Oscillation criteria for nonlinear fractional differential equation with damping term

Mustafa Bayram; Hakan Adiguzel; Aydin Secer

doi:10.1515/phys-2016-0012

Article Open Access

Oscillation criteria for nonlinear fractional differential equation with damping term

Mustafa Bayram , Hakan Adiguzel and Aydin Secer

Published/Copyright: April 28, 2016

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Open Physics Volume 14 Issue 1

Abstract

In this paper, we study the oscillation of solutions to a non-linear fractional differential equation with damping term. The fractional derivative is defined in the sense of the modified Riemann-Liouville derivative. By using a variable transformation, a generalized Riccati transformation, inequalities, and integration average techniquewe establish new oscillation criteria for the fractional differential equation. Several illustrative examples are also given.

Keywords: oscillation; oscillation criteria; fractional derivative; modified Riemann-Lioville derivative; damping term

PACS: 02.30.Hq; 02.30.Sa; 02.60.Lj; 02.90.+p

1 Introduction

Fractional differential equations are generalizations of classical differential equations of integer order and have recently proved to be valuable tools in the modelling of many phenomena in various fields of science and engineering. Apart from diverse areas of mathematics, fractional differential equations arise in rheology, viscoelasticity, chemical physics, electrical networks, fluid flows, control, dynamical processes in self-similar and porous structures, etc.; see, for example, [1–6]. Fractional derivatives have appeared in lots of work where they are used for better descriptions of material properties. Mathematical modelling based on enhanced rheological models naturally leads to differential equations of fractional order and to the necessity of the formulation of initial conditions to such equations.This growth in use has been caused by the intensive development of the theory of fractional calculus itself and its applications. The books on the subject of fractional integrals and fractional derivatives by Diethelm [7], Miller and Ross [8], Podlubny [9] and Kilbaset al. [10] summarise and organise much of the field of fractional calculus including many of the theories and applications of fractional differential equations. Many papers have studied some aspects of fractional differential equations. Most have focused on the existence of, methods for defining or stability of the solutions (or positive solutions) to nonlinear initial (or boundary) value problems for fractional differential equations (or systems) using nonlinear analysis techniques (fixed-point theorems, Leray-Schauder theory). We refer to [11–21] and the references cited therein.

Recently, research on the oscillation of various equations including differential equations, difference equations and dynamic equations on time scales, has been a hot topic the literature. A lot of effort has been committed to establishing new oscillation criteria for these equations; see the monographs [22, 23]. In these investigations, we notice that very little attention has been paid to the oscillation of fractional differential equations.

In 2006, a definition for a fractional derivative called the modified Riemann-Liouville derivative, was suggested by Jumarie [24] and its application have subsequently been studied by many researchers [25–28].

Recently, Qin and Zheng [29] established oscillation criteria for linear fractional differential equations with damping term of the form:

Dtα(a(t)Dtα(r(t)Dtαx(t)))+p(t)Dtα(r(t)Dtαx(t))+q(t)x(t)=0 t≥t0>0,0<α<1,

where Dtα(⋅) denotes the modified Riemann-Liouville derivative with respect to variable t.

Now, in this paper, we are concerned with the oscillation of fractional differential equations with damping term in the form of:

(1)Dtα(a(t)Dtα(r(t)Dtαx(t)))+p(t)Dtα(r(t)Dtαx(t))+q(t)f(x(t))=0, t≥t0>0, 0<α<1,

where: Dtα(⋅) denotes the modified Riemann-Liouville derivative with respect to the variable t; the function a ∈ C^a([t₀, ∞) , R₊); r ∈ C^2a ([t₀, ∞) ,R₊); p, q ∈ C([t₀, ∞), R₊); the function of f belongs to C(R, R); f (x) /x ≥ k > 0 for all x ≠ 0 and C^a denotes a continuous derivative of order α.

Some of the key properties of Jumarie’s modified Riemann-Liouville derivative of order are listed as follows:

(2)Dtαf(t)={1Γ(1−α)ddt∫0tf(t,ξ,α)dξ, 0<α<1(f(n)(t))(α−n),1≤n≤α≤n+1

(3)Dtα(f(t)g(t))=g(t)Dtαf(t)+f(t)Dtαg(t)

(4)Dtαf[g(t)]=fg′[g(t)]Dtαg(t)=Dtαf[g(t)](g′(t))α

(5)Dtαtβ=Γ(β+1)Γ(β+1−α)tβ−α,

where f (t, ξ, α) = (t − ξ)^−α (f (ξ) − f(0)).

