ON THE OSCILLATION OF THIRD ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS. Cairo University, Orman, Giza 12221, Egypt

a 1/α s)ds < Indian J. pre appl. Mah., 396): 491-507, December 2008 c Prined in India. ON THE OSCILLATION OF THIRD ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS SAID R. GRACE 1, RAVI P. AGARWAL 2 AND MUSTAFA F. AKTAS 3 1 Deparmen of Engineering Mahemaics, Facly of Engineering, Cairo Universiy, Orman, Giza 12221, Egyp e-mail: srgrace@eng.c.ed.eg 2 Deparmen of Mahemaical Sciences, Florida Insie of Technology, Melborne, FL 32901, U.S.A. e-mail: agarwal@fi.ed 3 Deparmen of Mahemaics, Gazi Universiy, Facly of Ars Sciences, Teknikokllar 06500 Ankara, Trkey e-mail: mfahri@gazi.ed.r Received 9 April 2008; afer final revision 29 Ocober 2008; acceped 21 November 2008) Some new crieria for he oscillaion of hird order fncional differenial eqaions of he form a) x ) ) α) + q)fx[g)]) = 0 a) x ) ) α) = q)fx[g)]) + p)hx[σ)]), where are esablished. Key words: Fncional differenial eqaion, oscillaion, nonoscillaion, comparison

492 SAID R. GRACE, RAVI P. AGARWAL AND MUSTAFA F. AKTAS 1. INTRODUCTION This paper deals wih he oscillaory behavior of solions of hird order nonlinear fncional differenial eqaions a) x )) α) + q)fx[g)]) = 0 1.1) a) x )) α) = q)fx[g)]) + p)hx[σ)]), 1.2) sbjec o he hypoheses: i) α is he raio of wo posiive odd inegers; ii) a, p, q : [ 0, ) 0, ) are coninos; iii) g, σ C 1 [ 0, ), IR), g) <, σ) >, g ) 0 σ ) 0 for 0 lim g) = ; iv) f, h C 1 IR, IR), xfx) > 0, xhx) > 0, f x) 0 h x) 0 for x 0 f xy) fxy) fx)fy) for xy > 0 1.3) h xy) hxy) hx)hy) for xy > 0. 1.4) By a solion of eqaion 1.1) respecively, eqaion 1.2)) is mean a fncion x : [T x, ) IR, T x 0 sch ha x), x ) a) x )) α ) are coninosly differeniable saisfy eqaion 1.1) respecively, eqaion 1.2)) on [T x, ). Or aenion will be resriced o solions x) of eqaions 1.1) 1.2) which saisfy sp{ x) : T } > 0 for any T T x. Sch a solion is said o be oscillaory if i has a seqence of zeros ending o infiniy, nonoscillaory oherwise. The oscillaory behavior of eqaions 1.1) 1.2) when α = 1, a) = 1 /or higher order fncional differenial eqaions has received a grea deal of aenion in he las hree decades. For recen conribions o his sdy we refer o reader [1-5, 8, 10] he references cied herein, however, he sdy of oscillaory behavior of eqaions 1.1) 1.2) when α 1 0 a 1/α s)ds = 1.5)

OSCILLATION OF THIRD ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 493 has received considerably less aenion, see [4-6, 9]. Therefore, he main prpose of his paper is o esablish some crieria for he oscillaion of eqaions 1.1) 1.2) when 0 a 1/α s)ds < 1.6) holds. In Secion 2, we presen oscillaion crieria for eqaion 1.1) Secion 3 is devoed o he sdy of he oscillaion of eqaion 1.2). The resls of his paper are presened in a form which is essenially new of high degree of generaliy. Some special cases of he obained resls nify improve many exising oscillaion crieria appeared in he lierare. 2. OSCILLATION OF EQUATION 1.1) Here we are ineresed o discss he siaion in which all solions of eqaion 1.1) are oscillaory. In order o prove or main resl we will compare he following firs order delay differenial ineqaliies wih he corresponding differenial eqaions y ) + q)fy[g)]) 0, 2.1) y ) q)fy[g)]) 0, 2.2) y ) + q)fy[g)]) = 0, 2.3) y ) q)fy[g)]) = 0. 2.4) We have he following lemma, which is given in [3, 7]. Lemma 2.1 If he ineqaliy 2.1) he ineqaliy 2.2)) has evenally posiive solion, hen he eqaion 2.3) he eqaion 2.4)) also has evenally posiive solion. Theorem 2.1 Sppose ha condiions i) iv), 1.3), 1.4) 1.6) hold assme ha here exiss a nondecreasing fncion η : [ 0, ) IR sch ha g) < η) < for 0. If boh firs order delay eqaions g) ) 1/α s z ) + c q)f ds) f z 1/α [g)] ) = 0 2.5) as) T for any consan c, 0 < c < 1 for all T 0, ) w ) + q)f A[g)] f [η ) g )] 1/α) f w 1/α [η)] ) = 0 2.6)

