A PRIORI BOUNDS AND PERIODIC SOLUTIONS FOR A CLASS OF PLANAR SYSTEMS WITH APPLICATIONS TO LOTKA–VOLTERRA

. The existence of periodic solutions for some planar systems is investigated. Applications are given to positive solutions for a class of Kolmogorov systems generalizing a predator - prey model for the dynamics of two species in a periodic enviroment.

1. Introduction. In this paper we study the first order Kolmogorov differential system in the (p, q)-plane p = pP(t, p, q) q = qQ(t, p, q) (1) in a case which generalizes the classical Lotka -Volterra models for the interaction of a prey population p(t) with a predator population q(t). Here and in the sequel, we set R + = [0, +∞), R + =]0, +∞) and denote by f the mean value of a T -periodic function f, that is f := T −1 T 0 f (t) dt. By |f | k , with 1 ≤ k ≤ ∞, we denote the L k -norm of a function f ∈ L k ([0, T ], R). A function f with values in R + is said to be non-negative, while we say that f is positive when its range is contained in R + . Similarly, a solution (p, q) of (1) is positive if it satisfies (p(t), q(t)) ∈ (R + ) 2 , for all t in its domain. We also use the following notation throughout, with respect to a continuous T -periodic function u(t) : u = max R u(t), u = min R u(t).
As is well known, the simplest example of (1) modeling a predator -prey interaction, is given by P(t, p, q) = a − c q, Q(t, p, q) = −d + e p, (with a, c, d, e positive constants), as proposed by Vito Volterra [30] in his pioneering work (see also [6]). Subsequently, many variants and extensions were considered by different authors (see, e.g. [2], [17], [23]). In particular, periodic or almost periodic nonautonomous Kolmogorov systems have been proposed in more recent years in order to take into account the possibility of time -varying parameters (see the references below). Here, as in [3]- [5], [8], [10]- [14], [19]- [22], [24], [26]- [29], [32] [34], we study the case in which the coefficients are periodic functions of the time variable having a common period T > 0. This situation usually occurs for the dynamics of populations living in a periodic enviroment, where a typical model of Lotka -Volterra prey -predator system takes the form with a, b, c, d, e, f continuous, a > 0, d > 0, b, f non-negative and c, e positive (see [10]). Note that with respect to the Volterra's equation, now some logistic terms are possibly added. Other examples of (2) are discussed in [4], [24], [29]. In particular, in [24], the authors refer to this choice of P and Q as motivated by the study of the dynamics of two species in a polluted enviroment.
In some of the above quoted papers conditions on the coefficients are imposed in order to obtain a positive periodic solution, or the asymptotic stability of that solution. A related problem is the existence of a compact attractor in (R + ) 2 for the solutions of (1) (a situation which is also termed as permanence or uniform persistence). Usually, for models like in (2), permanence or asymptotic stability of the positive periodic solution(s) is obtained under the basic assumption of a logistic growth for the prey population that is, requiring the positivity of b (further conditions have to be cosidered too). On the other hand, if one takes b ≡ 0 and f ≡ 0 in (2), so that with a, c, d, e positive and continuous functions of the same period, in order to have a generalization of the Volterra's model with periodic coefficients, then there is no hope to obtain a compact attractor in (R + ) 2 , as a simple analysis for the case of constant coefficients shows. In this situation, like in [12], [22], one could try to prove at least the existence of positive T -periodic solutions and find a priori bounds for such solutions. In particular, let us recall that in [14, Theorem 3] the following result was obtained for the special case when P = P(t, q) and Q = Q(t, p) in (1) : Let P ∞ , Q ∞ : R → R be continuous and T -periodic functions such that has at least one positive T -periodic solution (and all the positive T -periodic solutions of (4) are contained in a compact subset of (R + ) 2 ) provided that If, in addition, P(t, ·) is strictly decreasing or Q(t, ·) is strictly increasing, for all t, then there is an index m * ≥ 2 such that for every m ≥ m * , equation ( The plan of this article is the following. We prove a result of existence and boundedness of positive periodic solutions for (1) (Theorem 1) under some assumptions which are meaningful in view of the application to some "ecological" models. In particular, if we adapt our main result to the case of (2) we obtain a corollary which can be applied without any special assumption on the logistic coefficient b. Thus, we generalize and unify at the same time some classical [10] and recent [12], [14] existence results. We point out that the general assumptions we consider on P and Q allow us to apply our main result to conservative systems and dissipative systems as well. Thus we cannot guarantee existence of subharmonic solutions like in [12]- [14], nor the existence of a compact attractor in (R + ) 2 like in [7], [29] and other articles dealing with "persistence". This point will be illustrated and commented by means of some simple examples (see Remark 2). Our main existence result follows from topological degree via a priori bounds for the T -periodic solutions of a one-parameter family of equations in which system (1) will be embedded. More precisely, we deform continuously equation (1) to an Hamiltonian system of the form with u = log p, v = log q and (A, B) satisfying suitable assumptions for the validity of [14,Theorem 1]. In this manner, we can exploit some informations about the coincidence degree previously obtained in [14] and prove our result in a more direct way.
