Interval Oscillation Criteria for a Class of Nonlinear Fractional Differential Equations with Nonlinear Damping Term

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Interval Oscillation Criteria for a Class of

Nonlinear Fractional Differential Equations with

Nonlinear Damping Term

Qinghua Feng

∗

Abstract—In this paper, based on certain variable trans-formation, we apply the known (G’/G) method to seek exact solutions for three fractional partial differential equations: the space fractional (2+1)-dimensional breaking soliton equation-s, the time fractional Fokas equation, and the space-time fractional Kaup-Kupershmidt equation. The fractional derivative is defined in the sense of modified Riemann-liouville derivative. With the aid of mathematical software Maple, a number of exact solutions including hyperbolic function solu-tions, trigonometric function solusolu-tions, and rational function solutions for them are obtained.

Index Terms—(G’/G) method, fractional partial differential equation, exact solution, variable transformation.

I. INTRODUCTION

Fractional differential equations are generalizations of classical differential equations of integer order, and can find their applications in many fields of science and engineer-ing. In the literature, research on the theory of differential equations, integral equations and matrix equations include various aspects, such as the existence and uniqueness of solutions [1,2], seeking for exact solutions [3,4], numerical method [5-7]. Among these investigations, 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, biol-ogy, engineering, signal processing, systems identification, control theory, finance and fractional dynamics, and has attracted much attention of more and more scholars. For example, Bouhassoun [8] extended the telescoping decompo-sition method to derive approximate analytical solutions of fractional differential equations. Bijura [9] investigated the solution of a singularly perturbed nonlinear system fractional integral equations. Blackledge [10] investigated the applica-tion of a certain fracapplica-tional Diffusion equaapplica-tion, and applied it for predicting market behavior. In [11-18], the existence, uniqueness, stability of solutions, and numerical methods of fractional differential equations were investigated.

In these investigations, we notice that very little attention is paid to oscillation of fractional differential equations. Recent results in this direction include Chen’s work [19,20] and Zheng’s work [21]. In [19], Chen researched oscillation of the following fractional differential equation:

[r(t)(Dαy(t))η]′−q(t)f(

∫ ∞

t

(v−t)−αy(v)dv) = 0, t >0,

Manuscript received May 2, 2013; revised 26 June, 2013.

Q. Feng is with the School of Science, Shandong University Of Technol-ogy, Zibo, Shandong, 255049 China∗e-mail: fqhua@sina.com

where r, q are positive-valued functions, η is the quotient of two odd positive numbers, α ∈ (0,1), Dαy(t) denotes the Liouville right-sided fractional derivative of order α

of x, and Dα_y₍_t_{) =} ₋ 1

Γ(1−α)

d dt

∫_∞

t (ξ−t)−

α_y₍_ξ₎_dξ_.

Then in [20], under similar conditions to [19], some new oscillation criteria are established for the following fractional differential equation with damping term:

D1+αy(t)−p(t)Dαy(t) +q(t)f(

∫ _∞

t

(v−t)−αy(v)dv) = 0, t >0,

At the same time, in [21], Zheng researched oscillation of the following nonlinear fractional differential equation with damping term and more general form than the equations mentioned above:

[a(t)(Dαx(t))γ]′+p(t)(Dαx(t))γ

−q(t)f(

∫ _∞

t

(ξ−t)−αx(ξ)dξ) = 0, t∈[t0,∞).

In this paper, we are concerned with oscillation for a class of nonlinear fractional differential equations with nonlinear damping term as follows:

D_tα(r(t)k1(x(t), Dtαx(t))) +p(t)k2(x(t), Dαtx(t))D α tx(t)

+q(t)f(x(t)) = 0, t≥t0≥0, 0< α <1, (1)

whereDα

t(.)denotes the modified Riemann-liouville

deriva-tive [22] with respect to the variable t, the functions

r, q ∈ Cα_([_t

0,∞), R+), p ∈ Cα([t0,∞),[0,∞), and Cα

denotes continuous derivative of order α, the function f

is continuous satisfying f(x)/x ≥ K for some positive constant K and ∀x ̸= 0, k1 is continuously differentiable

satisfyingk2₁(u, v)≤Avk1(u, v)for some positive constant

A,∀v∈R\{0} and∀u∈R.

