Oscillation results for certain forced fractional difference equations with damping term

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1 Li Advances in Difference Equations 06) 06:70 DOI 0.86/s R E S E A R C H Open Access Oscillation results for certain forced fractional difference equations with damping term Wei Nian Li * * Correspondence: wnli@6.net Department of Mathematics, Binzhou University, Binzhou, Shandong 5660, P.R. China Abstract In this paper, we establish two sufficient conditions for the oscillation of forced fractional difference equations with damping term of the form +pt)) α xt)) + pt) α xt)+ft, xt)) = gt), t N 0, with initial condition α xt) t=0 = x 0,where0<α < is a constant, α x is the Riemann-Liouville fractional difference operator of order α of x,andn 0 = 0,,,... MSC: 6A; 9A; 9A Keywords: oscillation; forced fractional difference equation; damping term Introduction In the past few years, the theory of fractional differential equations and their applications have been investigated extensively. For example, see monographs 4. In recent years, fractional difference equations, which are the discrete counterpart of the corresponding fractional differential equations, have comparably gained attention by some researchers. Many interesting results were established. For instance, see papers 5 0 and the references therein. The oscillation theory as a part of the qualitative theory of differential equations and difference equations has been developed rapidly in the last decades, and there have been many results on the oscillatory behavior of integer-order differential equations and integer-order difference equations. In particular, we notice that the oscillation of fractional differential equations has been developed significantly in recent years. We refer the reader to and the references therein. However, to the best of author s knowledge, up to now, very little is known regarding the oscillatory behavior of fractional difference equations 8 0. Unfortunately, themainresultsofpaper 8areincorrect.Themainreason for the mistakes in 8 is an incorrect relation of t α ) and t α). In fact, noting the definition of t α) = Ɣt+) Ɣt+ α),itiseasytoobservethattα ) t α). In this paper we investigate the oscillation of forced fractional difference equations with damping term of the form +pt) ) α xt) ) + pt) α xt)+f t, xt) ) = gt), t N 0, ) 06 Li. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Li Advances in Difference Equations 06) 06:70 Page of 9 with initial condition α xt) t=0 = x 0,where0<α < is a constant, α x is the Riemann- Liouville difference operator of order α of x,andn 0 = 0,,,... Throughout this paper, we assume that A) pt) and gt) are real sequences, pt)>, f : N 0 R R,andxf t, x)>0for x 0, t N 0. Asolutionxt)oftheEq.) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory. Preliminaries In this section, we present some preliminary results of discrete fractional calculus. Definition. 7) Let ν >0.Theνth fractional sum f is defined by ν f t)= t ν t s ) ν ) f s), ) Ɣν) s=a where f is defined for s = a mod), ν f is defined for t =a+ν) mod), and t ν) = Ɣt+) Ɣt+ ν). The fractional sum ν f maps functions defined on N a = a, a +,a +,... to functions defined on N a+ν = a + ν, a + ν +,a + ν +,...,whereɣ is the gamma function. Definition. 7) Let μ >0andm <μ < m,wherem is a positive integer, m = μ. Set ν = m μ.theμth fractional difference is defined as μ f t)= m ν f t)= m ν f t), ) where μ is the ceiling function of μ. Lemma. 7) Let f be a real-valued function defined on N a, and let μ, ν >0.Then the following equalities hold: ν μ f t) = μ+ν) f t)= μ ν f t) ; 4) ν f t)= ν f t) Lemma.4 Let xt) be a solution of Eq.),and let Then t a)ν ) f a). 5) Ɣν) t +α Et)= t s ) α) xs), t N 0. 6) Et)=Ɣ α) α xt). 7) Proof Using Definition., it followsfrom 6)that t +α t α) Et)= t s ) α) xs)= t s ) α) ) xs) = Ɣ α) α) xt).

