ON THE OSCILLATION OF THREE DIMENSIONAL. Department of Electrical and Electronic Engineering Educators N. Heraklio, Athens, 14121, GREECE

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1 Dynamic Systems and Applications, 27, No ), ISSN: ON THE OSCILLATION OF THREE DIMENSIONAL α-fractional DIFFERENTIAL SYSTEMS G.E. CHATZARAKIS, M. DEEPA 2, N. NAGAJOTHI 3, AND V. SADHASIVAM 4 School of Pedagogical and Technological Education ASPETE) Department of Electrical and Electronic Engineering Educators N. Heraklio, Athens, 42, GREECE 2,3,4 Department of Mathematics Thiruvalluvar Government Arts College Affli. to Periyar University), Rasipuram Namakkal Dt. Tamil Nadu, INDIA ABSTRACT: In this article, we consider the three dimensional α-fractional nonlinear differential system of the form D α xt)) = at)f yt)), D α yt)) = bt)gzt)), D α zt)) = ct)hxt)), t, where 0 < α, D α denotes the Katugampola fractional derivative of order α. We establish some new sufficient conditions for the oscillation of the solutions of the differential system, using the generalized Riccati transformation and inequality technique. Examples illustrating the results are also given. AMS Subject Classification: 34A08, 34A34, 34K Key Words: oscillation, α-fractional derivative, nonlinear systems Received: Augus3, 208; Accepted: September 25, 208; Published: November 26, 208. doi: /dsa.v27i4.2 Dynamic Publishers, Inc., Acad. Publishers, Ltd. INTRODUCTION The problem of oscillation and nonoscillation of differential equations was first studied by C. Sturm in his seminal paper, published in 836. In the last decades, a number

2 874 G.E. CHATZARAKIS, M. DEEPA, N. NAGAJOTHI, AND V. SADHASIVAM of papers and research monographs have appeared on the theory and applications of oscillations. The problem has been studied by several authors [5, 9, 2, 28]. On the other hand, the qualitative theory of nonlinear systems of differential equations was introduced by Henry Poincare at the end of the nineteenth century. Considerable attention has been given to the oscillation of linear and nonlinear differential systems that have been studied by many authors [8,, 5, 7, 27, 29, 34]. Even though, the oscillation theory of classical differential system is well established, the progress in the oscillation of nonlinear fractional differential systems is relatively slower, due to the nonlocal behavior of fractional derivatives, involving singular kernels. Below, we review some of the applications of three-dimensional systems, starting with the system studied by Shirokorad [3]: ẋ = y fx), ẏ = z, ż = ρz kfx). The above system works as a flight controller, for an aircraft, but is also the classical scheme of a one-valve electronic generator. In chemical modeling, Robertson[25] proposed the system of differential equations, x = 0.04x +0 4 x 2 x 3, x 0) =, x 2 = 0.04x x x 2 x 3, x 2 0) = 0, x 3 = x 2 2, x 3 0) = 0. In 983, Rai et al. [24] modeled mutualism or inter-specific cooperation among three species, using the system ) du dt = γu u, L 0 +lx dx dt = αx x ) βxy K +mu, dy dt = y s+ cβx +mu Lorenz derived a simple model, for predicting the weather based on a simplified version of the Rayleigh-Bernard[20] convection fluids model. The Lorenz system has the form ẋ = σy x), ẏ = Rx y xz, ż = bz +xy. ).