For the reader's convenience, we recall now some basic tools in the frame of Mawhin's coincidence degree that we adapt to the present setting, borrowing notations and terminology from [25].
For each λ ∈ [0, 1], we can write (5) as a coincidence equation of the form Lz = N λ z, in the Banach space C T of the continuous and T -periodic functions R → R 2 , endowed with the | · | ∞ -norm. Here L : C T ⊃ dom L → C T is the linear differential operator z → z , while N λ : C T → C T is the superposition operator z → Z(·, z, λ) (if necessary we can extend Z by continuity and periodicity to R × R 2 × [0, 1] in order to have N λ defined on all C T ). For any ε ∈]0, 1[ we also define If we find a compact subset R of (R + ) 2 such that holds for some λ ∈ [0, 1], then, by excision, the coincidence degree D L (L−N λ , Ω(ε)) is well defined and constant with respect to ε for ε > 0 sufficiently small (see [25,Proposition II.1]). Hence, we can define the index A standard consequence of the invariance under homotopy and existence property of the degree (see [25, pages 15-16]) reads as follows.
See [25] and the references therein and [16] for various applications of Lemma 1. See also [25,Chapter 3] and [9] for concrete examples of computation of i λ (Z). We notice that assuming i 0 (Z) = 0, we implicitly suppose that i 0 (Z) is defined, that is, (6) holds for λ = 0.
2. Main results. Let P, Q : R × (R + ) 2 → R be continuous functions which are T -periodic in their first variable. Throughout this section we assume there is a continuous and T -periodic function p 0 : R → R + such that any possible positive satisfies Note that if (7) (7) is nonempty, compact and it has a minimum (cf. [32], [33] ). In this case, p 0 is precisely such a minimal positive function.
Let us define P 0 (t) := P(t, 0, 0) and assume further there are continuous and T -periodic functions A(·), P ∞ (·), Q 0 (·) and Q ∞ (·) such that lim sup If (7) has no positive T -periodic solutions, then we consider (11) to hold provided that Q(t, x, 0) ≥ Q ∞ (t) for sufficiently large x > 0. In this case, we can equivalently replace (11) by assuming lim inf x→+∞ Q(t, With the above positions, we can state the following.
and for each K > 0 there are M K > 0 and a continuous and T -periodic function Assume further that Then there is a compact set in (R + ) 2 containing all the possible positive T -periodic solutions of system (1) and (1) has at least one positive T -periodic solution, provided that Before starting with the proof of Theorem 1, we give a simple corollary which applies to the case when the function Q does not depend on the q-variable. A similar case was also considered in [15], [18], but we point out that here we don't assume any monotonicity condition like in [15], [18] on the functions P and Q. Suppose and also define Q 0 (t) := Q(t, 0, 0), so that (10) holds. Observe that now (11) writes as Then we have that assumption (12) is an obvious consequence of the continuity of Q, while for any K > 0 we can take M K = max [0,T ] p 0 and Q K = Q ∞ and thus have (13) satisfied. Then we can conclude with the following: (9), (16) and (14) hold. Then there is a compact set in (R + ) 2 containing all the possible positive T -periodic solutions of (15) and system (15) has at least one positive T -periodic solution.