494 SAID R. GRACE, RAVI P. AGARWAL AND MUSTAFA F. AKTAS are oscillaory, 0 1 as) for any T 0, where A) = s T T ) 1/α qv)f A[gv)] dv d) ds =, 2.7) a 1/α s)ds, hen eqaion 1.1) is oscillaory. PROOF : Le x) be a nonoscillaory solion of eqaion 1.1), say, x) > 0 x[g)] > 0 for 0 0. Now a) x )) α ) 0 for 0. There exiss a 1 0 sch ha x ) a) x )) α ) are of consan sign for 1. There are for possibiliies o consider: I) a) x )) α ) > 0 x ) > 0 for 1 ; II) a) x )) α ) > 0 x ) < 0 for 1 ; III) a) x )) α ) < 0 x ) < 0 for 1 ; IV) a) x )) α ) < 0 x ) > 0 for 1. The case IV) canno hold. In fac, if we le y) = a) x )) α for 1, hen we see ha y ) < 0 y ) < 0 for 1 hence lim y) =, which conradics he posiiviy of x ). Nex, we consider: Case I): Le y) = a) x )) α. Then, since y) > 0 y ) is decreasing on [ 1, ), we have Then, here exis a 2 1 y) y 1 ) = 1 y s) ds y ) 1 ). a consan b, 0 < b < 1 sch ha y) b y ) for 2. Now ) 1/α x ) b 1/α y )) 1/α for 2. a) Inegraing boh sides of he above ineqaliy from 2 x) b 1/α 2 o, we ge ) 1/α s ds) y )) 1/α for 2. as)

OSCILLATION OF THIRD ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 495 There exiss a 3 2 sch ha g) ) 1/α s x[g)] b ds) y [g )]) 1/α for 3, 2.8) as) where b = b 1/α. Using 2.8) 1.3) in eqaion 1.1), we have 2 y ) = q)fx[g)]) fb)q)f f y [g )]) 1/α) for 3. g) 2 ) 1/α s ds) as) Se z) = y ) > 0 for 3 in he above ineqaliy, o obain g) ) 1/α s z ) + fb)q)f ds) f z 1/α [g)] ) 0 for 3. as) 2 By Lemma 2.1, we see ha he eqaion 2.5) has an evenally posiive solion, which conradics he hypohesis. Case II): Since a) x )) α is increasing, for s 1, we obain a) x )) α as) x s)) α or as) x s)) α a) x )) α, which by inegraion from o implies x ) [ a ) x )) α] 1/α a 1/α s)ds. Ping ha z) = a) x )) α replacing by g), we find x[g)] [ a g )) x g ))) α] 1/α A g )) for 2 1 = z 1/α [g )] A g )) for 2. 2.9) Using 2.9) 1.3) in eqaion 1.1), we have ) z ) q)f Ag)) f z 1/α [g )] ) for 2. 2.10) Clearly, z) > 0, z ) < 0, z ) > 0 for 2. Then, for s 2, we ge zs) z) zs) = s z ) d z ) s),

