Uniqueness and existence of positive periodic solutions of functional di ﬀ erential equations

: In this paper, some new ﬁndings on the uniqueness and existence of positive periodic solutions to ﬁrst-order functional di ﬀ erential equations are presented. These equations have wide applications in a variety of ﬁelds. The most important feature of our argument is that we use the theory of Hilbert’s metric to prove the uniqueness of the positive periodic solution when q = − 1 and − 1 < q < 0. In addition, we also investigate the existence results of positive periodic solutions by applying a ﬁxed point theorem for completely continuous maps in a cone. Two examples demonstrate our ﬁndings.

In 1750, one of the earliest functional differential equation problems was the Euler's problem of finding a curve so that it resembles a shrinking line. In the past 50 years, many mathematicians are familiar with first-order functional differential equations (see [5][6][7][8][9][10][11][12][13][14][15][16][17]). They researched the same and critical question of whether these equations can support positive periodic solutions. Jiang, Wei and Zhang [8], Cheng and Zhang [15], and Wang [16] proved the existence of positive periodic solution for the first-order functional differential in their papers. In particular, Wang [16] studied the following equation x (t) = a(t)g(x(t))x(t) − λb(t) f (x(t − τ(t))). (1.2) Using a famous result of the fixed point index, the author derived the existence results of the positive periodic solution to Eq (1.2) (for any x > 0, f (x) > 0). Among others, the writer proved the connection between the open intervals (eigenvalue intervals) of the parameter λ and the asymptotic behaviors of the quotient f (x) x (at x → 0 and x → ∞) so that Eq (1.2) admits zero, one and multiple positive solutions. However, the criteria for the uniqueness of the positive periodic solution of Eq (1.2) have not been established.
In [6], Liu and Li analyzed the existence of the positive periodic solutions to the equation The authors first proved the existence of the positive periodic solution to Eq (1.3) by applying the eigenvalue theory in cones. In addition, by employing the theory of α-concave operator, they derived an excellent result regarding the uniqueness of the periodic solution to Eq (1.3). However, it is difficult to prove the uniqueness of positive periodic solution for Eq (1.1) when q = −1 and −1 < q < 0. Inspired by the pieces above, in our paper, we will construct two results for the uniqueness of positive periodic solutions to Eq (1.1) by employing the method of Hilbert's metric, which is completely different from that used in [5][6][7][8][9][10][11][12][13][14][15][16][17]. We establish uniqueness results, especially when q = −1 and −1 < q < 1 for Eq (1.1).
Hilbert [18] first considered Hilbert's metric on the foundations of geometry in 1895. Three noncollinear points have been modeled algebraically, in which the length of one side is equal to the sum of the other two sides. In 1957, Birkhoff [19] proved several extensions of Jentzsch's theorem on integral equations with positive kernels by employing Hilbert's metric. He gives some applications to the projective contraction theorem and simply Jentzsch's theorem. The earlier paper of Klein [20] also presented particular examples of Hilbert's metric. In the past time, the applications of Hilbert's metric were mentioned by plenty of authors, see [21][22][23].
Furthermore, we research the existence of the positive periodic solution to the following equation: where a(t), ψ(t) and g(t) > 0 are T -periodic functions and T 0 a(u)du > 0 (T > 0). Comparing with Wang [16] and Liu-Li [17], we here employ a completely different technique to treat Eq (1.4), for detail to see the proof of Theorem 4.1.
In Section 2, we discuss a large number of necessary definitions and lemmas related to Hilbert's metric, which are necessary to prove our main point. In Section 3, we state and prove the uniqueness results of the positive periodic solution to Eq (1.1) by applying the theory of Hilbert's metric. In Section 4, based on the fixed point theorem in a cone, we prove the existence of the positive periodic solution to Eq (1.4). In the last section, the results of our study are illustrated by two examples.

