Boundedness of positive solutions of a system of nonlinear delay differential equations

In this manuscript the system of nonlinear delay differential equations \begin{document}$\dot{x}_i(t) =\sum\limits_{j =1}^{n}\sum\limits_{\ell =1}^{n_0}α_{ij\ell} (t) h_{ij}(x_j(t-τ_{ij\ell}(t)))$\end{document} \begin{document}$-β_i(t)f_i(x_i(t))+ρ_i(t)$\end{document} , \begin{document}$t≥0$\end{document} , \begin{document}$1≤i ≤n$\end{document} is considered. Sufficient conditions are established for the uniform permanence of the positive solutions of the system. In several particular cases, explicit formulas are given for the estimates of the upper and lower limit of the solutions. In a special case, the asymptotic equivalence of the solutions is investigated.

1. Introduction. Nonlinear differential equations with delays frequently appear as model equations in physics, engineering, economics and biology. Next we recall some typical applications.
Compartmental systems are used to model many processes in pharmacokinetics, metabolism, epidemiology and ecology (see [19,20,24]). The nonlinear donorcontrolled compartmental systeṁ q i (t) = −k ii f i (q i (t)) + n j=1 j =i k ij f j (q j (t − τ ij )) + I i , i = 1, . . . , n was studied in [8,9]. Here q i (t) is the mass of the ith compartment at time t, k ij > 0 represent the transfer or rate coefficients, I i ≥ 0 is the inflow to the ith compartment, and in this model it is assumed that the intercompartmental flows are functions of the state of the donor compartments only in the form k ij f j (q j ) with some positive nonlinear function f j . The classical Hopfield neural networks [23] have been successfully used in many different fields of engineering applications, e.g., in signal processing, image processing, pattern recognition. In hardware implementations time delays appear naturally due to finite switching speed of the amplifiers, see, e.g., [3,13,17]. We recall the system T ij g j (u j (t − τ ij )) + I i , i = 1, . . . , n, which was studied in [17]. Here C i > 0, R i > 0 and I i are capacity, resistance, bias, respectively, T ij is the interconnection weight, and g i is a strictly monotone increasing nonlinear function with g i (0) = 0. In [11] the existence, uniqueness and global stability of asymptotically periodic solutions of the bidirectional associative memory (BAM) network p ji (t)f j (y j (t − τ ji )) + I i (t), i = 1, . . . , n, q ij (t)f i (x j (t − σ ij )) + J i (t), j = 1, . . . , k was examined.
In [12] the delay modelṘ was considered for the control of the secretion of the hormone testosterone. Here R(t), L(t) and T (t) are the concentrations of the gonadotropinreleasing hormone, luteinizing hormone and testosterone, respectively, r 1 , r 2 , d 1 , d 2 , d 3 are positive constants. Global stability of a positive equilibrium and oscillations of the solutions were investigated depending on the values of a parameter in the formula of the positive nonlinear function f . In [5] the two-dimensional systeṁ x(t) = r 1 (t) f 1 (y(t − τ 1 (t)) − x(t) , t ≥ 0 (1) y(t) = r 2 (t) f 2 (x(t − τ 2 (t)) − y(t) , t ≥ 0 (2) was considered as a special case of a more general two-dimensional system of nonlinear delay equations with distributed delays. Sufficient conditions were given implying that the solutions of the System (1)- (2) are permanent, i.e., there exist positive constants a, A, b and B such that a ≤ x(t) ≤ A and b ≤ y(t) ≤ B hold for t ≥ 0.
Populations are frequently modeled in heterogeneous environments due to, e.g., different food-rich patches, different stages of a species according to age or size. In such models time delays appear naturally due to the time needed for species to disperse from one patch to another. We recall here the n-dimensional Nicholson's blowflies systems with patch structurė where d i > 0, β i ≥ 0, a ij ≥ 0, τ i ≥ 0 for 1 ≤ i = j ≤ n, = 1, . . . , n 0 . Asymptotic behavior, permanence of the solutions was investigated, e.g., in, [4,7,16,26]. For the scalar case, this model reduces to the famous Nicholson's blowflies equation introduced in [18] to model the Australian sheep-blowfly population.