496 SAID R. GRACE, RAVI P. AGARWAL AND MUSTAFA F. AKTAS or zs) s) z )). Replacing s by g) η) respecively, we find z[g)] η) g)) z [η)]) for 3 2. 2.11) Using 2.11) 1.3) in 2.10), seing w) = z ), 3, we ge ) w ) q)f Ag)) f [η ) g )] 1/α) f w 1/α [η)] ) or ) w ) + q)f Ag)) f [η ) g )] 1/α) f w 1/α [η)] ) 0 for 3. By applying Lemma 2.1, we see ha he eqaion 2.6) has an evenally posiive solion, which is conradicion. Case III): For s 1, one can easily find as) x s)) α a) x )) α, or x s) a 1/α )x ) ) a 1/α s) ). Inegraing he above ineqaliy from o leing, we ge x) a 1/α ) x ) a 1/α s)ds = a 1/α ) x ) A) for 1 2.12) x) a 1/α 1 ) x 1 ) A) for 1. Now, here exis a 2 1 a consan c > 0 sch ha x[g)] c A[g)] for 2. 2.13) Using 2.13) 1.3) in eqaion 1.1), we ge a) x )) α) = q)fx[g)]) c q)f ) A[g)] for 2, 2.14) where c = fc). Inegraing 2.14) from 2 a) x )) α) + a2 )x 2 )) α ) o, we have ) c qs)f A[gs)] ds, 2

OSCILLATION OF THIRD ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 497 or a) x )) α) Inegraing 2.15) from 2 ) c qs)f A[gs)] ds for 2. 2.15) 2 o, we have a) x )) α c s 2 2 ) q)f A[g)] d ds, or 1 x ) c a) s 2 2 ) 1/α q)f A[g)] d ds), 2.16) where c = c) 1/α. Inegraing 2.16) from 2 o, sing he assmpion 2.7) we have > x 2 ) x 2 ) x) c dv as 2 1 v av) 2 s 2 ) ) 1/α q)f A[g)] d ds which is a conradicion. This complees he proof. We sae some known resls for he firs order fncional differenial eqaions y ) + Q)y[g)] = 0, 2.17) y ) Q)y[g)] = 0, 2.18) y ) + Q)F y[g)]) = 0, 2.19) y ) Q)F y[g)]) = 0. 2.20) Theorem A Le he fncions g, Q C[ 0, ), IR), Q) 0 evenally, g), g ) 0 for 0 lim g) =. i) If lim inf g) Qs) ds > 1, hen he eqaion 2.17) is oscillaory. e ii) If lim inf g) Qs) ds > 1, hen he eqaion 2.18) is oscillaory. e iii) If lim sp Qs) ds > 1, hen he eqaion 2.17) is oscillaory. g)

498 SAID R. GRACE, RAVI P. AGARWAL AND MUSTAFA F. AKTAS iv) If lim sp g) Qs) ds > 1, hen he eqaion 2.18) is oscillaory. Theorem B Le he fncions g, Q C[ 0, ), IR), Q) 0 evenally, g), g ) 0 for 0 lim g) =, F C 1 IR, IR), x F x) > 0, F x) 0. i) If F ) oscillaory. 0 as 0, lim sp Qs) ds > 0, hen he eqaion 2.19) is g) ii) If F ) oscillaory. iii) If ±0 0 as 0, lim inf g) Qs) ds > 0, hen he eqaion 2.20) is d F ) < Qs) ds =, hen he eqaion 2.19) is oscillaory. iv) If ± d F ) < Qs) ds =, hen he eqaion 2.20) is oscillaory. By combining Theorem A., Theorem B. Theorem 2.1. we have he following resl: Theorem 2.2 Le condiions i) iv), 1.6) 2.7) hold sppose ha here exiss a nondecreasing fncion η : [ 0, ) IR sch ha g) < η) < for 0. Eqaion 1.1) is oscillaory if one of he following condiions holds: I 1 ) f1/α ) k for 0 some k > 0 gs) lim sp qs) g) T ) 1/α d a) for any consan c, 0 < c < 1 every T 0, ) αds 1 lim sp qs) ηs) gs)) A[gs)] > k. 3 η) ) α ds > 1 c k 2 2.21) I 2 ) f1/α ) k for 0 some k > 0 gs) lim inf qs) g) T ) 1/α d a) ) α ds > 1 c e k 2 2.22)