Proof of Theorem 1. First of all, we adapt some of our assumptions to a form which is more directly applicable to the computations we have to perform below. Accordingly, from (8), (9) and the second inequality in (14) take a constant M > 0 and two continuous and T -periodic functions α(·) ≥ 0 andP ∞ (·) such that Then, let us set, for z = (x, y) ∈ (R + ) 2 , uniformly in t. Recall also that (14) gives Thus we are in the same situation as in [14,Theorem 3] for the perturbed conservative system p = pP(t, 0, q), q = qQ(t, p, 0), (which corresponds to z = Z(t, z, 0)) and therefore we know that all its positive T -periodic solutions are contained in a compact subset of (R + ) 2 and also (cf. [14, proof of Theorem 1]) i 0 (Z) = 1.
Hence, thanks to Lemma 1 we have only to find "a priori bounds" in (R + ) 2 for the T -periodic solutions of This result will be attained through some steps. From now on in the rest of the proof, (p, q) is any T -periodic solution to (17) for some λ ∈]0, 1]. Note that we assume p(t) > 0 and q(t) > 0 for all t ∈ R. Step Step 2. ∃ L > 0 : min p = p < L. According to (13), let us take a positive constant C * such that C * > sup{ |Q K | 1 : K > 0} and set Assume by contradiction that p(t) ≥ L, for all t ∈ R. If this is the case, we claim that we must have max R λq > H. Otherwise, if λq(t) ≤ H for all t, we obtain from the second equation in (17) which is absurd.
So we know that min λq ≤ q < M (from Step 1) and max λq > H (just proved). By the periodicity of λq we can find an open interval ]t 1 , Hence, integrating over I we obtain that is, H < M exp C * . This contradicts the choice of H and proves Step 2.
Step 3. ∃ D > 0 : max p = p < D. Indeed, we have just seen that there is t 0 ∈ [0, T ] such that p(t 0 ) < L. By the upper bound on P and the first equation in (17) we have that p (t)/p(t) ≤ α(t) and hence, Then, using the periodicity of p(·), we find p(t) < L exp |α| 1 := D, for all t ∈ R.
Step 4. ∃ C > 0 : max q = q < C. Since p(t) < D for all t ∈ R, by the assumption (12) we know that there is a positive constant E such that Q(t, p(t), λq(t)) ≤ E, for all t ∈ R. Then, using Step 1 and the second equation in (17), we obtain q < M exp(ET ) := C.
Step 5. ∃ δ > 0 : q > δ. Indeed, suppose by contradiction that for a sequence (p n , q n ) of positive T -periodic solutions of (17) for λ = λ n ∈]0, 1], one has q n → 0 uniformly in t ∈ R. Since from the first equation of (17) we have T 0 P((t, λ n p n (t), q n (t)) dt = 0, and T 0 P(t, 0, 0) dt = T P 0 > 0, it is straightforward to check that there is ε 0 > 0 such that max λ n p n ≥ ε 0 , for all n. Now, as λ n p n (t)/λ n p n (t) = p n (t)/p n (t) = P(t, λ n p n , q n (t)) ≤ α(t), we easily obtain that there is ε > 0 such that min λ n p n > ε for all n (use the periodicity of λ n p n (·) and take ε = ε 0 exp(−|α| 1 ) ). By Step 3 we know that ε ≤ λ n p n ≤ D, for all t ∈ R and each n. On the other hand, we have that 0 < q n (t) ≤ 1 for all t and each n sufficiently large, say n ≥ n * (as q n → 0). Hence, for all t ∈ R and n ≥ n * , |(d/dt)(λ n p n (t))| ≤ K, where Ascoli -Arzelá theorem now implies that (λ n p n ) n has a subsequence converging uniformly to a positive continuous and T -periodic function z(·). For notational convention, here and below, we denote the converging subsequence with the same symbols like the original one. Moreover, from λ n p n (t) = λ n p n (t)P(t, λ n p n (t), q n (t)), we obtain that z(·) is continuously differentiable and it satisfies z (t) = z(t)P(t, z(t), 0), for all t ∈ R. Hence z(·) is a positive T -periodic solution of (7) and from our assumptions we conclude that z(t) ≥ p 0 (t), for all t ∈ R. Recall now Step 3 according to which we know that 0 < p n (t) < D, for all t and n and |p n (t)| = |p n (t)P(t, λ n p n (t), q n (t))| ≤ K for all t and n ≥ n * . Hence, repeating the above argument, it is easy to see that there arep with 0 ≤p(t) ≤ D andλ ∈ [0, 1] such that, passing to the limit on a common subsequence of indexes (which will be common also to the subsequence converging to z), we finally have p n →p uniformly in R and λ n →λ. Then,λp(t) = z(t) for all t, so that 0 <λ ≤ 1 and thus p(t) = z(t)/λ ≥ p 0 (t), for all t ∈ R. On the other hand, from the second equation in (17) we have T 0 Q(t, p n (t), λ n q n (t)) dt = 0 and hence, passing to the limit on the above subsequence, we have T 0 Q(t,p(t), 0) dt = 0. But now we can use condition (11) according to which Q(t,p(t), 0) ≥ Q ∞ (t), and thus we obtain T 0 Q ∞ ≤ 0. This contradicts the last inequality in (14) and thus our result is proved.
