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Existence of positive periodic solutions for the p -Laplacian system
Applied Mathematics Letters 20 (2007) 696–701 www.elsevier.com/locate/aml
Existence of positive periodic solutions for the p-Laplacian systemI Jiebao Sun a , Yuanyuan Ke a,b , Chunhua Jin a,∗ , Jingxue Yin a a Department of Mathematics, Key Laboratory of Symbolic Computation and Knowledge Engineering of the Ministry of Education,
Jilin University, Changchun 130012, PR China b Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, PR China
Received 22 March 2006; received in revised form 11 July 2006; accepted 19 July 2006
Abstract In this work, we study the existence of positive periodic solutions for the p-Laplacian system. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Periodic boundary value; Positive periodic solutions; p-Laplacian; Topological degree; Existence
1. Introduction In this work we consider the following boundary value problem for the p-Laplacian system: (Φ p (u 0 (t)))0 + A(t) · u(t) = f (t, u(t)), u(0) = u(T ),
t ∈ I = [0, T ],
u (0) = u (T ), 0
0
(1.1) (1.2)
where p > 2, u = (u 1 , u 2 , . . . , u n ), Φ p (u) : Rn → Rn is given by (|u| p−2 u 1 , |u| p−2 u 2 , . . . , |u| p−2 u n ) and A(t) = (ai j (t))n×n is a matrix-valued function of size n × n. If v(t) = Φ p (u 0 (t)), where u(t) is a positive solution of (1.1) and (1.2), then for any t ∈ I , we have u 0 (t) = Φ −1 p (v(t)), v 0 (t) = f (t, u(t)) − A(t)u(t). The system models some phenomena in different physical and other natural sciences: non-Newtonian mechanics, nonlinear elasticity and glaciology, combustion theory, population biology, etc.; see [1–6]. As an example, in the population model, u(t) and v(t) are n-dimensional vectors of the population densities, Φ −1 p (v(t)) and f (t, u) represent some contributions of two kinds of species to each other, while the linear matrix A(t) represents the carrying capacity of the environment of u(t). Recently, problems concerning periodic solutions for the p-Laplacian have been considered by many authors; see for example [9,10,13]. Most authors study the problems governed by a single equation (see [11,12]) or consider the I This work was partially supported by the National Science Foundation of China, partially supported by the 985 project of Jilin University, NSFGD-06300481, and China Postdoctoral Science Foundation. ∗ Corresponding author. E-mail address: [email protected] (C. Jin).
c 2006 Elsevier Ltd. All rights reserved. 0893-9659/$ - see front matter doi:10.1016/j.aml.2006.07.010
J. Sun et al. / Applied Mathematics Letters 20 (2007) 696–701
697
problems for the special case p = 2 (see [14–16]). Our consideration is motivated by the results in [1] and [8]. In these papers, the authors studied the following ordinary differential equation with the periodic boundary value condition: 00 u (t) + a(t)u(t) = f (t, u(t)), t ∈ [0, T ], (1.3) u (k) (0) = u (k) (T ), k = 0, 1, where a(t) is a nonnegative T -periodic continuous function, f (t, u(t)) is nonnegative continuous and f (·, u) is also a T -periodic function for each u ∈ [0, +∞). By applying the fixed point index theory, the authors proved the existence of positive solutions of the problem (1.3). Problem (1.3) corresponds to the special case n = 1, p = 2 of the problem (1.1) and (1.2). In this work, we extend the results of [7] and [8]. It is not a simple extension since the dimension n ≥ 2 and the exponent p 6= 2. In one respect, due to p 6= 2, the problem is changed into a nonlinear problem, and the well-known Green’s function for the linear operator is no longer applicable. On the other hand, the most important aspect is that, unlike for the problem with n = 1 and p = 2, the upper boundedness of the solution in C 1 norm is needed to prove the existence of the solutions. All these difficulties require us to adopt new methods to solve the problem. Motivated by the ideas in [9], we will make some a priori estimates and use topological degree theory to obtain the existence of positive periodic solutions for (1.1) and (1.2). This work is organized as follows. In Section 2 we introduce some necessary preliminaries. In Section 3 we give the statement of our main result and its proof. 2. Preliminaries Assume that A(t) and f (t, u) satisfy the following conditions: (H1) For every i, j ∈ {1, . . . , n}, ai j : R+ → R+ is a T -periodic continuous function, A(t) is a positive semidefinite matrix and the spectral radius ρ(A(t)) is bounded uniformly. (H2) f : R+ × Rn+ → Rn+ is continuous and f (·, u) : R+ → Rn+ is also a T -periodic function for each u ∈ Rn+ . (H3) There existsPa constant M > 0, such that P for any |D| > M, there exists i ∈ {1, 2, . . . , n}, such that f i (t, D) > nj=1 ai j (t)D j or f i (t, D) < nj=1 ai j (t)D j , t ∈ I . Throughout this work, we denote as h·, ·i the inner product in Rn . For n ≥ 1 we set C = C(I, Rn+ ), C 1 = ∈ C|u(0) = u(T )}, C T1 = {u ∈ C 1 |u(0) = u(T ), u 0 (0) = u 0 (T )}, L p = L p (I, Rn+ ) and The norm in C and C T will be denoted by k · k0 , the norm in C 1 and C T1 by k · k1 . These norms are defined by
C 1 (I, Rn+ ), C T = {u W 2, p = W 2, p (I, Rn+ ).