The definition and some important properties for the Jumarie’s modified Riemann-Liouville derivative of orderα

are listed as follows (see also in [23-25]):

Dα_tf(t) =

    

1 Γ(1−α)

d dt

∫t

0(t−ξ)−

α₍_f₍_ξ₎₋_f₍₀₎₎_dξ,

0< α <1,

(f(n)₍_t₎₎(α−n)_{, n}_≤_{α < n}_{+ 1}_{, n}_≥₁_.

Dα_ttr= Γ(1 +r) Γ(1 +r−α)t

r−α_, ₍₂₎

Dαt(f(t)g(t)) =g(t)D α

tf(t) +f(t)D α

tg(t), (3)

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D_tαf[g(t)] =f_g′[g(t)]Dα_tg(t) =Dα_gf[g(t)](g′(t))α. (4)

As usual, a nonconstant continuable solution x(t)of Eq. (1) is called proper if sup

t_≥t0

| x(t) |> 0. A proper solution

x(t)of Eq. (1) is called oscillatory if it has arbitrarily large zeros, otherwise it is called non-oscillatory. Eq. (1) is called oscillatory if all its solutions are oscillatory.

We organize the next of this paper as follows. In Section 2, we establish some new interval oscillation criteria for Eq. (1) under the condition for f(x) without monotonicity, while oscillation criteria are established for Eq. (1) under the condition for f(x) with monotonicity in Section 3. We present some examples for the results established in Section 4. Some conclusions are presented at the end of this paper.

For the sake of convenience, in the next of this paper, we

denote ξ0 = t

α

0

Γ(1 +α), ξ =

tα

Γ(1 +α), R+ = (0,∞), and leth1, h2, H ∈C([ξ0,∞), R)satisfying

H(ξ, ξ) = 0, H(ξ, s)>0, ξ > s≥ξ0.

H has continuous partial derivatives∂H_∂ξ(ξ, s) and∂H_∂s(ξ, s) on[ξ0,∞)such that

∂H(ξ, s)

∂ξ =−h1(ξ, s) √

H(ξ, s),

∂H(ξ, s)

∂s =−h2(ξ, s) √

H(ξ, s), ξ > s≥ξ0.

II. INTERVAL OSCILLATION CRITERIA WITHf(x)NOT BEING MONOTONE

Lemma 2.1. Suppose thatk2(u, v) :R2→R2is continuous

and has the sign of v for all v ∈ R\{0} and all u ∈ R. If x(t) is a non-oscillatory solution of Eq. (1), then

x(t)D_tαx(t) < 0 for t ≥ t_∗, where t_∗ ≥ t0 is sufficiently

large.

Proof. Supposex(t)be a proper non-oscillatory solution of Eq. (1). Letr(t) =er(ξ), x(t) =xe(ξ), p(t) =pe(ξ), q(t) =

e

q(ξ), where ξ = tα

Γ(1 +α). Then by use of Eq. (2) we obtain Dα

tξ(t) = 1, and furthermore by use of the first

equality in Eq. (4), we have

D_tαr(t) =D_tαre(ξ) =re′(ξ)Dα_tξ(t) =er′(ξ).

Similarly we have Dα

tx(t) = xe′(ξ). So Eq. (1) can be

transformed into the following form:

(er(ξ)k1(xe(ξ),xe′(ξ)))′+pe(ξ)k2(ex(ξ),ex′(ξ))xe′(ξ)

+qe(ξ)f(xe(ξ)) = 0, ξ≥ξ0≥0. (5)

Then ex(ξ) be a proper non-oscillatory solution of Eq. (5). By [26, Lemma 1], we have ex(ξ)xe′(ξ) < 0, ξ ≥ ξ_∗, where ξ_∗ ≥ ξ0 is sufficiently large. Let ξ∗ =

tα_∗

Γ(1 +α). Thenx(t)Dα_tx(t)<0, t≥t_∗, and the proof is complete.

Theorem 2.1. Under the conditions of Lemma 2.1, furthermore, assume that there exists ρ ∈Cα([t0,∞), R+)

such that for any sufficiently large T ≥ ξ0, there exist

a, b, cwithT ≤a < c < bsatisfying 1

H(c, a)

∫ c

a

H(s, a)Kρe(s)eq(s)ds

+ 1

H(b, c)

∫ b

c

H(b, s)Kρe(s)qe(s)ds

> A

4H(c, a)

∫ c

a e

r(s)ρe(s)Q2₁(s, a)ds

+ A

4H(b, c)

∫ b

c e

r(s)ρe(s)Q2₂(b, s)ds, (6) where ρe(ξ) = ρ(t), eq(ξ) = q(t), er(ξ) = r(t),

Q1(s, ξ) = h1(s, ξ) − e

ρ′(s)

e ρ(s)

√

H(s, ξ), Q2(ξ, s) =

h2(ξ, s)−ρe

′₍_s₎

e ρ(s)

√

H(ξ, s),then Eq. (1) is oscillatory.

