Show Summary Details
In This Section

# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year

IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

CiteScore 2016: 0.62

SCImago Journal Rank (SJR) 2015: 0.521
Source Normalized Impact per Paper (SNIP) 2015: 1.233

Mathematical Citation Quotient (MCQ) 2015: 0.39

Open Access
Online
ISSN
2391-5455
See all formats and pricing
In This Section

# Oscillation of impulsive conformable fractional differential equations

• Corresponding author
• Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
• Email:
/ Sotiris K. Ntouyas
• Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece and Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Arabia
• Email:
Published Online: 2016-07-22 | DOI: https://doi.org/10.1515/math-2016-0044

## Abstract

In this paper, we investigate oscillation results for the solutions of impulsive conformable fractional differential equations of the form

$tkDαpttkDαxt+rtxt+qtxt=0,t≥t0,t≠tk,xtk+=akx(tk−),tkDαxtk+=bktk−1Dαx(tk−),k=1,2,….$

Some new oscillation results are obtained by using the equivalence transformation and the associated Riccati techniques.

MSC 2010: 34A08; 34A37; 34C10

## 1 Introduction

Fractional differential equations are generalizations of classical differential equations of integer order, and can find their applications in many fields of science and engineering. Research on the theory and applications of fractional differential and integral equations has been the focus of many studies due to their frequent appearance in various applications in physics, biology, engineering, signal processing, systems identification, control theory, finance and fractional dynamics, and has attracted much attention of more and more scholars. The books on the subject of fractional integrals and fractional derivatives by Diethelm [1], Miller and Ross [2], Podlubny [3] and Kilbas et al. [4] summarize and organize much of fractional calculus and many of theories and applications of fractional differential equations. Initial and boundary value problems, stability of solutions, explicit and numerical solutions and many other properties have obtained significant development [5]-[11].

The oscillation of fractional differential equations as a new research field has received significant attention, and some interesting results have already been obtained. We refer to [12]-[18] and the references quoted therein.

The definition of the fractional order derivative used is either the Caputo or the Riemann-Liouville fractional order derivative involving an integral expression and the gamma function. Recently, Khalil et al. [19] introduced a new well-behaved definition of local fractional derivative, called the conformable fractional derivative, depending just on the basic limit definition of the derivative. This new theory is improved by Abdeljawad [20]. For recent results on conformable fractional derivatives we refer the reader to [21]-[25].

Impulsive differential equations are recognized as adequate mathematical models for studying evolution processes that are subject to abrupt changes in their states at certain moments. Many applications in physics, biology, control theory, economics, applied sciences and engineering exhibit impulse effects, see [26]-[28].

Recently, in [29] Tariboon and Thiramanus considered the following second-order linear impulsive differential equation of the form

$atx′t+λxt′+ptxt=0,t≥t0,t≠tk,xtk+=bkx(tk−),x′tk+=ckx′(tk−),k=1,2,…,$(1)

where 0 ≤ t0 < t1 < … < tk < …, limk → ∞ tk = + ∞, aC([t0, ∞), (0, ∞)), pC([t0, ∞), ℝ), {bk}, {ck} are two known sequences of positive real numbers and λ ∈ ℝ. By using the equivalence transformation and the associated Riccati techniques, some interesting oscillation results were obtained.

In this paper, we investigate some new oscillation results for the solutions of impulsive conformable fractional differential equations of the form

$tkDαpttkDαxt+rtxt+qtxt=0,t≥t0,t≠tk,xtk+=akx(tk−),k=1,2,…,tkDαxtk+=bktk−1Dαx(tk−),k=1,2,…,$(2)

where ϕDα denotes the conformable fractional derivative of order 0 < α ≤ 1 starting from ϕ ∈ {t0,…, tk,…}, 0 ≤ t0 < t1 < …, < tk < …, limk→∞ tk = ∞, pC([t0, ∞), (0, ∞)), q, rC([t0, ∞), ℝ), {ak} and {bk} are two known sequences of positive real numbers and $x\left({t}_{k}^{+}\right)=\underset{\theta \to 0+}{lim}x\left({t}_{k}+\theta \right),x\left({t}_{k}^{-}\right)=\underset{\theta \to 0+}{lim}x\left({t}_{k}+\theta \right)$.

