AN APPLICATION OF THE DUAL VARIATIONAL PRINCIPLE TO A HAMILTONIAN SYSTEM WITH DISCONTINUOUS NONLINEARITIES

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1 Electronic Journal of Differential Equations, Vol. 2004(2004), No. 46, pp. 2. ISSN: URL: or ftp ejde.math.txstate.edu (login: ftp) AN APPLICATION OF THE DUAL VARIATIONAL PRINCIPLE TO A HAMILTONIAN SYSTEM WITH DISCONTINUOUS NONLINEARITIES CLAUDIANOR O. ALVES, DANIEL C. DE MORAIS FILHO, & MARCO AURELIO S. SOUTO Abstract. In this article, we study the existence of solutions to the Hamiltonian elliptic system with discontinuous nonlinearities u = au + bv + f(x, v), v = cu + av + g(x, u) on a bounded subset of R n, with zero Dirichlet boundary conditions. functions f and g have a finite number of jumping discontinuities. The. Introduction Differential equations with discontinuous nonlinearities play an important role in modelling problems in mathematical physics and in different applications in other fields. In this work, we give some contributions to the study of systems of equations with discontinuous nonlinearities. Our goal concerns finding non-trivial solutions to the system u = au + bv + f(x, v) v = cu + av + g(x, u) (.) u = v = 0 on, ( ) a b where R N, N 3 is a smooth bounded domain, A = is a matrix c a of real entries, f, g : R R are functions with a finite number of jumping discontinuities whose properties will be detailed later. In the scalar case, problems with discontinuous nonlinearities have been studied in the previous decades; see for example [3, 4, 6, 7, 8] and the references therein. These problems model many physical phenomena, [9, 0], such those of plasma physics in the case of the distribution of temperature in an electric arc for which the constitutive law contains a discontinuity [2, ]. Many physical interesting cases also arise when the discontinuity is of the form h(s θ)f(s) where h(s) = 0, for 2000 Mathematics Subject Classification. 35J50, 37K05, 34A34. Key words and phrases. Hamiltonian systems, discontinuous nonlinearities, dual variational principle. c 2004 Texas State University - San Marcos. Submitted January 26, Published March 3, Research partially supported by the Milênio Project/CNPq/Brazil.

2 2 C. O. ALVES, D. C. DE MORAIS FILHO, & M. A. S. SOUTO EJDE-2004/46 s 0 and h(s) =, for s > 0, is the Heaviside function, and f is a continuous function. Form the mathematical point of view we observe that the classical variational method can not be used directly to these problems since the Euler-Lagrange functionals associated to them are not of class C. In these cases, Convex Analysis and the Dual Variational Method are powerful tools to be employed. Motivated by the aforementioned facts in the scalar case, more specifically, by the papers [3] and [4], we study a class of hamiltonian systems with discontinuous nonlinearities such those introduced in (.). With this intention, we had to develop some specific techniques and procedures in order to apply the Dual Variational Method to attack the system case problems. This article is schemed as follows: In Section 2 we state some preliminaries definitions and results in order to state the main results (Theorems 3. and 3.2) in Section 3. In Section 4 we prove Theorem 3., and in Section 5 we give an application of this theorem. Finally in Section 6 we prove Theorem Preliminaries To state our main theorem we need to establish some notation. Let us consider a function f : R R discontinuous only at x = θ, but such that the limit from the right f(θ + ) and from the left f(θ ) exist, where f(θ ± ) = lim x θ ± f(x). Suppose that there is m > 0 such that the function ms + f(s) is strictly increasing. We set the interval T f,θ = [f(θ ), f(θ + )] and define the following multivalued function induced by filling in the jump of the discontinuous function ms + f(s) at x = θ: { f(s) + ms, s θ ˆf m (s) = T f,θ + mθ, s = θ. Now it is possible to define a single valued function (the inverse of ˆf m ) { θ, if t T f,θ + mθ f m (t) = s, with ms + f(s) = t, if t / T f,θ + mθ such that f m (t) = s if and only if t ˆf m ((s). ) It is easy to check that f m C(R). mu Let us go back to system (.). Add, m, n > 0, to both sides of (.), so nv that the right hand side functions become f m (v) := f(v) + mv, g n (u) := g(u) + nu which are strictly increasing. Following the previous discussion, we may define the functions ˆf m, ĝ n, and their respective inverses, f m, g n. For simplicity, we denote f = f m and g = g n.

