Oscillatory Mixed Di erential Systems

Transcription

1 Oscillatory Mixed Di erential Systems by José M. Ferreira Instituto Superior Técnico Department of Mathematics Av. Rovisco Pais 49- Lisboa, Portugal Sra Pinelas Universidade dos Açores Department of Mathematics R. Mãe de Deus Ponta Delgada, Portugal Abstract. In this work are obtained some criteria which guarantee the oscillatory behavior of the di erential system of mixed type x (t) = d [ ()] x (t r ()) + d [ ()] x (t + ()) ; where x (t) 2 R n ; r () () are real nonnegative continuous functions on [ ; ] ; () () are real n-by-n matrix valued function of bounded variation on [ ; ] : Key Words Phrases. Oscillatory, di erential system of mixed type. 2 Mathematics Subject Classi cation Numbers. 34K6, 34K. Reseach partially supported by FCT

2 Introduction In this note is considered the di erential di erence system of mixed type x (t) = A i x (t r i ) + B j x (t + j ) ; () i= where x (t) = [x (t) ; :::; x n (t)] T 2 R n ; the A i B j are n-by-n real matrices the r i j are positive real numbers. Equations of this kind can arise in the study on travelling waves in domains with nonlocal interactions, initiated in [], [2], developed more recently in [3], [4]. system System () can be looked in the more general framework of the mixed functional di erential x (t) = j= d [ ()] x (t r ()) + d [ ()] x (t + ()) ; (2) where r () () are real nonnegative continuous functions on [ ; ] ; () () are real n-by-n matrix valued functions of bounded variation on [ ; ] : Some interest on these equations can be noticed in the area of economic research, as it can be seen through [5], [6] references therein. The system () corresponds to have in (2), as step functions of the form () = H( i )A i ; () = H( j )B j ; (3) i= j= where H denotes the Heaviside function, < < ::: < p < < < ::: < q < ; r () () are any continuous positive functions on [ ; ] in manner that r ( i ) = r i j = j ; for i = ; :::; p j = ; :::; q: The aim of this work is to complement the study made in [7] on the oscillatory behavior of systems () (2). Considering the value krk = max fr() : g ; by a solution of (2) we mean a continuous function x : [ krk ; +[ R n ; which is di erentiable on ]; +[ satis es (2) for every t > : Taking the space C = C ([ krk ; kk] ; R n ) ; let 2 C: Considering the initial condition x () = () for 2 [ krk ; kk] ; the initial value problem associated to (2) may be ill-posed (see [8]). This leads to some questions concerning the admissibility of the function spaces for abstract Cauchy problems of functional di erential equations. On this purpose we call the attention of the reader to [9] []. 2

3 A set K R n is called a cone if for each u; v 2 K a ; b ; one has au + bv 2 K; u =2 K whenever u 2 Kn fg : Then following [7], considering an interval J [; +[ a continuous function f : J R n will be said nonoscillatory if there exists a T a closed cone K R n such that f (t) 2 Kn fg for all t T ; otherwise, f is called oscillatory. This de nition of oscillatory function seems to be a middle situation between the concepts of oscillatory componentwise weakly oscillatory functions introduced in []. A functional perspective of oscillatory function can also be referred. As equation (2) can induce a semidynamical system in the space C thought the relation x t () = x (t + ) for 2 [ krk ; kk] ; following [2] [3], a solution of (2) is said strongly nonoscillatory if there exists a function, '; of bounded variation on [ krk ; kk] a T > such that Z kk krk x (t + ) d' () ; for all t T: This means that, for every t T; x t is contained in some closed half-space of C: Otherwise that solution is said strongly oscillatory. For the delay case, corresponding to have = = in (2), independently of the kind of oscillatory behavior mentioned above, the system x (t) = d [ ()] x (t r ()) ; is oscillatory, that is, has all its solutions oscillatory, if only if the corresponding characteristic equation det I exp (r ()) d [ ()] = ; where I denotes the n-by-n identity matrix, has no real roots (see [], [2] [4]). However, in the regard of system with : [ x (t) = Z d [ (s)] x(t + s) (4) ; ] R nn of bounded variation, in [7] is given an example where the characteristic equation det I Z exp (s) d [ (s)] = ; 3