3 α-fractional DIFFERENTIAL SYSTEM 875 This system provides a specific example of chaotic dynamics, persisting all the time. Fractional order differential equations have been applied to study several physical phenomena such as Physics and Chemistry [viscoelastic systems, polymeric materials], Medicine and Pharmacology [neuronal dynamics, pharmacokinetics], Biology and Ecology [population evolution, illness propagation], Financial Mathematics and Economics [Black-Scholes equation, price formation], etc. In stock market analysis, fractional order models have recently been used to describe the probability distribution of log-prices in the long-time limit, which is useful to characterize the natural variability in prices in the long term. For the fundamental theory of fractional differential equations and the general background, we refer the reader to the papers and monographs in [, 6, 7, 0, 6, 9, 22, 23, 30, 33, 35]. The Caputo and Riemann-Liouville fractional derivatives are based on integral expressions and gamma functions which are nonlocal. In 204, though, Khalil et al.[4] introduced a new fractional derivative called the conformable derivative, using a limit definition analogous to that of standard derivative. Then the fractional version of the chain rule, exponential functions and integration by parts was developed in [2, 3]. The conformable derivative of Khalil was soon generalized by Katugampola to what in this paper, is referred as the Katugampola fractional derivative [4, 2, 3]. In 207, Sadhasivam et al. [26] studied the existence of solutions of three-dimensional fractional differential systems. In 2000, Spanikova et al.[32] investigated the oscillatory properties of three-dimensional differential systems of the neutral type. To the best of our knowledge, the oscillatory behavior of an α-fractional nonlinear three-dimensional differential system has been investigated yet. This paper attempts to deal with such type of system as it is yet, an unexplored area. This scarcity of work has led us to consider the following system D α xt)) = at)f yt)), D α yt)) = bt)gzt)), ) D α zt)) = ct)hxt)), t, where 0 < α, D α denotes the α-fractional derivative of order α with respect to t. Throughout this paper, we assume the following conditions: A ) at) C 2α [, ),R + ), bt) C α [, ),R + ), ct) C[, ),R + ), ct) is not identically zero on any interval of the form [T 0, ), where T 0 ; A 2 ) f C R,R),yfy) > 0,D α fy) K > 0, g C R,R), zgz) > 0,D α gz) L > 0, h CR,R),xhx) > 0 and hx) x M > 0 for x 0. By a solution of the system ), we mean a function

4 876 G.E. CHATZARAKIS, M. DEEPA, N. NAGAJOTHI, AND V. SADHASIVAM Ut) = xt),yt),zt)), which has the properties rt)d α xt) C 2α [T, ),R), pt)d α rt)d α xt)) C α [T, ),R) and satisfies system ) on [T, ). Denote by S the set of all solutions Ut) of ) which exist on some ray [T, ) [, ) and satisfy sup{ xt) + yt) + zt) : t T} > 0 for any T T. We assume tha) possesses such solution. A solution Ut) S is considered to be oscillatory if all the components are oscillatory, otherwise it will be called nonoscillatory. The system ) is called oscillatory if all its solutions are oscillatory otherwise it will be called nonoscillatory. The objective in this paper is to establish new oscillation criteria, for ), making use of generalized Riccati transformation and inequality technique. This paper is organized as follows. In Section 2, we review fundamental concepts on the α- fractional derivative. In Section 3, we establish some new conditions for the oscillatory behavior of the solutions of system ). In the final section, we present some illustrative examples on our results. 2. PRELIMINARIES The purpose of this section is to introduce some basic definitions of the Katugampola α-fractional derivatives and integrals which we will use throughout the paper. We begin with the following definition. Definition. [2] Let y : [0, ) R and t > 0. Then the fractional derivative of y of order α is given by D α yte ǫt α ) yt) y)t) := lim ǫ 0 ǫ for t > 0, 2) α 0,]. If y is α-differentiable in some 0,a), a > 0, and lim t 0 +Dα y)t) exists, then we define D α y)0) := lim t 0 +Dα y)t). The α-fractional derivative satisfies the following properties. Let α 0,] and f, g be α- differentiable at a point t > 0. Then p ) D α t n ) = nt n α for all n R. p 2 ) D α C) = 0 for all constant functions, ft) = C. p 3 ) D α fg) = fd α g)+gd α f). p 4 ) D α f g ) = gdα f) fd α g) g 2. p 5 ) D α f g)t) = D α fgt))d α g)t). p 6 ) If f is differentiable, then D α f)t) = αdf dt t).

