Oscillation of impulsive conformable fractional differential equations

Jessada Tariboon; Sotiris K. Ntouyas

doi:10.1515/math-2016-0044

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Oscillation of impulsive conformable fractional differential equations

Jessada Tariboon and Sotiris K. Ntouyas

Published/Copyright: July 22, 2016

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$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

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.

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

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

where 0 ≤ t₀ < t₁ < … < t_k < …, lim_{k → ∞}t_k = + ∞, a ∈ C([t₀, ∞), (0, ∞)), p ∈ C([t₀, ∞), ℝ), {b_k}, {c_k} 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 _ϕD^α denotes the conformable fractional derivative of order 0 < α ≤ 1 starting from ϕ ∈ {t₀,…, t_k,…}, 0 ≤ t₀ < t₁ < …, < t_k < …, lim_k→∞t_k = ∞, p ∈ C([t₀, ∞), (0, ∞)), q, r ∈ C([t₀, ∞), ℝ), {a_k} and {b_k} are two known sequences of positive real numbers and x(tk+)=limθ→0+x(tk+θ),x(tk−)=limθ→0+x(tk+θ).

Some new oscillation results are obtained, generalizing the results of [29] to impulsive conformable fractional differential equations. Note that if a_k = 1 for all k = 1,2,…, then x is continuous on [t₀, ∞). However, if a_k = b_k = 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 _ϕD^α f(t) = (t – ϕ)^1—αf′(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:

ϕDα(k1f+k2g)=k1φDα(f)+k2φDα(g);
ϕDα(t−ϕ)p=p(t−ϕ)p−α;
ϕDαλ=0for all constant functions f(t) = λ
ϕDα(fg)=fϕDαg+gϕDαf;
ϕDαfg=gϕDαf−fϕDαgg2for 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 J_k = [t_k, t_k+1), k = 0,1,2,… be subintervals of [t₀, ∞). PC([t₀, ∞) = {x: [t₀, ∞) → ℝ: x be continuous everywhere except at some t_k at which x(tk+) and x(tk−) exist and x(tk−)=x(tk)}.

Lemma 3.1

Let w ∈ PC([t₀, ∞), ℝ) be integrable on J_k, k = 0,1,2,… and z ∈ C([t₀, ∞), ℝ)⋂ PC([t₀, ∞), ℝ) such that

zt=exp∑t0ttkIαw(t):=exp(t0Iαwt1+t1Iαwt2+⋯+tkIαw(t)),(5)

for some t ∈ J_k, k = 0,1,2,…. Then

zt=exp∑t0ttkIαw(t)⟺tkDαzt=wtz(t),t∈Jk,k=0,1,2,.…(6)

Proof

Observe that z(t) > 0 for all t ∈ [t₀, ∞) and z(t₀) = 1. Assume that tkDαz(t)=w(t)z(t) holds. For t ∈ J₀, we have

t0Dα[exp(−toIαw(t))z(t)]=exp(−toIαw(t))[tkDαz(t)−w(t)z(t)]=0.

Taking the conformable fractional integral of order a to both sides of the above equation, we get

z(t)=exp(toIαw(t)).

In particular, z(t1)=exp(t0Iαw(t1)). For t ∈ J₁ and following the above process, we have

exp(−t1Iαw(t))z(t)−z(t1)=0,

which implies

z(t)=exp(t0Iαw(t1)+t1Iαw(t)),t∈J1.

Repeating the above method, for any t ∈ J_k, k = 0,1,2,…, we obtain (5).

On the other hand, for any t ∈ J_k, k = 0,1,2,…, we have

tkDαz(t)=exp∑t0ttkIαw(t)tkDα(tkIαw(t))=z(t)w(t).

This completes the proof. □

Let T ≥ t₀ be a positive constant. We denote tr=mink⁡{tk:T≤tk,k=0,1,2,…}.

Lemma 3.2

Let g, h ∈ C([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)

Proof

For t ∈ J_r, we have

trDαe−trIαg(t)z(t)=e−trIαg(t)h(t),

which leads to

zt=ztretrIαg(t)+etrIαg(t)trIαe−trIαgh(t).