kuk0 = max ku i k∞ , 1≤i≤n
kuk1 = kuk0 + ku 0 k0 ,
where ku i k∞ = supt∈I |u i (t)|. Furthermore, we define the norm in L p by !1 !1 p p n Z T n X X p p kuk L p = |u i (t)| dt = ku i k L p , where ku i k L p =
i=1
0
R T
|u i (t)| p dt
0
i=1
1
p
.
Definition 2.1. By a positive solution of (1.1) and (1.2), we mean a function u : I → Rn+ of class C 1 with Φ p (u 0 ) absolutely continuous, which satisfies (1.1) and (1.2) a.e. on I . Now we will introduce a useful lemma, which plays a fundamental role in the proof of the main theorem, and it was proved by Ra´ul Man´asevich and Jean Mawhin in [9]. Consider the periodic boundary problem (Φ p (u 0 ))0 = f (t, u, u 0 ), u(0) = u(T ),
t ∈ I,
u (0) = u (T ), 0
0
where the function f : I × Rn × Rn → Rn is assumed to be Carath´eodory.
(2.1) (2.2)
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J. Sun et al. / Applied Mathematics Letters 20 (2007) 696–701
Lemma 2.1. Assume that Ω is an open bounded set in C T1 such that the following conditions hold. (1) For each λ ∈ (0, 1) the problem (Φ p (u 0 ))0 = λ f (t, u, u 0 ),
u(t) = u(T ),
u 0 (t) = u 0 (T ),
has no solution on ∂Ω . (2) The equation Z 1 T F(a) := f (t, a, 0) dt = 0, T 0
(2.3)
(2.4)
has no solution on ∂Ω ∩ Rn . (3) The Brouwer degree deg(F, Ω ∩ Rn , 0) 6= 0.
(2.5)
Then problem (2.1) and (2.2) has a solution in Ω . 3. Main result and its proof Let Ω = {u ∈ W 2, p (I, Rn+ ); u(0) = u(T ), u 0 (0) = u 0 (T )}, where W 2, p (I, Rn+ ) is the usual Sobolev space. The following theorem is the main result of this work. Theorem 3.1. If the assumptions (H1 ), (H2 ) and (H3 ) hold, then the problem (1.1) and (1.2) admits at least one positive periodic solution u ∈ C 1 . Proof. Suppose u ∈ C 1 is a solution of the following problem: (|u 0 (t)| p−2 u 0 (t))0 = λ( f (t, u(t)) − A(t) · u(t)), u(0) = u(T ),
t ∈ I,
u (0) = u (T ). 0
0
(3.1) (3.2)
Next, we will show that u must be uniform C 1 bounded for λ ∈ [0, 1]. By the boundary value conditions, we have Z T p −ku 0 k L p = hu(t), (Φ p (u 0 (t)))0 i dt. 0
Multiplying (3.1) by u(t), and integrating the resulting relation on [0, T ], due to the periodicity of u(t), we can deduce Z T Z T 0 p ku k p = λ hA(t) · u(t), u(t)i dt − λ h f (t, u(t)), u(t)i dt 0
≤ λρ
n X
0
ku i k2L p T
1− 2p
i=1
≤ λρ M p T
1− 2p
kuk2L p
= λCkuk2L p ,
(3.3)
where ρ = maxt∈I ρ(A(t)), and M p is a constant that depends merely on p. In addition, by the periodic boundary value conditions we also have Z T Z T 0= (Φ p (u 0 (t)))0 dt = λ (3.4) ( f (t, u(t)) − A(t) · u(t)) dt. 0
0
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J. Sun et al. / Applied Mathematics Letters 20 (2007) 696–701
So we can confirm that for any i ∈ {1, 2, . . . , n}, there exists ti0 ∈ I , such that u i (ti0 ) ≤ M. Suppose the contrary; then there exists j ∈ {1, 2, . . . , n} such that u j (t) > M for any t ∈ I , which implies that |u(t)| > M for any t ∈ I . Recalling the condition (H3), there exists an i ∈ {1, . . . , n} such that ! Z T n X f i (t, u 1 (t), . . . , u n (t)) − ai j (t)u j (t) dt > 0 0
j=1
or Z
T
f i (t, u 1 (t), . . . , u n (t)) −
0
n X
! ai j (t)u j (t)
dt < 0.