Step 6. ∃ σ > 0 : q > σ. This follows from an easy inequality, as q > δ and |q /q| is bounded by Step 7. ∃ ε > 0 : p > ε. Indeed, if by contradiction there is a sequence (p n , q n ) of positive T -periodic solutions of (17) for λ = λ n ∈]0, 1], with p n → 0 uniformly in t ∈ R, then, as 0 < λ n q n (t) ≤ C for all t, arguing as in Step 5, by the Ascoli -Arzelá theorem and λ n q n (t) = λ n q n (t)Q(t, p n (t), λ n q n (t)), we can find a subsequence of (λ n q n ) n converging uniformly to a non-negative T -periodic functionỹ(·). Using T 0 Q(t, p n (t), λ n q n (t)) dt = 0 and passing to the limit on this subsequence, we obtain T 0 Q(t, 0,ỹ(t)) dt = 0. Then, as Q(t, 0,ỹ(t)) ≤ Q 0 (t) , for all t, we conclude that T 0 Q 0 ≥ 0, a contradiction to the third inequality in (14).
Step 8. ∃ η > 0 : p > η. This is like in Step 6. At the end, we have proved that any solution (p, q) of (17) satisfies 0 < η < p(t) < D and 0 < σ < q(t) < C, for all t ∈ R. On the other hand, as pointed out at the beginning of the proof, for λ = 0, we have already the result in [14]. So all the assumptions of Lemma 1 are satisfied and therefore equation (1) has at least one positive T -periodic solution. Moreover, we have proved that there is a compact set in (R + ) 2 which contains all the positive T -periodic solutions of (1).

Remark 1. (i) If P = P(t, q)
and Q = Q(t, p), many of the assumptions in Theorem 1 are vacuously satisfied. In this case, Theorem 1 is exactly the same like the first part of the statement in Theorem 3 of [14]. From this point of view, Theorem 1 generalizes to the Kolmogorov system (1) the existence result for positive harmonic solutions obtained in [14]. (ii) In the case that the equation p = P(t, p, 0) does not possess positive T -periodic solutions, then the choice of the function p 0 (·) is free. Hence, with respect to (11) we have only to require that Q(t, x, 0) ≥ Q ∞ (t) for x sufficiently large, say x ≥ M. (iii) The assumptions of Theorem 1 fit rather naturally when applied to prey-predator equations. To show this aspect, we present the following example. Example 1. Consider the equation where a, d : R → R and b, c, e, f : R → R + = [0, +∞) are continuous functions which are all periodic of a common period T > 0. We also suppose that Note that this is a typical assumption for the growth of the prey population p in absence of the predator q and the growth of q in absence of p. In order to apply Theorem 1 to equation (18) we observe that P and Q are like in (2). Now, let us consider the assumptions of our main result. We see that (12) is always true and (8) is trivially satisfied with A(t) = |a(t)|. Moreover P 0 = a and we can take Q 0 = −d, so that P 0 > 0 and Q 0 < 0 follow from (19). To have (9) satisfied by a suitable function P ∞ satisfying the second inequality in (14) it is sufficient to require c > 0 (20) which, as c ≥ 0, is the same like max c > 0 or c ≡ 0. On the other hand, if we assume either f ≡ 0 and e > 0, or min e > 0 (21) then either we are in the same situation like in Corollary 1 (case f ≡ 0), or for each K > 0 we can take M K = (1 + d + f K)/e and Q K = 1 in order to have (13) satisfied. Finally, it remains to consider equation (7) which now reads as An elementary analysis of this equation with a > 0 and b ≥ 0, shows that there are no positive T -periodic solutions if b ≡ 0 and there is exactly one positive and T -periodic solution w(·) which is globally asymptotically stable with respect to R + if b > 0. The explicit form of w is given by

In fact, z(t) = w(t) −1 is the unique T -periodic solution of the linear equation z = b(t) − a(t)z which can be solved explicitly.