BOUNDEDNESS OF POSITIVE SOLUTIONS OF A SYSTEM OF NONLINEAR DDES 811
The n-dimensional population model with patch structurė was introduced in [15], and the permanence of the positive solutions was investigated. Here all functions are nonnegative. It is a generalization of a scalar modified logistic equation with several delays introduced in [2]. Motivated by the above models, in this paper we consider a system of nonlinear delay differential equations of the forṁ where, f i , h ij , α ij , β i , ρ i and τ ij are nonnegative continuous functions. We associate the initial condition to our system, where τ > 0 and 0 ≤ τ ij (t) ≤ τ hold for t ≥ 0, 1 ≤ i, j ≤ n and 1 ≤ ≤ n 0 . We study positive solutions of the System (5), so we assume that Our main result, Theorem 2.4 below shows that, under certain conditions, the solutions of the initial value problem (IVP) (5) and (6) is uniformly permanent, i.e., there exist positive constants k 1 , . . . , k n , K 1 , . . . , K n , such that for any initial functions ϕ i ∈ C + , i = 1, . . . , n the corresponding solution satisfies Moreover, the constants k 1 , . . . , k n and K 1 , . . . , K n are given explicitly, as unique positive solutions of an associated nonlinear algebraic systems. As a consequence of the main result, we formulate conditions which imply that all the positive solutions converge to a constant limit (see Corollary 3.1 below). In Theorem 3.5, for nonlinear systems of the forṁ we give sufficient conditions which imply that the positive solutions are asymptotically equivalent, i.e., the difference of any two positive solutions tends to 0 as the time goes to ∞. Permanence of solutions of a differential equation model is especially important in mathematical biology and ecology [7,16,25]. Permanence of solutions of scalar delay population models was recently studied in [1,2,6,14,16]. In [21] we considered a scalar delay population model where r, h ∈ C(R + , R + ), g ∈ C(R + × C([−τ, 0], R), R + ), τ > 0 is fixed, and x t (s) = x(t + s) for s ∈ [−τ, 0]. This manuscript extends the method introduced for the scalar case in [21] to the nonlinear delay system (5). A key element of the proof of our Theorem 2.4 is a result proved in [22], where sufficient conditions are formulated implying that a certain nonlinear algebraic system associated to (5) has a unique positive solution (see Lemma 2.3 and Theorem 4.2 below). The structure of our paper is the following. In Section 2 we formulate our main results: Theorem 2.4 below gives estimates for the limit inferior and limit superior of the positive solutions of System (5). In Section 3 we show several corollaries of our main results and numerical examples. In Section 4 we give the proofs of our main results, and in Section 5 we summarize our conclusions and formulate some open questions.
2. Main results. We start this section by listing all conditions on the parameters of the IVP (5) and (6) will be used in the rest of the manuscript. τ > 0 is a fixed constant, and all delay functions are assumed to be uniformly bounded by τ .
(A 3 ) f i ∈ C(R + , R + ), 1 ≤ i ≤ n, are strictly increasing with f i (0) = 0 and f i are locally Lipschitz continuous; (A 4 ) h ij ∈ C(R + , R + ) are increasing, locally Lipschitz continuous, and h ij (u) > 0 for u > 0 and 1 ≤ i, j ≤ n; (A 5 ) ρ i ∈ C(R + , R + ) and for each i = 1, . . . , n, and n j=1 lim sup hii(u) is strictly increasing on the interval (0, ∞) or lim inf Clearly, under conditions (A 0 )-(A 5 ), the IVP (5) and (6) has a unique solution corresponding to any ϕ = (ϕ 1 , . . . , ϕ n ) ∈ C n + . This solution is denoted by x(ϕ) = (x 1 (ϕ), . . . , x n (ϕ)). Note that in [21] a scalar version of (5) was studied where, instead of the local Lipschitz-continuity, it was assumed that f i are such that for any nonnegative constants and L satisfying L = , one has Hence the solution studied in [21] was not necessary unique. It is easy to see that the locally Lipschitz-continuity of f i implies condition (11). We assume the locally Lipschitz-continuity of f i and h ij to simplify the presentation, but it can be omitted as in [21]. We note that assumption (A 3 ) is weaker than that used in the [2,14], where, investigating permanence of a scalar population model, it was assumed that the coefficient function β i is bounded below and above by positive constants.
The monotonicity assumptions of (A 6 ) for the ratios fi(u) hij (u) and hjj (u) hij (u) are crucial for using Lemma 2.3 below. This assumption allows us to include examples when some ratios are constants, and only some of these functions are strictly monotone. This week form of the condition will be important when we apply our main results to the population models (3) and (4) (see Corollary 3.7 and 3.8 below).
The following result implies that, under our conditions, the System (5) is persistent.
hii(u) is strictly increasing on (0, ∞) or l i > 0 and h ii (u) is strictly increasing on (0, ∞) ; and Then (i) the System (13) has a unique positive solution one has (iii) For any x = (x 1 , . . . , x n ) satisfying one has