4 has no real roots (4) has a nonoscillatory solution, whatever be the above sense we choose. Contrarily to the delay case, for the mixed type systems the real part of the spectral set fre : det () = g is not in general bounded above the solutions may not be exponentially bounded (see [7], [8] [4]). These di culties are overcame in [7], by assuming that: There exist a nonsingular matrix A 2 R nn a nondecreasing function : [ ; ] R such that (s) < () for all s 2 [ ; ) ; j A Ij () j A Ij (s) () (s) as s ; (H) where by jj (s) is meant the total variation on [ ; s] of any n-by-n matrix valued function of bounded variation on [ ; ] : The assumption (H) holds, for example, either if () is a nonsingular matrix; (H) or if there are a nonsingular matrix B 2 R nn ; s 2 [ ; ) ; a monotone function : [s ; ] R so that (s) B = (s) I for all s 2 [s ; ] (s) 6= () for all s 2 [s ; ) : (H2) The condition (H) means that the operator L : C ([ ; ] ; R n ) given by 7 Z d [ (s)] (s) is atomic at in the sense of [5], while through (H2), (H) may be satis ed in cases where L is nonatomic at. Comparatively to system (4), there is no gain in generality by considering the system (2). However, as we will see, there is some convenience in separating advances delays. For a matter of simplicity of brevity of exposition, we will assume that the delay function r () the advance function () are increasing on [ ; ] : However the methods we will use in the sections 2 can be applied, more generally, to delays advances both monotonous as well as to delays advances exhibiting some oscillations. On this subject, if is a real function on 4

5 [ ; ] ; whenever is said to be increasing or decreasing on a interval J [ ; ] ; we exclude the possibility of be constant on J: Therefore in order that be veri ed a similar condition to (H) we will suppose hereafter that the advance function () is such that () < () for every 2 [ ; [ ; that () ( ) is a nonsingular matrix. In the concerning of system () we will assume the delays advances ordered as r < ::: < r p ; < ::: < q ; that the matrix B q corresponding to the largest advance is non singular. Letting then () = I exp ( r ()) d [ ()] + exp ( ()) d [ ()] ; supposing that det () = ; for some 2 R; one easily sees that then for some u 2 R n n fg ; the eigenvector exp () u is a nonoscillatory solution of the system (2). Moreover, having in mind the rst de nition of oscillations given above, according to [7] one can summarize the main characteristics of that system in the following lemma. Lemma i) f 2 R : det () = g is bounded above; ii) Any nonoscillatory solution of (2) is exponentially bounded. iii) If (2) has a nonoscillatory solution then det () = has a real root. iv) (2) is oscillatory if only if det () = has no real roots. Therefore de ning M () = exp ( r ()) d [ ()] + exp ( ()) d [ ()] ; 5

6 the system (2) is oscillatory if only =2 (M ()) ; for every real : (5) Nonoscillatory solutions will exist, whenever det [I M ()] = (6) has at least a real root. As in many papers on the oscillatory theory of delay equations, matrix measures also constitute here a relevant tool for the analysis of the oscillatory behavior of (2) (4). For the de nition its main properties we will follow [6] [7]. Notice that, since for any matrix measure, one has Re (M ()) [ ( M ()) ; (M ())] ; if either (M ()) < ; for every real ; (7) or < ( M ()) ; for every real ; (8) then (5) is satis ed. 2 Continuos delays advances For a given function of bounded variation on [ ; ] ; we will associate the functions ; also of bounded variation, de ned, respectively, by () = () (); () = () ( ) ( 2 [ ; ]) : By we will mean the di erence () ( ) = ( ) = (): 6

7 Theorem 2 a) If for some matrix measure are decreasing, (9) are increasing, () then (2) is oscillatory independently of the advances. b) If for some matrix measure r () d ( ) () > e ; () ( ) ( ) are decreasing, (2) ( ) ( ) are increasing, (3) then (2) is oscillatory independently of the delays. () d ( ( )) () > e : (4) Proof. a) We rst notice that by the subadditivity property of a matrix measure we have (M ()) exp ( r ()) d [ ()] + exp ( ()) d [ ()] : (5) Therefore (M ()) () + () : As by (9) () () = ( ) () < ( ) ( ) = () = ( ) () < ( ) ( ) = ; then (M ()) < (7) is veri ed for = : On the other h by the properties of the matrix measures obtained in [7, Lemma 2.2, (i) (ii)], one has exp ( 8 < r ()) d [ ()] : R exp ( r ()) d ( ) () ; if > ; R exp ( r ()) d ( ) () ; if < ; (6) 8 < exp ( ()) d [ ()] : R exp ( ()) d ( ) () ; if > ; R exp ( ()) d ( ) () ; if < ; (7) 7