3 EJDE-2004/46 AN APPLICATION OF THE DUAL VARIATIONAL PRINCIPLE 3 Remarks. (i) The results proved in this paper also work for functions with a finite number of jumping discontinuities. Moreover, it completes the study made in [3] and [4] for the elliptic systems case. (ii) For simplicity we shall deal with the autonomous case, when f(x, v) = f(v) and g(x, u) = g(u). The stated general case follows with simple changes. Let us denote p(λ, A) = ( (λ a) ) 2 bc 0 m J m,n = n 0 A m,n = A J m,n We denote by λ j the eigenvalues of the eigenvalue problem (, H 0 ()), subjected to Dirichlet boundary condition, and by ϕ j its corresponding eigenfunctions. We define system (.) as resonant if p(λ, A) = 0 and nonresonant otherwise. 2.. The Linear Case. To apply the dual variational method to system (.), we have to study, for m, n > 0 to be picked up later, the system where f, g L 2 (). u = au + (b m)v + f(x) v = (c n)u + av + g(x) u = v = 0, on (2.) Theorem 2.. For small m, n > 0, system (2.) has a unique solution (u, v) H 0 () H 0 () for each pair (f, g) L 2 () L 2 (). Proof. The Laplacean operator subjected to Dirichlet boundary conditions is invertible and ( ) : L 2 () H 0 () H 2 () H 0 (). Thus, it is possible to define the operator where T Am,n : L 2 () L 2 () H 0 () H 0 () F = (f, g)) T Am,n F ( ) ( ) a( ) (m b)( ) T Am,n F = f (n c)( ) a( ). g Now, system (2.) is equivalent to U + T Am,n U = G, U = 0 on where U = (u, v) and G = (( ) f, ( ) g). The proof of the proposition relies on the following remarks: T Am,n is a continuous operator, since ( ) is. By the compact Sobolev embedding, the operator T Am,n : L 2 () L 2 () L 2 () L 2 () is compact. I + T Am,n is a one-to-one operator for small m and n.

4 4 C. O. ALVES, D. C. DE MORAIS FILHO, & M. A. S. SOUTO EJDE-2004/46 Indeed, suppose that (u, v) (0, 0) satisfy (2.) with f = g = 0. Then ( ) uϕ j dx, vϕ j dx := (X, Y ) (0, 0) for some j. Hence, multiplying both equations in (2.) by ϕ j and integrating by parts we achieve ( ) X (A m,n λ j I) = 0. Y Since (X, Y ) (0, 0), we have that p(λ j, A m,n ) = 0. However it is possible to find small m and n, such that p(λ j, A m,n ) 0. Hence (u, v) = (0, 0). The proposition follows from the Fredholm Alternative. The solution operator. The above proposition allows us defining the continuous operator (In fact, a compact operator) B : L 2 () L 2 () L 2 () L 2 () (ω, ω 2 ) (u, v) where B(ω, ω 2 ) = (u, v) if and only if for (u, v) H 0 () H 0 (), u = au + (b m)v + ω 2, v = (c n)u + av + ω,. 3. The Dual Variational Framework and the main Theorems (2.2) The use of the Dual Variational Principle allows us to find a solution for (.) as a minimum or as a critical point of a certain C functional associated with the system. Let the Hilbert space W := L 2 () L 2 () be endowed with the norm. (u, v) 2 W := u v 2 2 where u 2 2 = u 2 dx. We define the functional Ψ : W R by Ψ(ω, ω 2 ) = G(ω )dx + F (ω 2 )dx Bω, ω dx 2 where ω = (ω, ω 2 ), G(t) = t 0 g (σ)dσ, F (t) = t 0 f (σ)dσ, and Bω, η = (u η + u 2 η 2 )dx, if Bω = (u, u 2 ) and η = (η, η 2 ). Since A is symmetric, we have Bω, η = ω, Bη. With additional hypotheses on the non-linearities we assure that Ψ C (W, R). Hence, a straightforward calculation leads to Ψ (ω, ω 2 ), (η, η 2 ) = (g (ω )η + f (ω 2 )η 2 )dx Bω, η dx. (3.) We see that a pair (ω, ω 2 ) W is a critical point of Ψ if and only if B(ω, ω 2 ) = (g (ω ), f (ω 2 )). Therefore, if (u, v) = (g (ω ), f (ω 2 )), then ( u au + (m b)v, v + (n a)u cv) ( ˆf m (v), ĝ n (u)). (3.2) In our context, a pair (u, v) satisfying the above inclusion is said to be a solution to (.).

