Existence of three periodic solutions for a quasilinear periodic boundary value problem

: In this paper, we prove the existence of at least three periodic solutions for the quasilinear periodic boundary value problem


Introduction and main results
In this paper, we consider the quasilinear periodic boundary value problem where f (t, x) : R × R → R is an L 1 -Carathéodory function which is 1-periodic in t and λ is a positive parameter.
We need the following assumptions: (Q 1 ) p : R → (0, +∞) is a continuous function such that there exist two positive numbers M ≥ m and m ≤ p(x) ≤ M, ∀ x ∈ R.
It is well known, the second order Hamiltonian systems satisfying periodic boundary conditions is motivated by celestial mechanics(see [15]).Finding periodic solutions for the system is a classic problem.The authors of [5] and [7] have proved the existence of three periodic solutions for this system.We also want to point out that, in [5] and [7], the nonlinear terms in the differential equations of the problems studied there do not depend on the derivatives of the unknown functions, i.e., p(x ) ≡ 1, in (1.1).So we are interested in problem (1.1).
On the other hand, in [3], using the three critical points theorem of [11], Afrouzi and Heidarkhani established a three solutions result for the following quasilinear two point boundary value problem where f : [a, b] × R → R is an L 1 -Carathéodory function, h : R → (0, +∞) is a continuous function and λ > 0, a, b ∈ R, and extend the main result of [8] to problem (1.6).Inspired by the ideas of [3,8], we discuss the existence of three periodic solutions for problem (1.1).The aim of this paper is to establish some new criteria for problem (1.1) to have at least three periodic solutions by applying the three critical points theorem due to B. Ricceri.In addition, we give an example to illustrate the validity of our result.
. Assume that there exist three positive constants c, d and s with s < 2, g(2d)+g(−2d) (1.8) Then, there exist an open set Λ ∈ [0, +∞) and a positive number r 0 such that for every λ ∈ Λ, problem has at least three periodic solutions whose norms in Z are less than r 0 .
Remark 1.If we take p(x ) ≡ 1 in Theorem 1.2, then we can get the corresponding results of problem (1.4) as T = 1, n = 1.Inspecting the conditions of Theorem 3.1 in [5], it is not difficult to find that its hypothesis is different from that of our Theorem 1.2, so this is a new result.Furthermore, when α(t) and f (t, x) don't depend on t, we have the following autonomous version of Theorem 1.1.
We postpone the proofs to the next section and turn to give an example to illustrate the validity of Theorem 1.1. and where and This shows that (1.7) of Theorem 1.1 holds.Further, if γ(t) ≡ e 16 and s = 2  3 , then all the assumptions of Theorem 1.1 are satisfied.Hence, there exist an open interval Λ ∈ [0, +∞) and a positive number r 0 such that for every λ ∈ Λ, problem (1.1) has at least three periodic solutions whose norms in Z are less than r 0 .
Remark 2 If we take p(x ) ≡ 1 in Theorem 1.1, then we can get the corresponding results of problem (1.5) as T = 1, p = 2, n = 1 and µ = 0. Let F(t, x) and α(t) be the functions in Example 1 respectively, and then after a simple calculation, it is not difficult to verify that satisfy the conditions of Theorem 1 of [7].Moreover, Example 1 does not satisfy the assumptions of Theorem 1 of [7], so Theorem 1.1 represents a development of Theorem 1 of [7] in some sense.