5 α-fractional DIFFERENTIAL SYSTEM 877 Definition 2. [2] Let a 0 and t a. Also, let y be a function defined on a,t] and α R. Then, the α-fractional integral of y is given by I α a y)t) := if the Riemann improper integral exists. a yx) x αdx 3) 3. MAIN RESULTS In this section, we study the oscillatory behavior of the solutions of system ) under certain conditions. A 3 ) we will consider the case: s α ds =, s α ds =, ps) rs) where pt) = bt),rt) = at) valued continuous functions. and qt) = KLct), pt),rt) and qt) are positive real Before stating the main theorems, we present some results in the form of Lemmas that will facilitate the proofs of our main results. Lemma 3. If Ut) S is a nonoscillatory solution of ), then the component function xt) is always nonoscillatory. Proof. The proof follows from Lemma. in [8]. The next lemma will be used in our main results. Lemma 4. Suppose that A 3 ) holds. Then there exists a such that either I) xt) > 0,D α xt) > 0,D α rt)d α xt)) > 0 for t. or II) xt) > 0,D α xt) < 0,D α rt)d α xt)) > 0 for t. Proof. Let xt) be an eventually positive solution of ) on, ). Now, system ) can be reduced to the following nonlinear differential inequality )) D α bt) Dα at) Dα xt) +KLct)hxt)) 0, t, 4) which implies, D α pt)d α rt)d α xt)))+qt)hxt)) 0, t, 5)

6 878 G.E. CHATZARAKIS, M. DEEPA, N. NAGAJOTHI, AND V. SADHASIVAM From 5), we get D α pt)d α rt)d α xt))) 0 for t. Then pt)d α rt)d α xt)) is decreasing on, ). Suppose now, D α rt)d α xt)) 0 then rt)d α xt) is decreasing and there exists a constant l and t 2 such that pt)d α rt)d α xt)) l for t t 2. Integrating from t 2 to t, we get rt)d α xt) rt 2 )D α xt 2 ) l s α ds. 6) t 2 ps) Letting t and using A 3 ), we get rt)d α xt). Hence, there is an integer t 2 t 3 such that rt)d α xt) rt 3 )D α xt 3 ) < 0 for t t 3. Integrating from t 3 to t, we get xt) xt 3 )+rt 3 )D α xt 3 ) s α ds. 7) t 3 rs) As t, xt) by A 3 ). Which gives a contradiction to xt) > 0. We conclude that D α rt)d α xt)) > 0 and rt)d α xt) is increasing and we are led to I) or II). Lemma 5. Suppose that A 3 ) and Case I) of Lemma 4 hold. Then there exists a such that where λt) = s α ps) ds. D α xt) λt) rt) pt)dα rt)d α xt)) for t, 8) Proof. Let xt) be an eventually positive solution of). Consider CaseI) of Lemma 4 and inequality 5). From these two, we obtain D α xt) > 0, D α rt)d α xt)) > 0, and D α pt)d α rt)d α xt))) 0 for t. Thus, From this, we get 8). rt)d α xt) = r )D α x )+ pt)d α rt)d α xt)) The following theorem is our main result. s α ps)dα rs)d α xs)) ds ps) s α ps) ds for t. Theorem 6. Suppose that assumptions A ) A 3 ) hold. Furthermore, assume that there exists a positive function δ C α [0, );R + ) such that s α Mqs)δs) s)) 2 s α ) rs) 4 δ + ds =, 9) δs)λs)

7 α-fractional DIFFERENTIAL SYSTEM 879 and ξ α rξ) ξ ζ α rζ) ζ ) ) s α qs)ds dζ dξ =, 0) { } where δ s)) + = max 0,δ s). Then every solution of system ) is either oscillatory or lim xt) = 0. Proof. Suppose tha) has a nonoscillatory solution xt),yt),zt)) on [, ). From Lemma 3, xt) is always nonoscillatory. Without loss of generality, we shall assume that xt) > 0 for t, where is chosen so large that Lemma 4 and Lemma 5hold. Asimilarargumentcould be made, ifthe solutionxt) wereeventually negative. Suppose that Case I) of Lemma 4 holds for t. Define the generalized Riccati transformation wt) = δt) pt)dα rt)d α xt)), t. ) xt) Thus wt) > 0. Differentiating with respect to t and using 5), 8), we have D α wt) Dα δt) wt) Mδt)qt) λt) δt) δt)rt) w2 t), t. 2) Therefore, from p 6 ), we have the inequality w t)) t) δ + wt) Mt α qt)δt) t α λt) δt) δt)rt) w2 t). 3) Using the inequality, γ 2) γab γ A γ γ )B γ, 4) we get w t) Mt α qt)δt) 4 t α δ t)) 2 + rt) δt)λt) ). 5) Integrating both sides from to t, we get w s)ds Ms α qs)δs) ) 4 s α δ s)) 2 rs) + ds, δs)λs) which yields that s α Mqs)δs) s)) 2 s α ) rs) 4 δ + ds w ), δs)λs) for all large, which contradicts hypothesis 9).

