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
where q ∈ Rn and V ∈ C1(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
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
and in [12], Jin Li discussed the existence of periodic solutions for a prescribed mean curvature Rayleigh equation of the form
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
where f ∈ C1(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:
where p > 1 and φp : R → R is given by φp(s) = |s|p−2s for s ≠ 0 and φp(0) = 0, g ∈ C(R2,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:
and
where a, r ≥ 1; m1 and m2 are positive constants. Through the transformation, (1) is equivalent to the system
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
where p > 1 and φp : R → R is given by φp(s) = |s|p−2s for s ≠ 0 and φp(0) = 0, f ∈ C(R,R), g ∈ C(R2,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 2kT-periodic solutions for the following equation:
where k ∈ N, ek : R → R is a 2kT-periodic function such that
where ε0 ∈ (0, T) is a constant independent of k. In our approach,the existence of 2kT-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 C2kT = {u|u ∈ C(R,R), u(t + 2kT) = u(t)}, If the norms of are defined by respectively, then are all Banach spaces. Furthermore, for ϕ ∈ C2kT, , 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 provided that
(c) is a relative compact set of X,
(d) is a bounded set of Y,
where we denote X1 = KerL, Y2 = ImL, then we have the decompositions X = X1 ⊕ X2, Y = Y1 ⊕ Y2, let P : X → X1, Q : Y → Y1 are continuous linear projectors(meaning P2 = P and Q2 = Q), and
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 is said to be L-compact in . In addition, if the following conditions hold:
(h1) Lx ≠ λNx, ∀(x, λ) ∈ ∂Ω × (0, 1);
(h2) QNx ≠ 0, ∀x ∈ KerL ∩ ∂Ω;
(h3) deg{JQN,Ω∩KerL, 0} ≠ 0, where J : ImQ → KerL is a homeomorphism, then Lx = Nx has at least one solution in
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,+∞), and u ∈ C1(R,R) with u(t + ω) = u(t). Then
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
In order to study the existence of 2kT-periodic solutions for Eq.(1.2), for each k ∈ N, from (1.3) we observe that ek ∈ C2kT.
Lemma 2.4 ([18]). Suppose τ ∈ C1(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 ∈ C1(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 Clearly, σ0 ≥ 0 and 0 ≤ σ1 < 1.
Lemma 2.5 ([3]). Let be a 2kT-periodic function for each k ∈ N with
where A0, A1 and A2 are constants independent of k ∈ N. Then there exists a function u ∈ C1(R,Rn) such that for each interval [c, d] ⊂ R, there is a subsequence {ukj} of {uk}k∈N with uniformly on [c, d].
The system (4) is equivalent to the system
where
Let Xk = {ω = (u(t), v(t))⊤ ∈ C(R,R2), ω(t) = ω(t + 2kT)} and Yk = {ω = (u(t), v(t))⊤ ∈ C(R,R2), ω(t) = ω(t + 2kT)}, where the norm ||ω|| = max{|u|0, |v|0} with It is obvious that Xk and Yk are Banach spaces.
Now we define the operator
where D(L) = {ω|ω = (u(t), v(t))⊤ ∈ C1(R,R2), ω(t) = ω(t + 2kT)}.
Let Zk = {ω|ω = (u(t), v(t))⊤ ∈ C1(R,R × Bk), ω(t) = ω(t + 2kT)}, where Bk = {x ∈ R, |x| < 1, x(t) = x(t + 2kT)}. Define a nonlinear operator as follows:
where and Ω is an open and bounded set. Then problem (6) can be written as
we know
then ∀t ∈ R we have u′(t) = 0, v′(t) = 0, obviously u = a1 ∈ R, v = a2 ∈ R, thus KerL = R2, and it is also easy to prove that Therefore, L is a Fredholm operator of index zero.
Let
Let then it is easy to see that:
where
For all such that we have is a relative compact set of Xk, is a bounded set of Yk, so the operator N is L-compact in .
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.
[H1] There exists constants α and β > 0 such that
[H2] There exists constants m0 and m1 > 0 such that
[H3] e ∈ C(R,R) is a bounded function with e(t) ≠ 0 and
Remark 2.2. From (5), we can see that So if [H3] holds, then for each
3. Main results
In order to study the existence of 2kT-periodic solutions to system (6), we firstly study some properties of all possible 2kT-periodic solutions to the following system:
where (uk, vk)⊤ ∈ Zk ⊂ Xk. For each k ∈ N and all λ ∈ (0, 1], let Δ represent the set of all the 2kT-periodic solutions to the above system.