Therefore, we have two possibilities to discuss: b ≡ 0 and b ≡ 0. In the former case we can take p 0 an arbitrary large constant and have (11) and the last inequality in (14) fulfilled by a suitable choice of Q ∞ , as a consequence of (21). In the latter case, we can take p 0 = w and, using the fact that in this example Q(t, ·, 0) is nondecreasing, we can choose Q ∞ (t) = −d(t) + e(t)w(t). Thus we can conclude as follows.

Remark 2.
Consider again equation (18) in the case that the coefficients are non-negative constants, so that we have For the applicability of Corollary 2 to system (24) we have to assume Note that no condition on f is required.
In the former case, if also f = 0, we have the Volterra equation and for it we know that there is a unique equilibrium point z * in (R + ) 2 and there is a limit positive period T * of the orbits approaching z * such that any other periodic solution of (24) has period larger than T * and the period of the positive periodic solutions tends increasingly to infinity as their amplitudes increase (cf. [22], [31]). So, if T ≤ T * (with T * computable explicitly in terms of the coefficients a, c, d, e) we have z * as unique positive T -periodic solution of (24) and if T > T * we have nonconstant positive T -periodic solutions as well; on the other hand, we don't have a compact attractor in (R + ) 2 so that "permanence" does not occur. In the latter case, if also f = 0, we obtain that there is a unique equilibrium point z * in (R + ) 2 which is globally asymptotically stable with respect to its basin of attraction (R + ) 2 (cf. [17]). Thus "permanence" occurs, but, on the other hand, we don't have any other positive periodic solution (of any period). This shows that the conclusion we obtain with Theorem 1 cannot be reinforced with respect to prove the existence of a compact attractor in (R + ) 2 , like in [7] and [29] and, at the same time, we cannot prove multiplicity results of periodic solutions (harmonic or subharmonic) like in [12]- [14] and [22], as our model equation (1) contains both examples of conservative and dissipative systems.
As we already observed before, Theorem 1 is devised for the obtention of an existence result which is applicable to the case in which the dynamics of the prey population in absence of predator, given by equation (7), may be influenced or may not be influenced by some logistic growth term. Indeed, as shown in Example 1, for the corresponding single species equation (22) we can have b ≡ 0 or b ≡ 0 as well. Next we show how a simple change in the assumptions of Theorem 1 yields a related existence result where the presence of a self-inhibition coefficient in equation (7) can more effectively show its effects. Namely, we have the following. Proof.. Besides the positions at the beginning of the proof of Theorem 1, we also observe that from (8) and (25) we can find a constant B > 0 and a continuous and T -periodic functionÃ(·) such that Then, let use set, for z = (x, y) ∈ (R + ) 2 , Z(t, z, λ) := xP(t, λx, y), yQ(t, x, 2(λ − (1/2))y) , for 1/2 ≤ λ ≤ 1 xP(t, λx, y), yQ(t, x, 0) , for 0 ≤ λ ≤ 1/2 so that Z(t, z, 0) = xP(t, 0, y), yQ(t, x, 0) . Now we repeat almost verbatin the proof of Theorem 1 taking into account that instead of equation (17) we have to consider p = pP(t, λp, q), q = qQ(t, p, 2(λ − (1/2))q), for 1/2 ≤ λ ≤ 1 p = pP(t, λp, q), q = qQ(t, p, 0), for 0 < λ ≤ 1/2 (26) with (p(t), q(t)) ∈ (R + ) 2 , ∀ t ∈ R. The only point in which some change in the proof is needed is that ∃ L > 0 : min p = p < L. Indeed, if λ ∈ [1/2, 1] and p(t) ≥ 2B for all t ∈ R, then λp(t) ≥ B and from the first equation in (26) we have 0 = p /p = P(·, λp(·), q(·)) ≤ Ã < 0, a contradiction. On the other hand, if λ ∈]0, 1/2] and p(t) ≥ max p 0 for all t ∈ R, then from (11) and the last equation in (26) we have 0 = q /q = Q(·, p(·), 0) ≥ Q ∞ > 0, a contradiction. Hence the claim is proved for L = max{2B, p 0 }. The proof of all the other steps is just the same as above or needs only obvious changes with respect to that of Theorem 1.