Some new oscillation results are obtained, generalizing the results of [29] to impulsive conformable fractional differential equations. Note that if ak = 1 for all k = 1,2,…, then x is continuous on [t0, ∞). However, if ak = bk = 1 for all k = 1,2,…, then it does not guarantee that x′ is also continuous function, as by the definition of conformable fractional derivative (see Definition 2.1 below) we have

$tkDαxtk+=limε→0⁡x(tk++ε(tk+−tk)1−α)−xtk+ε,$

and

$tk−1Dαx(tk−)=limε→0⁡x(tk−+ε(tk−−tk−1)1−α)−x(tk−)ε.$

We organize this paper as follows: In Section 2, we present some useful preliminaries from fractional calculus. In Section 3, we prove some auxiliary lemmas. The main oscillation results are established in Section 4. Examples illustrating the results are presented in Section 5.

## 2 Conformable Fractional Calculus

In this section, we recall some definitions, notations and results which will be used in our main results.

Definition 2.1: [20] The conformable fractional derivative starting from a point ϕ of a function f: [ϕ,∞) → ℝ of order 0 < α ≤; 1 is defined by$ϕDαf(t)=limε→0⁡f(t+ε(t−φ)1−α)−f(t)ε,$(3)provided that the limit exists.

If f is differentiable then ϕDα f(t) = (tϕ)1—α f′(t). In addition, if the conformable fractional derivative of f of order α exists on [ϕ, ∞), then we say that f is α-differentiable on [ϕ, ∞).

It is easy to prove the following results.

Lemma 2.2: Let α ∈ (0,1], k1, k2, p, λ ∈ ℝ and functions f, g be α-differentiable on [ϕ, ∞). Then:

1. ${}_{\varphi }{D}^{\alpha }\left({k}_{1}f+{k}_{2}g\right)={k}_{1\phi }{D}^{\alpha }\left(f\right)+{k}_{2\phi }{D}^{\alpha }\left(g\right);$

2. ${}_{\varphi }{D}^{\alpha }\left(t-\varphi {\right)}^{p}=p\left(t-\varphi {\right)}^{p-\alpha };$

3. ${}_{\varphi }{D}^{\alpha }\lambda =0$ for all constant functions f(t) = λ

4. ${}_{\varphi }{D}^{\alpha }\left(fg\right)={f}_{{}_{\varphi }}{D}^{\alpha }g+{g}_{{}_{\varphi }}{D}^{\alpha }f;$

5. ${}_{\varphi }{D}^{\alpha }\left(\frac{f}{g}\right)=\frac{{g}_{{}_{\varphi }}{D}^{\alpha }f-{f}_{{}_{\varphi }}{D}^{\alpha }g}{{g}^{2}}$ for all functions g(t) ≠ 0.

Definition 2.3: ([20]). Let α ∈ (0, 1]. The conformable fractional integral starting from a point ϕ of a function f: [ϕ, ∞) → ℝ of order α is efined as$ϕIαf(t)=∫ϕt(s−ϕ)α−1f(s)ds.$(4)

Remark 2.4: If ϕ = 0, the definitions of the conformable fractional derivative and integral above will be reduced to the results in [19].

Definition 2.5: A nontrivial solution of Eq. (2) is said to be nonoscillatory if the solution is eventually positive or eventually negative. Otherwise, it is said to be oscillatory. Eq. (2) is said to be oscillatory if all solutions are oscillatory.

## 3 Auxiliary results

Let Jk = [tk, tk+1), k = 0,1,2,… be subintervals of [t0, ∞). PC([t0, ∞) = {x: [t0, ∞) → ℝ: x be continuous everywhere except at some tk at which $x\left({t}_{k}^{+}\right)$ and $x\left({t}_{k}^{-}\right)$ exist and $x\left({t}_{k}^{-}\right)=x\left({t}_{k}\right)\right\}$.

Lemma 3.1: Let wPC([t0, ∞), ℝ) be integrable on Jk, k = 0,1,2,… and zC([t0, ∞), ℝ)⋂ PC([t0, ∞), ℝ) such that$zt=exp∑t0ttkIαw(t):=exp(t0Iαwt1+t1Iαwt2+⋯+tkIαw(t)),$(5)for some tJk, k = 0,1,2,…. Then$zt=exp∑t0ttkIαw(t)⟺tkDαzt=wtz(t),t∈Jk,k=0,1,2,.…$(6)

Let Tt0 be a positive constant. We denote ${t}_{r}=\underset{k}{min}\left\{{t}_{k}:T\le {t}_{k},k=0,1,2,\dots \right\}$.

