Oscillation of nonlinear impulsive differential equations with piecewise constant arguments

Transcription

1 Electroic Joural of Qualitative Theory of Differetial Equatios 2013, No. 49, 1-12; Oscillatio of oliear impulsive differetial equatios with piecewise costat argumets Fatma KARAKOÇ 1, Arzu OGUN UNAL 2 ad Huseyi BEREKETOGLU 3 Departmet of Mathematics, Faculty of Sciece, Akara Uiversity, Akara, Turkey Abstract Existece ad uiqueess of the solutios of a class of first order oliear impulsive differetial equatio with piecewise costat argumets is studied. Moreover, sufficiet coditios for the oscillatio of the solutios are obtaied. Keywords: impulsive differetial equatio, piecewise costat argumet, oscillatio. AMS Subject Classificatio: 34K11, 34K Itroductio I this paper, we cosider a impulsive differetial equatio with piecewise costat argumets of the form with the iitial coditios x (t) + a (t) x (t) + x ([t 1]) f(x [t]) = 0, t, (1) x () = d x (), N = {0, 1, 2,...}, (2) x( 1) = x 1, x(0) = x 0, (3) where a : [0, ) R, f : R R are cotiuous fuctios, d : N R {1}, x () = x ( + ) x ( ), x ( + ) = lim x (t), x ( ) = lim x (t), [.] t + t deotes the greatest iteger fuctio, ad x 1, x 0 are give real umbers. Sice 1980 s differetial equatios with piecewise costat argumets have attracted great deal of attetio of researchers i mathematical ad some of the others fields i sciece. Piecewise costat systems exist i a widely expaded areas such as biomedicie, chemistry, mechaical egieerig, physics, etc. These kid of equatios such as Eq.(1) are similar i structure to those foud i certai sequetial-cotiuous models of disease dyamics [1]. I 1994, Dai ad Sig [2] studied the oscillatory motio of sprig-mass systems with subject to piecewise costat forces of the form f(x[t]) or f([t]). Later, they improved a aalytical ad umerical method for solvig liear ad oliear vibratio problems ad they showed that a fuctio f([n(t)]/n) is a good approximatio to the give cotiuous fuctio f(t) if N is sufficietly large [3]. 1 Correspodig Author. fkarakoc@akara.edu.tr, telephoe: +90(312) , fax: +90(312) aogu@sciece.akara.edu.tr 3 bereket@sciece.akara.edu.tr EJQTDE, 2013 No. 49, p. 1

2 This method was also used to fid the umerical solutios of a o-liear Froude pedulum ad the oscillatory behavior of the pedulum [4]. I 1984, Cooke ad Wieer [5] studied oscillatory ad periodic solutios of a liear differetial equatio with piecewise costat argumet ad they ote that such equatios are comprehesively related to impulsive ad differece equatios. After this work, oscillatory ad periodic solutios of liear differetial equatios with piecewise costat argumets have bee dealt with by may authors [6, 7, 8] ad the refereces cited therei. But, as we kow, oliear differetial equatios with piecewise costat argumets have bee studied i a few papers [9, 10, 11]. O the other had, i 1994, the case of studyig discotiuous solutios of differetial equatios with piecewise cotiuous argumets has bee proposed as a ope problem by Wieer [12]. Due to this ope problem, the followig liear impulsive differetial equatios have bee studied [13, 14]: { x (t) + a(t)x(t) + b(t)x([t 1]) = 0, t, x( + ) x( (4) ) = d x(), N = {0, 1, 2,...}, ad { x (t) + a(t)x(t) + b(t)x([t]) + c(t)x([t + 1]) = f(t), t, x () = d x(), N = {0, 1, 2,...}. Now, our aim is to cosider the Wieer s ope problem for the oliear problem (1)-(3). I this respect, we first prove existece ad uiqueess of the solutios of Eq. (1)-(3) ad we also obtai sufficiet coditios for the existece of oscillatory solutios. Fially, we give some examples to illustrate our results. 2. Existece of solutios Defiitio 1. It is said that a fuctio x : R + { 1} R is a solutio of Eq. (1)-(2) if it satisfies the followig coditios: (i) x(t) is cotiuous o R + with the possible exceptio of the poits [t] [0, ), (ii) x(t) is right cotiuous ad has left-had limit at the poits [t] [0, ), (iii) x(t) is differetiable ad satisfies (1) for ay t R +, with the possible exceptio of the poits [t] [0, ) where oe-sided derivatives exist, (iv) x() satisfies (2) for N. Theorem 1. The iitial value problem (1)-(3) has a uique solutio x(t) o [0, ) { 1}. Moreover, for t < + 1, N, x has the form x (t) = t a (s) ds y() y( 1)f(y()) t a (s) ds du, (5) EJQTDE, 2013 No. 49, p. 2

