Oscillatory behavior of solutions of certain fractional difference equations

(1)

R E S E A R C H

Open Access

Oscillatory behavior of solutions of certain

fractional difference equations

Hakan Adiguzel

1*

*_{Correspondence:}

[email protected] 1_{Department of Mechatronics}

Engineering, Istanbul Gelisim University, Istanbul, Turkey

Abstract

In this paper, we consider the oscillation behavior of solutions of the following fractional diﬀerence equation:

(

c(t)

(

a(t)

(

r(t)

αx(t)

)))

+q(t)G(t) = 0,

wheret∈Nt0+1–α,G(t) =

t–1+α

s=t0 (t–s– 1)

–α_x₍_s_{), and}

α_{denotes a Riemann–Liouville}

fractional diﬀerence operator of order 0 <

α

≤1. By using the generalized Riccati transformation technique, we obtain some oscillation criteria. Finally we give an example.

Keywords: Oscillation; Oscillation criteria; Fractional diﬀerence operator; Riemann–Liouville; Fractional diﬀerence equations; Riccati technique; Hardy inequalities

1 Introduction and preliminaries

Fractional differential (or difference) equations are a more general form of differential equations with integer order. And there is an increasing interest in the study of them due to some important contributions [1,2].

Many authors have been focused on various equations like ordinary and partial differen-tial equations [3–6], difference equations [7–9], dynamic equations on time scales [10–14], and fractional differential (difference) equations [15–31] obtaining some oscillation crite-ria. Recently, oscillation studies have become a very hot topic. That is why, we consider the following fractional difference equation:

c(t)a(t)r(t)αx(t)+q(t)G(t) = 0, (1)

wheret∈Nt0+1–α,G(t) =

t–1+α

s=t0 (t–s– 1)

(–α)_x₍_s_),_c₍_t_),_a₍_t_),_r₍_t_{), and}_q₍_t_{) are positive}

se-quences, andαdenotes the Riemann–Liouville fractional diﬀerence operator of order 0 <α≤1.

By a solution of Eq. (1), we mean a real-valued sequencex(t) satisfying Eq. (1) fort∈ Nt0. A solutionx(t) of Eq. (1) is called oscillatory if it is neither eventually positive nor

eventually negative, otherwise it is called non-oscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory.

(2)

Deﬁnition 1([32]) Letv> 0. Thevth fractional sumf is deﬁned by

–vf(t) = 1

Γ(v)

t–v

s=a

(t–s– 1)v–1f(s), (2)

wherefis deﬁned fors≡amod(1),–vf is deﬁned fort≡(a+v)mod(1), andt(v)₌ Γ(t+1)

Γ(t–v+1). The fractional sum –v_f _{maps functions deﬁned on N}

a to functions deﬁned on Na+v,

where Nt={t,t+ 1,t+ 2, . . .}.

Deﬁnition 2([32]) Letv> 0 andm– 1 <μ<m, wheremdenotes a positive integer,m=

μ. Setv=m–μ. Theμth fractional diﬀerence is deﬁned as

μf(t) =m–vf(t) =m–vf(t), (3)

whereμis the ceiling function ofμ.

Lemma 1([33]) Assume that A and B are nonnegative real numbers.Then

λABλ–1_–_Aλ_≤₍_λ_{– 1)}_Bλ ₍₄₎

for allλ> 1.

2 Main results

Throughout this paper, we denote

φ(t) =

t–1

s=t1

1

c(s); ϑ(t) =

t–1

s=t2 φ(s)

a(s); δ(t) =

t–1

s=t3 ϑ(s)

r(s).

For simpliﬁcation, we consider

γ+(s) =max

0,γ(s)

and

β+(s) =max0,β(s).

Lemma 2([28]) Let x(t)be a solution of Eq. (1),and let

G(t) =

t–1+α

s=t0

(t–s– 1)(–α)_x₍_s_), ₍₅₎

then

(3)

Lemma 3 Assume that x(t)is an eventually positive solution of Eq. (1).If

∞

s=t0

1

c(s)=

∞

s=t0

1

a(s)=

∞

s=t0

1

r(s)=∞, (7)

then we have two possible cases for t∈[t1,∞),t1>t0is suﬃciently large: Case 1 α_x₍_t_{) > 0}_,₍_r₍_t₎α_x₍_t_{)) > 0}_,₍_a₍_t₎₍_r₍_t₎α_x₍_t_{))) > 0}_or

Case 2 α_x₍_t_{) > 0}_,₍_r₍_t₎α_x₍_t_{)) < 0}_,₍_a₍_t₎₍_r₍_t₎α_x₍_t_{))) > 0}_.