8 880 G.E. CHATZARAKIS, M. DEEPA, N. NAGAJOTHI, AND V. SADHASIVAM Next, weconsidercaseii)oflemma4. Thatis, D α xt) < 0, D α rt)d α xt)) > 0 for t. Since xt) is positive and decreasing, there exists a lim xt) = k 0. Suppose that k > 0. Integrating 5) from t to, we have then, which implies, t ) ps)d α rs)d α xs)) ds pt)d α rt)d α xt)) D α rt)d α xt)) pt) t rt)d α xt)) t α pt) t t t s α qs)hxs))ds, 6) s α qs)hxs))ds, s α qs)hxs))ds, t s α qs)hxs))ds, Again integrating the above inequality from t to, we obtain rt)d α xt) ζ α ) s α qs)hxs))ds dζ, t pζ) ζ D α xt) ζ α ) s α qs)hxs))ds dζ, rt) pζ) ζ then, x t) t α rt) t ζ α pζ) ζ ) s α qs)hxs))ds dζ. Once again integrating this inequality from to, we get x ) ξ α rξ) From A 2 ) and xt) k, we get ξ ζ α pζ) ζ s α qs)hxs))dsdζdξ. 7) then, x ) M x ) Mk ξ α rξ) ξ ξ α rξ) ζ α pζ) ξ ζ ζ α pζ) s α qs)xs)dsdζdξ, ζ ) s α qs)dsdζ dξ, 8) which contradicts 0). Hence k=0. That is, xt) 0 as t and hence Ut) 0 as t. Let δt) = t, t, we yields the corollary.

9 α-fractional DIFFERENTIAL SYSTEM 88 Corollary 7. Suppose that A ) A 3 ) and 0) hold. Furthermore, assume that Ms α qs) 4 rs) s α λs) ) ds =. 9) Then every solution of ) is either oscillatory or tends to zero as t. Next, we extend the results of Theorem 6, using the Kamanev-type condition to the system under study. Theorem 8. Assume that A ) A 3 ) and 0) hold. Moreover, assume there exists a δ C α [0, );R + ) such that t t n t s) n s α Mqs)δs) δ s)) 2 s α rs) + 4δs)λs) ) ds =, 20) where δ s)) + as in Theorem 6. Then every solution of system ) is either oscillatory or lim xt) = 0. Proof. Assume that Case I) of Lemma 4 holds, for t. Proceeding as in the proof of Theorem 6, we obtain 5) for t. Multiplying by t s) n and integrating from to t, we get t s) n s α Mqs)δs) s)) 2 4 δ + Then t t n s α ) rs) ds δs)λs) t s) n w s)ds, 2) t n t )n w ). 22) t s) n s α Mqs)δs) s α δ s)) 2 + rs) )ds w ). 23) 4δs)λs) This contradicts 20). We follow the exact same procedure, for case II), as in Theorem 6. Theorem 9. Assume that A ) A 3 ), and 0) hold. Furthermore, assume that there exists a positive function δ C α [0, );R + ) such that t t n t s) n s α Mqs)δs) s α ) rs)δs) 4 λs)t s) n F2 t,s) ds =, 24) where Ft,s) = t s) n δ s)) + δs) nt s) n. Then every solution of system ) is either oscillatory or lim xt) = 0.