OSCILLATION OF THIRD ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 499 for any consan c, 0 < c < 1 any T 0, lim inf η) ) αds 1 qs)ηs) gs)) A[gs)] > e k. 3 I 3 ) f 1/α ) 0 as 0, lim sp g) qs)f gs) T ) 1/α d) ds > 0 for T 0, a) ) lim sp qs)f A[gs)] f [ηs) gs)] 1/α) ds > 0. η) I 4 ) ±0 d f 1/α ) <, qs)f gs) T ) 1/α d) ds =, for T 0, a) ) qs)f A[gs)] f [ηs) gs)] 1/α) ds =. When he condiion 1.5) holds, we can also eliminae he case III). Indeed, from 2.12) we have x) x 1 ) K 1 a 1/α s) ds,, where K = a 1/α 1 ) x 1 ) > 0, which conradics he posiiviy of x). Therefore, we have he following resl: Theorem 2.3 Sppose ha condiions i) iv), 1.3), 1.4) 1.5) hold assme ha here exiss a nondecreasing fncion η : [ 0, ) IR sch ha g) < η) < for 0. If boh firs order delay eqaions 2.5) 2.6) are oscillaory, hen eqaion 1.1) is oscillaory. The following example is illsraive. Example 2.1 Consider he hird order eqaion 4 x )) 3) + 1 ln xβ [ ] = 0, > 5, 2.23)

500 SAID R. GRACE, RAVI P. AGARWAL AND MUSTAFA F. AKTAS where β is he raio of wo posiive odd inegers, 0 < β 3. Here a) = 4 α = 3 so 0 a 1/α s)ds = 5 s 4/3 ds <. Also, g) = so, we le η) = 2 for all > 5. I is easy o check ha all condiions of Theorem 2.2 are saisfied hence eqaion 2.23) is oscillaory. Similarly, we see ha he eqaion is oscillaory by Theorem 2.2 wih α = 1. 4/3 x ) ) + 1 ln xβ [ ] = 0 3. OSCILLATION OF EQUATION 1.2) The main objecive of his secion is o esablish some crieria for he oscillaion of all solions of eqaion 1.2) of mixed nonlineariies argmens. Now, we we will compare he following ineqaliy wih he corresponding second order differenial eqaions y ) + q)fy[g)]) 0, 3.1) y ) + q)fy[g)]) = 0, 3.2) We have he following lemma, which is given in [3, 7]. Lemma 3.1 If he ineqaliy 3.1) has evenally posiive solion, hen he eqaion 3.2) also has evenally posiive solion. Theorem 3.1 Sppose ha condiions i) iv), 1.3), 1.4) 1.6) hold assme ha here exis nondecreasing fncions η, ρ, θ : [ 0, ) IR sch ha g) < η) < σ) > ρ) > θ) > for 0. If he firs order advanced eqaion ) σ) y ) p)h a 1/α s)ds h [ρ) θ)] 1/α) h y 1/α [θ)] ) = 0, 3.3) ρ) he firs order delay eqaion ) g) z ) + q)f a 1/α s)ds f [η) g)] 1/α) f z 1/α [η)] ) = 0, 3.4) he second order eqaion 0 w ) + q)f ) A[g)] fw 1/α [g)]) = 0, 3.5)

OSCILLATION OF THIRD ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 501 where A) = a 1/α s)ds, are oscillaory, hen eqaion 1.2) is oscillaory. PROOF : Le x) be a nonoscillaory solion of eqaion 1.2), say, x) > 0, x[g)] > 0 x[σ)] > 0 for 0 0. Clearly, a) x )) α ) 0 for 0. There exiss a 1 0 sch ha a) x )) α ) x ) are of fixed sign for 1. Then as earlier here are for possibiliies o consider: I) a) x )) α ) > 0 x ) > 0 for 1 ; II) a) x )) α ) > 0 x ) < 0 for 1 ; III) a) x )) α ) < 0 x ) < 0 for 1 ; IV) a) x )) α ) < 0 x ) > 0 for 1. Now, he case II) canno hold. In fac, if we le y) = a) x )) α for 1, hen we see ha y ) > 0 y ) > 0 for 1 hence lim y) = which conradics he negaiviy of x ). Nex, we shall consider: Case I): For s 1, we have a ) x )) α a s) x s)) α hence x ) [ a s) x s)) α] 1/α s a 1/α )d. Replacing s by σ) ρ) respecively, we find x[σ)] [a ρ )) x ρ ))) α] 1/α σ) a 1/α s)ds = y 1/α [ρ )] σ) ρ) ρ) a 1/α s)ds for 2 1, 3.6) where y) = a) x )) α. Using 3.6) 1.4) in eqaion 1.2), we obain ) σ) y ) p)hx[σ)]) p)h a 1/α s)ds h y 1/α [ρ)] ) for 2. 3.7) ρ) For s 2, we have y ) y s) = s y ) d

