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HOMOCLINIC SOLUTIONS FOR A PRESCRIBED MEAN CURVATURE RAYLEIGH p-LAPLACIAN EQUATION WITH A DEVIATING ARGUMENT†
HOMOCLINIC SOLUTIONS FOR A PRESCRIBED MEAN CURVATURE RAYLEIGH p-LAPLACIAN EQUATION WITH A DEVIATING ARGUMENT†
Journal of Applied Mathematics & Informatics. 2015. Sep, 33(5_6): 723-738
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : January 10, 2015
  • Accepted : January 30, 2015
  • Published : September 30, 2015
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FANCHAO KONG

Abstract
In this paper, the prescribed mean curvature Rayleigh p -Laplacian equation with a deviating argument is studied. By using Mawhin’s continuation theorem and some analysis methods, we obtain the existence of a set with 2 kT -periodic solutions for this equation and then a homoclinic solution is obtained as a limit of a certain subsequence of the above set. AMS Mathematics Subject Classification : 34C37.
Keywords
1. Introduction
In resent years, The existence of homoclinic solutions have been studied widely, especially for the Hamiltonian systems and the p -Laplacian systems(see [1 - 4] ). For example, in [1] , Lzydorek, M and Janczewska, J studied the homoclinic solutions for a class of the second order Hamiltonian systems as the following form
PPT Slide
Lager Image
where q Rn and V C 1 ( R × Rn,R ), V ( t, q ) = − K ( t, q ) + W ( t, q ) is T -periodic in t . And in [4] , Lu, SP studied the homoclinic solutions for a class of second-order p -Laplacian differential systems with delay of the form
PPT Slide
Lager Image
Nowadays, the prescribed mean curvature equation and its modified forms, which arises from some problems associated with differential geometry and physics such as combustible gas dynamics [5 - 7] have been studied widely. As researchers continue to study the prescribed mean curvature equation, the existence of the periodic solutions for the prescribed curvature mean equation attracts researchers’ attention and there are many papers about the existence of the periodic solutions for the prescribed curvature mean equation. For example, in [11] , Feng discussed the existence of periodic solutions of a delay prescribed mean curvature Li´enard equation of the form
PPT Slide
Lager Image
and in [12] , Jin Li discussed the existence of periodic solutions for a prescribed mean curvature Rayleigh equation of the form
PPT Slide
Lager Image
As is well known, a solution u ( t ) of Eq.(1.1) is named homoclinic (to 0) if u ( t ) → 0 and u' ( t ) → 0 as | t | → +∞. In addition, if u ≠ 0, then u is called a nontrivial homoclinic solution.
In [13] , Liang and Lu studied the homoclinic solution for the prescribed mean curvature Duffing-type equation of the form
PPT Slide
Lager Image
where f C 1 ( R,R ), p C ( R,R ), c > 0 is a given constant.
Recently, in [14] , Wang studied the periodic solution for the following prescribed mean curvature Rayleigh equation with a deviating argument of the form:
PPT Slide
Lager Image
where p > 1 and φp : R R is given by φp ( s ) = | s | p −2 s for s ≠ 0 and φp (0) = 0, g C ( R 2 , R ), e, τ C ( R,R ), g ( t + ω, x ) = g ( t, x ), f ( t + ω, x ) = f ( t, x ), f ( t , 0) = 0, e ( t + ω ) = e ( t ) and τ ( t + ω ) = τ ( t ). Under the assumptions:
PPT Slide
Lager Image
and
PPT Slide
Lager Image
where a, r ≥ 1; m 1 and m 2 are positive constants. Through the transformation, (1) is equivalent to the system
PPT Slide
Lager Image
By using Mawhin’s continuation theorem and given some sufficient conditions, the authors obtained that Eq.(1) has at least one periodic solution.
However, to the best of our knowledge, there are no papers about the studying of the homoclinic solutions for the prescribed mean curvature Rayleigh p -Laplacian equation. In order to solve this problem, in this paper, we consider the following the prescribed mean curvature Rayleigh p -Laplacian equation with a deviating argument
PPT Slide
Lager Image
where p > 1 and φp : R R is given by φp ( s ) = | s | p −2 s for s ≠ 0 and φp (0) = 0, f C ( R,R ), g C ( R 2 , R ), g is T -periodic in the first argument. e ( t ), τ ( t ) are continuous T -periodic function and T > 0 is a given constant.