j=1
This is a contradiction to (3.4). Then ∀i ∈ {1, 2, . . . , n} we have Z t 1 0 u i (s) ds ≤ M + T q ku i0 k L p , t ∈ I, |u i (t)| = u i (ti0 ) + ti
(3.5)
0
1
where q is the conjugate exponent of p. The above inequality implies ku i k0 ≤ M + T q ku i0 k L p . Also by u i (t) = Rt u i (ti0 ) + ti u i0 (s) ds, i = 1, 2, . . . , n and (3.3), we have 0
Z
t
0 ku i k L p ≤ ku i (ti0 )k L p + u i (s) ds
ti
p L
0
≤ MT
1 p
T
Z + 0
≤ MT ≤ MT
p ! 1 Z p t u i0 (s) ds dt ti 0
1 p
+ ku i0 k L p T
1 p
+ T λ p C p kuk Lp p .
Furthermore, we have kuk L p ≤ C M T
1
1 p
1
2
2 1 1 + T λ p C p kuk Lp p .
(3.6)
Because p > 2, the above inequality implies that kuk L p is bounded. Consequently, there exists a constant M0 > 0, such that kuk L p ≤ M0 . Combining this with (3.3) we have 1
2
1
2
ku 0 k L p ≤ (λC) p kuk Lp p ≤ (λC) p M0p = M1 , and then by (3.5) we can obtain 1
kuk0 ≤ M + T q M1 = M2 .
(3.7)
For any i ∈ {1, . . . , n}, by the boundary condition u i (0) = u i (T ), it is easy to see that there exists ti∗ ∈ I , such that u i0 (ti∗ ) = 0. Integrating (3.1) from ti∗ to t, we have 0 |u i (t)| p−2 u i 0 (t) ≤ |u 0 (t)| p−2 u i 0 (t) Z ! n t X = λ f i (s, u 1 (s), . . . , u n (s)) − ai j (s)u j (s) ds t∗ j=1 n √ X e + λρ n ≤ λT M i=1
≤ M3 ,
T
Z
|u i (t)| dt p
1p
1
Tq
0
(3.8)
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J. Sun et al. / Applied Mathematics Letters 20 (2007) 696–701
e = supt∈I,d ∈[0,M ] f i (t, d1 , . . . , dn ). So from (3.8) we have |u 0 (t)| ≤ |Φq (M3 )|, that is ku 0 k0 ≤ Φ −1 where M p (M3 ). i k 2 Combining this with (3.7), we obtain kuk1 ≤ M2 + Φ −1 p (M3 ) = M. That is, ku(t)k1 is bounded uniformly. Let Ω1 = {u ∈ Ω ∩ Rn : F(u) = 0}, where Z 1 T F(u) = ( f (t, u(t)) − A(t) · u(t)) dt. T 0
(3.9)
We will prove that for any D ∈ Ω1 , |D| ≤ M. For any D ∈ Ω1 , by F(D) = 0 we have Z 1 T ( f (t, D) − A(t) · D) dt = 0, T 0 that is for every i ∈ {1, . . . , n}, 1 T
T
Z
f i (t, D) −
0
n X
! ai j (t)D j
dt = 0.
(3.10)
j=1
We by (H3), there exists i 0 ∈ {1, 2, . . . , n}, such that f i0 (t, D) − Pn can confirm that |D| ≤ M. Otherwise, Pn a (t)D > 0 or f (t, D) − a (t)D i j j i i j j < 0, which implies that Fi 0 (D) < 0 or Fi 0 (D) > 0. That 0 j=1 0 j=1 0 is a contradiction to (3.10). Finally we will prove that the condition (3) of Lemma 2.1 is also satisfied. For any D ∈ Ω ∩ Rn , recalling (H3), if |D| > M, there exists i 0 ∈ {1, . . . , n}, such that f i0 (t, D) −
n X
ai0 j (t)D j > 0,
(3.11)
ai0 j (t)D j < 0.
