Positive periodic solutions for multiparameter nonlinear differential systems with delays

We establish several criteria for the existence of positive periodic solutions of the multi-parameter differential systems { u′(t) + a1(t)g1(u(t))u(t) = λb1(t)f (u(t – τ1(t)), v(t – ζ1(t))), v′(t) + a2(t)g2(v(t))v(t) =μb2(t)g(u(t – τ2(t)), v(t – ζ2(t))), where the functions g1,g2 : [0,∞) → [0,∞) are assumed to be unbounded. The analysis in the paper relies on the classical fixed point index theory. Our main findings improve and complement some existing results in the literature.

Moreover, for (x, y) ∈ E, we denote (x, y) = x + y , and write (x, y) Obviously, the first equation of (1.1) reduces in some special circumstances to and when λ = 0, g(u) ≡ 1, Eq. (1.2) becomes u (t) = a(t)u(t), which is famous in Malthusian population dynamics. In recent decades, (1.2) has also been extensively applied to describe various physiological processes emerging in practical applications, for instance, the production of blood cells, respiration, cardiac arrhythmias, etc. One may refer to [1][2][3][4][5][6] and references therein. Nevertheless, the research work in the above mentioned papers is mainly dependent on the condition that g(u) is positive and bounded, that is, there are constants L > l > 0 such that 0 < l ≤ g(u) ≤ L, u ∈ [0, ∞). Jin and Wang [7] have recently studied the spectral problem and they obtained some existence results on positive periodic solutions by means of the fixed point theory. It is worth noting the function e u is unbounded on [0, ∞). Since then, Eq. (1.2) has been extensively investigated under the more general case that g(u) is unbounded on [0, ∞), by applying the lower and upper solutions method, fixed point theory, and so on. See, for example, [7][8][9][10].
Besides, researchers have focused on the differential systems associated to (1.2), namely, One can see [11][12][13][14] for some related results. However, in [11][12][13], the authors have only dealt with the special case g i (u i ) ≡ 1, i = 1, 2, . . . , n. Indeed in that case, the Green's function corresponding to u i (t) = a i (t)u i (t) is simple, and some suitable cones could be easily constructed. Furthermore, system (1.3) investigated in above papers includes only one positive parameter λ. Hence, it will be interesting to study the multiparameter systems (1.1) with g i (i = 1, 2) being unbounded. On the other hand, what is worth mentioning is that Zhang et al. [14] considered system (1.1) for the special case g i ≡ 1, i = 1, 2, where nonlinearities f (u, v) and g(u, v) were assumed to be nondecreasing, and only the case f (0, 0) > 0, g(0, 0) > 0 was treated. Therefore, we want to know whether or not (1.1) has a positive periodic solution under more relaxed assumption f (0, 0) = 0, g(0, 0) = 0. In view of above reasons, we shall concentrate on the existence of positive periodic solutions for system (1.1) in the current paper, to further improve and generalize tho results in the literature. For this purpose, we assume Remark 1.1 For other research work on periodic solutions of functional differential equations and systems, we refer the readers to [15][16][17] and references therein.
The remainder of the paper is arranged as follows. In Sect. 2, we introduce some preliminaries needed in our proof. Section 3 is devoted to stating and proving our main findings. Meanwhile, some related results and remarks will be given.

Preliminaries
Recall that E = X 2 is the Banach space defined as in Sect. 1. We first give the following lemma.
Proof Multiplying the both sides of the first equation of (1.1) with e -t 0 a 1 (s)g 1 (u(s)) ds , we can obtain Integrating above equation from t to t + ω and by elementary calculation, we can easily get Let q > 0 be a fixed constant. Then we can establish a series of lemmas required in the subsequent discussion. Thus, and then simple estimation shows (2.1) holds for i = 1. The case i = 2 is similar.
Then it is not hard to verify η i (q) ∈ (0, 1), and accordingly, and for r > 0, Then P and K q are cones in E.
Proof For (u, v) ∈Ω r , we can deduce from Lemma 2.3 that
It is obvious that if (u, v) is a fixed point of the completely continuous operator T λ,μ in K q , then (u, v) is a positive periodic solution of (1.1). We conclude this section by giving the main tool employed in proving our main results. Lemma 2.5 ([18, 19]) Assume E is a Banach space and K ⊆ E is a cone. For r > 0, let K r = {u ∈ K : u < r} and ∂K r = {u ∈ K : u = r}. Suppose T :K r → K is a completely continuous operator satisfying Tu = u, u ∈ ∂K r . Then (i) If Tu < u , u ∈ ∂K r , then i(T,K r , K) = 1; (ii) If Tu > u , u ∈ ∂K r , then i(T,K r , K) = 0.
Furthermore, by a similar argument as above, it is not difficult to see that the results of Theorems 3.1-3.3 remain true for system (3.3).
Remark 3.2 It is worth remarking that, under some reasonable assumptions, the results of the paper are still valid for the more general coupled systems u i (t) + a i (t)g i u i (t) u i (t) = λ i b i (t)f i u 1 tτ i1 (t) , . . . , u n tτ in (t) , i = 1, 2, . . . , n and u i (t) = a i (t)g i u i (t) u i (t)λ i b i (t)f i u 1 tτ i1 (t) , . . . , u n tτ in (t) , i = 1, 2, . . . , n.