Proof: Suppose to the contrary that x(t) be a non-oscillatory solution of Eq. (1). Without loss of generality, we may assume thatx(t)>0 on[T0,∞), whereT0≥t0is

sufficiently large. Furthermore, by Lemma 2.1, there exists

t_∗≥T0such thatDtαx(t)<0, t≥t∗. So by the assumption

on Eq. (1) we havek1(x(t), Dαtx(t))≥ADtαx(t), t≥t∗.

Define

w(t) =ρ(t)r(t)k1(x(t), D

α tx(t))

x(t) . (7) Then fort≥t_∗, we have

Dtαw(t) =

Dtαρ(t)

ρ(t) w(t)−ρ(t)p(t)

k2(x(t), Dtαx(t))Dtαx(t) x(t)

−ρ(t)q(t)f(x(t))

x(t) −ρ(t)

r(t)k1(x(t), Dtαx(t))D α tx(t) x2(t)

= D

α tρ(t)

k2(x(t), Dαtx(t))D α tx(t) x(t)

−ρ(t)q(t)f(x(t))

x(t) −

w2(t)Dtαx(t) ρ(t)r(t)k1(x(t), Dαtx(t)) ≤ D

α tρ(t)

k2(x(t), Dαtx(t))D α tx(t) x(t)

−ρ(t)q(t)f(x(t))

x(t) −

w2(t)

Aρ(t)r(t). (8) Sincek2(x(t), Dαtx(t))Dαtx(t)≥0, furthermore we obtain

D_tαw(t)≤ −Kρ(t)q(t) +D

α tρ(t) ρ(t) w(t)−

w2(t)

Ar(t)ρ(t), t≥t∗. (9) Letw(t) =we(ξ). ThenDαtw(t) =we′(ξ), Dtαρ(t) =ρe′(ξ),

and (9) is transformed into the following form

e

w′(ξ)≤ −Kρe(ξ)qe(ξ) +ρe ′₍_ξ₎

e

ρ(ξ)we(ξ)−

e w2(ξ)

Aer(ξ)ρe(ξ), ξ ≥ξ∗, (10)

whereξ_∗= t

α

∗ Γ(1 +α).

Select a, b, c arbitrarily in [ξ_∗,∞) with b > c > a. Substituting ξ with s, multiplying both sides of (10) by

IAENG International Journal of Applied Mathematics, 43:3, IJAM_43_3_09

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H(ξ, s)and integrating it with respect to sfrom c to ξfor

ξ∈[c, b), we get that ∫ ξ

c

H(ξ, s)Kρe(s)qe(s)ds≤ − ∫ ξ

c

H(ξ, s)we′(s)ds

+

∫ ξ

c

H(ξ, s)ρe ′₍_s₎

e

ρ(s)we(s)ds−

∫ ξ

c

H(ξ, s) we

2₍_s₎

Aρe(s)er(s)ds

=H(ξ, c)we(c)−

∫ ξ

c

[( H(ξ, s)

Aeρ(s)er(s))

1/2_w_e₍_s₎

+1

2(Aρe(s)re(s))

1/2_Q

2(ξ, s)]2ds+

∫ ξ

c

Aρe(s)re(s)

4 Q

2 2(ξ, s)ds

≤H(ξ, c)we(c) +

∫ ξ

c

Aeρ(s)er(s)

4 Q

2

2(ξ, s)ds. (11)

Dividing both sides of the inequality (11) by H(ξ, c) and letξ→b−, we obtain

1

H(b, c)

∫ b

c

H(b, s)Kρe(s)eq(s)ds≤

e

w(c) + 1

H(b, c)

∫ b

c

Aρe(s)er(s)

4 Q

2

2(b, s)ds. (12)

On the other hand, substitutingξwiths, multiplying both sides of (10) byH(s, ξ)and integrating it with respect tos

fromξ tocfor ξ∈(a, c], we get that ∫ c

ξ

H(s, ξ)Kρe(s)qe(s)ds≤ − ∫ c

ξ

H(s, ξ)we′(s)ds

+

∫ c

ξ

H(s, ξ)ρe ′₍_s₎

e

ρ(s)we(s)ds−

∫ c

ξ

H(s, ξ) we

2₍_s₎

Aρe(s)er(s)ds

=−H(c, ξ)we(c)−

∫ c

ξ

[( H(s, ξ)