In order to study the homoclinic solution for Eq.(3), firstly, like in the work of Lzydorek and Janczewska in [1] , Rabinowitz in [2] , X. H. Tang and Li Xiao in [3] and Lu in [4] , the existence of a homoclinic solution for Eq.(3) is obtained as a limit of a certain sequence of 2 kT -periodic solutions for the following equation:
PPT Slide
Lager Image
where k N, ek : R R is a 2 kT -periodic function such that
PPT Slide
Lager Image
where ε 0 ∈ (0, T ) is a constant independent of k . In our approach,the existence of 2 kT -periodic solutions to Eq.(4) is obtained by applying Mawhin’s continuation theorem [16] .
The structure of the rest of this paper is as follows: Section 2, we state some necessary definitions and lemmas. Section 3, we prove the main result.
2. Preliminary
Throughout this paper, | · | will denote the absolute value and the Euclidean norm on R . For each k N , let C 2 kT = { u | u C ( R,R ), u ( t + 2 kT ) = u ( t )},
PPT Slide
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If the norms of
PPT Slide
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are defined by
PPT Slide
Lager Image
respectively, then
PPT Slide
Lager Image
are all Banach spaces. Furthermore, for ϕ C 2kT ,
PPT Slide
Lager Image
, where r ∈ (1,+∞).
In order to use Mawhin’s continuation theorem, we first recall it.
Let X and Y be two Banach spaces, a linear operator L : D ( L ) ⊂ X Y is said to be a Fredholm operator of index zero provided that
(a) ImL is a closed subset of Y,
(b) dimKerL = codimImL < ∞.
Let X and Y be two Banach spaces, Ω ⊂ X be an open and bounded set, and L : D ( L ) ⊂ X Y is a Fredholm operator of index zero, and continuous operator N : Ω ⊂ X Y is said to be L-compact in
PPT Slide
Lager Image
provided that
(c)
PPT Slide
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is a relative compact set of X,
(d)
PPT Slide
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is a bounded set of Y,
where we denote X 1 = KerL , Y 2 = ImL , then we have the decompositions X = X 1 X 2 , Y = Y 1 Y 2 , let P : X X 1 , Q : Y Y 1 are continuous linear projectors(meaning P 2 = P and Q 2 = Q ), and
PPT Slide
Lager Image
Lemma 2.1 (16). Let X and Y be two real Banach spaces, and Ω is an open and bounded set of X, and L : D ( L ) ⊂ X Y is a Fredholm operator of index zero and the operator
PPT Slide
Lager Image
is said to be L-compact in
PPT Slide
Lager Image
. In addition, if the following conditions hold:
( h 1 ) Lx λNx , ∀( x, λ ) ∈ Ω × (0, 1);
( h 2 ) QNx ≠ 0, ∀ x KerL Ω;
( h 3 ) deg { JQN ,Ω∩ KerL , 0} ≠ 0, where J : ImQ KerL is a homeomorphism, then Lx = Nx has at least one solution in
PPT Slide
Lager Image
Lemma 2.2 ( [4] ). Let s C ( R,R ) with s ( t + ω ) ≡ s ( t ) and s ( t ) ∈ [0, ω ], ∀ t R . Suppose p ∈ (1,+∞),
PPT Slide
Lager Image
and u C 1 ( R,R ) with u ( t + ω ) = u ( t ). Then
PPT Slide
Lager Image
Lemma 2.3. If u : R R is continuously differentiable on R, a > 0, μ > 1 and p > 1 are constants, then for every t R, the following inequality holds
PPT Slide
Lager Image
In order to study the existence of 2 kT-periodic solutions for Eq.(1.2), for each k N, from (1.3) we observe that ek C 2kT .
PPT Slide
Lager Image
Lemma 2.4 ( [18] ). Suppose τ C 1 ( R,R ) with τ ( t + ω ) ≡ τ ( t ) and τ' ( t ) < 1, ∀ t ∈ [0, ω ]. Then the function t τ ( t ) has an inverse μ ( t ) satisfying μ C ( R,R ) with μ ( t + ω ) ≡ μ ( t ) + ω , ∀ t ∈ [0, ω ].