Variational setting and proof of Theorems
For the reader's convenience, we first recall here the three critical points theorem of [12] and Proposition 3.1 of [13].Assume that there are r > 0 and x 0 , x 1 ∈ Z such that Φ(x 1 ) = Ψ(x 1 ) = 0, Φ(x 0 ) > r, and sup Then, for each β ∈ R satisfying ) is continuous for almost every t ∈ [0, T ]; (C 3 ) for every ρ > 0 there exists a function Next, we establish the variational setting for problem (1.1).Throughout the sequel, the Sobolev space Z is defined by .
Clearly, Z is a Hilbert space and Z * = Z, where Z * is the dual space of Z. Setting we have g (y) = y 0 p(ξ)dξ, and g (y) = p(y), for every y ∈ R.
Proof.From (1.2) and (1.3), we have for every x ∈ Z, which implies that Φ is well-defined in Z.
Taken that x, y ∈ Z and {a n } ∈ R\{0} with lim n→+∞ a n = 0.By the mean value theorem of differential calculus, for a.e.t ∈ [0, 1], we can see that there exist u n (t) and v n (t) such that Then, by (1.2) and (1.3), if n is large enough, we have for almost every t ∈ [0, 1].Again using the Lebesgue's theorem, from the continuity of g and the arbitrariness of {a n }, we know that Φ is Gâteaux differentiable in Z with If we fix x, y, z ∈ Z with z ≤ 1.Let u(t) be such that for every t ∈ [0, 1].Thus, by the Hölder inequality, (1.2), (1.3) and (2.4), we have for every x, y ∈ Z and Φ is Lipschitzian.Finally, from (1.2) and (2.4), we know that g is convex.Noticing that α(t)x 2 is convex in x, we have Φ is convex in Z.
For every x ∈ Z, put Carathéodory function, we know that Ψ is a well-defined and Gâteaux differentiable functional with for every x, y ∈ Z. Since the embeddings Z → L q (q ≥ 1) and Z → L ∞ are compact (See R. A. Adams [16]), we have Ψ : Z → Z * is a continuous and compact operator.
Next, we consider the functional I : Z → R defined by for every x ∈ Z, where λ > 0. Clearly, I is Gâteaux differentiable.If x ∈ Z is a critical point for I, we have for each y ∈ Z.This implies that g • x has a weak derivative which equals α(t)x(t) − λ f (t, x(t)) and is thus continuous, so g • x is C 1 ([0, 1]).Since g is an invertible C 1 -function, it follows that x is also in for all t ∈ [0, 1].Hence we conclude that x is a solution of problem (1.1) belongs to C 2 ([0, 1]).
Proof of Theorem 1.1 Consider the functional I(x) = Φ(x) − λΨ(x) for every x ∈ Z and λ > 0. From Proposition 2.4, we know that Φ is well-defined, Gâteaux differentiable and convex functional in Z, and Φ is a Lipschitzian operator, which implies that Φ is a sequentially weakly lower semicontinuous via Theorem 1.2 of [15].Further, we claim that Φ admits a continuous inverse on Z * .In fact, by (1.2), (1.3), Proposition 2.3 and (2.6), we have for all x, y ∈ Z, which shows that Φ is uniformly monotone in Z.Put y = 0, then we have min{m, which shows that Φ is coercive in Z. Since Φ is a Lipschitzian operator, Φ is hemicontinuous in Z.By Theorem 26.A of [14] we can see that Φ admits a continuous inverse on Z * .From the estimation formula in the proof of Proposition 2.4 Φ(x) ≤ 1 2 max{M, α 1 } x 2 , we see that Φ is bounded on each bounded subset of Z.
On the other hand, as we saw in above, Ψ : Z → R a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact.Based on the previous discussion of I, we know that the critical points of I = Φ − λΨ in Z are the solutions of problem (1.1).Therefore, we only need to verify that both assumptions of Lemma 2.1 are valid.
From (1.2) and (1.3), it follows that for all x ∈ Z.By assumption (ii), we see that lim and so (2.1) of Lemma 2.1 holds.Next, We want to prove the validity of (2.2) in Lemma 2.1 by using Proposition 2.2.For x ∈ Z, taking into account Thus, for each r > 0, we can obtain it is easy to see that the assumptions of Theorem 1.1 hold.So, the proof is complete.

Conclusions
Periodic solutions of Hamiltonian systems are important in applications.For second order Hamiltonian systems or p-Hamiltonian systems subject to periodic boundary conditions, there are many works reported on the existence of three periodic solutions.But the results on the multiplicity of periodic solutions of quasilinear periodic boundary value problem are very rare.In this paper, we study a quasilinear second order differential equation involving periodic boundary condition.Using a three critical points theorem obtained by B. Ricceri, we establish some new existence theorem of at least three periodic solutions for the quasilinear periodic boundary value problem (1.1) under appropriate hypotheses.In addition, we give an example to illustrate the validity of our results.