Therefore, (8) is true for [t_r, t_r+1). Now assume that (8) holds for t ∈ [t_r, t_r+n) for some integer n > 1. Then, for t ∈ [t_r+n, t_r+n+1), it follows from (7) that

z(t)=z(tr+n)etr+nIαg(t)+etr+nIαg(t)tr+nIαe−tr+nIαgh(t).

Using (8), we deduce that

zt=z(tr)exp∑rr+nIαg+∑rr+nIαe−Iαghexp∑rr+nIαgetr+nIαg(t)+etr+nIαg(t)tr+nIαe−tr+nIαgh(t)=ztrexp∑rtIαg+∑rr+nIαe−Iαghexp∑rtIαg+etr+nIαg(t)tr+nIαe−tr+nIαgh(t)=ztrexp∑rtIαg+∑rtIαe−Iαghexp∑rtIαg,

which gives (8) for t ∈ [t_r, t_r+n+1). The proof is completed. □

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, thenEq. (2)is oscillatory, where C_k = b_k/a_k, d_k = p(t_k)r(t_k)(a_k – b_k)/a_k, for k = 1,2,3,…, and

p∗(t)=∏tr≤tk<t1ckp(t),(10)

r∗(t)=r(t)−2p(t)∑tr≤tk<t∏tk<tj<tcjdk,(11)

q∗(t)=∏tr≤tk<t1ckq(t)+1p(t)∑tr≤tk<t∏tk≤tj<tcjdk2−r(t)∑tr≤tk<t∏tk≤tj<tcjdk.(12)

Proof

Without loss of generality, we assume that Eq. (2) has an eventually positive solution x. This means that there exists a constant T ≥ t₀ such that x(t) > 0 for all t ≥ T. For any t ∈ J_k with t_k ≥ t_r ≥ T, setting

ut=p(t)tkDαetkIαr(t)x(t)etkIαr(t)x(t),(13)

it is easy to verify that

ut=p(t)(tkDαx(t)+r(t)x(t))x(t),t∈Jk.

Therefore, we obtain

tkDαu(t)=tkDαp(t)tkDαetkIαr(t)x(t)etkIαr(t)x(t)−p(t)tkDαetkIαr(t)x(t)2etkIαr(t)x(t)2=tkDαetkIαr(t)p(t)tkDαx(t)+r(t)x(t)etkIαr(t)x(t)−u2(t)p(t)=tkDαp(t)tkDαx(t)+r(t)x(t)x(t)+r(t)p(t)tkDαx(t)+r(t)x(t)x(t)−u2(t)p(t)=−q(t)+r(t)u(t)−u2(t)p(t),

which leads to

tkDαut−rtut+u2tpt+qt=0,t≥tr,t≠tk.(14)

For t=tk+≥tr, k = 1,2,…, we have

u(tk+)=p(tk+)tkDαx(tk+)+r(tk+)x(tk+)x(tk+)=p(tk)bktk−1Dαx(tk−)+r(tk)akx(tk−)akx(tk−)=bkakp(tk)tk−1Dαx(tk−)+r(tk)x(tk−)x(tk−)+p(tk)r(tk)ak−bkak=cku(tk−)+dk.

Define a function v(t) for t ≥ t_r by

vt=∏tr≤tk<t1ckut∑tr≤tk<t∏tk≤tj<tcjdk.

For each t_n ≥ t_r, we see that

vtn+=∏tr≤tk≤tn1ckutn+−∑tr≤tk≤tn∏tk≤tj≤tncjdk=∏tr≤tk≤tn1ckcnutn−+dn−∑tr≤tk<tn∏tk<tj<tncjcndk−dn=∏tr≤tk≤tn1ckutn−−∑tr≤tk<tn∏tk<tj<tncjdk=v(tn−),

which implies that v(t) is continuous on (t_r, ∞).