10 882 G.E. CHATZARAKIS, M. DEEPA, N. NAGAJOTHI, AND V. SADHASIVAM Proof. For case I) of Lemma 4, we proceed as in the proof of Theorem 6, to obtain 3), for t. Multiplying by t s) n and integrating from to t and using 22), we get Mt s) n s α qs)δs)ds t ) n w )+ nt s) n )ws)ds Then, from 4), we have that Hence t s) nδ s)) + δs) t s) nsα λs) rs)δs) w2 s)ds. 25) Mt s) n s α qs)δs)ds t ) n w ) + 4 s α rs)δs) t s) n λs) ) 2 s)) t s) nδ + nt s) n ds. 26) δs) t t n t s) n s α Mqs)δs) s α ) rs)δs) 4 λs)t s) n F2 t,s) ds w ), which contradicts 24). For Case II), we use the same arguments, as in the proof of Theorem 6. Let D 0 = {t,s) : t > s }, D = {t,s) : t s }. The function H CD,R) is said to belong to the class R, if T ) Ht,t) = 0 for t, Ht,s) > 0 for t,s) D 0. T 2 ) H has a continuous nonpositive partial derivative on D 0 with respect to s such that ht,s) Ht,s) = H s t,s). Theorem 0. Assume that A ) A 3 ), and 0) hold. Furthermore, assume that there exists a function δ C α [0, );R + ) and H R such that Ht, ) Ht,s)s α Mqs)δs) s α ) rs)δs) 4 λs)ht,s) G2 t,s) ds =, 27) where Gt,s) = Ht,s) δ s)) + δs) ht,s) Ht,s). Then every solution of system ) is either oscillatory or xt) 0 as t.

11 α-fractional DIFFERENTIAL SYSTEM 883 Proof. Proceeding as in the proof of Theorem 6, for case I), we obtain 3), for t. Multiplying by Ht,s) and integrating from to t, we get MHt,s)s α qs)δs)ds Ht, )w ) + H δ s t,s)ws)+ht,s) Ht, )w )+ s)) + ws) sα λs) δs) Ht,s) δ s)) + ht,s) Ht,s) δs) )) rs)δs) w2 s) ) ws) ds. 28) Ht,s) sα λs) rs)δs) w2 s)ds. 29) Using 4), we get Ht, ) Ht,s)s α Mqs)δs) which contradicts 27). The proof for Case II) is as in Theorem 6. s α ) rs)δs) 4 λs)ht,s) G2 t,s) ds w ), 30) The following result establishes a modified oscillation criterion, excluding condition 27). Theorem. Assume that all the conditions of Theorem 6 hold, except from 27). Let ) Ht, s) 0 < inf lim inf, 3) s Ht, ) and Moreover, let some φ C[, );R + ) such that MHt,s)s α qs)δs)ds <. 32) Ht, ) and for all T large enough Ht,T) T s α φ 2 s)λs) ds =, 33) rs)δs) Ht,s)s α Mqs)δs)

12 884 G.E. CHATZARAKIS, M. DEEPA, N. NAGAJOTHI, AND V. SADHASIVAM s α ) rs)δs) 4 λs)ht,s) G2 t,s) ds φt), 34) where Ht,s) and Gt,s) are defined in Theorem 0. Then every solution of system ) is either oscillatory or xt) 0 as t. Proof. Proceedingasin Theorem0, forcasei), weget 30), with = T. Therefore From 34), we obtain Ht,T) T Ht,s)s α Mqs)δs) s α ) rs)δs) 4 λs)ht,s) G2 t,s) ds wt). φt) wt) for all T, 35) and φ ) liminf Ht, ) Ht,s)s α Mqs)δs) s α ) rs)δs) 4 λs)ht,s) G2 t,s) ds. By 32), 4Ht, ) s α rs)δs) λs)ht,s) G2 t,s)ds <. 36) Now, we define the functions χt) and Ψt) as χt) = Gt,s)ws)ds 37) Ht, ) and Then, from 29), it follows that s α λs) Ψt) = Ht,s) Ht, ) rs)δs) w2 s)ds. 38) Ψt) χt) w ) Ht,s)s α Mqs)δs)ds < w ), 39) Ht, )