502 SAID R. GRACE, RAVI P. AGARWAL AND MUSTAFA F. AKTAS since y ) is increasing or y) s)y s) y) 1/α s) 1/α y s)) 1/α. Replacing s in he above ineqaliy by ρ) θ) respecively, we ge y 1/α [ρ)] ρ) θ)) 1/α y [θ)]) 1/α for 3 2. 3.8) Using 3.8) 1.4) in 3.7) leing z) = y ) for 3, we have ) σ) z ) p)h a 1/α s)ds h [ρ) θ)] 1/α) h z 1/α [θ)] ) for 3. ρ) By Lemma 2.1, we arrive a he desired conradicion. Case III): As in he proof of Theorem 2.1 Case III), we obain 2.12). There exiss a 2 1 sch ha x[g)] A[g)]w 1/α [g)] for 2, 3.9) where w) = a) x )) α > 0. Using 3.9) 1.3) in eqaion 1.2), we have w ) + q)fa[g)])f w 1/α [g)] ) 0 for 2. Using Lemma 3.1, we see ha he eqaion 3.5) has an evenally posiive solion, which conradics he hypohesis. Case IV) : For 1, we have ) a) x) = x 1 ) + x s)ds a 1/α s)ds x )) α) 1/α. 1 1 There exiss a 2 1 sch ha ) g) x[g)] a 1/α s)ds y 1/α [g)] for 2, 3.10) 1 where y) = a) x )) α for 2. Using 3.10) 1.3) in eqaion 1.2), we ge ) g) y ) q)f a 1/α s)ds f y 1/α [g)] ) for 2. 3.11) 1

OSCILLATION OF THIRD ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 503 have or Clearly, y ) is negaive increasing for 2. Therefore, for s 2, we y) ys) = s y )d y ) s), ys) s) y )). Replacing s in he above ineqaliy by g) η) respecively, we ge y[g)] η) g)) y [η)]) for 3 2. 3.12) Using 3.12) 1.3) in 3.11), leing z) = y ) > 0 for 3, we have ) g) z ) + q)f a 1/α s)ds f [η) g)] 1/α) f z 1/α [η)] ) 0 for 3. 1 By Lemma 2.1, we arrive a he desired conradicion. This complees he proof. From Theorems A., B. Theorem 3.1., he following resl is immediae. Theorem 3.2 Le condiions i) iv), 1.3), 1.4) 1.6) hold sppose ha here exis nondecreasing fncions η, ρ, θ : [ 0, ) IR sch ha g) < η) < σ) > ρ) > θ) > for 0. Eqaion 1.2) is oscillaory if ) qs)f A[gs)] ds =, 3.13) one of he following condiions holds: II 1 ) f1/α ) k h1/α ) h for 0 some k, h > 0 lim sp θ) α σs) h 3 ps) ρs) θs)) a τ)dτ) 1/α ds > 1 ρs) ) α gs) lim sp k 3 qs) ηs) gs)) a 1/α τ)dτ ds > 1 for 0 0. η) 0 II 2 ) f1/α ) k h1/α ) h for 0 some k, h > 0 lim inf h3 θ) σs) ps) ρs) θs)) a 1/α τ) dτ ρs) ) α ds > 1 e

504 SAID R. GRACE, RAVI P. AGARWAL AND MUSTAFA F. AKTAS II 3 ) II 4 ) f 1/α ) ±0 lim inf lim sp k3 η) ) α gs) qs) ηs) gs)) a 1/α τ)dτ ds > 1 e. 0 as 0 0 as, h 1/α ) θ) ) σs) ps)h a 1/α τ) dτ h [ρs) θs)] 1/α) ds > 0 ρs) ) gs) lim sp qs)f a 1/α τ)dτ f [ηs) gs)] 1/α) ds > 0. η) 0 d f 1/α ) < ± 0 d h 1/α ) <, ) σs) ps)h a 1/α τ) dτ h [ρs) θs)] 1/α) ds = ρs) ) gs) qs)f a 1/α τ)dτ f [ηs) gs)] 1/α) ds =. 0 When he condiion 1.5) holds, we eliminae he case III) from Theorem 3.1, we obain Theorem 3.3 Sppose ha condiions i) iv), 1.3), 1.4) 1.5) hold assme ha here exis nondecreasing fncions η, ρ, θ : [ 0, ) IR sch ha g) < η) < σ) > ρ) > θ) > for 0. If he firs order advanced eqaion 3.3) he firs order delay eqaion 3.4) are oscillaory, hen eqaion 1.2) is oscillaory. The following example is illsraive. Example 3.1 Consider he hird order mixed eqaion 6 x )) 3) = 1 ) 2 β x β [ ] + 2γ/3 x γ [2], 5. 3.14) Here, a) = 6, α = 3, q) = 1/ ) 2 β p) = 2γ/3, β γ are he raios of posiive odd inegers wih 0 < β 3 γ. Also g) = σ) = 2. For 5, we le η) = 2, ρ) = 5/3 θ) = 4/3.