Aρe(s)re(s))

1/2_e

w(s)

+1

2(Aρe(s)re(s))

1/2_Q

1(s, ξ)]2ds+

∫ c

ξ

Aρe(s)re(s)

4 Q

2 1(s, ξ)ds

≤ −H(c, ξ)we(c) +

∫ c

ξ

Aρe(s)re(s)

4 Q

2

1(s, ξ)ds. (13)

Dividing both sides of the inequality (13) by H(c, ξ) and letting ξ→a+_{, we obtain}

1

H(c, a)

∫ c

a

H(s, a)Kρe(s)qe(s)ds≤

−we(c) + 1

H(c, a)

∫ c

a

Aρe(s)er(s)

4 Q

2

1(s, a)ds. (14)

A combination of (12) and (14) yields

1

H(c, a)

∫ c

a

H(s, a)Kρe(s)qe(s)ds

+ 1

H(b, c)

∫ b

c

H(b, s)Kρe(s)qe(s)ds

≤ A

4H(c, a)

∫ c

a e

+ A

4H(b, c)

∫ b

c e

r(s)ρe(s)Q22(b, s)ds,

which contradicts to (6). So the proof is complete.

Theorem 2.2. Under the conditions of Theorem 2.1, furthermore, suppose (6) does not hold. If for anyl≥ξ0,

lim

ξ→∞sup ∫ ξ

l [

H(s, l)Kρe(s)eq(s)−Aer(s)ρe(s)

4 Q

2 1(s, l)

] ds >0

(15)

and

lim

l [

H(ξ, s)Keρ(s)eq(s)−Aer(s)eρ(s)

4 Q

2 2(ξ, s)

] ds >0,

(16)

then Eq. (1) is oscillatory.

Proof: For anyT ≥ξ0, leta=T. In (15) we choosel=a.

Then there existsc > asuch that ∫ c

a [

H(s, a)Kρe(s)qe(s)−Aer(s)ρe(s)

4 Q

2 1(s, a)

] ds >0.

(17) In (16) we choosel=c. Then there existsb > csuch that

∫ b

c [

H(b, s)Kρe(s)qe(s)−Aer(s)ρe(s)

4 Q

2 2(b, s)

] ds >0.

(18) Combining (17) and (18) we obtain (6). The conclusion thus comes from Theorem 2.1, and the proof is complete.

In Theorems 2.1-2.2, if we chooseH(ξ, s) = (ξ−s)λ, ξ≥ s ≥ ξ0, where λ > 1 is a constant, then we obtain the

following two corollaries.

Corollary 2.1. Under the conditions of Theorem 2.1, if for any sufficiently largeT ≥ξ0, there exista, b, c with

T ≤a < c < bsatisfying 1

(c−a)λ

∫ c

a

(s−a)λKρe(s)qe(s)ds

+ 1

(b−c)λ

∫ b

c

(b−s)λKρ(s)eq(s)ds

> A

4(c−a)λ

∫ c

a e

r(s)eρ(s)(s−a)λ−2

( λ+ρe

′₍_s₎ e

ρ(s)(s−a)

)2

ds

+ A

4(b−c)λ

∫ b

c e

r(s)ρe(s)(b−s)λ−2

( λ−ρe

′₍_s₎

e

ρ(s)(b−s)

)2

ds,

(19) then Eq. (1) is oscillatory.

Corollary 2.2. Under the conditions of Theorem 2.2, if for anyl≥ξ0,

lim

l

[(s−l)λKρe(s)qe(s)

−Are(s)ρe(s)

4 (s−l)

λ₋2₍_λ₊ρe′(s)

e

ρ(s)(s−l))

2_]_{ds >}₀

(20)

and

lim

l

[(ξ−s)λKρe(s)qe(s)

−Aer(s)ρe(s)

4 (ξ−s)

λ₋2

(λ−ρe

′₍_s₎

e

ρ(s)(ξ−s))

2

]ds >0, (21)

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Theorem 2.3. Under the conditions of Theorem 2.1, furthermore, suppose (6) does not hold. If for any T ≥ξ0,

there exist a, b with b > a ≥ T such that for any

u ∈ C[a, b], u′(t) ∈ L2_[_{a, b}_]_, _u₍_a_{) =} _u₍_b_{) = 0}_{, the}

following inequality holds: ∫ b

a

[u2(s)Kqe(s)ρe(s)