13 α-fractional DIFFERENTIAL SYSTEM 885 so that Ψt) χt)) w ). Thus, there exists an increasing sequence {t k } k=0 with t k as k so that From this We have to show that Ψt) χt)) = lim Ψt k) χt k )). k Ψt k ) χt k ) < w ), k =,2,... 40) s α λs) rs)δs) w2 s)ds <. 4) If not, s α λs) rs)δs) w2 s)ds =. 42) By 3), there is a constant η > 0 satisfying Ht, s) Ht, ) > η > 0 for all s for large t. 43) Let γ > 0 be an arbitrary constant. Then from 42), assuming t 2 is very large, we get s α λs) rs)δs) w2 s)ds γ η for all t t 2. Therefore, for t, we have Ψt) = Ht, ) Ht, ) γ η Ht, ) Ht, s) s Ht, s) s s s τ α λτ) rτ)δτ) w2 τ)dτ ds τ α λτ) rτ)δτ) w2 τ)dτ ds Ht, s) ds = γ Ht, ) s η Ht, ).

14 886 G.E. CHATZARAKIS, M. DEEPA, N. NAGAJOTHI, AND V. SADHASIVAM By 43), Ψt) γ. Since γ is arbitrary, Ψt) as t. Therefore, Ψt k ) as k. In view of 40), and for large k, since w )/Ψt k ) 0 as k. Thus By 44), lim χt k) =, 44) k χt k ) Ψt k ) w) Ψt k ) c, χt k ) > c for some 0 < c < for all large k, Ψt k ) On the other hand, using Holders inequality, for k=,2,... χ 2 t k ) lim =. 45) k Ψt k ) and k χt k ) = Ft k,s)ws)ds Ht k, ) = Ht k, ) k Htk,s)ws) s α λs) rs)δs) ) ) 2 2 rs)δs) Gt k,s) s α λs) Htk,s) ds t k Ht k,s)w 2 s α ) λs) t k s) Ht k, ) rs)δs) ds 2 rs)δs) G 2 ) t k,s) s α λs) Ht k,s) ds 2 Ht k, ) ) t k 2 Ψt k )Ht k, ) rs)δs) G 2 ) t k,s) s α λs) Ht k,s) ds 2 From 45), which gives χ 2 t k ) Ψt k ) Ht k, ) lim k Ht k, ) Ht, ) k k rs)δs) s α λs) rs)δs) G 2 t k,s) s α λs) Ht k,s) ds. s α rs)δs) λs) G 2 t k,s) ds =, Ht k,s) G 2 t,s) ds =, Ht,s)

15 α-fractional DIFFERENTIAL SYSTEM 887 which contradicts to 36). Hence 4) holds. From 35), we have s α φ 2 s)λs) rs)δs) ds s α w 2 s)λs) ds <, 46) rs)δs) which contradicts 33). For case II), we proceed as in the proof of Theorem 6, for that case. 4. EXAMPLES In this section, we present some examples to illustrate the effectiveness of our results. Example 2. Consider the system of α-fractional differential equations D α xt)) = α fyt)), D α yt)) = α gzt)), 47) D α zt)) = hxt)), t. Here at) = α, bt) = α, ct) =, fy) = y 3, gz) = z+z 2 ) and hx) = x. It is easyto see that D α fy) = 3y α y 2 3ǫ α = K > 0, D α gz) = z α +3z 2 ) > z α ǫ α = L > 0 for some ǫ > 0, hx)/x = = M > 0, qt) = 3ǫ 4 2α, pt) = rt) = α and λt) = t. If we take δt) = then δ t) = 0. Consider Also from 0), the inner integral becomes ζ s α Mqs)δs) s)) 2 s α ) rs) 4 δ + ds δs)λs) = limsup 3s α ǫ 4 2α ds as t. s α qs)ds = ζ s α 3ǫ 4 2α ds. All the conditions of Theorem 6 are satisfied. Hence every solution of 47) is either oscillatory or tends to zero. Example 3. Consider the α-fractional differential system D 2 xt)) = te 2t fyt)),