BOUNDEDNESS OF POSITIVE SOLUTIONS OF A SYSTEM OF NONLINEAR DDES 815
We use the following notations in our main theorem: We note that (A 2 ), (A 5 ) and Lemma 2.
Corollaries and examples. In this section we present several corollaries to our main result and illustrative numerical examples.

ISTVÁN GYŐRI, FERENC HARTUNG AND NAHED A. MOHAMADY
Now, we study a special form of (5). We consider the IVṖ with the initial condition where and are satisfied. Therefore Theorem 2.4 has the following consequence. (23) and (24) hold. Then, for any initial function (21) and (22) satisfies where (x * 1 , . . . , x * n ) is the unique positive solution of the algebraic system and (x * 1 , . . . , x * n ) is the unique positive solution of the algebraic system respectively, where m ij , m ij , l i and l i are defined in (17) and (18) We remark that the condition (23) in Corollary 3.2 can be weakened.
Example 3.3. Consider the following system of nonlinear differential equations in three dimensions, for t ≥ 0,

BOUNDEDNESS OF POSITIVE SOLUTIONS OF A SYSTEM OF NONLINEAR DDES 817
Note that the conditions of Corollary 3.2 are satisfied for (27). So, we see from is the unique positive solution of the algebraic system We solve the System (28) numerically by the fixed point iteration We compute the sequence (x 3 ) = (0, 0, 0). The first ten terms of this sequence are displayed in Table 1. We can observe that the sequence is convergent, and its limit is is the unique positive solution of the algebraic system We solve the System (30) numerically by a fixed point iteration defined similarly to (29) from the starting value (0, 0, 0). The numerical results can be seen in Table  2. We conclude that (x * 1 , x * 2 , x * 3 ) = (6.7840 . . . , 11.1161 . . . , 8.7126 . . .). Therefore Corollary 3.2 yields We plotted the numerical solution of (27) in Figure 1 corresponding to the constant initial functions (ϕ 1 (t), ϕ 2 (t), ϕ 3 (t)) = (2.5, 6, 2.5) and (ϕ 1 (t), ϕ 2 (t), ϕ 3 (t)) = (3.5, 8, 4). The horizontal lines in Figure 1 correspond to the upper and lower bounds listed in (31), respectively. We also observe that the difference of the components of the two solutions converges to zero, i.e., the positive solutions are asymptotically equivalent. The numerical results demonstrate the theoretical bounds (31).

BOUNDEDNESS OF POSITIVE SOLUTIONS OF A SYSTEM OF NONLINEAR DDES 821
Since ε > 0 can be arbitrary small, we get max 1≤i≤n z i (∞) = 0 and consequently lim t→∞ z i (t) = 0, 1 ≤ i ≤ n. Hence the proof is completed.
Example 3.6. Consider the following system of nonlinear differential equations in two dimensions, for t ≥ 0, Note that the conditions of Theorem 3.
We plotted the numerical solution of (40) in Figure 2 corresponding to the initial functions (ϕ 1 (t), ϕ 2 (t)) = (3, 2), (ϕ 1 (t), ϕ 2 (t)) = (7, 7) and (ϕ 1 (t), ϕ 2 (t)) = (9, 10). The horizontal lines in Figure 2 correspond to the upper and lower bounds listed in (44), respectively. We also observe that the difference of the components of every two solutions converges to zero, i.e., the positive solutions are asymptotically equivalent which coincide (3.5) in Theorem 3.5.    Next, we consider again the population model (4): with the initial condition We assume that ϕ = (ϕ 1 , ϕ 2 , . . . , The permanence of positive solutions of (45) was investigated in [15] for the case when the delays in the model can be unbounded. Next, we show that, for the bounded delay case, our Theorem 2.4 gives permanence of the positive solutions for this model under weak conditions. We note that we do not need the boundedness of the functions λ i , a ij , µ i and κ i which was assumed in [15].