8 Then letting > ; by (5), (6) (7) we have (M ()) So, (M ()) < for every > : Let now < : Analogously one obtains (M ()) Therefore, for every < one has exp ( r ()) d ( ) () (8) exp ( ()) d ( ) () : exp ( r ()) d ( ) () (9) + exp ( ()) d ( ) () ; (M ()) exp ( r ()) d ( ) () : Then noticing that exp u eu for every u; () implies that for every < : Thus (7) is satis ed. (M ()) e r () d ( ) () < ; b) Analogous arguments will enable us to conclude (8). Now by the subadditivity property of a matrix measure we have ( M ()) ( exp ( r ())) d [ ()] + ( exp ( ())) d [ ()] ; (2) Hence for = ; (2) (3) imply that ( M ()) ( ) + ( ) = ( ( )) () + ( ( )) () < ; then < ( M ()) : 8

9 By the properties of the matrix measures expressed in [7, Lemma 2.2, (iii) (iv)], the following inequalities hold: 8 < ( exp ( r ())) d [ ()] : 8 < ( exp ( ())) d [ ()] : R exp ( r ()) d ( ( )) () ; if > ; R exp ( r ()) d ( ( )) () ; if < ; R exp ( ()) d ( ( )) () ; if > ; R exp ( ()) d ( ( )) () ; if < ; (2) (22) Thus for < one has by (2), (2) (22) ( M ()) exp ( r ()) d ( ( )) () (23) + exp ( ()) d ( ( )) () : So, by (2) (3) it is ( M ()) < :for every < : For > ; one obtains analogously ( M ()) exp ( r ()) d ( ( )) () (24) exp ( ()) d ( ( )) () ; Therefore, for every > one has as before through (4) ( M ()) e exp ( ()) d ( ( )) () () d ( ( )) () < : Thus (8) is satis ed. Remark 3 Adding to the parts a) b) of the Theorem 2, respectively, the conditions r () d ( ) () < e ; (25) r () d ( ( )) () < e ; (26) 9

10 identical arguments lead us to the same conclusions for monotonous delays advances. In that way one obtains an extension of [8, Theorem ] for monotonic delays advances. With respect to the results obtained in this work, for systems we are very much more restricted by the properties of the matrix measures. Remark 4 We notice that since () is satis ed if Similarly, from one has (4) veri ed if r () d ( ) () () min 2[ ;] r () ; () min r () < e : 2[ ;] () d ( ( )) () ( ) min 2[ ;] () ; ( ) min 2[ ;] () < e : In the regard of the system (), for a given matrix measure ; following [] we consider the measures relatively to a given ordered family of matrices (C ; :::; C m ): jx Xj (C ) = (C ); (C j ) = C k C k for j = 2; :::; p; k= (C p ) = (C p ); (C j ) C k k= k=j k=j+ Observe that by the subadditivity of a matrix measure one easily sees that C k A for j = ; :::; p : Moreover if, as in (3), for 2 [ then (C j ) (C j ); (C j ) (C j ); j = ; :::; m: (27) ; ] ; () = P m k= H( i)c i ; with < < ::: < m < ; ( ) () = mx ( H( i )) C i i=

11 mx ( ) () = H( i )C i are step functions with jumps at k equal, respectively, to mx mx kx (C k ) = C i C i ; (C k ) = C i i= i=k+ i=k i= In the same way, by considering the sequence ( C ; :::; C m ) one has mx ( ( )) () = ( H( i )) ( C i ) i= mx ( ( )) () = H( i ) ( C i ) ; which are step functions with jumps at k equal to, respectively, mx mx ( C k ) = C i C i ; ( C k ) = i= i=k+ kx C i i= i= i=k kx C i : kx C i : i= Therefore if for every k = ; :::; m one has (C k ) < ; then is increasing if for every k = ; :::; m it is (C k ) < then is decreasing. Thus the following corollary holds. Corollary 5 a) Let (A i ) < ; (A i ) < ; for every i = ; :::; p; (B j ) < ; (B j ) < ; for every j = ; :::; q: If r i (A i ) < e (28) i= then () is oscillatory, independently of the advances < ::: < q : b) Let ( A i ) < ; ( A i ) < ; for every i = ; :::; p; ( B j ) < ; ( B j ) < ; for every j = ; :::; q: If j ( B j ) < e (29) then () is oscillatory, independently of the delays r < ::: < r q : j=