Proof From the hypothesis, there existst1such thatx(t) > 0 on [t1,∞), so thatG(t) > 0 on [t1,∞), and from Eq. (1), we have

c(t)a(t)r(t)αx(t)= –q(t)G(t) < 0. (8)

Then c(t)(a(t)(r(t)α_x₍_t_{))) is an eventually non-increasing sequence on [}_t_1,_∞_{). We}

know thatα_x₍_t_),₍_r₍_t₎α_x₍_t_{)), and}₍_a₍_t₎₍_r₍_t₎α_x₍_t_{))) are eventually of one sign. For} t2>t1is suﬃciently large, we claim that(a(t)(r(t)αx(t))) > 0 on [t2,∞). Otherwise, assume that there exists suﬃciently larget3>t2 such that(a(t)(r(t)αx(t))) < 0 on [t3,∞). For [t3,∞) and there exists a constantl1> 0, we have

a(t)r(t)αx(t)≤– l1

c(t)< 0.

Hence, there exist a constantl2> 0 and suﬃciently larget4>t3such that

r(t)αx(t)≤– l2

a(t)< 0. (9)

Then there exist a constantl3> 0 and suﬃciently larget5>t4such that

αx(t)≤– l3

r(t),

that is,

G(t)≤–Γ(1 –α)l3

r(t) < 0.

By (7), we obtain limt→∞G(t) = –∞. This is a contradiction. If (r(t)αx(t)) < 0,

then αx(t) > 0 due to _s∞₌_t₀ _r1₍_s₎ =∞. If (r(t)αx(t)) > 0, then αx(t) > 0 due to

(a(t)(r(t)α_x₍_t_{))) > 0. So, the proof is complete.}

Lemma 4 Assume that x(t)is an eventually positive solution of Eq. (1),which satisﬁes Case1of Lemma3.Then

a(t)r(t)αx(t)≥c(t)a(t)r(t)αx(t)

t–1

s=t0

1

(4)

If there exists a positive sequenceφsuch that,for t∈[t1,∞),

φ(t)

c(t)t_s–1₌_t₀ _c₍1_s₎ –φ(t)≤0,

where t1is suﬃciently large,then a(t)(r(t)αx(t))/φ(t)is a non-increasing sequence on [t1,∞)and

r(t)αx(t)≥r(t)αx(t)a(t)

φ(t)

t–1

s=t1 φ(s)

a(s).

Furthermore,if there exists a positive sequenceϑand t2>t1is suﬃciently large such that,

for t∈[t2,∞),

ϑ(t)

a(t)

φ(t) t–1

s=t2 φ(s)

a(s)

–ϑ(t)≤0,

then r(t)α_x₍_t_)/_ϑ₍_t₎_{is a non-increasing sequence on}_[_t_2,_∞₎_and

G(t)≥G(t)r(t)

ϑ(t)

t–1

s=t2 ϑ(s)

r(s).

Suppose also that there exists a positive sequenceδand t3>t2is suﬃciently large such that,

for t∈[t3,∞),

δ(t)

r(t)

ϑ(t) t–1

s=t2 ϑ(s)

r(s)

–δ(t)≤0.

Then G(t)/δ(t)is a non-increasing sequence on[t3,∞).

Proof Assume that x is an eventually positive solution of Eq. (1). Then we have that

(r(t)α_x₍_t_{)) > 0 and}₍_c₍_t₎₍_a₍_t₎₍_r₍_t₎α_x₍_t_{)))) < 0 on [}_t_0,_∞_{). So,}

a(t)r(t)αx(t)=a(t0)r(t0)αx(t0)

+

t–1

s=t0

c(s)(a(s)(r(s)α_x₍_s₎₎₎ c(s)

≥c(t)a(t)r(t)αx(t)

t–1

s=t0

1

c(s),

and then

a(t)(r(t)α_x₍_t₎₎ φ(t)

=(a(t)(r(t)

α_x₍_t₎₎₎_φ₍_t_{) –}_a₍_t₎₍_r₍_t₎α_x₍_t₎₎_φ₍_t₎

φ(t)φ(t+ 1)

≤(a(t)(r(t)αx(t))) φ(t)φ(t+ 1)

φ(t)

(5)

Hence,a(t)(r(t)α_x₍_t_))/_φ₍_t_{) is a non-increasing sequence on [}_t_1,_∞_{) where}_t

1>t0is suf-ﬁciently large. Then we have

r(t)αx(t) =r(t1)αx(t1) +

So the proof is complete.