> 1 by (48), and
hjj (u) hij (u) is strictly decreasing on (0, ∞), for all j = i. Hence conditions (A 6 ) (i), (ii) and (iii) are satisfied, and we can apply Theorem 2.4 (i) to the System (51). Therefore we get the lower estimates lim inf where (x * 1 , . . . , x * n ) is the unique positive solution of the algebraic system (49). Similarly, we can get the upper estimates lim sup where (x * 1 , . . . , x * n ) is the unique positive solution of the algebraic system (50). Now, we consider a time-dependent version of the n-dimensional Nicholson's blowflies system (3) for t ≥ 0: where τ > 0, ϕ = (ϕ 1 , ϕ 2 , . . . , ϕ n ) ∈ C n + , b i , a ij , d i ∈ C(R + , R + ), and σ i ∈ C(R + , R + ) with 0 ≤ σ i (t) ≤ τ for t ≥ 0, 1 ≤ i = j ≤ n, = 1, . . . , n 0 . The the persistence and permanence of the autonomous system (3) was investigated in [16]. Unfortunately, our method does not work for this population model, since the function ue −u is not monotone increasing, and so condition (A 4 ) of our main Theorem 2.4 is not satisfied for (53). But we can apply our method to get an upper bound of the limit superior of the solutions of (53). We formulate this result next.

BOUNDEDNESS OF POSITIVE SOLUTIONS OF A SYSTEM OF NONLINEAR DDES 825
Corollary 3.8. Assume b i , a ij , d i ∈ C(R + , R + ), and σ i ∈ C(R + , R + ) with 0 ≤ σ i (t) ≤ τ for t ≥ 0, 1 ≤ i = j ≤ n and = 1, . . . , n 0 . Moreover, we assume that, for all 1 ≤ i, j ≤ n, and Then, for any initial function ϕ = (ϕ 1 , ϕ 2 , . . . , ϕ n ) ∈ C n + , the solution x(ϕ)(t) = (x 1 (ϕ)(t), . . . , x n (ϕ)(t)) of the IVP (53) and (54) satisfies where (x * 1 , . . . , x * n ) is the unique positive solution of the algebraic system where m ii := lim sup Proof. All conditions of Lemma 2.1 hold for the System (53), therefore it implies that x i (ϕ)(t) > 0 for t ≥ 0 and i = 1, . . . , n. We have ue −u ≤ H(u) for u ≥ 0, therefore (53) yieldṡ By comparison theorem of differential inequalities, we have x i (t) ≤ y i (t) for t ≥ 0, i = 1, . . . , n, where y i (t) is the positive solution of the differential equatioṅ with the initial condition Next, we check that (A 0 )-(A 6 ) of Theorem 2.4 are satisfied for the System (59). First note that we can rewrite (59) in the form (5) with Thus, by our assumptions (55), (56) and (57), we can see that conditions (A 0 )-(A 5 ) hold. To check condition (A 6 ), we observe that is increasing and hjj (u) hij (u) is strictly decreasing on (0, ∞), for all j = i. Hence conditions (A 6 ) (i), (iv) and (v) are satisfied, and we can apply Theorem 2.4 (ii) to the System (59). Therefore we get the upper estimates lim sup is the unique positive solution of the algebraic system (58). 4. Proof of the main results. In this section we provide the proofs of our main results formulated in Section 2. First we recall the following result from [21]. Lemma 4.1. (see [21]) Let 1 ≤ i ≤ n be fixed and y(T, y 0 , c, β i , f i )(t) be the solution of the IVPẏ where c ≥ 0, β i satisfies condition (A 1 ) and f i satisfies condition (A 3 ). Then for any T ≥ 0, y 0 > 0, and c ≥ 0 the corresponding solution y(T, y 0 , c, β i , f i )(t) of the IVP (61) and (62) is uniquely defined on [T, ∞), moreover we have (i) c > 0 and 0 < y But from the comparison theorem of the differential inequalities (see [10]), we have where y i (t) = y(0, ϕ i (0), 0, β i , f i )(t), 1 ≤ i ≤ n is the unique positive solution of (61) with c = 0 and with the initial condition Lemma 4.1 yields y i (t) > 0, for all t ≥ 0. Then at t = t 1 we get x i (t 1 ) ≥ y i (t 1 ) > 0, 1 ≤ i ≤ n, which contradicts our assumption that min{x 1 (t 1 ), . . . , x n (t 1 )} = 0. Hence x i (t) > 0, 1 ≤ i ≤ n for t ∈ R + .
, then let K i > 0 be such that Thus there exists T i > 0 such that Also, there exists c i > 0 such that and hence (64) holds for such i. Therefore (64) is satisfied, for all i = 1, . . . , n, with T = max{T 1 , . . . , T n } and c = min{c 1 , . . . , c n }. Now, in virtue of (64), either x i (t) > c for all t ≥ 0, 1 ≤ i ≤ n, or there exists t 2 ∈ (T, ∞) such that min{x 1 (t 2 ), . . . , x n (t 2 )} = c and x i (t) > c for t ∈ [0, t 2 ), 1 ≤ i ≤ n. In this case at least one of the values of x 1 (t 2 ), . . . , x n (t 2 ) is equal to c. Assume, e.g., that x 1 (t 2 ) = c, thenẋ 1 (t 2 ) ≤ 0. On the other hand, the monotonicity of h 1j and (64) yield thaṫ which is a contradiction, sinceẋ 1 (t 2 ) ≤ 0. Hence x 1 (t) > c for all t ≥ 0. Similarly, we can show that x i (t) > c, for all t ≥ 0, 2 ≤ i ≤ n, and therefore (63) holds. Now we show that We claim that there exist T > 0 and M > 0 such that the following inequalities are satisfied, for every i = 1, . . . , n,