Theorem 3.1. Assume that conditions [H1]-[H3] hold,
and there exists a positive constant d0 such that
where
then for each k ∈ N, if (u, v)⊤ ∈ Δ, there are positive constants ρ1, ρ2, ρ3 and ρ4 which are independent of k and λ, such that
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
In view of [H1] and [H2] and by Hölder inequality, we get
Furthermore,
and by Lemma 2.4,
It follows from Remark 2.1 that
Substituting (9) into (8) and combining with Remark 2.2, we can obtain
which yields
Multiplying the second equation of (7) by u(t) and integrating from −kT to kT, we have
From the equality above, we have
Since and in view of [H1], [H2] and Lemma 2.2, we can get
By applying (9) to (11), we have
From the inequality above, we can see that
and
Substituting (10) into (13), we get
Since it is easy to see that there exists a constant d0 such that
Substituting (14) into (10), we obtain
It follows from Lemma 2.2 that
In view of (14) and (15), we have
then we get
Clearly, ρ1 is independent of k and λ. Furthermore, substituting (14) and (15) into (12), we can see that
Multiplying the second equation of (7) by v′(t) and integrating from −kT to kT, we have
From the first equation of (7), we can see that thus
Substituting (19) into (18) and in view of [H2], we get
It follows from (14) and (16) that
Applying the Lemma 2.2 again, we have
then combining (17) and (20) gives
It follows from that
Clearly, ρ2 is independent of k and λ.
Clearly, ρ3 is independent of k and λ. Let define then from the second equation of (7), we can obtain
and also ρ4 is independent of k and λ. Therefore, From (16), (22), (23) and (24), we know ρ1, ρ2, ρ3 and ρ4 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 2kT-periodic solution (uk(t), vk(t))⊤ in Δ ⊂ Xk such that
whereρ1, ρ2, ρ3 and ρ4 are constants defined by Theorem 3.1.
Proof. In order to use Lemma 2.1, for each k ∈ N, we consider the following system:
where Let Ω1 ⊂ Xk represent the set of all the 2kT-periodic solutions of system (25). Since (0, 1) ⊂ (0, 1], then Ω1 ⊂ Δ, where Δ is defined by Theorem 3.1. If (u, v)⊤ ∈ Ω1, by using Theorem 3.1, we have
Let Ω2 = {ω = (u, v)⊤ ∈ KerL,QNω = 0}, if (u, v)⊤ ∈ Ω2, then (u, v)⊤ = (a1, a2)⊤ ∈ R2(constant vector) and we can see that
i.e.,
Multiplying the second equation of (26) by a1 and combining with [H2], we have
Thus
Now, if we define it is easy to see that Ω1 ∪ Ω2 ⊂ Ω. So, condition (h1) and condition (h2) of Lemma 2.1 are satisfied. In order to verify the condition (h3) of Lemma 2.1, let
where J : ImQ → KerL is a linear isomorphism, J(u, v) = (v, u)⊤. From assumption [H1] and [H2], we have
Hence,
So, the condition (h3) of Lemma 2.1 is satisfied. Therefore, by using Lemma 2.1, we see that Eq.(6) has a 2kT-periodic solution (uk, vk)⊤ ∈ Ω. Obviously, (uk, vk)⊤ is a 2kT-periodic solution to Eq.(2) for the case of λ = 1, so (uk, vk)⊤ ∈ Δ. Thus, by using Theorem 3.1, we have
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 2kT-periodic solution (uk, vk)⊤ to Eq.(2) with
where ρ1, ρ2, ρ3, ρ4 are constants independent of k ∈ N. And uk(t) is a solution of (2), so
with implies that vk(t) is continuously differentiable for t ∈ R. Also, from (27), we have |vk|0 ≤ ρ2 < 1. It follows that is continuously differentiable for t ∈ R, i.e.,
By using (27) again and combining with φq(s) = |s|q−2s for s ≠ 0, then we have
Clearly, ρ5 is a constant independent of k ∈ N. From Lemma 2.5, we can see that there is a function u0 ∈ C1(R,Rn) such that for each interval [a, b] ⊂ R, there is a subsequence {ukj} of {uN}k∈N with uniformly on [a, b]. In the following, we show that u0(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 > j0, [−kjT, kjT − ε0] ⊂ [a − α, b + α]. So, for j > j0, from (3) and (26) we see that
Then from (29) we can have
Since uniformly for t ∈ [a, b] and is continuous differentiable for t ∈ [a, b], we can have
Considering that a, b are two arbitrary constants with a < b, it is easy to see that u0(t), t ∈ R is a solution to system (1).
Now, we prove u0(t) → 0 and u'(t) → 0 as |t| → ∞.
Since
Clearly, for every i ∈ N if kj > i, then by (14) and (15), we have
Let i → +∞, j → +∞, we have
and then
as r → +∞. So, by using Lemma 2.3 as |t| → +∞, we obtain
Finally, we will proof
From (27), we know
Then, we have
If (32) does not hold, then there exist and a sequence {tk} such that
and
Then, for we can have
It follows that
which contradicts (30), thus (32) holds. Clearly, u0(t) ≠ 0, otherwise e(t) ≡ 0, which contradicts assumption (H3). Hence the conclusion of Theorem 3.3 holds. □
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