Lemma 3.2: Let g, hC([t0, ∞), ℝ) be two given functions. The linear conformable fractional differential equation$tkDαz(t)−g(t)z(t)=h(t),t∈Jk,k=r,r+1,r+2,…,$(7)has a solution given by$zt=ztrexp∑rtIαg+∑rtIαe−Iαghexp∑rtIαg.$(8)

## 4 Main results

Theorem 4.1: If the following conformable fractional differential equation$tkDαp∗(t)tkDαy(t)+r∗(t)y(t)+q∗(t)y(t)=0,t>t0,$(9)is oscillatory, then Eq. (2) is oscillatory, where Ck = bk/ak, dk = p(tk)r(tk)(akbk)/ak, for k = 1,2,3,…, and$p∗(t)=∏tr≤tk(10)$r∗(t)=r(t)−2p(t)∑tr≤tk(11)$q∗(t)=∏tr≤tk(12)

Theorem 4.2: Assume that g, hC([t0, ∞), ℝ) and fC([t0, ∞), ℝ+) If$∑r∞Iαe−Iαgh=∞and∑r∞IαeIαg1f=∞,$then$tkDαfttkDαxt+gtxt+htxt=0,t>t0,$(17)is oscillatory.

Theorem 4.3: If$∑r∞Iαe−Iαr∗q∗=∞and∑r∞IαeIαr∗1p∗=∞,$(21)where functions p*, r* and q* are defined by (10)-(12), respectively, then Eq. (2) is oscillatory.

If ak = bk = ek for all k = 1,2,3,… then ck = 1 and dk = 0 for k = 1,2,3,… and (2) can be written as

$tkDαp(t)tkDαx(t)+r(t)x(t)+q(t)x(t)=0,t≥t0,t≠tk,x(tk+)=ekx(tk−),k=1,2,…,tkDαx(tk+)=ektk−1Dαx(tk−),k=1,2,….$(22)

Theorem 4.4: Assume that ak = bk for all k = 1,2,3,…. Eq. (22) is oscillatory, if and only if$tkDαp(t)tkDαy(t)+r(t)y(t)+q(t)y(t)=0,t>t0,$(23)is oscillatory.

Corollary 4.5: If$∑r∞Iαe−Iαrq=∞and∑r∞IαeIαr1p=∞,$then Eq. (22) is oscillatory.

## 5 Examples

Example 5.1: Consider the following impulsive conformable fractional differential equation$kD23[t]kD23x(t)+1[t]x(t)+(t+2)3t4=0,t∈(k,k+1),x(k+)=kx(k−),kD23x(k+)=(k+1)k−1D23x(k−),k=1,2,3,....$(24)Here α = 2/3, p(t) = [t], r(t) = 1/[t], t > 0, where [·] denotes the greatest integer function, $q\left(t\right)=\left(t+2\right){3}^{{t}^{4}}$, ak = k, bk = k + 1, k = l, 2, 3,…. We find ck = bk/ak = (k + 1/k), dk = p(tk)r(tk)(akbk)/ak = –1/k. Let tr ∈ (m, m + 1] for some integer m > 1. Then we have$∏tr≤tk<[t]+1⁡1ck=∏tr≤tk<[t]+1⁡kk+1=m+1m+2⋅m+2m+3⋅⋅⋅⋅⋅[t][t]+1=m+1[t]+1.$and$∑tr≤tk<[t]∏tkObserve that 1 – ([t])/(m + 1) < 0. Hence, by direct computation, we have$∑r∞I23q∗exp−I23r∗=∑r∞I23m+1[t]+1(t+2)3t4+1[t]1−[t]m+12−1[t]1−[t]m+1×exp−I231[t]−2[t]1−[t]m+1≥∑r∞I23(m+1)3t4expI231[t]1−2[t]m+1=∞,$and$∑r∞I231p∗expI23r∗=∑r∞I23[t]+1m+1⋅1[t]expI231[t]−2[t]1−[t]m+1≥1m+1∑r∞I23expI231[t]2[t]m+1−1=∞.$Therefore, by Theorem 4.3, we deduce that Eq. (24) is oscillatory.

Example 5.2: Consider the following impulsive conformable fractional differential equation$kD231t2+4kD23x(t)+tx(t)+54t+3=0,t∈(k,k+1),x(k+)=3k+14k+2x(k−),kD23x(k+)=3k+14k+2k−1D23x(k−),k=1,2,3,....$(25)Here $p\left(t\right)=1/\sqrt{{t}^{2}+4},r\left(t\right)=t,q\left(t\right)={5}^{4t+3},\alpha =2/3$. We find that ${}_{k}{I}^{2/3}r\left(t\right){|}_{t=k+1}=\left(\left(3k\right)/2\right)+\left(3/5\right)$.For each tr ∈ (m, m + 1], m > 1, we have$∑r∞I2354t+3exp−I23t=∞,$and$∑r∞I23t2+4expI23t=∞.$Applying Corollary 4.5, we obtain that Eq. (25) is oscillatory.

## Acknowledgement

This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-GOV-59-31.

