Three positive periodic solutions of second order nonlinear neutral functional differential equations with delayed derivative

This paper deals with the existence of three positive periodic solutions for a class of second order neutral functional differential equations involving the delayed derivative term in nonlinearity (x(t) – cx(t – δ))′′ + a(t)g(x(t))x(t) = λb(t)f (t, x(t), x(t – τ1(t)), x′(t – τ2(t))). By utilizing the perturbation method of positive operator and Leggett–Williams fixed point theorem, a group of sufficient conditions are established.


Introduction
In the present work, we study the existence of three positive periodic solutions for the second order neutral functional differential equation of the form where λ > 0 is a positive parameter, c, δ are constants, and |c| < 1. a(t), b(t) are nonnegative ω-periodic continuous functions, τ i (t), i = 1, 2, are continuous ω-periodic functions, f : R × [0, +∞) 2 × R → [0, +∞) is a continuous function, and f (t, u, v, w) is ω-periodic with respect to t, g ∈ C([0, +∞), [0, +∞)). Neutral functional differential equations have a wide range of applications in the field of physics, biology, economics, and so on, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14] for more details. In [15], the authors pointed out that the growth of single or multiple species was mainly affected by seasonal changes (especially cyclical changes) and time lags. So it is important to study the periodic solutions of such models. The issues of the existence of positive periodic solutions of neutral functional differential equations have received more attention in recent years, see [7][8][9][10][11][12][13][14]. The existence of positive periodic solutions for first order neutral functional differential equations has been studied by many authors, see [7][8][9][10][11][12] and the references therein. But the research results on the case of second order are more seldom.
In [13], the authors studied the existence, multiplicity, and nonexistence of positive periodic solutions of second order neutral functional differential equations of the form where λ > 0 is a positive parameter, c, δ are constants, and |c| < 1, a(t), b(t) are nonnegative ω-periodic continuous functions. But the nonlinear term does not contain the derivative term.
Recently, Li [14] discussed the existence and nonexistence of positive ω-periodic solutions of second order neutral functional differential equations with delayed derivative in nonlinear term by using the positive operator perturbation method and the fixed point index theory are ωperiodic functions. But he did not consider the multiplicity of the positive periodic solutions.
Motivated by the above mentioned results, in this work, by using a different method, we mainly study the existence and multiplicity of positive periodic solutions for a class of second order neutral nonlinear functional differential equations with delayed derivative of the form (1).
Let C ω (R) be the Banach space of all continuous ω-periodic functions endowed with the norm x C = max t∈[0,ω] |x(t)|, C 1 ω (R) be the Banach space of all continuous differentiable ω-periodic functions with the norm x C 1 = x C + x C . In general, for n ∈ N, C n ω (R) represents the Banach space of all nth order continuous differentiable ω-periodic functions. Let C + ω (R) = C ω (R, [0, ∞)) be a nonnegative function cone in C ω (R). The main results of the present paper are summarized as follows: (i) We establish the existence (and uniqueness) of ω-periodic solutions for the corresponding linear second order neutral functional differential equation See Lemma 5. (ii) We provide the strong positive estimate and C 1 -estimate of the periodic solution operator by using the positive operator perturbation method, see Lemma 6. (iii) Let where σ and C 0 will be specified later. We define an operator Q λ which maps K into itself and prove that the operator Q λ has at least three positive fixed points by using Leggett-Williams fixed point theorem, see Theorem 1. In this paper, we always assume that and ω-periodic in t; g ∈ C([0, ∞), [0, ∞)); (H2) a, b ∈ C + ω (R), b := 1 ω ω 0 b(s) ds > 0, and τ i ∈ C ω (R), i = 1, 2; (H3) there exist two positive constants d and D satisfying 0 < d ≤ a(t)g(x(t)) ≤ D < ( π ω ) 2 for any t ∈ [0, ω], x ∈ C + ω (R).

Preliminaries
Firstly, let 0 < M < ( π ω ) 2 . We consider the second order linear ordinary differential equation By Lemma 2.1 of [14], the following lemma is obtained. where And the operator T λ :

linear completely continuous operator.
For the sake of brevity, let β = √ M and denote Then Proof Let h ∈ C + ω (R). By (3) and (4), we have For ∀τ ∈ R, noticing that Consequently, by (5), In order to prove the existence of ω-periodic solutions of equation (1), we consider the corresponding linear neutral functional differential equation Define a linear operator A : Then Lemma 3 ([11, 12, 14]) If |c| = 1, then the operator A, defined by (7), has a linear bounded inverse operator A -1 on C ω (R) given by Lemma 4 If |c| < σ , then for any y ∈ K , we have Proof For any y ∈ K , by virtue of Lemma 3, we have Then the proof of Lemma 4 is complete.
Proof Define an operator B λ : where G(y(t)) = ca(t)g((A -1 y)(t))(A -1 y)(tδ). Then equation (8) can be rewritten as By Lemma 1, we have It follows from (9) that Consequently, Combining this fact with T λ ≤ λ M , we have Then T λ B λ < 1 because of |c| < d D+d . Hence the operator I -T λ B λ has a bounded inverse operator (I -T λ B λ ) -1 which can be expressed by Therefore, operator equation (11) has a unique ω-periodic solution y ∈ C 2 ω (R) expressed by Let z = T λ h for any h ∈ C + ω (R). By Lemma 2, we get z ∈ K . Then we have Hence B λ z(t) ≥ 0 for any t ∈ R, that is, B λ z ∈ C + ω (R). Then applying Lemma 2 again, (T λ B λ )z = T λ B λ z ∈ K . Consequently, (T λ B λ ) n z ∈ K for ∀n ∈ N. By boundedness of the linear operator T λ : This completes the proof of Lemma 5.
At the end of this section, we introduce a fixed point theorem, which will be used in the proof of our main result.
Let (X, · ) be a real Banach space and K be a cone in X. A map ρ is called a nonnegative continuous concave function on K if ρ : K → [0, +∞) is continuous and for all x, y ∈ K and t ∈ [0, 1].
For each λ ∈ (λ 1 , λ 2 ) and y ∈ K , denote F by then F : K → C + ω (R) is continuous. We define a mapping Q λ by By Lemma 6, Q λ : K → K is completely continuous. Then ρ is a nonnegative continuous concave function on K and ρ(y) ≤ y C , ∀y ∈ K R .
For any y ∈ K R and λ ∈ (λ 1 , λ 2 ), by (4), (14), (15), Lemmas 4 and 5, we have Hence Q λ y C ≤ R and Q λ is completely continuous on K R . We now verify that condition (b) of Lemma 7 holds. Indeed, if y ∈ K r , we have Hence, Q λ y C < r.
In the end, we prove that condition (c) of Lemma 7 holds. Let y ∈ K(ρ, r 1 , R) and Q λ y C > r 2 . We prove ρ(Q λ y) > r 1 . It follows from (15) that Therefore, Now, all the conditions of Lemma 7 are satisfied. By Lemma 7, Q λ has at least three positive fixed points y 1 , y 2 , and y 3 satisfying y 1 ∈ K r , y 2 ∈ y ∈ K(ρ, r 1 , R) : ρ(y) > r 1 , Then equation (1) has at least three positive ω-periodic solutions: This completes the proof.
Example 1 We consider the positive 2π -periodic solutions for the second order neutral differential equation where λ > 0 is a constant. Corresponding to equation (1) Therefore, it is easy to verify that conditions (H2)-(H5) are satisfied.