−Aer(s)ρe(s)(u′(s) +1 2u(s)

e ρ′(s)

e ρ(s))

2_]_{ds >}₀_, ₍₂₂₎

Proof: Suppose to the contrary that x(t) be a non-oscillatory solution of Eq. (1). Without loss of generality, we may assume thatx(t)>0 on[T0,∞), whereT0≥t0 is

sufficiently large. Similar to the proof of Theorem 2.1, we obtain (10). Select a, b arbitrarily in [ξ_∗,∞) with b > a. Substituting ξ with s, multiplying both sides of (10) by

u2(s), integrating it with respect to sfromatob and using

u(a) =u(b) = 0, we get that ∫ b

a

u2(s)Kqe(s)ρe(s)ds≤ − ∫ b

a

u2(s)we′(s)ds

− ∫ b

a

u2(s) we

2₍_s₎

Aer(s)ρe(s)ds+

∫ b

a

u2(s)we(s)ρe ′₍_s₎

e ρ(s)ds

= 2

∫ b

a

u(s)u′(s)we(s)ds− ∫ b

a

u2(s) we

2₍_s₎

Aer(s)ρe(s)ds

+

∫ b

a

u2(s)we(s)ρe ′₍_s₎

e ρ(s)ds

=−

∫ b

a {[

√

1

Aer(s)ρe(s)u(s)we(s)

−√Aer(s)ρe(s)(u′(s) +1 2u(s)

e ρ′(s)

e ρ(s))]

2

+Aer(s)ρe(s)(u′(s) +1 2u(s)

e ρ′(s)

e ρ(s))

2_}_ds.

Moreover, ∫ b

a

[u2(s)Kqe(s)eρ(s)−Aer(s)ρe(s)(u′(s) +1 2u(s)

e ρ′(s)

e ρ(s))

2

]ds≤0,

(23)

which contradicts to the assumption (22). So every solution of Eq. (1) is oscillatory, and the proof is complete.

III. INTERVAL OSCILLATION CRITERIA WITHf(x)BEING MONOTONE

Theorem 3.1. Under the conditions of Lemma 2.1, further-more, assume f ∈C1_[_{R, R}_] _satisfying _f′₍_x₎_≥ _{µ >} ₀ _for

x ̸= 0, and for any sufficiently large T ≥ ξ0, there exist

a, b, cwithT ≤a < c < bsuch that

1

H(c, a)

∫ c

a

H(s, a)ρe(s)eq(s)ds+ 1

H(b, c)

∫ b

c

H(b, s)eρ(s)eq(s)ds

> A

4µH(c, a)

∫ c

a e

+ A

4µH(b, c)

∫ b

c e

r(s)ρe(s)Q2₂(b, s)ds, (24) where ρ,e q,e er, Q1, Q2 are defined as Theorem 2.1, then

Eq. (1) is oscillatory.

Proof. Suppose to the contrary that x(t) be a non-oscillatory solution of Eq. (1). Without loss of generality, we may assume thatx(t)>0 on[T0,∞), whereT0≥t0is

sufficiently large. Furthermore, by Lemma 2.1, there exists

t_∗≥T0such thatDtαx(t)<0, t≥t∗. So by the assumption

on Eq. (1) we havek1(x(t), Dαtx(t))≥ADtαx(t), t≥t∗.

Define

w(t) =ρ(t)r(t)k1(x(t), D

α tx(t))

f(x(t)) . (25) Then fort≥t_∗, we have

D_tαw(t) = D

α tρ(t)

k2(x(t), Dtαx(t))D α tx(t) f(x(t))

−ρ(t)q(t)−ρ(t)r(t)k1(x(t), D

α

tx(t))f′(x(t))D α tx(t) f2(x(t))

= D

α tρ(t)

k2(x(t), Dαtx(t))Dαtx(t) f(x(t))

−ρ(t)q(t)− w

2₍_t₎_f′₍_x₍_t₎₎_Dα tx(t) ρ(t)r(t)k1(x(t), Dαtx(t)) ≤ Dtαρ(t)

k2(x(t), Dαtx(t))D α tx(t) f(x(t))

−ρ(t)q(t)− µw

2₍_t₎

Aρ(t)r(t)

≤ Dαtρ(t)

ρ(t) w(t)−ρ(t)q(t)−

µw2(t)

Aρ(t)r(t). (26) We notice that (26) is similar to (9). So by similar process from (9) to the end in Theorem 2.1, we can deduce the desired result.