16 888 G.E. CHATZARAKIS, M. DEEPA, N. NAGAJOTHI, AND V. SADHASIVAM D 2 yt)) = te 2t gzt)), 48) D 2 zt)) = thxt)), t t0. Here α = 2, at) = te 2t, bt) = te 2t, ct) = t, fy) = y, gz) = z and hx) = x. It is easy to see that D 2fy) = y 2 ǫ = K > 0, D 2gz) = z 2 ǫ = L > 0 for some ǫ > 0, hx)/x = = M > 0. From A 3 ), we have s α ps) ds = s 2 s 2 e 2s ds <. Some conditions of Theorem 6 are not satisfied, infact A 3 ) fails to hold. Thus, xt),yt),zt)) = e t,e t,e t ) is a nonoscillatory solution of 48). Example 4. Consider the α-fractional differential system D α xt)) = fyt)), D α yt)) = gzt)), 49) D α zt)) = hxt)), t. Here at) = bt) = ct) =, fy) = y, gz) = z and hx) = x 2. It is easy to see that D α fy) = y α ǫ α = K > 0, D α gz) = z α ǫ α = L > 0, hx)/x = x 2 x ǫ 2 ǫ = M > 0 for some 0 < ǫ <, qt) = ǫ 2 2α, pt) = rt) =. If we take δt) = then δ t) = 0 and n=2. Consider t n t s) n = limsup s α Mqs)δs) s)) 2 s α ) rs) 4 δ + ds δs)λs) t ǫ t 2 t s) 2 s α 2 ǫ 2 2α ds as t. ǫ That is, all conditions of Theorem 8 are satisfied. Therefore, every solution of 49) is either oscillatory or tends to zero. For example, is such a solution. xt),yt),zt)) = sin α tα ),cos α tα ),sin α tα )) Example 5. Consider the α-fractional differential system D 3 xt)) = t 2 3 fyt)), D 3 yt)) = t 2 3 gzt)), 50)

17 α-fractional DIFFERENTIAL SYSTEM 889 D 3 zt)) = t 2 3 hxt)), t t0. Here at) = bt) = ct) = t 2 3, fy) = y, gz) = z and hx) = x 2. It is easy to see that D 3fy) = y 2 3 ǫ 2 3 = K > 0, D 3gz) = z 2 3 ǫ 2 3 = L > 0, hx)/x = x 2 x ǫ 2 ǫ = M > 0 for some0 < ǫ <, qt) = ǫ 3t 4 2 3, pt) = rt) =, t 2 3 λt) = t and Ft,s) = 2t s). If we take δt) = then δ t) = 0 and n=2. Consider = limsup t t n t s) n s α Mqs)δs) s α ) rs)δs) 4 λs)t s) n F2 t,s) ds t 2 as t. t s) 2 s 3 ǫ 2 ǫ s ǫ s 3 s 2 3 4t s) 2 4s )t s) 2 ) ds That is, all conditions of Theorem 9 are satisfied. Therefore, every solution of 50) is either oscillatory or tends to zero. For example, xt),yt),zt)) = sint,cost,sint) is such a solution. Remark 6. Inthe previousexample, forthechoiceofht,s) = ln t s )n andht,s) = n s ln t s )n 2 ), onecanverifyalltheconditionsoftheorem0aresatisfied. Thus, every solution of system 50) is either oscillatory or tends to zero. Example 7. Consider the system of α-fractional differential equations D 4 xt)) = fyt)), 4 D 4 yt)) = gzt)), 5) 4 D 4 zt)) = hxt)), t t 0. 4 Here at) = bt) = ct) =, fy) = y, gz) = z and hx) = x 2. 4 It is easy to verify that D 4fy) = y 3 4 ǫ 3 4 = K > 0, D 4gz) = z 3 4 ǫ 3 4 = L > 0, hx)/x = x 2 x ǫ 2 ǫ = M > 0 for some 0 < ǫ <, qt) = ǫ 3 2 t 4, pt) = rt) = t4, λt) = lnt/ ). If we choose δs) = s2 t s) then δ s) = 2st 2 t s), Gt,s) = 2 3 s t s)2, Ht,s) = t s du θu)) n, n=2, θu) = and φs) =. Now, MHt,s)s α qs)δs)ds Ht, )