## References

• [1]

Diethelm K., The Analysis of Fractional Differential Equations, Springer, Berlin, 2010.

• [2]

Miller K.S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.

• [3]

Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999.

• [4]

Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

• [5]

Ahmad B., Ntouyas S.K., Alsaedi A., New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv. Difference Equ., 2011, Art. ID 107384, 11 pp.

• [6]

Alsaedi A., Ntouyas S.K., Agarwal R.P., Ahmad B., On Caputo type sequential fractional differential equations with nonlocal integral boundary conditions, Adv. Difference Equ., 2015, 2015:33.

• [7]

Ahmad B., Ntouyas S.K., Tariboon J., Fractional differential equations with nonlocal integral and integer-fractional-order Neumann type boundary conditions, Mediterr. J. Math., DOI 10.1007/s00009-015-0629-9. [Crossref]

• [8]

Bai Z.B., Sun W., Existence and multiplicity of positive solutions for singular fractional boundary value problems, Comput. Math. Appl., 2012, 63, 1369-1381.

• [9]

Su Y., Feng Z., Existence theory for an arbitrary order fractional differential equation with deviating argument, Acta Appl. Math., 2012, 118, 81-105.

• [10]

Diethelm K., Ford N.J., Analysis of fractional differential equations, J. Math. Anal. Appl., 2002, 265, 229-248.

• [11]

Galeone L., Garrappa R., Explicit methods for fractional differential equations and their stability properties, J. Comput. Appl. Math., 2009, 288, 548-560.

• [12]

Grace S.R., Agarwal R.P., Wong J.Y., Zafer A., On the oscillation of fractional differential equations, Frac. Calc. Appl. Anal., 2012, 15, 222-231.

• [13]

Chen D., Oscillation criteria of fractional differential equations, Adv. Difference Equ., 2012, 33, 1-18.

• [14]

Han Z., Zhao Y., Sun Y., Zhang C., Oscillation for a class of fractional differential equation, Discrete Dyn. Nat. Soc., 2013, Art. ID 390282, 1-6.

• [15]

Feng Q., Meng F., Oscillation of solutions to nonlinear forced fractional differential equations, Electron. J. Differential Equations, 2013, 169, 1-10.

• [16]

Liu T., Zheng B., Meng F., Oscillation on a class of differential equations of fractional order, Math. Probl. Eng., 2013, Art. ID 830836, 1-13.

• [17]

Chen D., Qu P., Lan Y., Forced oscillation of certain fractional differential equations, Adv. Difference Equ., 2013, 2013:125.

• [18]

Wang Y.Z., Han Z.L., Zhao P., Sun S.R., On the oscillation and asymptotic behavior for a kind of fractional differential equations, Adv. Difference Equ., 2014, 2014:50.

• [19]

Khalil R., Al Horani M., Yousef A., Sababheh M., A new definition of fractional derivative, J. Comput. Appl. Math., 2014, 264, 65-70.

• [20]

Abdeljawad T., On conformable fractional calculus, J. Comput. Appl. Math., 2015, 279, 57-66.

• [21]

Anderson D., Ulness D., Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 2015, 10, 109-137.

• [22]

Batarfi H., Losada J., Nieto J.J., Shammakh W., Three-point boundary value problems for conformable fractional differential equations, J. Func. Spaces, 2015, Art. ID 706383, 1-6.

• [23]

Abdeljawad T., Al Horani M., Khalil R., Conformable fractional semigroups of operators, J. Semigroup Theory Appl., 2015, Art. ID 7.

• [24]

Abu Hammad M., Khalil R., Fractional Fourier series with applications, Amer. J. Comput. Appl. Math., 2014, 4, 187-191.

• [25]

Abu Hammad M., Khalil R., Abel’s formula and Wronskian for conformable fractional differential equations, Internat. J. Diff. Equ. Appl., 2014, 13, 177-183.

• [26]

Lakshmikantham V., Bainov D.D., Simeonov P.S., Theory of impulsive differential equations, World Scientific, Singapore-London, 1989.

• [27]

Samoilenko A.M., Perestyuk N.A., Impulsive Differential Equations, World Scientific, Singapore, 1995.

• [28]

Benchohra M., Henderson J., Ntouyas S.K., Impulsive Differential Equations and Inclusions, vol. 2, Hindawi Publishing Corporation, New York, 2006.

• [29]

Tariboon J., Thiramanus P., Oscillation of a class of second-order linear impulsive differential equations, Adv. Difference Equ., 2012, 2012:205.

Accepted: 2016-06-21

Published Online: 2016-07-22

Published in Print: 2016-01-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455, Export Citation