Denote t_n,α = (t — t_n)^1–α for some t ∈ J_n and t_n ≥ t_r. From the definition of conformable fractional derivative, for t ∈ J_n we obtain

tnDαvt=limε→0∏tr≤tk<t+εtn,α1cku(t+εtn,α)−∑tr≤tk<t+εtn,α∏tk<tj<t+εtn,αcjdk−∏tr≤tk<t1cku(t)−∑tr≤tk<t∏tk<tj<tcjdk/ε=limε→0∏tr≤tk<t+εtn,α1ckut+εtn,α−∏tr≤tk<t1ckut/ε=∏tr≤tk<t1cklimε→0⁡ut+εtn,α−utε=∏tr≤tk<t1cktnDαut.

From (14), we have

tnDαvt=∏tr≤tk<t1ckrtut−u2tpt−qt=∏tr≤tk<t1ckrtvt∏tr≤tk<tck+∑tr≤tk<t∏tk<tj<tcjdk−1ptvt∏tr≤tk<tck+∑tr≤tk<t∏tk<tj<tcjdk2−qt=rtvt+rt∑tr≤tk<t∏tr≤tj≤tkdkcj−∏tr≤tk<tckv2tpt−2vtpt∑tr≤tk<t∏tk<tj<tcjdk−1pt∏tr≤tk<t1ck∑tr≤tk<t∏tk<tj<tcjdk2−∏tr≤tk<t1ckq(t),

by using the fact that

∏tr≤tk<t1ck∑tr≤tk<t∏tk<tj<tcjdk=∑tr≤tk<t∏tr≤tj≤tkdkcj.

Setting

ξt=exp∑rttkIασt,t≥tr,

where

σt=−rt+∏tr≤tk<tckvtpt+2pt∑tr≤tk<t∏tk<tj<tcjdk,

we have ξ(t) > 0 for t ≥ t_r and from Lemma 3.1 that

tkDαξt=ξtσt=ξt−rt+∏T≤tk<tckvtpt+2pt∑T≤tk<t∏tk<tj<tcjdk.

Hence

ξtvt=∏tr≤tk<t1ckp(t)tkDαξt+∏T≤tk<t1ckrtptξt−2∑tr≤tk<t∏T≤tj≤tkdkcjξt=p∗ttkDαξt+r∗tξt.(15)

Applying the conformable fractional derivative of order α on any interval Jk to the above equation, we get

tkDαξtvt=ξ(t)tkDαvt+v(t)tkDαξt=ξttkDαvt+vtσt=ξtrt∑tr≤tk<t∏T≤tj≤tkdkcj−∏tr≤tk<t1ckqt−1pt∏tr≤tk<t1ck∑tr≤tk<t∏tk<tj<tkcjdk2.(16)

From (15) and (16), we obtain

tkDαp∗ttkDαξt+r∗tξt+q∗tξt=0,t≥tr.

This means that ξ(t) is an eventually positive solution of Eq. (9) which is a contradiction. In the same way, we can prove that Eq. (9) could not have an eventually negative solution. Therefore, the solution of Eq. (9) is oscillatory. This completes the proof. □

Theorem 4.2

Assume that g, h ∈ C([t₀, ∞), ℝ) and f ∈ C([t₀, ∞), ℝ₊) If

∑r∞Iαe−Iαgh=∞and∑r∞IαeIαg1f=∞,

then

tkDαfttkDαxt+gtxt+htxt=0,t>t0,(17)

is oscillatory.

Proof

Let a function x be a nonoscillatory solution of Eq. (17). Then we can assume that there exists a constant T ≥ t₀ such that x(t) > 0 for all t ≥ T. For the case x(t) < 0, t ≥ T, the proof is similar and we omit it.

Using the Riccati transform for conformable fractional differential equation

wt=f(t)tkDαetkIαg(t)xtetkIαg(t)xt,t∈Jk,k≥r,(18)

we get that the equation

tkDαw(t)−g(t)w(t)+1f(t)w2(t)+h(t)=0,(19)

has a solution w(t) on [t_r, ∞). From Lemma 3.2, we get

w(t)=w(tr)exp∑rtIαg−∑rtIαe−Iαgw2fexp∑rtIαg−∑rtIαe−Iαghexp∑rtIαg.(20)

Since ∑r∞Iα(e−Iαgh)=∞, then there exists a fixed point ƞ > t_r such that

w(tr)exp∑rtIαg−∑rtIαe−Iαghexp∑rtIαg<0,∀t∈η,∞.