Preliminaries
In this section, we review plenty of crucial lemmas and definitions, which will help us to prove our primary results.  [24]) Let P be a convex closed set, and P ∅. The following conditions must be met for P to be considered a cone: (i) if a ∈ P, λ > 0, then λa ∈ P; (ii) if a ∈ P and −a ∈ P, then a = 0, i.e., 0 ∈ P. A cone P defines a partial ordering in X by a ≤ b (a, b ∈ P) if and only if b − a ∈ P.
Let the set of all the interior points of P be P • . We identify that There is no doubt in our minds that   [26]) If the norm is monotonic with respect to P, then A is said to be increasing (decreasing).

Definition 2.5. (See Definition 3.2 of [25]) If A is positive and
A is said to be positive homogeneous of degree α in P • .
Lemma 2.5. (See [27]) Supposed that X is a Banach space, A : X → X is an operator. There is Then there exists a unique fixed point u 0 ∈ X of T . Alternatively, Lemma 2.6. We make the following assumptions: (1) the norm is monotonic with respect to P, (2) A : P • → P • is an operator and positive homogeneous of degree q in P • , where |q| ∈ (0, 1), Then A has a unique fixed point in P • .
Proof. If 0 < q < 1, then T is an increasing operator and positive homogeneous of degree q. By It shows that Next, we will show that the (P ∩ U, d) is a Banach space. Let {a n } be a Cauchy sequence in (P ∩ U, d), for all ε > 0, N 1 > 0, we have d(a n , a m ) < ε (m, n > N 1 ).
Then there exists N 2 > 0 so that 1 − ι < m(a n /a m ) ≤ 1, 1 ≤ M(a n /a m ) < 1 + ι (n, m > N 2 ). Thus In addition, by Lemma 2.3, there is no doubt about the fact that a m − a n ≤ 2(e d(a m ,a n ) − 1), (2.4) if m → n, then a m − a n → 0. There exists a 0 ∈ X so that a n → a 0 (n → ∞). Let n → ∞ and m be not changed for (2.3). Then we have hence a 0 ∈ P • . Obviously, a 0 = 1, then a 0 ∈ P ∩ U. We see that .
Set a * = Aa 1 1 1−q a 1 . Then a * ∈ P • , and Hence there is a fixed point a * of A in P • . In addition, if there exists b * ∈ P • so that so κ = 1 and a * = b * . To put it another way, a * is a unique fixed point for 0 < q < 1. If −1 < q < 0, then A is decreasing. Based on Eq (2.1), we have Let A 2 a = Aa Aa , a ∈ P. Then A 2 : P ∩ U → P ∩ U, and for all a, b ∈ P ∩ U, Banach's contraction mapping theorem indicates that there is a fixed point of A 2 , i.e., there exists a 2 ∈ P ∩ U such that A 2 a 2 = a 2 . Now we assert that there is a fixed point a * of A. Set a * = Aa 2 1 1−q a 2 . Then Consequently, d(a * , b * ) = 0, i.e., a * = δb * (δ > 0). Then we have a * = Aa * = A(δb * ) = δ q Ab * = δ q b * .
Obviously, δ = 1 and a * = b * . All in all, A has a unique fixed point a * for −1 < q < 0.

Uniqueness of the positive periodic solution
In this section, we consider the uniqueness of the positive periodic solution to Eq (1.1) by employing the method of Hilbert's metric.
It is well known that Eq (1.1) is equal to the following equation Then X is a real Banach space. Set P = {u|u ∈ X, u ≥ 0}.
Then P is a cone of X.
Define the interior of P by The norm in X is defined by u = sup 0≤t≤T |u(t)|. Together with Definition 2.5, we note that the operator T is positive homogeneous of degree q in P • .
Then T is an increasing mapping. For any y 1 , y 2 ∈ P and y 1 > y 2 > 0, we have Thus the norm is monotonic with respect to P by Lemma 2.3. By Lemma 2.6, T has a unique fixed point in P • , which implies that Eq (1.1) has a unique positive periodic solution in P • for 0 < q < 1.
If −1 < q < 0, y q 1 > y q 2 , then T y 1 − T y 2 > 0, that is T is decreasing. Combine the above proof, T has a unique fixed point in P • , that is to say, Eq (1.1) has a unique positive periodic solution in P • for −1 < q < 0.
There is no doubt that α = 1 andx =ȳ. So Eq (3.5) has a unique positive periodic solution.