BOUNDEDNESS OF POSITIVE SOLUTIONS OF A SYSTEM OF NONLINEAR DDES 829
max 0≤t≤T x i (t) < M and (66) The second relation of (66) holds if Using (10), there exists a µ i > 0 such that then there exists an δ > 0 such that Thus there exist T i > 0 and V 1i > 0 such that Moreover, using (9), there exists a V 2i > 0 such that is equal to M . Assume, e.g., that x 1 (t 3 ) = M , thenẋ 1 (t 3 ) ≥ 0. On the other hand, using (66) and the monotonicity of h 1j , we havė which is a contradiction, sinceẋ 1 (t 3 ) ≥ 0. Hence x 1 (t) < M, for all t ≥ 0. Similarly, we can show that x i (t) < M, for all t ≥ 0, 2 ≤ i ≤ n, and therefore (65) holds. Now, we consider the system of nonlinear algebraic equations where γ i ∈ C(R + , R), g ij ∈ C(R + , R + ), 1 ≤ i, j ≤ n.
In [22] sufficient conditions were given to guarantee the existence of a unique positive solution of (68). We recall this result next.
We note that uniqueness of Theorem 4.2 was proved in [22] under the condition that (D*) for each 1 ≤ i, j ≤ n, γj (u) gij (u) is strictly monotone increasing on (u * j , ∞), assuming g ij (u) > 0 for u > 0. We note that the uniqueness of the positive solutions of (68) also holds if we assume (D) instead of (D*), and the proof is an obvious extension of that presented in [22].
The left hand side of (70) is increasing and the right hand side of (70) is decreasing, moreover, assumption (H 2 ) yields that either the left hand side or the right hand side is a strictly monotone function. Therefore, condition (A) of Theorem 4.2 holds, if we show and If l i > 0 and h ii (0) = 0, then (71) follows, since the left hand side of (71) is always finite, since fi(u) hii(u) is monotone increasing. If l i > 0 and h ii (0) > 0, then the right hand side of (71) is finite and positive, but lim u→0 + fi(u) hii(u) = 0 using (A 3 ). If l i = 0, then assumption (14) yields (71). Relation (72) follows immediately from (15). Hence condition (A) is satisfied.
To check condition (B), we see that g ij (u) := m ij h ij (u), 1 ≤ i = j ≤ n, and g ii (u) = 0 are increasing on R + , and relation (69) is monotone increasing on (0, ∞), by (A 4 ) and (H 1 ). By assumption (H 3 ), there exists i = j such that γj (u) gij (u) is strictly monotone increasing on (0, ∞), and so condition (D) is satisfied. Hence the System (13) has a unique positive solution. Now we prove (ii). From (16) we have Assumption (A 3 ) and (14) yield that there exists a small u * such that and , which implies and hence the proof of (ii) is completed. The proof of part (iii) is similar to that of part (ii), so it is omitted here. Now, we are ready to prove the main result of the manuscript.
Proof of Theorem 2.4. In the proof we will use the notations  (5) we geṫ or equivalentlẏ where C i (T ) := n j=1 m ij (T )h ij (x j (T )) + l i (T ). From (83) and the comparison theorem of differential inequalities we get