18 890 G.E. CHATZARAKIS, M. DEEPA, N. NAGAJOTHI, AND V. SADHASIVAM Hence, = limsup t ǫ 2 t ) 2 t s) 2 s 4 ǫ 3 2 s 4 ǫ s α φ 2 s)λs) rs)δs) limsup ds = limsup t ) 2 t 3 0 s 2 t s) 2ds <. s 4 lns/ )t s) 2 ds s 4 s 2 lns/ )ds as t and Ht,T) = T t T) 2 Ht,s)s α Mqs)δs) s α ) rs)δs) 4 λs)ht,s) G2 t,s) ds T T t s) 2 s ǫ 4 2 ǫ 3 2 s s 2 4 ǫ t s) 2 s 4s s 2 ) 4 t s) 2 4 4lns/ )t s) 2 s 2t s)4 ds t ǫ ǫ2 = t T) 2 )s+ s ) ln t s ) ds ǫ ǫ2 2t T) 2 ) T sds = φt) as t. That is, all conditions of Theorem are satisfied. Therefore, every solution of 5) is either oscillatory or tends to zero. For example, is such a solution. xt),yt),zt)) = sinlnt),coslnt),sinlnt)) REFERENCES [] S. Abbas, M. Benchohra and G.M. N Guerekata, Topics in fractional differential equations, Springer, Newyork, 202. [2] T. Abdeljawad, On conformable fractional calculus, J. Compu. Appl. Math., )

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20 892 G.E. CHATZARAKIS, M. DEEPA, N. NAGAJOTHI, AND V. SADHASIVAM [9] C. Milici and G. Draganescu, Introduction to fractional calculus, Lambert academic publishing, 206. [20] A.K. Nandakumaran, P.S. Datti and Raju K. George, Ordinary differential equations, principles and applications, Cambridge University press, 207. [2] S. Padhi and S. Pati, Theory of third order differential equations, Electronic journal of differential equations, Springer, New Delhi, 204. [22] I. Podlubny, Fractional differential equations, Academic press, San Diego, Calif, USA, 999. [23] S. Harikrishnan, P. Prakash, and J.J. Nieto, Forced oscillation of solutions of a nonlinear fractional partial differential equation, Applied Mathematics and computation, ) 4-9. [24] B. Rai, H.I. Freedman and J.F. Addicott, Analysis of three spieces model of mutualism in predator-prey and competitive systems, Mathematical biosciences, ) [25] H.H. Robertson, The solution of a set of reaction rate equations, Academic press, 966) [26] V. Sadhasivam, J. Kavitha and M. Deepa, Existence of solutions of three-dimensional fractional differential systems, Applied mathematics, 8 207) [27] V. Sadhasivam, M. Deepa and N. Nagajothi, On the oscillation of nonlinear fractional differential systems, International journal of advanced research in science, engineering and technology, 4 ) 207) [28] S.H. Saker, Oscillation theory of delay differential and difference equations, VDM Verlag Dr.Muller Aktiengesellschaft and Co, USA, 200. [29] H. Samajova, B. Ftorek and E. Spanikova, Asymptotic character of non-oscillatory solutions to functional differential systems, Electronic journal of qualitative theory of differential equations, ) -3. [30] S.G. Samko, A.A. Kilbas and O.I. Mirchev, Fractional integrals and derivatives, Theory and applications, Gordon and breach science publishers, Singapore, 993. [3] B.V. Shirokorad, On the existence of a cycle provided that conditions for absolute stability of a three-dimensional system are not fulfilled, Avtomatika i Telemekhanika, 9 958) [32] E. Spanikova and Zilina, Oscillatory properties of solutions of three-dimensional differential systems of neutral type, Czechoslovak mathematical journal, 50 25) 2000) [33] Vasily E. Tarasov, Fractional dynamics, Springer, 2009.

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