(3.12)
j=1
or f i0 (t, D) −
n X j=1
In the following we assume (3.11). As for the case of (3.12), the proof is similar, so we can omit it. Let Ω2 = {D ∈ Ω ∩ Rn : λ(D − ξ ) + (1 − λ)F(D) = 0, λ ∈ [0, 1]}, where ξ ∈ Rn+ with 0 < |ξ | < M. We show that Ω2 is bounded by M. Suppose the contrary; there exists D ∈ Ω2 with |D| > M. Hence there exists i 0 ∈ {1, . . . , n} such that (3.11) holds, which implies that Fi0 (D) > 0. Furthermore, we have λ(Di0 − ξi0 ) + (1 − λ)Fi0 (D) > 0. Obviously, this contradicts the definition of Ω2 . That is for any D ∈ Ω2 , we always have |D| ≤ M. From the definition of F, it is easy to see that F : Rn+ → Rn+ is completely continuous. Let h λ (D) = λ(D − ξ ) + (1 − λ)F(D), and define Ω ∗ := {u ∈ Ω : kuk1 < M + M}. Then clearly h λ (∂Ω ∗ ∩ Rn ) 6= 0 for any λ ∈ [0, 1]. By virtue of the invariance property of homotopy, we obtain deg(F, Ω ∗ ∩ Rn , 0) = deg(h 0 , Ω ∗ ∩ Rn , 0) = deg(h 1 , Ω ∗ ∩ Rn , 0) = deg(I, Ω ∗ ∩ Rn , ξ ) = 1. Up to now, we have proved that the conditions (1), (2) and (3) in Lemma 2.1 are all satisfied. Therefore, the problem (1.1), (1.2) admits a solution in Ω ∗ by Lemma 2.1. The proof is completed.
J. Sun et al. / Applied Mathematics Letters 20 (2007) 696–701
As a simple example for Theorem 3.1, we consider the system 2π p−2 0 2 + cos t 0 (|u| u1) f 1 (t, u) u1 T + 2π 2π 2π u 2 = f 2 (t, u) , (|u| p−2 u 2 )0 t + cos t 1 − sin t sin T T T u 1 (0) = u 1 (T ), u 01 (0) = u 01 (T ), u 2 (0) = u 2 (T ), u 02 (0) = u 02 (T ),
701
(3.13) (3.14)
where t ∈ I and 2π 4 4 2 2 t u + u − u − u + 2 − cos 2 1 2 f 1 (t, u) 1 T . = 2π f 2 (t, u) t (u 1 + 1)2 + (u 2 + 1)2 − sin T It is easy to see the coefficient matrix satisfies condition (H1) and f i (t, u)(i = 1, 2) satisfies condition (H2). Notice that f i (t, u) → +∞ when |u| → +∞; thus condition (H3) holds. Hence, by Theorem 3.1, problems (3.13) and (3.14) have positive periodic solutions.
Remark 3.1. Indeed, for the case Φ p (s) = (Φ p1 (s1 ), . . . , Φ pn (sn )), with, for each i = 1, 2, . . . , n, pi > 2, and Φ pi : R+ → R+ is the one-dimensional pi -Laplacian, if (H1), (H2), (H3) hold, then the problem (1.1) and (1.2) admits at least one positive periodic solution. References [1] J.I. Diaz, F. de Thelin, On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM J. Math. Anal. 25 (4) (1994) 1085–1111. [2] R. Glowinski, J. Rappaz, Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology, Math. Model. Numer. Anal. 37 (1) (2003) 175–186. [3] U. Janfalk, On certain problem concerning the p-Laplace operator, Likping Studies in Sciences and Technology, Dissertations, 326, 1993. [4] S. Oruganti, J. Shi, R. Shivaji, Diffusive logistic equation with constant yield harvesting, I: Steady-states, Trans. Amer. Math. Soc. 354 (9) (2002) 3601–3619. [5] M. Ramaswamy, R. Shivaji, Multiple positive solutions for classes of p-Laplacian equations, Differential Integral Equations 17 (11–12) (2004) 1255–1261. [6] J.D. Murray, Mathematical Biology, Scientific Publications, Beijing, 1998. [7] F. Li, Z. Liang, Existence of positive periodic solutions to nonlinear second order differential equations, Appl. Math. Lett. 18 (11) (2005) 1256–1264. [8] Y. Li, Positive periodic solutions of nonlinear second order differential equations, Acta Math. Sinica 45 (3) (2002) 481–488. [9] R. Man´asevich, J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. Differential Equations 145 (1998) 367–393. [10] P. Yan, Nonresonance for one-dimensional p-Laplacian with regular restoring, J. Math. Anal. Appl 285 (2003) 141–154. [11] X. Yang, Existence of periodic solutions for quasilinear differential equations, Appl. Math. Comput. 153 (2004) 231–237. [12] X. Yang, Multiple periodic solutions of a class of p-Laplacian, J. Math. Anal. Appl. 314 (2006) 17–29. [13] Y. Wang, W. Ge, Existence of periodic solutions for nonlinear differential equations with a p-Laplacian-like operator, Appl. Math. Lett. 19 (2006) 251–259. [14] J. Henderson, H.B. Thompson, Multiple symmetric positive solutions for a second order boundary value problem, Proc. Amer. Math. Soc. 128 (2000) 2373–2379. [15] R.I. Avery, Three symmetric positive solutions for a second-order boundary value problem, Appl. Math. Lett. 13 (3) (2000) 1–7. [16] R.A. Khan, Existence and approximation of solutions of second order nonlinear Neumann problems, Electron. J. Differential Equations 2005 (3) (2005) 1–10.
