT p < 1, we can get some positive constant M1 such that for all t ∈ R, |x (t)|M1 , in view of (8), which implies that for all t ∈ R, |x(t)| 21 T M 1 < M2 .
(11)
Hence taking r = 1/2T M 1 + M1 + 1, we have that the set of all possible T-periodic solutions of Eq. (5) is a subset of B. T On the other hand, it is clear that, in our case F (a) = −1/T 0 g(t, a) dt. From (A2 ) it follows that F (−r)F (r) < 0. In consequence we can apply Manasevich–Mawhin continuation theorem to deduce that Eq. (5) has at least one solution ¯ This completes the proof. in B. Theorem 2. Let (A1 ) hold. Suppose there exist positive constants K, and M such that: (A2 ) xg(t, x) < 0 for |x| > 0 and t ∈ R. (A3 ) 22−p MT p < 1, g(t, x) M|x|p−1 + K, for x 0 and t ∈ R. Then Eq. (1) has a unique T-periodic solution. Remark. To prove the existence of at least one T-periodic solution of Eq. (1) we need only (A2 ), (A3 ) in Theorem 1, respectively, (A2 ), (A3 ) in Theorem 2. 3. An example As an application, let us consider the following equation: (p x (t)) + 100x (t) − g(t, x) = cos t, (12) √ where p = 2, g(t, x)=−1/(100 +cos2 t)|x(t)|p−2 x(t) for all t ∈ R, x > 0, and g(t, x)=−x 33 (t) for all t ∈ R, x 0. We can easily check the conditions (A1 ).(A3 ) hold. By Theorem 1, Eq. (12) has a unique 2-periodic solution. √ Remark 1. Since p = 2, one can easily to see that all the results in [1–9] and the references therein are cannot be applicable to (12) to obtain the existence and uniqueness of 2-periodic solutions. This implies that the results of this paper are essentially new. Acknowledgments The author is grateful to the referee for his or her suggestions on the first draft of the paper. References [1] R.E. Gaines, J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Mathematics, vol. 568, Springer, Berlin, New York, 1977. [2] X. Huang, Z. Xiang, On the existence of 2-periodic solution for delay Duffing equation x (t) + g(t, x(t − )) = p(t), Chin. Sci. Bull. 39 (3) (1994) 201–203. [3] Y. Li, Periodic solutions of the Liénard equation with deviating arguments, J. Math. Res. Exp. 18 (4) (1998) 565–570 (in Chinese).
F. Zhang, Y. Li / Nonlinear Analysis: Real World Applications 9 (2008) 985 – 989
989
[4] S. Lu, W. Ge, Some new results on the existence of periodic solutions to a kind of Rayleigh equation with a deviating argument, Nonlinear Anal. TMA 56 (2004) 501–514. [5] S. Lu, Z. Gui, On the existence of periodic solutions to p-Laplacian Rayleigh differential equation with a delay, J. Math. Anal. Appl. 325 (2007) 685–702. [6] R. Manásevich, J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. Differential Equations 145 (1998) 367–393. [7] G.Q. Wang, S.S. Cheng, A priori bounds for periodic solutions of a delay Rayleigh equation, Appl. Math. Lett. 12 (1999) 41–44. [8] G.Q. Wang, J. Yan, On existence of periodic solutions of the Rayleigh equation of retarded type, Int. J. Math. Math. Sci. 23 (1) (2000) 65–68. [9] M. Zong, H. Liang, Periodic solutions for Rayleigh type p-Laplacian equation with deviating arguments, Appl. Math. Lett. 206 (2007) 43–47.