3 where y() = x () ad the sequece {y()} 1 is the uique solutio of the differece equatio +1 1 y( + 1) = a (s) ds 1 d +1 y() y( 1)f(y()) +1 (6) a (s) ds du, 0 with the iitial coditios y( 1) = x 1, y(0) = x 0. (7) Proof. Let x (t) x(t) be a solutio of (1)-(2) o t < + 1. Eq. (1)-(2) is rewritte i the form x (t) + a (t) x (t) = x ( 1) f (x()), t < + 1. (8) From (8), for t < + 1 we obtai x (t) = t a (s) ds x () x ( 1) f(x()) t a (s) ds du. O the other had, if x 1 (t) is a solutio of Eq.(1)-(2) o 1 t <, the we get x 1 (t) = t 1 (9) a (s) ds (10) x ( 1) x ( 2) f(x( 1)) t 1 1 a (s) ds du. Usig the impulse coditios (2), from (9) ad (10), we obtai the differece equatio +1 1 x( + 1) = a (s) ds 1 d +1 x() x( 1)f(x()) +1 a (s) ds du. EJQTDE, 2013 No. 49, p. 3

4 Cosiderig the iitial coditios (7), the solutio of Equatio (6) ca be obtaied uiquely. Thus, the uique solutio of (1)-(3) is obtaied as (5). Theorem 2. The problem (1)-(3) has a uique backward cotiuatio o (, 0] give by (5)-(6) for Z {0}. 3. Oscillatory solutios Defiitio 2. A fuctio x (t) defied o [0, ) is called oscillatory if there exist two real valued sequeces {t } 0, {t } 0 [0, ) such that t +, t + as + ad x (t ) 0 x (t ) for N where N is sufficietly large. Otherwise, the solutio is called ooscillatory. Remark 1. Accordig to Defiitio 2, a piecewise cotiuous fuctio x : [0, ) R ca be oscillatory eve if x (t) 0 for all t [0, ). Defiitio 3. A solutio {y } 1 of Eq.(6) is said to be oscillatory if the sequece {y } 1 is either evetually positive or evetually egative. Otherwise, the solutio is called o-oscillatory. Theorem 3. Let x (t) be the uique solutio of the problem (1)-(3) o [0, ). If the solutio y(), 1, of Eq. (6) with the iitial coditios (7) is oscillatory, the the solutio x (t) is also oscillatory. Proof. Sice x (t) = y () for t =, the proof is clear. Remark 2. We ote that eve if the solutio y(), 1, of the Eq. (6) with the iitial coditios (7) is ooscillatory, the solutio x(t) of (1)-(3) might be oscillatory. I the followig theorem give a ecessary ad sufficiet coditio for the existece of ooscillatory solutio x(t), whe the solutio of differece equatio (6)-(7) is ooscillatory. Theorem 4. Let {y } 1 be a ooscillatory solutio of Eq. (6) with the iitial coditios (7). The the solutio x(t) of the problem (1)-(3) is ooscillatory iff there exist a N N such that y() t y( 1) > f(y()) a (s) ds du, t < + 1, > N. (11) Proof. Without loss of geerality we may assume that y() = x() > 0, y( 1) = x( 1) > 0 for > N. If x(t) is ooscillatory, the x(t) > 0, t > T N. So coditio (11) is obtaied from (5) easily. Now, let us assume that (11) is true. We should show that x(t) is ooscillatory. For cotradictio, let x(t) be oscillatory. Therefore there exist sequeces {t k } k 0, {t k } k 0 such that t k +, t k + as k + ad EJQTDE, 2013 No. 49, p. 4

