EXISTENCE AND MULTIPLICITY OF POSITIVE PERIODIC SOLUTIONS FOR A CLASS OF SECOND ORDER DAMPED FUNCTIONAL DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS

By using the Krasnoselskii fixed point theorem, sufficient conditions are obtained for the existence and multiplicity of positive periodic solutions for a class of second order damped functional differential equations with multiple delays. Our results are a further expansion of the previous research results.


Introduction
For the following equation where f ∈ C(R/TZ, (0, +∞)), there are many results [3,18] on the periodic solution of this equation. However, the systems controlled by feedback loops in engineering, predator-prey models in ecosystems [8,12], and value laws in economics in real life all have the influence of delay factors, so the research on functional differential equations has already stepped into a climax period [1,15,17]. At the same time, many research methods have been considered, such as the upper and lower solutions method and monotone iterative technique [10,16], fixed point theorems [11,13,21] and so on [5,9,14,19,20].
Jiang et al. [10] studied the following periodic problem where f ∈ C(R 3 , R), τ ∈ C(R, [0, +∞)), and they are T -periodic functions. They established the existence results of T -periodic solutions by using monotone iterative technique.
However, for many problems in real life, we only need to consider the properties of its positive periodic solution. In [21], Wu obtained the existence and multiplicity of the solutions to the following equation where a ∈ C(R/TZ, (0, +∞)), f ∈ C (R/TZ) × [0, +∞) n+1 , [0, +∞) , τ i (t) ∈ C(R/TZ, R), and a(t) satisfies the condition that 0 < a(t) < π 2 T for every t ∈ R.
Li et al. studied the following equation in [13] x ′′ + a(t) where a ∈ C(R/TZ, (0, +∞)), , they obtained the existence of positive periodic solution by using the first eigenvalue corresponding to the relevant linear operator and fixedpoint index theory in cones.
In [11], Kang et al. considered the following equation with damped term . They obtained the existence and multiplicity of positive periodic solutions when the coefficients h(t), a(t) and g(t) satisfy Motivated by the above papers, in this paper, we study the existence, multiplicity of positive periodic solutions for the following equation Three highlights should be pointed out. Firstly, compared with the equation studied in [13,21], we add the damping term h(t)x ′ . Secondly, different from [11], the equation we studied has multiple delays. Thirdly, we relax the restrictions for the coefficients h(t) and a(t) in [11].

Preliminaries
If the unique solution of linear equation associated to periodic boundary conditions is trivial, then it is nonresonant. By Fredholm's alternative theorem, we know that when (2.1)-(2.2) is nonresonant, has a unique solution and it can be expressed as where G(t, ξ) is the Green's function of (2.1)-(2.2). Next we assume that: In general, condition (A0) is difficult to establish. However, through the antimaximum principle established by Hakl and Torres (see [7]), Chu, Fan and Torres obtained that (A0) is true in [2]. Describe the above criterion by defining the following function Lemma 2.1 (Corollary 2.6, [7]). If a(t) ̸ ≡ 0 and the following two inequalities and are satisfied, where [a(ξ)] + = max{a(ξ), 0}. Then (A0) holds.
Define operator: Therefore, the fixed point of the operator equation x = Q λ x is the T -periodic solution of (1.1).

Lemma 2.2. Q λ : P → P is completely continuous and
Then, according to the Arscoli-Arzele theorem, Q λ is completely continuous. The proof is completed. 4,6]). Let X be a Banach space and P be a close convex cone in X.
is a completely continuous operator. Assume that Q satisfies one of the following conditions: Then Q has at least one fixed point in P ∩ (Ω 2 \ Ω 1 ).

Proof. Still define
( Thus, we have On the other hand, there exists Then, for ϕ ∈ P ∩ ∂Ω r2 , we can obtain Thus, by Lemma 2.4(ii), the operator Q λ has at least one fixed point in P ∩(Ω r2 \Ω 1 ), that is, (1.1) has at least a positive T -periodic solution for Thus, for all ϕ ∈ P ∩ ∂Ω r1 , we have 0 ≤ Φ(t) ≤ r 1 , that is Thus, we have Then, for ϕ ∈ P ∩ ∂Ω r2 , we can obtain Thus, by Lemma 2.4(i), the operator Q λ has at least one fixed point in P ∩ (Ω r2 \ Ω 1 ), which is the positive T -periodic solution of (1.1) for The proof is completed.