3 Li Advances in Difference Equations 06) 06:70 Page of 9 Therefore, Et)=Ɣ α) α) xt)=ɣ α) α xt). The proof of Lemma.4 is complete. Lemma.5 6) Let μ R \...,,. Then ν t μ) = Ɣμ +) Ɣμ + ν +) tμ+ν). 8) Main results In this section, we establish the oscillation results for Eq. ). Theorem. Suppose that, for t 0 N 0, lim inf t α t s ) α ) M + gξ)vξ) <0 9) Vs) and lim sup t s ) α ) M + gξ)vξ) >0, 0) Vs) t α where M is a constant, and t ) Vt)= +ps). ) Then every solution xt) of Eq.) isoscillatory. Proof Suppose to the contrary that there is a nonoscillatory solution xt)of Eq.)which has no zero in N t0 = t 0, t 0 +,t 0 +,...Thenxt)>0orxt)<0fort N t0. Case. xt)>0,t N t0. Noting assumption A), from Eq. )wehave +pt) ) α xt) ) + pt) α xt)= f t, xt) ) + gt)<gt). ) Therefore, using the fundamental property of and noting the definition of Vt), we get α xt) ) Vt) ) = α xt) ) Vt +)+ α xt) Vt) = α xt) ) +pt) ) Vt)+ α xt)pt)vt) < gt)vt). ) Summing both sides of )fromt 0 to t,weobtain α xt) ) Vt)< α xt 0 ) ) Vt 0 )+ gs)vs)=m + gs)vs),

4 Li Advances in Difference Equations 06) 06:70 Page 4 of 9 where M = α xt 0 ))Vt 0 ), that is, α xt)< M Vt) + Vt) gs)vs). 4) Applying the α operator to inequality 4), we have α α xt)< α M Vt) + Vt) gs)vs). 5) On the one hand, applying Lemma. to the left-hand side of 5), we obtain α α xt)= α α) xt) = α α) xt) tα ) Ɣα) x 0 = xt) x 0 Ɣα) tα ). 6) On the other hand, using Definition., it followsfromtheright-hand side of 5)that α M Vt) + gs)vs) Vt) = t α Ɣα) t s ) α ) Combining 5)-7), we get M Vs) + Vs) gξ)vξ). 7) xt)< x 0 Ɣα) tα ) + t α t s ) α ) M Ɣα) Vs) + gξ)vξ). 8) Vs) It follows from 8)that Ɣα)t α xt)<x 0 t α ) t α t α + t α t s ) α ) M Vs) + gξ)vξ). 9) Vs) By using the Stirling formula 0 lim we obtain Ɣt)t ε =, ε >0, Ɣt + ε) lim t α t α ) = lim t α Ɣt +) Ɣt + α +) = lim t α tɣt) t + α)ɣt + α))

5 Li Advances in Difference Equations 06) 06:70 Page 5 of 9 = lim t t + α Ɣt)t α Ɣt + α)) =. 0) From 0), taking then limit as t in 9), we have lim inf t α xt), which contradicts with xt)>0. Case. xt)<0,t N t0. By assumption A), from Eq. )wehave +pt) ) α xt) ) + pt) α xt)= f t, xt) ) + gt)>gt). ) Therefore, α xt) ) Vt) ) > gt)vt). ) Summing both sides of )fromt 0 to t,weobtain α xt) ) Vt)> α xt 0 ) ) Vt 0 )+ gs)vs)=m + gs)vs), where M = α xt 0 ))Vt 0 ), that is, α xt)> M Vt) + Vt) gs)vs). ) Using the procedure of the proof of Case, we conclude that Ɣα)t α xt)>x 0 t α ) t α t α + t α t s ) α ) M Vs) + Vs) By 0), taking the limit as t in 4), we have lim sup t α xt), gξ)vξ). 4) which contradicts with xt) <0. The proofof Theorem. is complete. Theorem. Suppose that, for t 0 N 0, lim inf Vs) M + gξ)vξ) = 5) and lim sup Vs) M + gξ)vξ) =, 6)