12 Proof. a) By the comments above, (9) () are satis ed. On the other h () is in this case equivalent to (28). The part b) follows analogously. As a consequence of this corollary, by (27) we conclude the following statements. Corollary 6 a) Let (A i ) < for every i = ; :::; p; (B j ) < for every j = ; :::; q: If r i (A i ) < e ; (3) i= then () is oscillatory, independently of the advances < ::: < q : b) Let ( A i ) < for every i = ; :::; p; ( B j ) < for every j = ; :::; q: If j ( B j ) < e ; (3) then () is oscillatory, independently of the delays r < ::: < r q : j= Remark 7 Assuming that r < ::: < r p ; < ::: < q ; by the Remark 4, the conditions (3) (3) in the Corollary 6 can be replaced, respectively, by r A i < i= j= B j A < e : We illustrate the Corollary 5 through the following example using the well-known matrix measure of a matrix C = [c jk ] 2 M n (R) given by 8 9 < (C) = max 6k6n : c kk + X = jc jk j ; : j6=k Example 8 Let the system () with r = =8; r 2 = =4; r 3 = =2 A = B = 2 ; A 2 = ; B 2 = ; A 3 = 3 2 :

13 Using ; we have (A ) = (A ) = max f 4; 2g = 2; (A 2 ) = (A + A 2 ) (A ) = max f 3; 5g + 2 = ; (A 3 ) = (A + A 2 + A 3 ) (A + A 2 ) = max f 9; g + = 8; (A 3 ) = (A 3 ) = max f 4; 6g = 4; (A 2 ) = (A 2 + A 3 ) (A 3 ) = max f 5; 7g + 4 = ; (A ) = (A + A 2 + A 3 ) (A 2 + A 3 ) = 9 + = 8; (B ) = (B ) = max f 6; 3g = 3; (B 2 ) = (B + B 2 ) (B ) = max f 8; 6g + 3 = 3; (B 2 ) = (B 2 ) = max f 2; g = ; (B ) = (B + B 2 ) (B 2 ) = max f 8; 6g + = 5: Since 3X r i (A i ) = i= 4 2 = 3 4 < e ; by Corollary 5 a), the correspondent () is oscillatory independently of the advances < 2 : 3 Di erentiable delays advances In this section we will assume further that r() () are di erentiable functions on [ ; ] : In the regard of the next theorem we observe that for a given function : [ ; ] R nn of bounded variation a matrix measure ; through integrations by parts the following relations hold for k = ; : exp ( r ()) d ( k ) () = P k () (32) + exp ( r ()) ( k ) () dr () ; 3

14 exp ( ()) d ( k ) () = Q k () (33) exp ( ()) ( k ) () d () ; where P = exp ( r ( )) ; P = exp ( r ()) ; Q = exp ( ( )) ; Q = exp ( ()) : Analogously Theorem 9 a) Let exp ( r ()) d ( ( k )) () = P k ( ) (34) + exp ( r ()) ( ( k )) () dr () ; exp ( ()) d ( ( k )) () = Q k ( ) (35) exp ( ()) ( ( k )) () d () ; ( ) () < ( ) () < ; on ] ; ] (36) ( ) () < ( ) () < ; on [ ; [ : (37) If er ( ) () + ( ) () dr () < ; (38) then (2) is oscillatory independently of the advances () : b)let ( ( )) () < ( ( )) () < ; on ] ; ] (39) ( ( )) () < ( ( )) () < ; on [ ; [ : (4) If e ( ) ( ) + ( ( )) () d () < ; (4) then (2) is oscillatory independently of the delays r () : 4

15 Proof. i) Conditions (36) (37) imply () = ( ()) = ( ( )) < ; () = ( ()) = ( ( )) < ; by (5), (7) is satis ed for = : In the same way, as by (39) (4) ( ) = ( ( )) () = ( ( )) ( ) < ( ) = ( ) () = ( ) ( ) < ; through (2), one concludes that (8) is veri ed for = : ii) Letting > from (8), (32) (33) one obtains (M ()) exp ( r ()) () + + exp ( ( )) () + Then by (36) (37), we conclude that (M ()) < < : exp ( r ()) ( ) () dr () (42) exp ( ()) ( ) () d () : Analogously, when < from the inequality (9) we obtain through (32) (33), (M ()) exp ( r ( )) () + exp ( ()) () exp ( r ()) ( ) () dr () (43) exp ( ()) ( ) () d () : Therefore (M ()) exp ( r ( )) () exp ( r ()) ( ) () dr () ; since exp ( r ()) exp ( r ( )) er ( ) we have then by (38) Thus (7) holds. (M ()) er ( ) () + ( ) () dr () < : (44) 5