5 x (t k ) 0 x (t k ). Let k = [t k ]. It is clear that k + as k +. So, from (5) we get x (t k ) = t k k a (s) ds t k y( k ) y( k 1)f(y( k )) k k a (s) ds du. Sice y( k ) > 0, y( k 1) > 0 ad x(t k ) 0 we obtai y( k ) t k y( k 1) f(y( k)) a (s) ds du, k t k < k + 1 k k which is a cotradictio to (11). If y() = x() < 0, y( 1) = x( 1) < 0 for > N, the the proof is doe by similar method. Theorem 5. Suppose that 1 d > 0 for N ad there exist a M > 0 such that f(x) M for x (, ) ad lim sup (1 d ) +1 1 The, all solutios of Eq. (6) are oscillatory. a (s) ds du > 1 M. (12) Proof. We prove that the existece of evetually positive (or egative) solutios leads to a cotradictio. Let y() be a solutio of Eq. (6). Assume that y() > 0, y( 1) > 0, y( 2) > 0 for > N, where N is sufficietly large. From (6) (1 d )y() 1 a (s) ds = y( 1) y( 2)f(y( 1)) 1 Sice y( 2) > 0 ad f(y( 1)) > 0, we have (1 d )y() 1 1 a (s) ds du. a (s) ds < y( 1). (13) EJQTDE, 2013 No. 49, p. 5

6 By usig iequality (13) ad Eq. (6), we obtai +1 y() 1 (1 d )f(y()) > y() y( 1)f(y()) +1 = (1 d +1 )y( + 1) 1 +1 a (s) ds du a (s) ds du a (s) ds. (14) Sice y() > 0, y( + 1) > 0, 1 d +1 > 0 ad f(x) M, from (14), we get +1 1 u M lim sup (1 d ) a (s) ds du, which is a cotradictio to (12). The proof is the same i case of existece of a evetually egative solutio. Corollary 1. Uder the hypotheses of Theorem 5, all solutios of (1)-(2) are oscillatory. Remark 3. If f(x) = b, b > 0 is a costat fuctio, the we have a liear equatio i the form { x (t) + a(t)x(t) + bx([t 1]) = 0, t, x( + ) x( (15) ) = d x(), N = {0, 1, 2,...}, which is a special case of (4). I this case, coditio (12) reduces to the followig coditio +1 lim sup (1 d )b a (s) ds du > 1, which is stated i [13] for b(t) b > 0. 1 Now, cosider followig oimpulsive equatio 1 x (t) + a (t) x (t) + x ([t 1]) f(x [t]) = 0, (16) where a : [0, ) R, f : R R are cotiuous fuctios. Corollary 2. Assume that there exists a costat M > 0 such that f(x) M. If +1 lim sup a (s) ds du > 1 M, 1 EJQTDE, 2013 No. 49, p. 6

7 the all solutios of Eq. (16) are oscillatory. Theorem 6. Assume that ad f(x) M > 0, (17) 1 d K > 0, = 0, 1, 2,..., (18) KM < lim if exp a(s)ds lim if The, all solutios of Eq. (6) are oscillatory. +1 exp( a(s)ds)du <. (19) Proof. Let y() be a solutio of Eq. (6). Assume that y() > 0, y( 1) > 0 for > N, where N is sufficietly large. From Eq. (6), we have Let w = (1 d +1 )y( + 1) exp +1 a(s)ds =y() y( 1)f(y()) +1 y() y( 1). Sice w > 0, we cosider two cases: a (s) ds du. (20) Case 1. Let lim if w =. The from (20), we have +1 1 (1 d +1 )w +1 exp a(s)ds + M +1 a (s) ds du. (21) w Takig the iferior limit o both sides of iequality (21), we get 1 lim if(1 d +1) lim if w +1 +M lim if 1 lim w if which is a cotradictio to the case; +1 lim if exp +1 a (s) ds du, a(s)ds lim if w =. So, we cosider the secod Case 2. Let 0 lim if w <. Dividig Eq. (20) by y( 1), we have +1 y() y( + 1) = (1 d +1 ) y( 1) y( 1) a (s) ds +1 +f(y()) a (s) ds du, EJQTDE, 2013 No. 49, p. 7