OSCILLATION OF THIRD ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS 505 I is easy o check ha all condiions of Theorem 3.2 for differen vales of β γ are saisfied hence eqaion 3.14) is oscillaory. We noe ha here are many sfficien condiions for he oscillaion of eqaion 3.5) raher han 3.13), see [1 3]. Nex, from he proof of Theorem 3.1 Case III), we obain he second order ineqaliy w ) + q)fa[g)])f w 1/α [g)] ) 0 for 2 3.15) for w) = a) x )) α. Since w ) > 0 w) > 0 for 2, here exis a 3 2 a consan k, 0 < k < 1 sch ha w[g)] kg)w [g)] for 3. 3.16) Using 3.16) 1.3) in 3.15), we obain ) v ) + c q) f A[g)] f g 1/α ) ) f v 1/α [g)] ) 0 for 3, 3.17) where c = fk 1/α ) v) = w ) for 3. Nex, we can combine eqaions 3.4) 3.17) in one, say, 0 y ) + Q)f y 1/α [η)] ) = 0, 3.18) where { ) g) Q) = min q)f a 1/α s)ds f [η) g)] 1/α) ), c q)f A[g)] f g 1/α ) )} for any consan c, 0 < c < 1 all 0. Ths, from Theorem 3.1 we ge he following corollary: Corollary 3.1 Le he hypoheses of Theorem 3.1 hold le eqaions 3.4) 3.5) be replaced by eqaion 3.18). Then eqaion 1.2) is oscillaory. In a similar way, we can combine eqaions 2.5) 2.6) in one resae Theorem 2.1. The deails are lef o he reader. 4. REMARKS 1. Condiions 1.3) 1.4) may be discarded if we le fx) = x β hx) = x γ, where β γ are he raios of posiive odd inegers.

506 SAID R. GRACE, RAVI P. AGARWAL AND MUSTAFA F. AKTAS 2. We noe ha condiions 2.21) 2.22) respecively can be replaced by lim sp k 2 qs) g) gs) T T a) ) 1/α d ) α ds > 1 lim inf k2 g) qs) gs) T T a) ) 1/α d ) α ds > 1 e. 3. The resls of Secion 3 can be obained for eqaions of he form a) x )) α) = q)fx[g)]), where he argmen g) is of mixed ype, say, g) = sin. The deails are lef o he reader. 4. The resls of his paper are exendable o neral eqaions of he form ) d 2 α ) d a) [x) + c)x[α)]] + q)fx[g)]) = 0 d 2 d ) d 2 α ) d a) [x) + c)x[α)]] d 2 d = q)fx[g)]) + p)hx[σ)]), where c), α) C[ 0, ), IR) reader. lim α) =. The deails are lef o he I will be ineresing o derive differen oscillaion crieria for he oscillaion of eqaions 1.1) 1.2) via oher echniqes raer han hose presened here. REFERENCES 1. R. P. Agarwal, S. R. Grace D. O Regan, Oscillaion Theory for Difference Fncional Differenial Eqaions, Klwer, Dordrech, 2000. 2. R. P. Agarwal, S. R. Grace D. O Regan, Oscillaion Theory for Second Order Linear, Half linear, Sperlinear Sblinear Dynamic Eqaions, Klwer, Dordrech, 2002. 3. R. P. Agarwal, S. R. Grace D. O Regan, Oscillaion Theory for Second Order Dynamic Eqaions, Taylor & Francis, London, 2003.

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