Theorem 3.2. Under the conditions of Theorem 3.1, furthermore, suppose (24) does not hold. If for anyl≥ξ0,

lim sup

ξ→∞ ∫ ξ

l [

H(s, l)ρe(s)eq(s)−Aer(s)ρe(s)

4µ Q

2 1(s, l)

] ds >0

(27) and

lim sup

ξ→∞ ∫ ξ

l [

H(ξ, s)ρe(s)qe(s)−Are(s)ρe(s)

4µ Q

2 2(ξ, s)

] ds >0,

(28) then Eq. (1) is oscillatory.

The proof of Theorem 3.2 is similar to Theorem 2.2, and we omit it here.

In Theorems 3.1-3.2, if we chooseH(ξ, s) = (ξ−s)λ, ξ≥ s ≥ ξ0, where λ > 1 is a constant, then we obtain the

following two corollaries, which are similar to Corollaries

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2.1-2.2.

Corollary 3.1. Under the conditions of Theorem 3.1, if for any sufficiently largeT ≥ξ0, there exist a, b, cwith

T ≤a < c < bsatisfying

1

(c−a)λ

∫ c

a

(s−a)λρe(s)eq(s)ds

+ 1

(b−c)λ

∫ b

c

(b−s)λρ(s)qe(s)ds

> A

4µ(c−a)λ

∫ c

a e

r(s)ρe(s)(s−a)λ−2

( λ+eρ

′₍_s₎ e

ρ(s)(s−a)

)2

ds

+ A

4µ(b−c)λ

∫ b

c e

r(s)ρe(s)(b−s)λ−2

( λ−ρe

′₍_s₎ e

ρ(s)(b−s)

)2

ds,

(29)

Corollary 3.2. Under the conditions of Theorem 3.2, if for any l≥ξ0,

lim

ξ_→∞sup ∫ ξ

l

−Aer(s)ρe(s)

4µ (s−l)

λ−2₍_λ₊ρe′(s)

e

ρ(s)(s−l))

2_]_{ds >}₀ ₍₃₀₎

and

lim

l

[(ξ−s)λKρe(s)eq(s)

−Are(s)ρe(s)

4µ (ξ−s)

λ−2₍_λ₋ρe′(s)

e

ρ(s)(ξ−s))

2_]_{ds >}₀_, ₍₃₁₎

Theorem 3.3. Under the conditions of Theorem 3.1, furthermore, suppose (24) does not hold. If for anyT ≥ξ0,

there exist a, b with b > a ≥ T such that for any

u ∈ C[a, b], u′(t) ∈ L2_[_{a, b}_]_, _u₍_a_{) =} _u₍_b_{) = 0}_{, the}

following inequality holds: ∫ b

a

[u2(s)qe(s)ρe(s)

−Are(s)ρe(s) µ (u

′₍_s_{) +}1 2u(s)

e ρ′(s)

e ρ(s))

2_]_{ds >}₀_, ₍₃₂₎

IV. APPLICATIONS

Example 1. Consider the nonlinear fractional differential equation with damping term

Dαt(sin

2

( t

α

Γ(1 +α))e −x2₍_t₎

Dαtx(t)) +t

2

x4(t)(Dαtx(t))

2

+

x(t)(1 +x2(t)) = 0, t≥2, 0< α <1. (33) In fact, if we set in Eq. (1) t0 = 2, r(t) =

sin2( tα

Γ(1 +α)), p(t) =t

2_{, q}₍_t₎_≡₁_{, k}

1(x(t), Dαtx(t)) =

e−x2₍_t₎

Dα

tx(t), k2(x(t), Dtαx(t)) = x4(t)Dtαx(t), f(x) = x+x3_{, then we obtain (33).}

So k2

1(x(t), Dαtx(t)) = e−2x

2₍_t₎ (Dα

tx(t))2 ≤ e−x2(t)₍_Dα

tx(t))2, which implies A = 1. Furthermore, f′(x) = 1 + 3x2 _≥ ₁_{, t} _≥₀_{, and then} _µ _{= 1}_{. Since} _ξ ₌

tα

Γ(1 +α), thener(ξ) =r(t) = sin

2₍ tα

Γ(1 +α)) = sin

2_ξ_.

In (32), lettingρe(s)≡1, a= 2kπ, b= 2kπ+π, u(s) = sins, then u(a) = u(b) = 0. Considering eq(s) ≡ 1, we obtain

∫ 2kπ+π

2kπ (

sin2s−sin2scos2s)ds=

∫ 2kπ+π

2kπ

sin4sds >0.