6 Li Advances in Difference Equations 06) 06:70 Page 6 of 9 where M is a constant, and V t) is defined by ). Then every solution xt) of Eq. ) is oscillatory. Proof Suppose to the contrary that there is a nonoscillatory solution xt) ofeq.) that has no zero in N t0.thenxt)>0orxt)<0fort N t0. Case. xt) >0,t N t0. As in the proof of Case in Theorem., weobtain4). By Lemma.4 it follows from 4)that Et)< Ɣ α) M + Vt) gs)vs). 7) Summing both sides of 7)fromt 0 to t,wehave Et)<Et 0 )+Ɣ α) M + gξ)vξ). 8) Vs) Letting t in 8), we obtain a contradiction with Et)>0. Case. xt)<0,t N t0.asintheproofofcaseintheorem., weobtain). By Lemma.4 it follows from )that Et)> Ɣ α) M + Vt) gs)vs). 9) Summing both sides of 9)fromt 0 to t,wehave Et)>Et 0 )+Ɣ α) M + gξ)vξ). 0) Vs) Letting t in 0), we obtain a contradiction with Et)<0.Thiscompletestheproof of Theorem.. 4 Examples In this section, we conclude from the following two examples that the assumptions of Theorem.and Theorem.cannot be dropped. Example 4. Consider the following fractional difference equation: xt) ) + ) xt)+ Ɣt + ) tɣt) xt)= Ɣ ), t N 0, ) 9 with the initial condition xt) t=0 =0. Here α =, pt)=, f t, xt)) = Ɣt+ ) tɣt) xt), gt)= Ɣ ) 9.Itiseasytoseethat t ) t Vt)= +pt) = s= s= = ) t, gt)= Ɣ ) >0. 9

7 Li Advances in Difference Equations 06) 06:70 Page 7 of 9 Therefore, we have t t s ) ) M + gξ)vξ) Vs) ξ= t ) = t s ) s ) Ɣ M + ) ) ξ 9 ξ= >0, ) which shows that condition 9)ofTheorem. does not hold. It is not difficult to see that xt)=t ) is a nonoscillatory solution of Eq. ). Indeed, on the one hand, using Lemma.5,weobtain xt)= t ) = t )) Ɣ = +) Ɣ + +)t ) = )t Ɣ ) ) ) = Ɣ t = ) Ɣ + ) ) ) and xt) ) = t )) =0. 4) On the other hand, we have xt)=t ) = Ɣt +) Ɣt + ) = tɣt) Ɣt + 5) ). Combining )-5), we conclude that xt)=t ) is a solution of Eq. ). Example 4. Consider the following fractional difference equation: xt) ) + ) xt)+ Ɣt + ) tɣt) xt)= Ɣ ), t N 0, 6) 6 with the initial condition xt) t=0 =0. Here α =, pt)=, f t, xt)) = Ɣt+ ) tɣt) xt), gt)= Ɣ ) 6.Obviously, t ) t Vt)= +pt) = s= s= = ) t, gt)= Ɣ ) >0. 6