Thus (20) implies

w(t)<−∑rtIαe−Iαgw2fexp∑rtIαg,∀t∈η,∞.

Let ƞ ≤ t ∈ J_j for some j. We observe that

∑rtIαe−Iαgw2fexp∑rtIαg=etjIαg(t)∑tr≤tk≤tj−1tkIαe−tkIαg(t)w2ftk+1exp∑tr≤tk≤tj−1tkIαg(tk+1)+tjIαe−tjIαgw2f(t):=etjIαg(t)u(t).

Then we have w(t)<−etjIαg(t)u(t) and

tjDαu(t)=e−tjIαg(t)w2tft>etjIαg(t)u2tft,t∈Jj,

which yields for t = t_j+1

−1u(tj+1)+1u(tj)>tjIαg(t)etjIαg(t)1f(tj+1),

and also

−1u(tj+2)+1u(tj+1)>tj+1Iαetj+1Iαg1f(tj+2),….

Summing up the above inequalities, we obtain

−1u∞+1utj>∑tj∞IαeIαg1f.

Therefore,

∑r∞IαeIαg1f=∞<1u(tj)<∞,

which is a contradiction. Hence, the solution x is oscillatory. This completes the proof. □

Theorem 4.3

∑r∞Iαe−Iαr∗q∗=∞and∑r∞IαeIαr∗1p∗=∞,(21)

where functions p^*, r^* and q^* are defined by (10)-(12), respectively, thenEq. (2)is oscillatory.

If a_k = b_k = e_k for all k = 1,2,3,… then c_k = 1 and d_k = 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 a_k = b_k 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.

Proof

From Theorem 4.1, we only need to prove that if Eq. (22) is oscillatory, then (23) is oscillatory.

Assume that the function y is an eventually positive solution of Eq. (18) such that y(t) > 0 for all t ≥ T ≥ t₀. We define

xt=∏tr≤tk≤teky(t),t≥tr≥T.

Then, we obtain x(t) > 0 for t ≥ t_r and

x(tn+)=∏tr≤tk≤tneky(tn+)=en∏tr≤tk≤tneky(tn)=enx(tn),

for some n > r. In addition, for t ∈ J_n, n > r, we have

tnDαx(t)=∏tr≤tk<tektnDαy(t),

and for t = t_n,

tnDαx(tn+)=∏tr≤tk≤tnektnDαy(tn+)=en∏tr≤tk≤tnektn−1Dαy(tn−)=entn−1Dαx(tn−).

For any t ∈ J_k, k ≥ r, we have

tkDαp(t)tkDαx(t)+r(t)x(t)=tkDαp(t)∏tr≤tk<tektkDαy(t)+r(t)∏tr≤tk<teky(t)=∏tr≤tk<tektkDαp(t)tkDαy(t)+r(t)y(t)=−∏tr≤tk<tekq(t)y(t)=−q(t)x(t).

Therefore,

tkDαp(t)tkDαx(t)+r(t)x(t)+q(t)x(t)=0,t≠tn,t>T.

This means that x is an eventually positive solution of Eq. (22) which is a contradiction. The proof is completed. □

Corollary 4.5

∑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(t)=(t+2)3t4, a_k = k, b_k = k + 1, k = l, 2, 3,…. We find c_k = b_k/a_k = (k + 1/k), d_k = p(t_k)r(t_k)(a_k – b_k)/a_k = –1/k. Let t_r ∈ (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]∏tk<tj<[t]cjdk=−∑tr≤tk<[t]∏tk<tj<[t]j+1j1k=−1m+1⋅m+3m+2⋅m+4m+3⋅⋅⋅⋅⋅[t][t]+1+1m+2⋅m+4m+3⋅m+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

∑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(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 t_r ∈ (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.

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Received: 2016-5-24

Accepted: 2016-6-21

Published Online: 2016-7-22

Published in Print: 2016-1-1

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.