Existence of positive periodic solutions for the p-Laplacian systemI Jiebao Sun a , Yuanyuan Ke a,b , Chunhua Jin a,∗ , Jingxue Yin a a Department of Mathematics, Key Laboratory of Symbolic Computation and Knowledge Engineering of the Ministry of Education,
Jilin University, Changchun 130012, PR China b Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, PR China
Received 22 March 2006; received in revised form 11 July 2006; accepted 19 July 2006
Abstract In this work, we study the existence of positive periodic solutions for the p-Laplacian system. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Periodic boundary value; Positive periodic solutions; p-Laplacian; Topological degree; Existence
1. Introduction In this work we consider the following boundary value problem for the p-Laplacian system: (Φ p (u 0 (t)))0 + A(t) · u(t) = f (t, u(t)), u(0) = u(T ),
t ∈ I = [0, T ],
u (0) = u (T ), 0
0
(1.1) (1.2)
where p > 2, u = (u 1 , u 2 , . . . , u n ), Φ p (u) : Rn → Rn is given by (|u| p−2 u 1 , |u| p−2 u 2 , . . . , |u| p−2 u n ) and A(t) = (ai j (t))n×n is a matrix-valued function of size n × n. If v(t) = Φ p (u 0 (t)), where u(t) is a positive solution of (1.1) and (1.2), then for any t ∈ I , we have u 0 (t) = Φ −1 p (v(t)), v 0 (t) = f (t, u(t)) − A(t)u(t). The system models some phenomena in different physical and other natural sciences: non-Newtonian mechanics, nonlinear elasticity and glaciology, combustion theory, population biology, etc.; see [1–6]. As an example, in the population model, u(t) and v(t) are n-dimensional vectors of the population densities, Φ −1 p (v(t)) and f (t, u) represent some contributions of two kinds of species to each other, while the linear matrix A(t) represents the carrying capacity of the environment of u(t). Recently, problems concerning periodic solutions for the p-Laplacian have been considered by many authors; see for example [9,10,13]. Most authors study the problems governed by a single equation (see [11,12]) or consider the I This work was partially supported by the National Science Foundation of China, partially supported by the 985 project of Jilin University, NSFGD-06300481, and China Postdoctoral Science Foundation. ∗ Corresponding author. E-mail address: [email protected] (C. Jin).
c 2006 Elsevier Ltd. All rights reserved. 0893-9659/$ - see front matter doi:10.1016/j.aml.2006.07.010
J. Sun et al. / Applied Mathematics Letters 20 (2007) 696–701
697
problems for the special case p = 2 (see [14–16]). Our consideration is motivated by the results in [1] and [8]. In these papers, the authors studied the following ordinary differential equation with the periodic boundary value condition: 00 u (t) + a(t)u(t) = f (t, u(t)), t ∈ [0, T ], (1.3) u (k) (0) = u (k) (T ), k = 0, 1, where a(t) is a nonnegative T -periodic continuous function, f (t, u(t)) is nonnegative continuous and f (·, u) is also a T -periodic function for each u ∈ [0, +∞). By applying the fixed point index theory, the authors proved the existence of positive solutions of the problem (1.3). Problem (1.3) corresponds to the special case n = 1, p = 2 of the problem (1.1) and (1.2). In this work, we extend the results of [7] and [8]. It is not a simple extension since the dimension n ≥ 2 and the exponent p 6= 2. In one respect, due to p 6= 2, the problem is changed into a nonlinear problem, and the well-known Green’s function for the linear operator is no longer applicable. On the other hand, the most important aspect is that, unlike for the problem with n = 1 and p = 2, the upper boundedness of the solution in C 1 norm is needed to prove the existence of the solutions. All these difficulties require us to adopt new methods to solve the problem. Motivated by the ideas in [9], we will make some a priori estimates and use topological degree theory to obtain the existence of positive periodic solutions for (1.1) and (1.2). This work is organized as follows. In Section 2 we introduce some necessary preliminaries. In Section 3 we give the statement of our main result and its proof. 2. Preliminaries Assume that A(t) and f (t, u) satisfy the following conditions: (H1) For every i, j ∈ {1, . . . , n}, ai j : R+ → R+ is a T -periodic continuous function, A(t) is a positive semidefinite matrix and the spectral radius ρ(A(t)) is bounded uniformly. (H2) f : R+ × Rn+ → Rn+ is continuous and f (·, u) : R+ → Rn+ is also a T -periodic function for each u ∈ Rn+ . (H3) There existsPa constant M > 0, such that P for any |D| > M, there exists i ∈ {1, 2, . . . , n}, such that f i (t, D) > nj=1 ai j (t)D j or f i (t, D) < nj=1 ai j (t)D j , t ∈ I . Throughout this work, we denote as h·, ·i the inner product in Rn . For n ≥ 1 we set C = C(I, Rn+ ), C 1 = ∈ C|u(0) = u(T )}, C T1 = {u ∈ C 1 |u(0) = u(T ), u 0 (0) = u 0 (T )}, L p = L p (I, Rn+ ) and The norm in C and C T will be denoted by k · k0 , the norm in C 1 and C T1 by k · k1 . These norms are defined by
C 1 (I, Rn+ ), C T = {u W 2, p = W 2, p (I, Rn+ ).