8 Li Advances in Difference Equations 06) 06:70 Page 8 of 9 Therefore, we have M + gξ)vξ) Vs) s= s= ξ= = M s Ɣ + ) ) ξ 6 ξ= >0. 7) Thus, condition 5)ofTheorem. does not hold. In fact, we can easily verify that xt)= t ) is a nonoscillatory solution of Eq. 6). Competing interests The author declares that there are no competing interests. Author s contributions The author read and approved the final manuscript. Acknowledgements This work is supported by the National Natural Science Foundation of China 09708). The author thanks the referees very much for their valuable comments and suggestions on this paper. Received: June 05 Accepted: March 06 References. Podlubny, I: Fractional Differential Equations. Academic Press, San Diego 999). Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam 006). Abbas, S, Benchohra, M, N Guérékata, GM: Topics in Fractional Differential Equations. Springer, New York 0) 4. Zhou, Y: Basic Theory of Fractional Differential Equations. World Scientific, Singapore 04) 5. Miller, KS, Ross, B: Fractional difference calculus. In: Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, May 988. Ellis Horwood Ser. Math. Appl., pp Horwood, Chichester 989) 6. Atici, FM, Eloe, PW: A transform method in discrete fractional calculus. Int. J. Difference Equ. ), ) 7. Atici, FM, Eloe, PW: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 7, ) 8. Atici, FM, Eloe, PW: Linear systems of fractional nabla difference equations. Rocky Mt. J. Math. 4, ) 9. Atici, FM, Wu, F: Existence of solutions for nonlinear fractional difference equations with initial conditions. Dyn. Syst. Appl., ) 0. Atici, FM, Şengül, S: Modeling with fractional difference equations. J. Math. Anal. Appl. 69, -9 00). Deekshitulu, GVSR, Mohan, JJ: Solutions of perturbed nonlinear nabla fractional difference equations of order 0<α <. Math. Æterna, ). Goodrich, CS: On a discrete fractional three-point boundary value problem. J. Differ. Equ. Appl. 8, ). Goodrich, CS: Existence of a positive solution to a system of discrete fractional boundary value problems. Appl. Math. Comput. 7, ) 4. Abdeljawad, T: On Riemann and Caputo fractional differences. Comput. Math. Appl. 6, ) 5. Cheng, J, Chu, Y: Fractional difference equations with real variable. Abstr. Appl. Anal. 0, Article ID9859 0) 6. Chen, F, Liu, Z: Asymptotic stability results for nonlinear fractional difference equations. J. Appl. Math. 0, Article ID ) 7. Cheng, J-F, Chu, Y-M: On the fractional difference equations of order ; q). Abstr. Appl. Anal. 0, Article ID ) 8. Marian, SL, Sagayaraj, MR, Selvam, AGM, Loganathan, MP: Oscillation of fractional nonlinear difference equations. Math. Æterna, ) 9. Sagayaraj, MR, Selvam, AGM, Loganathan, MP: On the oscillation of nonlinear fractional difference equations. Math. Æterna 4, ) 0. Alzabut, JO, Abdeljawad, T: Sufficient conditions for the oscillation of nonlinear fractional difference equations. J. Fract. Calc. Appl. 5, ). Chen, DX: Oscillation criteria of fractional differential equations. Adv. Differ. Equ. 0, 0). Chen, DX: Oscillatory behavior of a class of fractional differential equations with damping. Univ. Politeh. Bucur. Sci. Bull., Ser. A 75, ). Grace, SR, Agarwal, RP, Wong, PJY, Zafer, A: On the oscillation of fractional differential equations. Fract. Calc. Appl. Anal. 5, - 0) 4. Zheng, B: Oscillation for a class of nonlinear fractional differential equations with damping term. J. Adv. Math. Stud. 6, ) 5. Han, Z, Zhao, Y, Sun, Y, Zhang, C: Oscillation for a class of fractional differential equation. Discrete Dyn. Nat. Soc. 0, Article ID 908 0)

9 Li Advances in Difference Equations 06) 06:70 Page 9 of 9 6. Qi, C, Cheng, J: Interval oscillation criteria for a class of fractional differential equations with damping term. Math. Probl. Eng. 0, Article ID ) 7. Chen, D, Qu, P, Lan, Y: Forced oscillation of certain fractional differential equations. Adv. Differ. Equ. 0, 5 0) 8. Wang, Y, Han, Z, Sun, S: Comment on On the oscillation of fractional-order delay differential equations with constant coefficients Commun. Nonlinear Sci. 9) 04) Commun. Nonlinear Sci. Numer. Simul. 6, ) 9. Yang, J, Liu, A, Liu, T: Forced oscillation of nonlinear fractional differential equations with damping term. Adv. Differ. Equ. 05, 05) 0. Öğrekçi, S: Interval oscillation criteria for functional differential equations of fractional order. Adv. Differ. Equ. 05, 05). Prakash, P, Harikrishnan, S, Benchohra, M: Oscillation of certain nonlinear fractional partial differential equation with damping term. Appl. Math. Lett. 4, ). Li, WN: Forced oscillation criteria for a class of fractional partial differential equations with damping term. Math. Probl. Eng. 05, Article ID ). Li, WN: On the forced oscillation of certain fractional partial differential equations. Appl. Math. Lett. 50, )