16 iii) For < one has by (23) (34) (35) ( M ()) exp ( r ( )) ( ) (45) exp ( r ()) ( ( )) () dr () + exp ( ()) ( ) So, by (39) (4) it is ( M ()) < for every < : For > ; one obtains analogously by (24), (34) (35) exp ( ()) ( ( )) () d () : ( M ()) exp ( r ()) ( ) (46) + exp ( r ()) ( ( )) () dr () + exp ( ( )) ( ) + exp ( ()) ( ( )) () d () Therefore, since exp ( ()) exp ( ( )) e ( ) one has for every > ( M ()) e ( ) ( ) + ( ( )) () d () : Then by (4) ( M ()) < (8) is veri ed. In the regard of (), one obtains the following corollary. Corollary a) If with r < ::: < r p ; kx `X A i i= j= A i i=k j=` B j B j A < ; k = ; :::; p; ` = ; :::; q; (47) A < ; k = ; :::; p; ; ` = ; :::; q; (48) er i= Xp A i + k= i=k+ A i (r k+ r k ) < ; (49) 6

17 then (4) is oscillatory independently of the advances j 2 R + such that < ::: < q : b) With < ::: < q ; if kx A i i= A i i=k `X j= j=` B j B j A < ; k = ; :::; p; ` = ; :::; q; (5) A < ; k = ; :::; p; ` = ; :::; q; (5) Xq A B j j= k= j=k+ B j A ( k+ k ) < ; (52) then (4) is oscillatory independently of the delays r j 2 R + such that r < ::: < r p : Proof. a) Notice that, ( ) () = H( i )A i ; ( ) () = ( H( i )) A i ; i= ( ) () H( j )B j A ; i= ( ) () H( j ) B j A ; j= j= so (36) (37) are ful lled if (47) (48) hold. On the other h as ( ) () dr () = k=2 A i (r k r k ) ; i=k (49) implies (38). b) Now one has ( ( )) () = ( ( )) () = H( i ) ( A i ) ; i= ( H( i )) ( A i ) ; i= ( ( )) () H( j ) ( B j ) A ; j= ( ( )) () H( j ) ( B j ) A ; j= 7

18 so (39) (4) are ful lled if (5) (5) hold. On the other h as ( ( )) () d () = k=2 B j ( k k ) i=k (52) implies (4). The following example illustrates this corollary by using the matrix measure of a matrix C = [c jk ] given by 8 9 < (C) = max 6j6n : c jj + X = jc jk j ; : k6=j Example Considering the system () with r = =; r 2 = =2 A = B = Using the matrix measure ; we have ; A 2 = ; B 2 = ; ; (A ) = max f ; 7g = ; (A 2 ) = max f ; 4g = ; (A + A 2 ) = max f 4; 3g = 4; (B ) = max f 4; 3g = 3; (B + B 2 ) = max f 7; g = 7; (B 2 ) = max f 3; 2g = 2: Since er 2X A i i= + (A ) (r 2 r ) = 2e < ; by the Corollary a) the correspondent () is oscillatory independently of the advances < < 2 : 8

19 4 Positive delays advances If the delay advance functions, r () () ; are both di erentiable positive one can still obtain the following theorem. Theorem 2 a) Let (36) (37) hold. If + ln (r ( ) j ()j) + r ( ) e ( ) () d ln r () > (53) then (2) is oscillatory independently of the advances () : b) Let (39) (39) hold. If + ln ( ( ) j ( )j) + ( ) e ( ( )) () d ln () > ; (54) then (2) is oscillatory independently of the delays r () : Proof. a) One can follow i) ii) of the proof of the Theorem 9. Now the inequality (44) can be written as (M ()) exp ( r ( )) () r () exp ( r ()) ( ) () d ln r () ; since u exp ( u) e ; one obtains (M ()) exp ( r ( )) () e ( ) () d ln r () : As the function f () = exp ( r ( )) () attains its maximum at (53) implies that ln (r ( ) j ()j) = r ( ) f ( ) = + ln (r ( ) j ()j) ; r ( ) (M ()) < 9

20 for every < : b) Analogously, following i) iii) of the proof of the Theorem 9, from the inequality (46) one obtains for > ; ( M ()) exp ( ( )) ( ) + () exp ( ()) ( ( )) () d ln () exp ( ( )) ( ) e ( ( )) () d ln () : Taking the function g () = exp ( ( )) ( ) + ; its maximum is obtained at As (54) is equivalent to = g ( ) = ln ( ( ) j ( )j) ( ) + ln ( ( ) j ( )j) : ( ) g ( ) < e ( ( )) () d ln () ; one concludes then that ( M ()) < ; for every real > : As before, the following corollary can be stated. Corollary 3 a) If with r < ::: < r p ; (47) (48) hold + ln r Xp A i + r e i= k= i=k+ A i ln r k+ r k > (55) then () is oscillatory independently of the advances j 2 R + such that < ::: < q : b) With < ::: < q ; if (5) (5) hold Xq AA + B j j= k= j=k+ B j A ln k+ k > ; (56) then () is oscillatory independently of the delays r j 2 R + such that r < ::: < r p : 2

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