8 which yields w (1 d +1 )w w +1 + M a (s) ds du. a (s) ds (22) +1 Let lim if w = W, lim if exp a(s)ds = A, +1 lim if exp( a(s)ds)du = B. Takig the iferior limit o both sides of iequality (22), we have W lim if(1 d +1)W 2 A + MB (23) Now, from (18), there are two subcases: (i) If lim if(1 d +1) =, the we obtai a cotradictio from (23). (ii) If lim if(1 d +1) <, the from (23) we have or AKW 2 W + MB 0, [ ( KA W 1 ) ] 2 + 4MBKA 1 2KA 4K 2 A 2 0, which cotradicts to (19). So Eq. (6) caot have a evetually positive solutio. Similarly, existece of a evetually egative solutio leads us a cotradictio. Thus all solutios of (6) are oscillatory. Corollary 3. Uder the hypotheses of Theorem 6, all solutios of (1)-(2) are oscillatory. Corollary 4. Assume that f(x) M > 0, ad M < lim if exp a(s)ds lim if the all solutios of (16) are oscillatory. +1 exp( a(s)ds)du <. Remark 4. If f(x) = b, b > 0 is a costat, ad d 0 for all N, the Eq.(1)-(2) reduces to the liear oimpulsive equatio x (t) + a(t)x(t) + bx([t 1]) = 0, (24) which is the same as Eq.(1) with b(t) b i [7]. I this case, coditios (12) ad (19), respectively, correspod to coditios (2) ad (8) i [7]. EJQTDE, 2013 No. 49, p. 8

9 Equatio (24) is also special case of Eq.(1.1) i [9]. I this case, coditio (19) reduces to coditio (2.3) i [9] with b(t) b. Moreover, if a(t) a (costat), the coditio (19) reduces to the coditio b > ae a 4(e a 1) which is kow as the best possible for the oscillatio [7, 9]. Remark 5. I the case of f(x) b, a(t) a, d d, N, a, b, d are costats, Theorem 9 i [13] ca be applied to Eq. (1)-(2) to obtai existece of periodic solutios. Cosider the followig equatio. { x (t) + ax(t) + bx([t 1]) = 0, t, x( + ) x( ) = dx(), N = {0, 1, 2,...}. (25) Corollary 5. Let 1 d > K > 0. A ecessary ad sufficiet coditio for every oscillatory solutio of Eq.(25) to be periodic with period k is ae a ( (1 d) e a = b ad a = l 2(1 d) cos 2πm ), (26) 1 k where m ad k are relatively prime ad m = 1, 2,..., [(k 1)/4]. 4. Examples I this sectio, we give some examples to illustrate our results. Example 1. Let us cosider the followig differetial equatio x (t) + x (t) + (x 2 [t] + 1)x ([t 1]) = 0, t, (27) x () = e 1 x (), N, (28) e which is a special case of (1)-(2) with a(t) = 1, f(x) = x 2 +1, d = e 1 e, N. It is easily checked that the Eq. (27)-(28) satisfies all hypotheses of Theorem 6. Thus every solutio of equatio (27)-(28) is oscillatory. The solutio x (t) of Eq. (27)-(28) with the iitial coditios x( 1) = 0, x (0) = for = 0, 1, 2, 3, 4 is demostrated i Figure 1. Example 2. Cosider the equatio x (t) + x (t) + x ([t 1]) = 0, t, (29) x () = 1 x (), N, (30) 2 that is a special case of Eq. (1)-(2) with a(t) = 1, f(x) = 1 ad d = 1 2, N. Sice all hypotheses of Theorem 5 are satisfied, every solutio of Eq.(29)-(30) is oscillatory. Ideed, the solutio x (t) of Eq.(29)-(30) is i the form x (t) = e t+ [y() y( 1) ( e t 1 ) ], t < + 1, (31) EJQTDE, 2013 No. 49, p. 9