Existence of the positive periodic solution
In this section, we will analyze the existence of the positive periodic solution for Eq (1.4). Let X = C[0, T ] denote a real Banach space with the norm y = sup 0≤t≤T |y(t)|.
We specify the operator A : P → P by As is well known, Eq (4.1) has a fixed pointŷ ∈ P (ŷ > 0) if and only ifŷ is the positive periodic solution to Eq (1.4). We verify the existence of the fixed point for Eq (4.1) by employing the fixed point theorem in a cone. Lemma 4.1. (See [26]) Let Γ 1 , Γ 2 be open bounded sets of P with 0 ∈ Γ 1 and Γ 1 ⊂ Γ 2 . Supposed that T : P ∩ (Γ 2 \ Γ 1 ) → P is complete continuous, and it satisfies at least one of the following requirements: (H 1 ) If there exists u 0 ∈ P \ {0} so that a − T a du 0 for all a ∈ P ∩ ∂Γ 2 and all d ≥ 0; T a µx for all a ∈ P ∩ ∂Γ 1 and all µ ≥ 1.
(H 2 ) If there exists u 0 ∈ P \ {0} so that a − T a du 0 for all a ∈ P ∩ ∂Γ 1 and all d ≥ 0; T a µx, for all a ∈ P ∩ ∂Γ 2 and all µ ≥ 1.
Then there is a fixed point of T in P ∩ (Γ 2 \ Γ 1 ).
(3) there exists 0 < α < 1 so that As a result, (4.1) has and only has one positive periodic solution.
Proof. If there exists ε 0 > 0 so that y − Ay 0, f or y ∈ P with 0 < y ≤ ε 0 . (4.2) Otherwise, there is a fixed point in P will be accurate.
On the other hand, from (4), there exist a > 0 and y 0 > 0 such that f (y) ≤ ay α * , f or y ≥ y 0 .
In light of Lemma 4.1, there is a fixed point of A in P ∩ (Γ R \ Γ r ).

Conclusions
This study sets out to verify the uniqueness and existence of positive periodic solutions for Eq (1.1). It contributes to our solving of other mathematical problems and practical problems. Our study shows that using the theory of Hilbert's metric to prove the uniqueness of positive periodic solution for Eq (1.1), as well as employing the fixed point theorem in a cone to prove the existence of positive periodic solution for Eq (1.4).
We employ the theory of Hilbert's metric to prove the uniqueness of the positive periodic solution for Eq (1.4) when f (y(t − ψ(t))) = y q (t − ψ(t)). When 0 < q < 1, q = −1 and −1 < q < 0, the uniqueness results are verified based on the theory of Hilbert's metric. The major limitation of this study is that it is difficult to directly verify the uniqueness of the positive periodic solution for Eq (1.4) by applying the theory of Hilbert's metric.

Two examples
This section illustrates our conclusions in Section 3 with two examples. Example 6.1. Suppose that X, P, P • , and S have the same meanings with Section 3, we think about the equation Proof. In this example, a(t) = 1 π , h(t) = sin 2 t, ψ(t) = π/1000, and q = 1 3 .
As we all know, Eq It is easy to see that A is an increasing operator which is positive homogeneous of degree 1 3 in P • . Moreover, the norm is monotonic with respect to P.
In a word, A has a unique fixed point in P • , which shows Eq (6.1) has a unique positive periodic solution. Next, we will verify that there is a fixed point of Eq (6.4).
Clearly, A is a decreasing operator and positive homogeneous of degree − 1 2 in P • . What's more, the norm is monotonic with respect to P • .
As a result, Eq (6.3) has a unique solution in P • .