Theorem 1 Assume that(7)holds and there exists a positive sequenceγ such that,for all suﬃciently large t,

lim

If there exist positive sequencesβ,λsuch that,for all suﬃciently large t,

λ(t)

(6)

and

Proof Suppose to the contrary thatx(t) is a non-oscillatory solution of Eq. (1). Then, with-out loss of generality, we may assume that there is a solutionx(t) of Eq. (1) such thatx(t) > 0 on [t0,∞), wheret0 is sufficiently large. From Lemma3,x(t) satisfies Case 1 or Case 2. Firstly, let Case 1 hold. Then we define the following function:

ω(t) =γ(t)c(t)(a(t)(r(t)

From Lemma4, we obtain

(7)

Settingλ= 2,A= (γ_c₍(_tt₎))1/2ω(t+1)

φ(t+1), andB= 1 2(

c(t)

γ(t))1/2γ+(t) using Lemma1, we obtain

ω(t)≤–γ(t)q(t) Γ(1 –α)

ϑ(t)φ(t+ 1)

t–1

s=t2 ϑ(s)

r(s) _t_–1

s=t1 φ(s)

a(s)

+ c(t) 4γ(t)

γ+(t)2.

Summing both sides of the above inequality fromt3tot– 1, we get

t–1

s=t3

Γ(1 –α)γ(s)q(s)

ϑ(s)φ(s+ 1)

s–1

u=t2 ϑ(u)

r(u) _s_–1

u=t1 φ(u)

a(u)

–c(s)(γ+(s)) 2

4γ(s)

≤ω(t3) –ω(t)≤ω(t3).

This contradicts (10). Now we consider Case 2. Then we deﬁne the following function:

ω2(t) =β(t)r(t)

α_x₍_t₎

G(t) .

Then

ω2(t) =β(t)ω(t+ 1)

β(t+ 1)+β(t)

r(t)αx(t)

G(t)

=β(t)ω(t+ 1)

β(t+ 1)+β(t)

(r(t)α_x₍_t₎₎_G₍_t_{) –}_r₍_t₎α_x₍_t₎_G₍_t₎ G(t)G(t+ 1)

=β(t)ω(t+ 1)

β(t+ 1)+β(t)

(r(t)α_x₍_t₎₎ G(t+ 1) –β(t)

r(t)α_x₍_t₎_G₍_t₎ G(t)G(t+ 1) .

Hence we have

G(t) =G(t1) +Γ(1 –α)

t–1

s=t1

r(s)αx(s)

r(s)

≥Γ(1 –α)r(t)αx(t)

t–1

s=t1

1

r(s).

That is,

G(t)

r(t)t_s–1₌_t₁ _r₍1_s₎ ≥Γ(1 –α)

α

x(t) =G(t)

and

G(t)

λ(t) =

G(t)λ(t) –G(t)λ(t)

λ(t)λ(t+ 1)

≤ G(t) λ(t)λ(t+ 1)

λ(t)

r(t)t_s–1₌_t₁ _r₍1_s₎ –λ(t) ≤0.

Thus we haveG(t)/λ(t) is eventually non-increasing and

G(t)

G(t+ 1)≥

λ(t)

(8)

(9)

Settingλ= 2,A= (Γ(1–_r₍α_t₎)β(t))1/2ω2(t+1)

Summing both sides of the above inequality fromt2tot– 1, we have

t–1

which contradicts (12). So, the proof is complete.

Theorem 2 Let(7)hold.Assume that there exists a positive sequenceγ such that,for all suﬃciently large t,

lim

Proof Suppose to the contrary thatx(t) is a non-oscillatory solution of (1). Then, without loss of generality, we may assume that there is a solutionx(t) of Eq. (1) such thatx(t) > 0 on [t0,∞) wheret0is sufficiently large. From Lemma3,x(t) satisfies Case 1 or Case 2. Firstly, let Case 1 hold. Then we define the following function:

π(t) =γ(t)c(t)(a(t)(r(t)

From Lemma4, we obtain

(10)

or

t–1

which contradicts (14). And the proof of Case 2 is the same as that of Theorem1and hence

is omitted. This completes the proof.