5 EJDE-2004/46 AN APPLICATION OF THE DUAL VARIATIONAL PRINCIPLE 5 The following are our main results. Theorem 3. (The nonresonant case). Suppose that f and g are real functions with jumping nonlinearities at θ and ξ, respectively. If (ω, ω 2 ) W is a minimum for Ψ, p(λ, A) > 0 (3.3) holds and then (u, v) := (g (ω ), f (ω 2 )) u(x) = au(x) + bv(x) + f(v(x)) v(x) = cu(x) + av(x) + g(u(x)) a.e. for x, i. e. (u, v) is a strong solution for system (.). Theorem 3.2 (A resonant case). Let f(v) = αv + a(v) and g(u) = βu + b(u), be real functions with jumping nonlinearities at θ and ξ, respectively, such that 0 < α, β ; α, β = λ 2 (3.4) If then the system has a strong solution. lim a(v) = v ± a±, a < 0 < a + lim b(u) = u ± b±, b < 0 < b +. (3.5) (0, 0) / (T f,θ, T g,ξ ) (3.6) u = αv + a(v), v = βu + b(u), u = v = 0, on (3.7) Regarding the proofs of the previous theorems, following our approach, we will see that the variational techniques lead to a pair (u, v) H 0 () H 0 () such that (3.2) holds. Hence, to assure that (u, v) is in fact a solution for (.) it suffices to prove that the sets ξ = {x : u(x) = ξ} and θ = {x : v(x) = θ} have zero Lebesgue measure. These sets play an important role when inverting the functions ˆf m, ĝ n. For the well known example in the scalar case u = h(θ v)f(v), u = 0, on (3.8) by a result of Stampacchia [5], u = 0 a.e. on θ. Therefore, the set θ does not play any important role in this case, and follows immediately that the found wanted critical point u is a strong solution for (3.8), since h(0) = 0. However due to the coupling of similar equations in the system case, this fact does not occur and others procedures are demanded. Theorem 3. may be applied to study these cases.

6 6 C. O. ALVES, D. C. DE MORAIS FILHO, & M. A. S. SOUTO EJDE-2004/46 4. Proof of Theorem 3. As mentioned before, the proof consists in showing that the sets θ and ξ have zero Lebesgue measure. Firstly, let us prove that ξ = 0, where is the Lebesgue measure. Let T = T g,ξ + nξ = [c, c 2 ], T + = [c, /2 (c + c 2 )], T = T T +, where c = g(ξ ) + nξ, c 2 = g(ξ + ) + nξ and ± = {x ξ : ω (x) T ± }. Define the function χ L 2 () by, x + χ(x) =, x (4.) 0, x ξ. For t > 0, small enough, we have that ω (x) + tχ(x) T, a.e. for x ξ. Since (ω, ω 2 ) E is a minimum for Ψ, we have that d dt Ψ(ω + θtχ, ω 2 ) 0 for some 0 < θ <. Hence, by (3.) Ψ (ω + θtχ, ω 2 ), (χ, 0) = g (ω + θtχ)χ dx uχdx θt B(χ, 0), (χ, 0) dx 0. On the other hand g (ω + θtχ)χ dx = g (ω + θtχ)χ dx = ξ ξ χ dx ξ (4.2) and uχ dx = uχ dx = ξ ξ χ dx. ξ Accordingly, the difference between the first two members of the right hand side of (4.2) is zero and thus B(χ, 0), (χ, 0) dx 0. (4.3) For t < 0, replacing χ by χ in the afore steps we get that B(χ, 0), (χ, 0) dx 0. (4.4) Consequently, by (4.3) and (4.4) Let B(χ, 0) = (u, v ). Then by (2.2) and (4.5) B(χ, 0), (χ, 0) dx = 0. (4.5) u = au + (b m) v, v = (c n) u + av + χ, u = v = 0, on χu dx = 0. (4.6)