Therefore, Eq. (33) is oscillatory by Theorem 3.3.

Example 2. Consider the nonlinear fractional differential equation with damping term

Dαt (

( t

α

Γ(1 +α)) 2

3 1

1 +x2(t)D

α tx(t)

)

+ ln(5 +t2) cos2t

(Dtαx(t))2+

x(t)(2 +x2(t))

1 +x2(t) = 0, t≥5, 0< α <1. (34) In fact, if we set in Eq. (1) t0 = 5, r(t) =

( tα Γ(1 +α))

2

3, p(t) = ln(5 +t2), q(t)≡1,

k1(x(t), Dαtx(t)) = 1

1 +x2(t)D

α

tx(t), k2(x(t), Dαtx(t)) =

cos2(t)(Dα_tx(t))2, f(x) =x(2 +x

2₎

1 +x2 , then we obtain (34).

So k2

1(x(t), Dαtx(t)) = (_{1 +}_x12₍_t₎)2(Dαtx(t))2 ≤

1 1 +x2(t)(D

α

tx(t))2, which implies A = 1. Furthermore, e

r(ξ) =r(t) = ( tα Γ(1 +α))

2 3 =ξ

2 3.

We notice that it is complicated in obtaining the lower-bound of f′(x), while one can easily see f(x)/x ≥1. So

K = 1, and in (20)-(21), after letting ρe(s) ≡ 1, λ = 2, consideringeq(s)≡1, we obtain

lim

l

−Aer(s)ρe(s)

4 (s−l)

λ−2₍_λ₊ρe′(s)

e

ρ(s)(s−l))

2_]_ds

= lim

l [

(s−l)2−s23

]

ds=∞

and

lim

l

[(ξ−s)λKρe(s)qe(s)

−Aer(s)ρe(s)

4 (ξ−s)

λ−2₍_λ₋ρe′(s)

e

ρ(s)(ξ−s))

2_]_ds

= lim

ξ_→∞sup ∫ ξ

l [

(ξ−s)2−s23

]

ds=∞.

So according to Corollary 2.2 we deduce that Eq. (34) is oscillatory.

Remark. We note that the oscillatory character of the two examples above are not deducible from previously known oscillation criteria in the literature.

IAENG International Journal of Applied Mathematics, 43:3, IJAM_43_3_09

(6)

V. CONCLUSIONS

We have established some new interval oscillation criteria for a class of nonlinear fractional differential equations with nonlinear damping term. As one can see, the variable transformation used inξis very important, which ensures that certain fractional differential equations can be turned into an-other ordinary differential equations of integer order, whose oscillation criteria can be established by generalized Riccati transformation, inequality and integration average technique. Finally, we note that this approach can also be applied to research oscillation for other fractional differential equations involving the modified Riemann-liouville derivative.

ACKNOWLEDGMENT

The author would like to thank the anonymous reviewers very much for their valuable suggestions on improving this paper.

REFERENCES

[1] A. M. A. Abou-El-Ela, A. I. Sadek and A. M. Mahmoud, “ Existence and Uniqueness of a Periodic Solution for Third-order Delay Differ-ential Equation with Two Deviating Arguments,”IAENG International Journal of Applied Mathematics, vol. 42, no.1, pp. 1-6, Feb. 2012. [2] A. Moumeni and L. S. Derradji, “ Global Existence of Solution for

Reaction Diffusion Systems,”IAENG International Journal of Applied Mathematics, vol. 42, no.2, pp. 1-7, May. 2010.

[3] M. Danish, Shashi. Kumar and S. Kumar, “ Exact Solutions of Three Nonlinear Heat Transfer Problems,”Engineering Letters, vol. 19, no. 3, pp. 1-6, Aug. 2011.

[4] T. Hasuike, “ Exact and Explicit Solution Algorithm for Linear Pro-gramming Problem with a Second-Order Cone,”IAENG International Journal of Applied Mathematics, vol. 41, no. 3, pp. 1-5, Aug. 2011. [5] Y. Warnapala, R. Siegel and J. Pleskunas, “ The Numerical Solution

of the Exterior Boundary Value Problems for the Helmholtz’s Equa-tion for the Pseudosphere,”IAENG International Journal of Applied Mathematics, vol. 41, no. 2, pp. 1-6, May. 2011.