Throughout this paper, besides τ being a periodic function with period T , we suppose in addition that τ C 1 ( R,R ) with τ' ( t ) < 1, ∀ t ∈ [0, T ].
Remark 2.1. From the above assumption, one can find from Lemma 2.4 that the function ( t τ ( t )) has an inverse denoted by μ ( t ). Define
PPT Slide
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Clearly, σ 0 ≥ 0 and 0 ≤ σ 1 < 1.
Lemma 2.5 ( [3] ). Let
PPT Slide
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be a 2kT-periodic function for each k N with
PPT Slide
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where A 0 , A 1 and A 2 are constants independent of k N. Then there exists a function u C 1 ( R,Rn ) such that for each interval [ c, d ] ⊂ R, there is a subsequence { ukj } of { uk } k N with
PPT Slide
Lager Image
uniformly on [ c, d ].
The system (4) is equivalent to the system
PPT Slide
Lager Image
where
PPT Slide
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Let Xk = { ω = ( u ( t ), v ( t )) C ( R,R 2 ), ω ( t ) = ω ( t + 2 kT )} and Yk = { ω = ( u ( t ), v ( t )) C ( R,R 2 ), ω ( t ) = ω ( t + 2 kT )}, where the norm || ω || = max{| u | 0 , | v | 0 } with
PPT Slide
Lager Image
It is obvious that Xk and Yk are Banach spaces.
Now we define the operator
PPT Slide
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where D ( L ) = { ω | ω = ( u ( t ), v ( t )) C 1 ( R,R 2 ), ω ( t ) = ω ( t + 2 kT )}.
Let Zk = { ω | ω = ( u ( t ), v ( t )) C 1 ( R,R × Bk ), ω ( t ) = ω ( t + 2 kT )}, where Bk = { x R , | x | < 1, x ( t ) = x ( t + 2 kT )}. Define a nonlinear operator
PPT Slide
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as follows:
PPT Slide
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where
PPT Slide
Lager Image
and Ω is an open and bounded set. Then problem (6) can be written as
PPT Slide
Lager Image
we know
PPT Slide
Lager Image
then ∀ t R we have u′ ( t ) = 0, v′ ( t ) = 0, obviously u = a 1 R, v = a 2 R , thus KerL = R 2 , and it is also easy to prove that
PPT Slide
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Therefore, L is a Fredholm operator of index zero.
Let
PPT Slide
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Let
PPT Slide
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then it is easy to see that:
PPT Slide
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where
PPT Slide
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For all
PPT Slide
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such that
PPT Slide
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we have
PPT Slide
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is a relative compact set of Xk ,
PPT Slide
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is a bounded set of Yk , so the operator N is L -compact in
PPT Slide
Lager Image
.
For the sake of convenience, we list the following assumption which will be used by us in studying the existence of homoclince solutions to the Eq.(3) in Section 3.
[ H 1 ] There exists constants α and β > 0 such that
PPT Slide
Lager Image
[ H 2 ] There exists constants m 0 and m 1 > 0 such that
PPT Slide
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[ H 3 ] e C ( R,R ) is a bounded function with e ( t ) ≠ 0 and
PPT Slide
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Remark 2.2. From (5), we can see that
PPT Slide
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So if [ H 3 ] holds, then for each
PPT Slide
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3. Main results
In order to study the existence of 2 kT -periodic solutions to system (6), we firstly study some properties of all possible 2 kT -periodic solutions to the following system:
PPT Slide
Lager Image
where ( uk , vk ) Zk Xk . For each k N and all λ ∈ (0, 1], let Δ represent the set of all the 2 kT -periodic solutions to the above system.