kuk0 = max ku i k∞ , 1≤i≤n
kuk1 = kuk0 + ku 0 k0 ,
where ku i k∞ = supt∈I |u i (t)|. Furthermore, we define the norm in L p by !1 !1 p p n Z T n X X p p kuk L p = |u i (t)| dt = ku i k L p , where ku i k L p =
i=1
0
R T
|u i (t)| p dt
0
i=1
1
p
.
Definition 2.1. By a positive solution of (1.1) and (1.2), we mean a function u : I → Rn+ of class C 1 with Φ p (u 0 ) absolutely continuous, which satisfies (1.1) and (1.2) a.e. on I . Now we will introduce a useful lemma, which plays a fundamental role in the proof of the main theorem, and it was proved by Ra´ul Man´asevich and Jean Mawhin in [9]. Consider the periodic boundary problem (Φ p (u 0 ))0 = f (t, u, u 0 ), u(0) = u(T ),
t ∈ I,
u (0) = u (T ), 0
0
where the function f : I × Rn × Rn → Rn is assumed to be Carath´eodory.
(2.1) (2.2)
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J. Sun et al. / Applied Mathematics Letters 20 (2007) 696–701
Lemma 2.1. Assume that Ω is an open bounded set in C T1 such that the following conditions hold. (1) For each λ ∈ (0, 1) the problem (Φ p (u 0 ))0 = λ f (t, u, u 0 ),
u(t) = u(T ),
u 0 (t) = u 0 (T ),
has no solution on ∂Ω . (2) The equation Z 1 T F(a) := f (t, a, 0) dt = 0, T 0
(2.3)
(2.4)
has no solution on ∂Ω ∩ Rn . (3) The Brouwer degree deg(F, Ω ∩ Rn , 0) 6= 0.
(2.5)
Then problem (2.1) and (2.2) has a solution in Ω . 3. Main result and its proof Let Ω = {u ∈ W 2, p (I, Rn+ ); u(0) = u(T ), u 0 (0) = u 0 (T )}, where W 2, p (I, Rn+ ) is the usual Sobolev space. The following theorem is the main result of this work. Theorem 3.1. If the assumptions (H1 ), (H2 ) and (H3 ) hold, then the problem (1.1) and (1.2) admits at least one positive periodic solution u ∈ C 1 . Proof. Suppose u ∈ C 1 is a solution of the following problem: (|u 0 (t)| p−2 u 0 (t))0 = λ( f (t, u(t)) − A(t) · u(t)), u(0) = u(T ),
t ∈ I,
u (0) = u (T ). 0
0
(3.1) (3.2)
Next, we will show that u must be uniform C 1 bounded for λ ∈ [0, 1]. By the boundary value conditions, we have Z T p −ku 0 k L p = hu(t), (Φ p (u 0 (t)))0 i dt. 0
Multiplying (3.1) by u(t), and integrating the resulting relation on [0, T ], due to the periodicity of u(t), we can deduce Z T Z T 0 p ku k p = λ hA(t) · u(t), u(t)i dt − λ h f (t, u(t)), u(t)i dt 0
≤ λρ
n X
0
ku i k2L p T
1− 2p
i=1
≤ λρ M p T
1− 2p
kuk2L p
= λCkuk2L p ,
(3.3)
where ρ = maxt∈I ρ(A(t)), and M p is a constant that depends merely on p. In addition, by the periodic boundary value conditions we also have Z T Z T 0= (Φ p (u 0 (t)))0 dt = λ (3.4) ( f (t, u(t)) − A(t) · u(t)) dt. 0
0
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J. Sun et al. / Applied Mathematics Letters 20 (2007) 696–701
So we can confirm that for any i ∈ {1, 2, . . . , n}, there exists ti0 ∈ I , such that u i (ti0 ) ≤ M. Suppose the contrary; then there exists j ∈ {1, 2, . . . , n} such that u j (t) > M for any t ∈ I , which implies that |u(t)| > M for any t ∈ I . Recalling the condition (H3), there exists an i ∈ {1, . . . , n} such that ! Z T n X f i (t, u 1 (t), . . . , u n (t)) − ai j (t)u j (t) dt > 0 0
j=1
or Z
T
f i (t, u 1 (t), . . . , u n (t)) −
0
n X
! ai j (t)u j (t)
dt < 0.
