An exact expression of positive periodic solution for a first-order singular equation

The main purpose of this paper is to obtain an exact expression of the positive periodic solution for a first-order differential equation with attractive and repulsive singularities. Moreover, we prove the existence of at least one positive periodic solution for this equation with an indefinite singularity by applications of topological degree theorem, and give the upper and lower bounds of the positive periodic solution.


Introduction
A variety of population dynamics and physiological processes can be described by the following singular differential equation: with periodic boundary value condition where a(t) and b(t) ∈ C(R, R) are ω-periodic functions, ρ and ω are positive constants. According to the literature [1][2][3], we say that Eq. (1.1) has an attractive singularity if b(t) < 0, repulsive singularity if b(t) > 0 and indefinite singularity if b(t) may change sign. Moreover, we say that Eq. (1.1) has a strong singularity if ρ ≥ 1 and a weak singularity if 0 < ρ < 1.
As is well known, singular equations have a wide range of applications in many fields, and the existence of positive ω-periodic solutions to singular equations plays a significant role in solving many practical problems. There is a good amount of work on periodic solutions for singular equations (see [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references cited therein). In 2003, Agarwal and O'Regan [4] provided some results on positive ω-periodic solutions of Eq. (1.1), where b(t) > 0 (i.e., repulsive singularity). After that, Chu and Nieto [10] in 2008 proved the existence of positive ω-periodic solutions for equation (1.1) with repulsive singularity and impulses by applications of Leray-Schauder alternative principle. Recently, Wang [15] and Chen et al. [7] discussed the existence of positive ω-periodic solutions for the following singular differential equation: where a 1 (t) and b 1 (t) ∈ C(R, [0, +∞)) are ω-periodic functions with ω 0 a 1 (t) dt > 0 and ω 0 b 1 (t) dt > 0, the nonlinear term f has an attractive singularity or repulsive singularity. Their proofs were based on Krasnoselskii's fixed point theorem in a cone, the Leray-Schauder degree and the upper and lower solutions method.
It is worth pointing out that the above results are only related to the existence of positive ω-periodic solutions of first-order differential equations with attractive singularity or repulsive singularity, but the exact expression of ω-periodic solutions is not involved.
First, in this paper we give an exact expression of positive ω-periodic solution for Eq. (1.1) with attractive and repulsive singularities.

Proof of Theorem 1.1
In this section, an exact expression of solution for Eq. (1.1) with periodic boundary value condition is given.
First, we change the variable x = u α , where α = 1 ρ+1 and 0 < α < 1. Then Eq. (1.1) is converted into the following form: with periodic boundary value condition Proof of Theorem 1.1 Applying method of variation of constant, solution of Eq. (2.1) can be written as the following form: Integrating the above equation over the interval [0, t], here t ∈ [0, ω], we get . (2.6) From the condition ω 0 a(t) dt · b(t) > 0, it is easy to see that x(t) is positive for all t ∈ R. The proof is completed.

Proof of Theorem 1.2
For convenience, we recall the topological degree theorem by Mawhin [19].
Thus condition (iii) of Lemma 3.1 holds. Therefore, Lx + Nx = 0 has at least one solution in 1 , which means Eq. (1.1) has at least one positive ω-periodic solution x(t) with .