Theorem 3.1. Assume that conditions [H 1 ]-[H 3 ] hold,
PPT Slide
Lager Image
and there exists a positive constant d 0 such that
PPT Slide
Lager Image
where
PPT Slide
Lager Image
then for each k N, if ( u, v ) ∈ Δ, there are positive constants ρ 1 , ρ 2 , ρ 3 and ρ 4 which are independent of k and λ, such that
PPT Slide
Lager Image
Proof . For each k N , if ( u, v ) ∈ Δ, it must satisfy the system (7). Multiplying the second equation of (7) by u′ ( t ) and integrating from − kT to kT , we have
PPT Slide
Lager Image
In view of [ H 1 ] and [ H 2 ] and by Hölder inequality, we get
PPT Slide
Lager Image
Furthermore,
PPT Slide
Lager Image
and by Lemma 2.4,
PPT Slide
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It follows from Remark 2.1 that
PPT Slide
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Substituting (9) into (8) and combining with Remark 2.2, we can obtain
PPT Slide
Lager Image
which yields
PPT Slide
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Multiplying the second equation of (7) by u(t) and integrating from −kT to kT, we have
PPT Slide
Lager Image
From the equality above, we have
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
and in view of [ H 1 ], [ H 2 ] and Lemma 2.2, we can get
PPT Slide
Lager Image
By applying (9) to (11), we have
PPT Slide
Lager Image
From the inequality above, we can see that
PPT Slide
Lager Image
and
PPT Slide
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Substituting (10) into (13), we get
PPT Slide
Lager Image
Since
PPT Slide
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it is easy to see that there exists a constant d 0 such that
PPT Slide
Lager Image
Substituting (14) into (10), we obtain
PPT Slide
Lager Image
It follows from Lemma 2.2 that
PPT Slide
Lager Image
In view of (14) and (15), we have
PPT Slide
Lager Image
then we get
PPT Slide
Lager Image
Clearly, ρ 1 is independent of k and λ . Furthermore, substituting (14) and (15) into (12), we can see that
PPT Slide
Lager Image
Multiplying the second equation of (7) by v′ ( t ) and integrating from − kT to kT , we have
PPT Slide
Lager Image
From the first equation of (7), we can see that
PPT Slide
Lager Image
thus
PPT Slide
Lager Image
Substituting (19) into (18) and in view of [ H 2 ], we get
PPT Slide
Lager Image
It follows from (14) and (16) that
PPT Slide
Lager Image
Applying the Lemma 2.2 again, we have
PPT Slide
Lager Image
then combining (17) and (20) gives
PPT Slide
Lager Image
It follows from
PPT Slide
Lager Image
that
PPT Slide
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Clearly, ρ 2 is independent of k and λ .
PPT Slide
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Clearly, ρ 3 is independent of k and λ . Let define
PPT Slide
Lager Image
then from the second equation of (7), we can obtain
PPT Slide
Lager Image
and also ρ 4 is independent of k and λ . Therefore, From (16), (22), (23) and (24), we know ρ 1 , ρ 2 , ρ 3 and ρ 4 are constants independent of k and λ . Hence the conclusion of Theorem 3.1 holds. □
Theorem 3.2. Assume that the conditions of Theorem 3.1 are satisfied . Then, for each k N, system (7) has at least one 2 kT-periodic solution ( uk ( t ), vk ( t )) in Δ ⊂ Xk such that
PPT Slide
Lager Image
where ρ 1 , ρ 2 , ρ 3 and ρ 4 are constants defined by Theorem 3.1.