j=1
This is a contradiction to (3.4). Then ∀i ∈ {1, 2, . . . , n} we have Z t 1 0 u i (s) ds ≤ M + T q ku i0 k L p , t ∈ I, |u i (t)| = u i (ti0 ) + ti
(3.5)
0
1
where q is the conjugate exponent of p. The above inequality implies ku i k0 ≤ M + T q ku i0 k L p . Also by u i (t) = Rt u i (ti0 ) + ti u i0 (s) ds, i = 1, 2, . . . , n and (3.3), we have 0
Z
t
0 ku i k L p ≤ ku i (ti0 )k L p + u i (s) ds
ti
p L
0
≤ MT
1 p
T
Z + 0
≤ MT ≤ MT
p ! 1 Z p t u i0 (s) ds dt ti 0
1 p
+ ku i0 k L p T
1 p
+ T λ p C p kuk Lp p .
Furthermore, we have kuk L p ≤ C M T
1
1 p
1
2
2 1 1 + T λ p C p kuk Lp p .
(3.6)
Because p > 2, the above inequality implies that kuk L p is bounded. Consequently, there exists a constant M0 > 0, such that kuk L p ≤ M0 . Combining this with (3.3) we have 1
2
1
2
ku 0 k L p ≤ (λC) p kuk Lp p ≤ (λC) p M0p = M1 , and then by (3.5) we can obtain 1
kuk0 ≤ M + T q M1 = M2 .
(3.7)
For any i ∈ {1, . . . , n}, by the boundary condition u i (0) = u i (T ), it is easy to see that there exists ti∗ ∈ I , such that u i0 (ti∗ ) = 0. Integrating (3.1) from ti∗ to t, we have 0 |u i (t)| p−2 u i 0 (t) ≤ |u 0 (t)| p−2 u i 0 (t) Z ! n t X = λ f i (s, u 1 (s), . . . , u n (s)) − ai j (s)u j (s) ds t∗ j=1 n √ X e + λρ n ≤ λT M i=1
≤ M3 ,
T
Z
|u i (t)| dt p
1p
1
Tq
0
(3.8)
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J. Sun et al. / Applied Mathematics Letters 20 (2007) 696–701
e = supt∈I,d ∈[0,M ] f i (t, d1 , . . . , dn ). So from (3.8) we have |u 0 (t)| ≤ |Φq (M3 )|, that is ku 0 k0 ≤ Φ −1 where M p (M3 ). i k 2 Combining this with (3.7), we obtain kuk1 ≤ M2 + Φ −1 p (M3 ) = M. That is, ku(t)k1 is bounded uniformly. Let Ω1 = {u ∈ Ω ∩ Rn : F(u) = 0}, where Z 1 T F(u) = ( f (t, u(t)) − A(t) · u(t)) dt. T 0
(3.9)
We will prove that for any D ∈ Ω1 , |D| ≤ M. For any D ∈ Ω1 , by F(D) = 0 we have Z 1 T ( f (t, D) − A(t) · D) dt = 0, T 0 that is for every i ∈ {1, . . . , n}, 1 T
T
Z
f i (t, D) −
0
n X
! ai j (t)D j
dt = 0.
(3.10)
j=1
We by (H3), there exists i 0 ∈ {1, 2, . . . , n}, such that f i0 (t, D) − Pn can confirm that |D| ≤ M. Otherwise, Pn a (t)D > 0 or f (t, D) − a (t)D i j j i i j j < 0, which implies that Fi 0 (D) < 0 or Fi 0 (D) > 0. That 0 j=1 0 j=1 0 is a contradiction to (3.10). Finally we will prove that the condition (3) of Lemma 2.1 is also satisfied. For any D ∈ Ω ∩ Rn , recalling (H3), if |D| > M, there exists i 0 ∈ {1, . . . , n}, such that f i0 (t, D) −
n X
ai0 j (t)D j > 0,
(3.11)
ai0 j (t)D j < 0.