Example 1 (continued). Consider again equation (18) where a, d : R → R and b, c, e, f : R → R + = [0, +∞) are continuous functions which are all periodic of a common period T > 0. We now suppose that Since b > 0, we know that equation (7) which corresponds to (22) has a unique positive and T -periodic solution w(·) which is globally asymptotically stable with respect to R + . Hence, Theorem 2 yields the following consequence.
Then the same conclusion of Theorem 1 holds for system (18).
Observe that we don't require any condition on f ≥ 0. We also notice that even if there is some overlapping between Corollary 2 and Corollary 3, as well as between Theorem 1 and Theorem 2, nevertheless it is possible to give examples showing that these results are independent each other. With this respect, let us choose arbitrarily a : R → R and b, c : R → R + continuous, T -periodic and such that a > 0, b > 0 and c > 0 and consider the positive and T -periodic function w(·) which comes from the solvability of (22). Now define  (18) previously obtained by Cushing [10] (see also [4], [7], [24], [29] for related conditions). Of course, we don't claim that we do better, since in these articles more informations on the dynamics of the solutions are given, on the other hand, our result improves the above quoted ones with respect to the existence and boundedness of T -periodic solutions. Indeed, to make a comparison with some of the preceding results, let us observe that if t ∈ R is a point of maximum or a point of minimum of the solution w(t) of (22), then w (t) = 0 and therefore a(t) = b(t)w(t). Thus, if we further suppose that a(t) > 0 and b(t) > 0 for all t ∈ [0, T ] (a condition which we don't need in our results but we assume here for the next discussion), then we have Hence (28) is satisfied provided that Therefore, a special case of Corollary 3 yields the existence of positive T -periodic solutions for equation (18), if (29)

{e(t)/b(t)}
and, taking the mean value on w /w = a − bw, we also obtain w b = a . Hence (28) is satisfied provided that Therefore, a special case of Corollary 3 yields the existence of positive T -periodic solutions for equation (18), if (30)  Corollary 3 may be complemented with a necessary condition for the existence of positive periodic solutions for equation (18). Namely, we have:  (22). Moreover, if (p,q) is any positive T -periodic solution of Proof.. Let (p,q) be a positive T -periodic solution of (18), so that we can consider p as a solution of equation Now, let v(·) be the solution of (22) with v(s) =p(s), where s ∈ [0, T ]. We know that v(t) > 0 for all t ≥ s , as v(s) > 0. Sinceã(t) ≤ a(t) for all t ≥ s , by a standard result on differential inequalities (cf. [20] ) we know thatp(t) ≤ v(t) for all t ≥ s . Furthermore, as (
From here, by induction, it is easy to see that the sequence (v(s + nT )) n is strictly increasing (arguing like in the proof of the classical Massera's theorem). On the other hand, v(·) is a solution of (22) and we already know that w(·) is the globally asymptotically stable T -periodic solution of (22), so that we obtain that lim n→+∞ v(s + nT ) = w(s).
We have thus proved that and therefore (28) holds. Thus we have proved the necessary condition. For the sufficient condition, just apply Corollary 3.

Remark 4. (i)
In all the article we have confined ourselves to show the applicability of Theorem 1 and Theorem 2 only to the model equation (18). Our choice has been motivated by the interest in finding some simple conditions on the coefficients for the validity of the main results as well as to the possibility of making a comparison with some preceding papers dealing with the prey -predator Lotka -Volterra equations. It is clear that the range of applicability of our theorems is much wider and in particular there are various Kolmogorov systems of "ecological" interest where our results can be successfully employed.
(ii) In the proofs of Theorem 1 and Theorem 2 we use only integral conditions on P and Q. Thus it is not difficult to extend all the above results by relaxing the continuity assumptions on P and Q to Caratheodory conditions (see [20], [25] ). Of couse, some of the hypotheses we used would require now a slight change, like for instance to replace systematically "for all" t with "for almost every" t in the inequalities concerning P and Q, to relax the regularity of the limiting functions A, P 0 , P ∞ , Q 0 , Q ∞ and Q K , or to bound some sup{. . . } by an appropriate L 1function instead of a constant (in particular, it will be necessary to modify suitably condition (12) ).