[6] J. V. Lambers, “ Solution of Time-Dependent PDE Through Component-wise Approximation of Matrix Functions,”IAENG Inter-national Journal of Applied Mathematics, vol. 41, no. 1, pp. 1-10, Feb. 2011.

[7] A. V. Kamyad, M. Mehrabinezhad and J. Saberi-Nadjafi, “ A Numerical Approach for Solving Linear and Nonlinear Volterra Integral Equations with Controlled Error,”IAENG International Journal of Applied Math-ematics, vol. 40, no. 2, pp. 1-6, May. 2010.

[8] A. Bouhassoun, “ Multistage Telescoping Decomposition Method for Solving Fractional Differential Equations,”IAENG International Jour-nal of Applied Mathematics, vol. 43, no. 1, pp. 1-7, Feb. 2013. [9] A. M. Bijura, “ Systems of Singularly Perturbed Fractional Integral

Equations II,”IAENG International Journal of Applied Mathematics, vol. 42, no. 4, pp. 1-6, Nov. 2012.

[10] J. M. Blackledge, “ Application of the Fractional Diffusion Equation for Predicting Market Behaviour,” IAENG International Journal of Applied Mathematics, vol. 40, no. 3, pp. 1-29, Aug. 2010.

[11] J. C. Trigeassou, N. Maamri, J. Sabatier and A. Oustaloup, ”A Lyapunov approach to the stability of fractional differential equations,” Signal Process., vol. 91, pp. 437-445, 2011.

[12] W. Deng, ”Smoothness and stability of the solutions for nonlinear fractional differential equations,”Nonlinear Anal., vol. 72, pp. 1768-1777, 2010.

[13] Y. Zhou, F. Jiao and J. Li, ”Existence and uniqueness for p-type fractional neutral differential equations,”Nonlinear Anal., vol. 71, pp. 2724-2733, 2009.

[14] L. Galeone and R. Garrappa, ”Explicit methods for fractional differ-ential equations and their stability properties,”J. Comput. Appl. Math., vol. 228, pp. 548-560, 2009.

[15] A. Saadatmandi and M. Dehghan, ”A new operational matrix for solving fractional-order differential equations,”Comput. Math. Appl., vol. 59, pp. 1326-1336, 2010.

[16] F. Ghoreishi and S. Yazdani, ”An extension of the spectral Tau method for numerical solution of multi-order fractional differential equations with convergence analysis,”Comput. Math. Appl., vol. 61, pp. 30-43, 2011.

[17] J. T. Edwards, N. J. Ford and A. C. Simpson, ”The numerical solution of linear multi-term fractional differential equations: systems of equations,”J. Comput. Appl. Math., vol. 148, pp. 401-418, 2002. [18] M. Muslim, ”Existence and approximation of solutions to fractional

differential equations,”Math. Comput. Model., vol. 49, pp. 1164-1172, 2009.

[19] D. X. Chen, ”Oscillation criteria of fractional differential equations,” Adv. Differ. Equ., vol. 2012, Art. 33, pp. 1-18, 2012.

[20] D. X. Chen, ”OSCILLATORY BEHAVIOR OF A CLASS OF FRAC-TIONAL DIFFERENTIAL EQUATIONS WITH DAMPING,”U.P.B. Sci. Bull. Series A, vol. 75, no. 1, pp. 107-118, 2013.

[21] B. Zheng, ”OSCILLATION FOR A CLASS OF NONLINEAR FRAC-TIONAL DIFFERENTIAL EQUATIONS WITH DAMPING TERM,” J. Adv. Math. Stu., vol. 6, no. 1, pp. 107-115, 2013.

[22] G. Jumarie, ”Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Comput. Math. Appl., vol. 51, pp. 1367-1376, 2006.

[23] S. Zhang and H. Q. Zhang, ”Fractional sub-equation method and its applications to nonlinear fractional PDEs,”Phys. Lett. A, vol. 375, pp. 1069-1073, 2011.

[24] S. M. Guo, L. Q. Mei, Y. Li and Y. F. Sun, ”The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics,”Phys. Lett. A, vol. 376, pp. 407-411, 2012.

[25] B. Zheng, ”(G’/G)-Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics,” Com-mun. Theor. Phys. (Beijing, China), vol. 58, pp. 623-630, 2012. [26] S. P. Rogovchenko and Y. V. Rogovchenko, ” Oscillation theorems for

differential equations with a nonlinear damping term,” J. Math. Ann. Appl., vol. 279, pp. 121-134, 2003.