(3.12)
j=1
or f i0 (t, D) −
n X j=1
In the following we assume (3.11). As for the case of (3.12), the proof is similar, so we can omit it. Let Ω2 = {D ∈ Ω ∩ Rn : λ(D − ξ ) + (1 − λ)F(D) = 0, λ ∈ [0, 1]}, where ξ ∈ Rn+ with 0 < |ξ | < M. We show that Ω2 is bounded by M. Suppose the contrary; there exists D ∈ Ω2 with |D| > M. Hence there exists i 0 ∈ {1, . . . , n} such that (3.11) holds, which implies that Fi0 (D) > 0. Furthermore, we have λ(Di0 − ξi0 ) + (1 − λ)Fi0 (D) > 0. Obviously, this contradicts the definition of Ω2 . That is for any D ∈ Ω2 , we always have |D| ≤ M. From the definition of F, it is easy to see that F : Rn+ → Rn+ is completely continuous. Let h λ (D) = λ(D − ξ ) + (1 − λ)F(D), and define Ω ∗ := {u ∈ Ω : kuk1 < M + M}. Then clearly h λ (∂Ω ∗ ∩ Rn ) 6= 0 for any λ ∈ [0, 1]. By virtue of the invariance property of homotopy, we obtain deg(F, Ω ∗ ∩ Rn , 0) = deg(h 0 , Ω ∗ ∩ Rn , 0) = deg(h 1 , Ω ∗ ∩ Rn , 0) = deg(I, Ω ∗ ∩ Rn , ξ ) = 1. Up to now, we have proved that the conditions (1), (2) and (3) in Lemma 2.1 are all satisfied. Therefore, the problem (1.1), (1.2) admits a solution in Ω ∗ by Lemma 2.1. The proof is completed.
J. Sun et al. / Applied Mathematics Letters 20 (2007) 696–701
As a simple example for Theorem 3.1, we consider the system 2π p−2 0 2 + cos t 0 (|u| u1) f 1 (t, u) u1 T + 2π 2π 2π u 2 = f 2 (t, u) , (|u| p−2 u 2 )0 t + cos t 1 − sin t sin T T T u 1 (0) = u 1 (T ), u 01 (0) = u 01 (T ), u 2 (0) = u 2 (T ), u 02 (0) = u 02 (T ),
701
(3.13) (3.14)
where t ∈ I and 2π 4 4 2 2 t u + u − u − u + 2 − cos 2 1 2 f 1 (t, u) 1 T . = 2π f 2 (t, u) t (u 1 + 1)2 + (u 2 + 1)2 − sin T It is easy to see the coefficient matrix satisfies condition (H1) and f i (t, u)(i = 1, 2) satisfies condition (H2). Notice that f i (t, u) → +∞ when |u| → +∞; thus condition (H3) holds. Hence, by Theorem 3.1, problems (3.13) and (3.14) have positive periodic solutions.
Remark 3.1. Indeed, for the case Φ p (s) = (Φ p1 (s1 ), . . . , Φ pn (sn )), with, for each i = 1, 2, . . . , n, pi > 2, and Φ pi : R+ → R+ is the one-dimensional pi -Laplacian, if (H1), (H2), (H3) hold, then the problem (1.1) and (1.2) admits at least one positive periodic solution. References [1] J.I. Diaz, F. de Thelin, On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM J. Math. Anal. 25 (4) (1994) 1085–1111. [2] R. Glowinski, J. Rappaz, Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology, Math. Model. Numer. Anal. 37 (1) (2003) 175–186. [3] U. Janfalk, On certain problem concerning the p-Laplace operator, Likping Studies in Sciences and Technology, Dissertations, 326, 1993. [4] S. Oruganti, J. Shi, R. Shivaji, Diffusive logistic equation with constant yield harvesting, I: Steady-states, Trans. Amer. Math. Soc. 354 (9) (2002) 3601–3619. [5] M. Ramaswamy, R. Shivaji, Multiple positive solutions for classes of p-Laplacian equations, Differential Integral Equations 17 (11–12) (2004) 1255–1261. [6] J.D. Murray, Mathematical Biology, Scientific Publications, Beijing, 1998. [7] F. Li, Z. Liang, Existence of positive periodic solutions to nonlinear second order differential equations, Appl. Math. Lett. 18 (11) (2005) 1256–1264. [8] Y. Li, Positive periodic solutions of nonlinear second order differential equations, Acta Math. Sinica 45 (3) (2002) 481–488. [9] R. Man´asevich, J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. Differential Equations 145 (1998) 367–393. [10] P. Yan, Nonresonance for one-dimensional p-Laplacian with regular restoring, J. Math. Anal. Appl 285 (2003) 141–154. [11] X. Yang, Existence of periodic solutions for quasilinear differential equations, Appl. Math. Comput. 153 (2004) 231–237. [12] X. Yang, Multiple periodic solutions of a class of p-Laplacian, J. Math. Anal. Appl. 314 (2006) 17–29. [13] Y. Wang, W. Ge, Existence of periodic solutions for nonlinear differential equations with a p-Laplacian-like operator, Appl. Math. Lett. 19 (2006) 251–259. [14] J. Henderson, H.B. Thompson, Multiple symmetric positive solutions for a second order boundary value problem, Proc. Amer. Math. Soc. 128 (2000) 2373–2379. [15] R.I. Avery, Three symmetric positive solutions for a second-order boundary value problem, Appl. Math. Lett. 13 (3) (2000) 1–7. [16] R.A. Khan, Existence and approximation of solutions of second order nonlinear Neumann problems, Electron. J. Differential Equations 2005 (3) (2005) 1–10.