7 EJDE-2004/46 AN APPLICATION OF THE DUAL VARIATIONAL PRINCIPLE 7 Claim. Let u, v H 0 () and χ L 2 () satisfy (4.6). If (3.3) holds, then u = v = 0. Proof. Multiplying the first equation in (4.6) by v, the second by u respectively, and integrating by parts we achieve vdx 2 = c n u 2 b m, for small m, n > 0. (4.7) Hence both c n and b m have the same signs. By the first equation in (4.6) λ u 2 dx u 2 dx = a u 2 dx + (b m) u v. Therefore, if b m > 0 λ By (4.7) and (4.8), and thus u 2 dx a u 2 dx + (b m) u 2 v 2. (4.8) (λ a) u 2 dx (b m) (c n) (b m) u 2 dx { (λ a) } (b m) (c n) u 2 dx 0. But (λ a) > (b m) (c n), for small m and n, since p(λ, A) > 0. This yields that u = 0. Then v = 0. When b m < 0 the proof follows analogous steps. By the above claim, we infer that u v 0 and hence χ 0. By the definition of the function (4.) we conclude that ξ = 0. reasoning applies to show that θ = 0. A similar 5. An application of Theorem 3. Our application of Theorem 3. consists of coercive functional bounded from below and associated to system (.). It is well known that this kind of functional has a minimum. Theorem 5.. Let f and g satisfy and define Suppose that and that (3.3) holds. H0 () H0 (). f(s) c + k s, g(s) c 2 + k 2 s (5.) 0 < K := max{k, k 2 } < λ (A) : (λ a) + max{ b, c }. p(λ, A) a < λ, bc > 0 (5.2) Then, system (.) possesses a strong solution (u, v)

8 8 C. O. ALVES, D. C. DE MORAIS FILHO, & M. A. S. SOUTO EJDE-2004/46 Proof. By (5.) G(w )dx 2K w 2 2 C w 2, (5.3) F (w 2 )dx 2K w C w 2 2. (5.4) Let us estimate Bw, w dx. Let us first suppose that b, c > 0. By (2.2), Poincaré inequality and Hölder inequality we have Since λ > a, by the above inequalities (λ a) u 2 (b m) v 2 + w 2 2 (λ a) v 2 (c n) u 2 + w 2. p(λ, A m,n ) b m u 2 w 2 λ a λ a w w 2 w 2 2 p(λ, A m,n ) c n v 2 w 2 2 λ a λ a w w 2 w 2 2. If m, n are such that p(λ, A m,n ) > 0, inequalities (5.5) yields (5.5) u 2 w 2 + v 2 w 2 2 (λ a) + max{ b m, c n } p(λ, A m,n ) ( w w 2 2 ) 2. Denoting we see that a(m, n) = (λ a) + max{ b m, c n } p(λ, A m,n ) Bw, w dx a(m, n) ( w w 2 2 2). (5.6) If b, c < 0 a similar reasoning leads to the last inequality. Observe that lim m,n 0 a(m, n) = / λ. Choosing m, n such that K, by (5.3), (5.4) and (5.6) we get λ < a(m, n) < Ψ(w, w 2 ) ( w w 2 2 a(m, n) ( 2K 2) w w 2 2 ) 2 C ( w 2 + w 2 ). 2 Thus, Ψ(w) is bounded from below and coercive and hence it has a minimum (w, w 2 ). 6. Proof of Theorem 3.2 In this resonant case, the operator B is compact and defined as { u = w 2 B(w, w 2 ) = (u, v) (6.) v = w The proof follows from the next two theorems. Firstly let us prove that the functional Ψ associated to the system in this case has a linking at the origin.

