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Open Mathematics

formerly Central European Journal of Mathematics

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

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Volume 14, Issue 1 (Jan 2016)

Issues

Oscillation of impulsive conformable fractional differential equations

Jessada Tariboon
  • 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
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ 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
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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,tt0,ttk,xtk+=akx(tk),tkDαxtk+=bktk1Dαx(tk),k=1,2,.

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

Keywords: Fractional differential equations; Impulsive differential equations; Conformable fractional derivative; Oscillation

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

atxt+λxt+ptxt=0,tt0,ttk,xtk+=bkx(tk),xtk+=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,tt0,ttk,xtk+=akx(tk),k=1,2,,tkDαxtk+=bktk1Dαx(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(tk+)=limθ0+x(tk+θ),x(tk)=limθ0+x(tk+θ).

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+=limε0x(tk++ε(tk+tk)1α)xtk+ε,

and

tk1Dαx(tk)=limε0x(tk+ε(tktk1)1α)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.

[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)=limε0f(t+ε(tφ)1α)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.

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

  1. ϕDα(k1f+k2g)=k1φDα(f)+k2φDα(g);

  2. ϕDα(tϕ)p=p(tϕ)pα;

  3. ϕDαλ=0 for all constant functions f(t) = λ

  4. ϕDα(fg)=fϕDαg+gϕDαf;

  5. ϕDαfg=gϕDαffϕDαgg2 for all functions g(t) ≠ 0.

([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(sϕ)α1f(s)ds.(4)

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

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(tk+) and x(tk) exist and x(tk)=x(tk)}.

Let wPC([t0, ∞), ℝ) be integrable on Jk, k = 0,1,2,… and zC([t0, ∞), ℝ)⋂ PC([t0, ∞), ℝ) such that

zt=expt0ttkIαw(t):=exp(t0Iαwt1+t1Iαwt2++tkIαw(t)),(5)

for some tJk, k = 0,1,2,…. Then

zt=expt0ttkIαw(t)tkDαzt=wtz(t),tJk,k=0,1,2,.(6)

Let Tt0 be a positive constant. We denote tr=mink{tk:Ttk,k=0,1,2,}.

Let g, hC([t0, ∞), ℝ) be two given functions. The linear conformable fractional differential equation

tkDαz(t)g(t)z(t)=h(t),tJk,k=r,r+1,r+2,,(7)

has a solution given by

zt=ztrexprtIαg+rtIαeIαghexprtIαg.(8)

4 Main results

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)=trtk<t1ckp(t),(10)

r(t)=r(t)2p(t)trtk<ttk<tj<tcjdk,(11)

q(t)=trtk<t1ckq(t)+1p(t)trtk<ttktj<tcjdk2r(t)trtk<ttktj<tcjdk.(12)

Assume that g, hC([t0, ∞), ℝ) and fC([t0, ∞), ℝ+) If

rIαeIαgh=andrIαeIαg1f=,

then

tkDαfttkDαxt+gtxt+htxt=0,t>t0,(17)

is oscillatory.

If

rIαeIαrq=andrIαeIαr1p=,(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,tt0,ttk,x(tk+)=ekx(tk),k=1,2,,tkDαx(tk+)=ektk1Dαx(tk),k=1,2,.(22)

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.

If

rIαeIαrq=andrIαeIαr1p=,

then Eq. (22) is oscillatory.

5 Examples

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)k1D23x(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(t)=(t+2)3t4, 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

trtk<[t]+11ck=trtk<[t]+1kk+1=m+1m+2m+2m+3[t][t]+1=m+1[t]+1.

and

trtk<[t]tk<tj<[t]cjdk=trtk<[t]tk<tj<[t]j+1j1k=1m+1m+3m+2m+4m+3[t][t]+1+1m+2m+4m+3m+5m+4[t][t]1++1[t]2[t][t]1+[t][t]1=1[t]m+1.

Observe that 1 – ([t])/(m + 1) < 0. Hence, by direct computation, we have

rI23qexpI23r=rI23m+1[t]+1(t+2)3t4+1[t]1[t]m+121[t]1[t]m+1×expI231[t]2[t]1[t]m+1rI23(m+1)3t4expI231[t]12[t]m+1=,

and

rI231pexpI23r=rI23[t]+1m+11[t]expI231[t]2[t]1[t]m+11m+1rI23expI231[t]2[t]m+11=.

Therefore, by Theorem 4.3, we deduce that Eq. (24) is oscillatory.

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+2k1D23x(k),k=1,2,3,....(25)

Here p(t)=1/t2+4,r(t)=t,q(t)=54t+3,α=2/3. We find that kI2/3r(t)|t=k+1=((3k)/2)+(3/5).

For each tr ∈ (m, m + 1], m > 1, we have

rI2354t+3expI23t=,

and

rI23t2+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.

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About the article

Received: 2016-05-24

Accepted: 2016-06-21

Published Online: 2016-07-22

Published in Print: 2016-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0044.

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© 2016 Nazwisko, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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