Theorem 3 Let(7)hold.Assume that there exists a positive sequenceγ such that,for all suﬃciently large t,

(11)

Proof Suppose to the contrary thatx(t) is a non-oscillatory solution of (1). Then, without loss of generality, we may assume that there is a solutionx(t) of Eq. (1) such thatx(t) > 0 on [t0,∞), wheret0is sufficiently large. From Lemma3,x(t) satisfies Case 1 or Case 2. Firstly, let Case 1 hold. Then we define the following function:

ν(t) =γ(t)c(t)(a(t)(r(t)

α_x₍_t₎₎₎

G(t) .

Fort∈[t0,∞), we get

ν(t) =γ(t)ν(t+ 1)

γ(t+ 1)+γ(t)

c(t)(a(t)(r(t)α_x₍_t₎₎₎ G(t)

=γ(t)ν(t+ 1)

α(t+ 1)–γ(t)

q(t)G(t)

G(t+ 1)

–γ(t)c(t)(a(t)(r(t)

α_x₍_t₎₎₎_G₍_t₎

G(t)G(t+ 1) .

From Lemma4, we have

G(t)≥ 1

r(t)

a(t)

φ(t)

t–1

s=t1 φ(s)

a(s)

t–1

s=t0

1

c(s)

a(t) c(t)

a(t)r(t)αx(t)

and

G(t)

G(t+ 1)≥

δ(t)

δ(t+ 1).

Thus we obtain

ν(t)≤γ+(t)

ν(t+ 1)

γ(t+ 1)–γ(t)p(t)

δ(t)

δ(t+ 1)

– γ(t)

r(t)φ(t)

t–1

s=t1 φ(s)

a(s)

t–1

s=t0

1

c(s)

ν2(t+ 1)

γ2₍_t_{+ 1)}.

Then, settingλ= 2,

A=

γ(t)

r(t)φ(t)

t–1

s=t1 φ(s)

a(s)

t–1

s=t0

1

c(s) 1/2

ν(t+ 1)

γ(t+ 1), and

B=1 2

r(t)φ(t)

γ(t)t_s–1₌_t₁ φ_a₍(_ss₎)_st–1₌_t₀ _c₍1_s₎ 1/2

γ+(t)

using Lemma1, we obtain

ν(t)≤–γ(t)q(t) δ(t)

δ(t+ 1)+

(12)

which contradicts (15). The proof of Case 2 is the same as that of Theorem1and hence is

omitted. This completes the proof.

3 Applications

Example1 Consider the following fractional diﬀerence equation fort≥2:

3+αx(t) +t–2

Thus, (16) is oscillatory from Theorem1.

Acknowledgements

The author is grateful to the scholars who provided the literature sources.

Funding

Not applicable.

Availability of data and materials

Not applicable.

Competing interests

The author declares that they have no competing interests.

Authors’ contributions

(13)

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 2 October 2018 Accepted: 26 November 2018 References

1. Diethelm, K.: The Analysis of Fractional Diﬀerential Equations. Springer, Berlin (2010)

2. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Diﬀerential Equations. Elsevier, Amsterdam (2006)

3. Ö ˘grekçi, S.: Interval oscillation criteria for second-order functional diﬀerential equations. SIGMA36(2), 351–359 (2018) 4. Ö ˘grekçi, S., Misir, A., Tiryaki, A.: On the oscillation of a second-order nonlinear diﬀerential equations with damping.

Miskolc Math. Notes18(1), 365–378 (2017)

5. Ö ˘grekçi, S.: New interval oscillation criteria for second-order functional diﬀerential equations with nonlinear damping. Open Math.13(1), 239–246 (2015)

6. Sadhasivam, V., Kavitha, J., Nagajothi, N.: Oscillation of neutral fractional order partial diﬀerential equations with damping term. Int. J. Pure Appl. Math.115(9), 47–64 (2017)

7. Hasil, P., Veselý, M.: Oscillation and non-oscillation criteria for linear and half-linear diﬀerence equations. J. Math. Anal. Appl.452(1), 401–428 (2017)

8. Hasil, P., Veselý, M.: Oscillation constants for half-linear difference equations with coefficients having mean values. Adv. Differ. Equ.2015, 210 (2015)