9 EJDE-2004/46 AN APPLICATION OF THE DUAL VARIATIONAL PRINCIPLE 9 Theorem 6.. Suppose that (3.4) and (3.5) hold. Then Ψ has a linking at the origin, i.e., there exist two subspaces Z, V W (:= L 2 () L 2 () ) such that W = Z V, dim Z < and there exists (z, z 2 ) Z such that lim Ψ(tz, tz 2 ) = (6.2) t + inf V Ψ > (6.3) Proof. For short notation we shall consider m = n = 0. By (3.4) and (3.5) it follows that G(w ) = w2 2β p(w ) F (w 2 ) = w2 2 2α q(w 2). Where (w, w 2 ) = w W, p and q are functions with linear growth such that (6.4) lim p(s) = lim q(s) = ±. (6.5) s ± s ± Accordingly, by (6.4), we have that Ψ(w, w 2 ) = ( w 2 2 β + w2 ) 2 dx (p(w ) + q(w 2 ))dx Bw, w dx (6.6) α 2 Let us choose (w, w 2 ) = (aφ, bφ ) := (z, z 2 ), where a, b R will be picked up later. If B(z, z 2 ) = (u, v), then Bz, z dx = 2ab φ 2 λ dx, z = (z, z 2 ). (6.7) Hence, if a 0, b 0 > 0 is chosen such that using (3.4) we have (a 0 b 0 ) 2 = β α [ (z 2 2 β + z2 ) ] 2 dx Bz, z dx = [a 2 0 α 2 = 2 ] λ β + b2 0 α 2a 0b 0 ( a 0 b 0 ) 2 β α φ 2 φ 2 = 0 (6.8) Therefore, by (6.5) and the above result, we obtain Ψ(tz, tz 2 ) = (p(tz ) + q(tz 2 ))dx, as t +. (6.9) Before proving (6.3) we need some remarks. Observe that we may decompose W = Z V. Where Z = (z, z 2 ) (the space generated by (z, z 2 ) W ) and V = {(rφ + v, sφ + v 2 ) : v, v 2 φ and r, s R, a 0 r + b 0 s = 0}.

10 0 C. O. ALVES, D. C. DE MORAIS FILHO, & M. A. S. SOUTO EJDE-2004/46 Claim. If v = (v, v 2 ) E, and v, v 2 φ, then Bv, v dx 2 λ 2 v 2 v 2 2. (6.0) Indeed, it follows from (6.) that if Bv = (u, v), then Bv, v dx = uv + vv 2 = 2 u v. If {φ j } is the orthonormal base of L 2 () composed of eigenvalues of the Laplacean we may write v = j=2 α jφ j and hence v 2 = u = k=2 α2 kφ j j=2 α j φ j v = αk 2φ k k=2 λ k Using that λ < λ 2 λ and the liner product properties it follows that u v v 2 v 2 2. (6.) λ 2 Thus, (6.0) is proved. Now, by (6.0), if v = (v, v 2 ) φ φ we obtain that ( v β + v α )dx Bv, v 2 2 [( v 2 ) 2 2 v 2 v ( v 2 2 ) 2 ] β λ 2 α since the discriminant of this quadric for is negative, because λ 2 < = 2 αβ λ 2, there exists k > 0 such that it is greater or equal to k( v v 2 2 2). Thus ( v β + v 2 2 ) 2 Bv, v k( v v 2 2 α 2 2). (6.2) Let (rφ + v, sφ + v 2 ) V. Therefore, by (3.4) and (6.2), Ψ(rφ + v, sφ + v 2 ) = [ ( r 2 2 β + s2 ) ] φ 2 α dx B(rφ, sφ ), (rφ, sφ ) dx + [ ( v 2 2 β + v 2 2 ) ] φ 2 α dx Bv, v dx (p(rφ + v ) + q(sφ + v 2 )dx [ ( r 2 2 β 2rs + s2 ) ( v φ 2 λ α dx + k 2 + v 2 2)] (p(rφ + v ) + q(sφ + v 2 )dx λ j and since r β s α inf V Ψ >. 0 and the functions p and q have linear growth we have that