Proof . In order to use Lemma 2.1, for each k N , we consider the following system:
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Let Ω 1 Xk represent the set of all the 2 kT -periodic solutions of system (25). Since (0, 1) ⊂ (0, 1], then Ω 1 ⊂ Δ, where Δ is defined by Theorem 3.1. If ( u, v ) ∈ Ω 1 , by using Theorem 3.1, we have
PPT Slide
Lager Image
Let Ω 2 = { ω = ( u, v ) KerL,QNω = 0}, if ( u, v ) ∈ Ω 2 , then ( u, v ) = ( a 1 , a 2 ) R 2 (constant vector) and we can see that
PPT Slide
Lager Image
i.e.,
PPT Slide
Lager Image
Multiplying the second equation of (26) by a 1 and combining with [ H 2 ], we have
PPT Slide
Lager Image
Thus
PPT Slide
Lager Image
Now, if we define
PPT Slide
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it is easy to see that Ω 1 ∪ Ω 2 ⊂ Ω. So, condition ( h 1 ) and condition ( h 2 ) of Lemma 2.1 are satisfied. In order to verify the condition ( h 3 ) of Lemma 2.1, let
PPT Slide
Lager Image
where J : ImQ KerL is a linear isomorphism, J ( u, v ) = ( v, u ) . From assumption [ H 1 ] and [ H 2 ], we have
PPT Slide
Lager Image
Hence,
PPT Slide
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So, the condition ( h 3 ) of Lemma 2.1 is satisfied. Therefore, by using Lemma 2.1, we see that Eq.(6) has a 2 kT -periodic solution ( uk, vk ) ∈ Ω. Obviously, ( uk, vk ) is a 2 kT -periodic solution to Eq.(2) for the case of λ = 1, so ( uk, vk ) ∈ Δ. Thus, by using Theorem 3.1, we have
PPT Slide
Lager Image
Hence the conclusion of Theorem 3.2 holds. □
Theorem 3.3. Suppose that the conditions in Theorem 3.1 hold, then Eq.(1) has a nontrivial homoclinic solution.
Proof . From Theorem 3.2, we see that for each k N , there exists a 2 kT -periodic solution ( uk, vk ) to Eq.(2) with
PPT Slide
Lager Image
where ρ 1 , ρ 2 , ρ 3 , ρ 4 are constants independent of k N . And uk ( t ) is a solution of (2), so
PPT Slide
Lager Image
with
PPT Slide
Lager Image
implies that vk ( t ) is continuously differentiable for t R . Also, from (27), we have | vk | 0 ρ 2 < 1. It follows that
PPT Slide
Lager Image
is continuously differentiable for t R , i.e.,
PPT Slide
Lager Image
By using (27) again and combining with φq ( s ) = | s | q −2 s for s ≠ 0, then we have
PPT Slide
Lager Image
Clearly, ρ 5 is a constant independent of k N . From Lemma 2.5, we can see that there is a function u 0 C 1 ( R,Rn ) such that for each interval [ a, b ] ⊂ R , there is a subsequence { ukj } of { uN } k N with
PPT Slide
Lager Image
uniformly on [ a, b ]. In the following, we show that u 0 ( t ) is just a homoclinic solution to Eq.(4).
For all a, b R with a < b , there must be a positive integer j such that for j > j 0 , [− kjT, kjT ε 0 ] ⊂ [ a α, b + α ]. So, for j > j 0 , from (3) and (26) we see that
PPT Slide
Lager Image
Then from (29) we can have
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
uniformly for t ∈ [ a, b ] and
PPT Slide
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is continuous differentiable for t ∈ [ a, b ], we can have
PPT Slide
Lager Image
Considering that a, b are two arbitrary constants with a < b , it is easy to see that u 0 ( t ), t R is a solution to system (1).
Now, we prove u 0 ( t ) → 0 and u' ( t ) → 0 as | t | → ∞.
Since
PPT Slide
Lager Image
Clearly, for every i N if kj > i , then by (14) and (15), we have
PPT Slide
Lager Image
Let i → +∞, j → +∞, we have
PPT Slide
Lager Image
and then
PPT Slide
Lager Image
as r → +∞. So, by using Lemma 2.3 as | t | → +∞, we obtain
PPT Slide
Lager Image
Finally, we will proof
PPT Slide
Lager Image
From (27), we know
PPT Slide
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Then, we have
PPT Slide
Lager Image
If (32) does not hold, then there exist
PPT Slide
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and a sequence { tk } such that
PPT Slide
Lager Image
and
PPT Slide
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Then, for
PPT Slide
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we can have
PPT Slide
Lager Image
It follows that
PPT Slide
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which contradicts (30), thus (32) holds. Clearly, u 0 ( t ) ≠ 0, otherwise e ( t ) ≡ 0, which contradicts assumption ( H 3 ). Hence the conclusion of Theorem 3.3 holds. □
BIO
Fanchao Kong received M.Sc. from Anhui Normal University. His research interest focuses on Boundary Value Problems.
Department of Mathematics, Anhui Normal University, Wuhu 241000, Anhui, P.R. China.
e-mail: fanchaokong88@sohu.com
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