9. Sugie, J., Tanaka, M.: Nonoscillation theorems for second-order linear diﬀerence equations via the Riccati-type transformation. Proc. Am. Math. Soc.145(5), 2059–2073 (2017)

10. Zhang, C., Agarwal, R.P., Bohner, M., Li, T.: Oscillation of fourth-order delay dynamic equations. Sci. China Math.58(1), 143–160 (2015)

11. Grace, S.R., Agarwal, R.P., Sae-Jie, W.: Monotone and oscillatory behavior of certain fourth order nonlinear dynamic equations. Dyn. Syst. Appl.19(1), 25–32 (2010)

12. Grace, S.R., Bohner, M., Sun, S.: Oscillation of fourth-order dynamic equations. Hacet. J. Math. Stat.39, 545–553 (2010) 13. Grace, S.R., Argawal, R.P., Pinelas, S.: On the oscillation of fourth order superlinear dynamic equations on time scales.

Dyn. Syst. Appl.20, 45–54 (2011)

14. Li, T., Thandapani, E., Tang, S.: Oscillation theorems for fourth-order delay dynamic equations on time scales. Bull. Math. Anal. Appl.3, 190–199 (2011)

15. Liu, T., Zheng, B., Meng, F.: Oscillation on a class of diﬀerential equations of fractional order. Math. Probl. Eng.2013, Article ID 830836 (2013)

16. Qin, H., Zheng, B.: Oscillation of a class of fractional diﬀerential equations with damping term. Sci. World J.2013, Article ID 685621 (2013)

17. Ogrekci, S.: Interval oscillation criteria for functional diﬀerential equations of fractional order. Adv. Diﬀer. Equ.2015, 3 (2015)

18. Muthulakshmi, V., Pavithra, S.: Interval oscillation criteria for forced fractional diﬀerential equations with mixed nonlinearities. Glob. J. Pure Appl. Math.13(9), 6343–6353 (2017)

19. Chen, D.-X.: Oscillation criteria of fractional diﬀerential equations. Adv. Diﬀer. Equ.2012, 33 (2012)

20. Zheng, B.: Oscillation for a class of nonlinear fractional diﬀerential equations with damping term. J. Adv. Math. Stud.

6(1), 107–115 (2013)

21. Xu, R.: Oscillation criteria for nonlinear fractional diﬀerential equations. J. Appl. Math.2013, Article ID 971357 (2013) 22. Secer, A., Adiguzel, H.: Oscillation of solutions for a class of nonlinear fractional diﬀerence equations. J. Nonlinear Sci.

Appl.9(11), 5862–5869 (2016)

23. Abdalla, B.: On the oscillation ofq-fractional diﬀerence equations. Adv. Diﬀer. Equ.2017, 254 (2017)

24. Abdalla, B., Abodayeh, K., Abdeljawad, T. Alzabut, J.: New oscillation criteria for forced nonlinear fractional diﬀerence equations. Vietnam J. Math.45(4), 609–618 (2017)

25. Alzabut, J.O., Abdeljawad, T.: Suﬃcient conditions for the oscillation of nonlinear fractional diﬀerence equations. J. Fract. Calc. Appl.5(1), 177–187 (2014)

26. Bai, Z., Xu, R.: The asymptotic behavior of solutions for a class of nonlinear fractional diﬀerence equations with damping term. Discrete Dyn. Nat. Soc.2018, Article ID 5232147 (2018)

27. Chatzarakis, G.E., Gokulraj, P., Kalaimani, T., Sadhasivam, V.: Oscillatory solutions of nonlinear fractional diﬀerence equations. Int. J. Diﬀer. Equ.13(1), 19–31 (2018)

28. Sagayaraj, M.R., Selvam, A.G.M., Loganathan, M.P.: On the oscillation of nonlinear fractional diﬀerence equations. Math. Æterna4, 220–224 (2014)

29. Selvam, A.G.M., Sagayaraj, M.R., Loganathan, M.P.: Oscillatory behavior of a class of fractional diﬀerence equations with damping. Int. J. Appl. Math. Res.3(3), 220–224 (2014)

30. Li, W.N.: Oscillation results for certain forced fractional diﬀerence equations with damping term. Adv. Diﬀer. Equ.

2016, 70 (2016)

31. Sagayaraj, M.R., Selvam, A.G.M., Loganathan, M.P.: Oscillation criteria for a class of discrete nonlinear fractional equations. Bull. Soc. Math. Serv. Stand.3(1), 27–35 (2014)