11 EJDE-2004/46 AN APPLICATION OF THE DUAL VARIATIONAL PRINCIPLE Theorem 6.2. With the hypotheses of Theorem 3.2, the functional Ψ has a critical point (w, w 2 ) W at the level c. We also have that if (u, v) := B(w, w 2 ), then (u, v) is a strong solution for system (3.7). Proof. Since Ψ has a linking geometry, if we prove that it satisfies (P S) c condition, by the Saddle Point, [, Theorem.2], it has a critical point. Let (w n, w 2 n) W be a sequence such that for some c R, Ψ(w n, w 2 n) c R, (6.3) Ψ (w n, w 2 n) 0 in W (6.4) We have that wn = t nφ + vn and wn 2 = t 2 nφ + vn 2 for some real sequences t n and t 2 n. Substituting these sequences in (6.2) and (6.3) and using (3.5) we have the boundedness of (wn, wn). 2 Therefore, up to subsequences, we may suppose that wn w and wn 2 w 2 in L 2 () for some w and w 2 L 2 (). Since B is a compact operator, B(wn, wn) 2 B(w, w 2 ) := (u, v). (6.5) By (6.4) we achieve that Thus g (w n) u and f (w 2 n) v in L 2 () and a.e.. (6.6) w n(x) g(u(x)) a.e. ξ (6.7) w 2 n(x) f(v(x)) a.e. θ g (w n(x)) u(x) = ξ ξ f (w 2 n(x)) v(x) = θ θ. (6.8) Our next step is proving that g (wn) g (w ) in L 2 ( ξ ) G(wn)dx G(w )dx ξ ξ f (wn) 2 f (w 2 ) in L 2 ( θ ) F (wn)dx 2 F (w 2 )dx. θ θ (6.9) (6.20) ξ = 0 (6.2) θ = 0. Hence, using the above assertions in (6.3) and (6.4), we have that Ψ (w, w 2 ) = 0 and Ψ(w, w 2 ) = c. Proof of (6.9). Firstly, since g(s) = βs + b(s) with b bounded, we have that wn(x) c g (wn(x)) + c 2, and by (6.6), wn(x) c h(x) for some h L 2 (). By (6.7) and Lebesgue Theorem we have that wn g(u) in L 2 ( ξ ). But wn w, which implies that wn w in L 2 ( ξ ). Since g is asymptotically linear, assertion (6.9) follows. Proof of (6.2). With the hypothesis (3.6) we may suppose that T g,ξ = [c, c 2 ] with c > 0. (The case c 2 < 0 follows in the same way). Let { h ξ, x ξ (x) = 0, x ξ.

12 2 C. O. ALVES, D. C. DE MORAIS FILHO, & M. A. S. SOUTO EJDE-2004/46 By a result of Stamppachia [5] already mentioned in the Introduction, w = 0 a.e. θ and w 2 = 0 a.e. ξ. But, since (6.7) holds we also have that u(x) = g (0) a.e. θ and by the same reasoning, v(x) = f (0) a.e. ξ. Hence, by (6.) we also have that w = 0 a.e. ξ and w 2 = 0 a.e. θ. On the other hand wndx = wn, h ξ wn, h ξ = ξ w dx. ξ (6.22) Since g is strictly monotonic, we see that lim inf wn(x) c a.e. ξ. But as wn h L 2 (), Fatou s Lemma and (6.22) imply that 0 = lim inf wn(x)dx c dx = c ξ ξ ξ and consequently (6.2) holds. The remaining part of the proof follows by making the respective changes. References [] G. Cimatti, A nonlinear elliptic eigenvalue problem for the Elenbaas equation, Boll. U.M.I. (5) 6-B (979), [2] G. Sherman, A free boundary problem, SIAM Rev. 2 (960), [3] A. Ambrosetti & M. Badiale, The dual variational principle and elliptic problems with discontinuous nonlinearities, J. Math. Anal. Appl (989), [4] A. Ambrosetti & R.E.L. Turner, Some discontinuous variational problems, Diff. and Int. Eq. Vol, n.3 (988), [5] G. Stampacchia, Le problème de Dirichlet pour les èquations elliptique du second ordre a coefficients discontinuous, Ann. Inst. Fourier 5 (965), [6] D.Arcoya & M. Calahorrano, Some discontinuous Problems with a Quasilinear Operator, J. Math. Anal. Appl. Vol. 87, No. 3 (994), [7] S. Hu, N. Kourogenis and N.S. Papageorgiou, Nonlinear elliptic eigenvalue problems with discontinuities, J. Math. Anal. Appl. Vol. 233 (999), [8] S.Carl & S. Heikkila, Elliptic equations with discontinuous nonlinearities, Nonlinear Analisys. vol 29 (998), [9] C. J. Amick & L. E. Fraenkel, The uniqueness of Hills spherical vortex, Arch. Rat. Mech. Anal. vol 2 (985), 9-9. [0] L. E. Fraenkel & M. S. Berger, A global theory of vortex rings in an ideal fluid, Acta Math. vol 32 (974), 4-5. [] P. H. Rabinowitz, Some minimax Theorems and applications to nonlinear partial differential equations, Nonlinear Analysis: A collection of papaer in honor or Erich N. Rothe (978), Departamento de Matemática e Estatística, Universidade Federal de Campina Grande, Cx Postal 0044, Campina Grande (PB), Brazil address, C. O. Alves: coalves@dme.ufpb.br address, D. C. de Morais Filho: daniel@dme.ufpb.br address, M. A. S. Souto: marco@dme.ufpb.br