10 x t Figure 1: Oscillatory solutios of Eq. (27)-(28) with the iitial coditios x( 1) = 0, x (0) = where y() is the solutio of the followig liear differece equatio y( + 2) 2e 1 y( + 1) + (2 2e 1 )y() = 0, (32) which has the complex characteristic roots λ 1,2 = 1 e [1 ± i 1 2e + 2e 2 ]. So, Eq. (32) has oly oscillatory solutios. Hece from Corollary 1, Eq. (29)- (30) has oly oscillatory solutios too. The solutio x (t), = 0, 1,...11, of (29)-(30) with the iitial coditios x( 1) = 2e 2 2e 1/(2 2e), x(0) = 0 is give i Figure 2. x t 1 2 Figure 2: Oscillatory solutios of Eq. (29)-(30) with the iitial coditios x( 1) = 2e 2 2e 1/(2 2e), x(0) = 0 EJQTDE, 2013 No. 49, p. 10

11 Example 3. Fially we cosider the equatio x (t) + (l 2)x (t) + l x ([t 1]) = 0, t, (33) x () = x (), N. (34) Sice a(t) = l 2, f(x) = l ad d = , N, verify the hypotheses of Theorem 5, all solutios of Eq. (33)-(34) are oscillatory. O the other had, Sice Eq. (33)-(34) satisfies the hypotheses of Corollary 5, all solutios of (33)- (34) are periodic with period 5. This fact ca be see i Figure 3. x t Figure 3: Oscillatory solutios of Eq. (33)-(34) with the iitial coditios x( 1) = /4, x(0) = 0. Ackowledgemet. We would like to thak to referee for his/her valuable commets. Refereces [1] S. Buseberg ad K.L. Cooke. Models of vertically trasmitted diseases with sequetial-cotiuous dyamics. I: Noliear Pheomea i Mathematical Scieces, V. Lakshmikatham (Ed.), Academic Press, (1982) [2] L. Dai ad M. C. Sigh, O oscillatory motio of sprig mass systems subject to piecewise costat forces. J. Soud Vib., 173 (1994) [3] L. Dai ad M. C. Sigh, A aalytical ad umerical method for solvig liear ad oliear vibratio problems. It. J. Solids Struct. 34 (1997) EJQTDE, 2013 No. 49, p. 11

12 [4] L. Dai ad M. C. Sigh, Periodic, quasiperiodic ad chaotic behavior of a drive Froude pedulum. It. J. Noliear Mech. 33 (1998) [5] K. L. Cooke ad J. Wieer, Retarded differetial equatios with piecewise costat delays. J. Math. Aal. Appl., 99 (1984) [6] A. R. Aftabizadeh ad J. Wieer, Oscillatory properties of first order liear fuctioal differetial equatios. Applicable Aal., 20 (1985) [7] A. R. Aftabizadeh, J. Wieer ad J.-M. Xu, Oscillatory ad periodic solutios of delay differetial equatios with piecewise costat argumet. Proc. of America Math. Soc., 99 (1987) [8] J. Wieer ad A. R. Aftabizadeh, Differetial equatios alterately of retarded ad advaced type. J. Math. Aal. Appl. 129 (1988) [9] J. H. She ad I. P. Stavroulakis, Oscillatory ad ooscillatory delay equatios with piecewise costat argumet. J. Math. Aal. Appl. 248 (2000) [10] L. A. V. Carvalho ad K. L. Cooke, a oliear equatio with piecewise cotiuous argumet. Differetial ad Itegral Equatios 1 (1988) [11] A. Cabada ad J. B. Ferreiro, First order differetial equatios with piecewise costat argumets ad oliear boudary value coditios. J. Math. Aal. Appl. 380 (2011) [12] J. Wieer, Geeralized Solutios of Fuctioal Differetial Equatios. World Scietific, Sigapore, [13] F. Karakoc, H. Bereketoglu ad G. Seyha, Oscillatory ad periodic solutios of impulsive differetial equatios with piesewise costat argumet. Acta Appl. Math. 110 (2010) [14] H. Bereketoglu, G. Seyha ad A. Ogu, Advaced impulsive differetial equatios with piecewise costat argumets. Math. Model. Aal. 15 (2010) (Received February 13, 2013) EJQTDE, 2013 No. 49, p. 12