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

In the rest of this paper, we denote for the sake of convenience:

ξ = t^α /Γ (1 + α); ξi=tiα/Γ(1+α);i=0,1,2,3,4,5a (t) = ã̃ (ξ), r(t) = r̃ (ξ), x (t) = x̃ (ξ), p (t) = p̃ (ξ), q (t) = q̃ (ξ); δ˜1(ξ,ξi)=∫ξiξ(1/A(s)a˜(s))ds;δ1(t,ti)=δ˜1(ξ,ξi); δ˜2(ξ,ξi)=∫ξiξ(δ˜1(s,ξi)/r˜(s))ds;δ2(t,ti)=δ˜2(ξ,ξi); A(ξ)=exp(∫ξ0ξ(p˜(s)/a˜(s)ds)).

Let h₁, h₂, H ∈ C ([ξ₀, ∞), R) satisfy

(6)H(ξ,ξ)=0, H(ξ,s)>0, ξ>s≥ξ0

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

(7)∂H(ξ,s)∂ξ=−h1(ξ,s)H(ξ,s)

(8)∂H(ξ,s)∂s=−h2(ξ,s)H(ξ,s), ξ>s≥ξ0.

This paper is organized next as follows: in Section 2, we establish new oscillation criteria for (1) using the Riccati transformation, inequalities and the integration average technique and in Section 3, we present some examples that apply the results established. Finally, we give a conclusion.

2 Oscillatory criteria

Lemma 1

Assume x (t) is an eventually positive solution of (1), and

(9)∫ξ0∞1Asa~sds=∞

(10)∫t0∞αtα−1Γ1+αrtdt=∞

(11)∫ξ0∞1r˜(ζ)∫ζ∞1A(τ)a˜(τ)∫τ∞A(s)q˜(s)dsdτdζ=∞.

Then, there exist a sufficiently large T such thatDtα(r(t)Dtαx(t))>0on [T, ∞) and either Dtαx(t)>0on [T, ∞) or lim_{_t→∞}x (t) = 0.

Proof

Suppose x (t) is an eventually positive solution of (1). Let a (t) = ã̃ (ξ), r(t) = r̃ (ξ), x (t) = x̃ (ξ), p (t) = p̃ (ξ), q (t) = q̃ (ξ) where ξ = t^α /Γ (1 + α). Then by using (5), we obtain Dtαξ(t)=1, and furthermore, by use of the first equality in (4), we have

(12)Dtαa(t)=Dtαa˜(ξ)=a˜′(ξ)Dtαξ(t)=a˜′(ξ).

Similarly we have Dtαr(t)=r˜′(ξ),Dtαx(t)=x˜′(ξ), Dtαp(t)=p˜′(ξ),Dtαq(t)=q˜′(ξ). So, (1) can be transformed into the following form:

(13)[a˜(ξ)(r˜(ξ)x˜′(ξ))′]′+p˜(ξ)(r˜(ξ)x˜′(ξ))′+q˜(ξ)f(x˜(ξ))=0, ξ≥ξ0>0.

Then x̃ (ξ) is an eventually positive solution of (13), and there exists ξ₁ >ξ₀ such that x̃ (ξ) > 0 on [ξ₁, ∞). So, f (x̃ (ξ)) > 0 and we have

[A(ξ)a˜(ξ)(r˜(ξ)x˜′(ξ))′]′=A(ξ)[a˜(ξ)(r˜(ξ)x˜′(ξ))′]′+A′(ξ)a˜(ξ)(r˜(ξ)x˜′(ξ))′=A(ξ){[a˜(ξ)(r˜(ξ)x˜′(ξ))′]′+p˜(ξ)(r˜(ξ)x˜′(ξ))′}.

Therefore, we get

(14)Aξa~ξ r~ξx~′ξ′′=−Aξq~ξfx~ξ<0,ξ≥ξ1.

Then, A(ξ) ã(ξ)(r̃(ξ)x̃′(ξ))′ is strictly decreasing on [ξ₁, ∞) , thus we know that (r̃(ξ)x̃′(ξ))′ is eventually of one sign. For ξ₂ > ξ₁ is sufficiently large, we claim (r̃(ξ)x̃′(ξ))′ > 0 on [ξ₂, ∞). Otherwise, assume that there exists a sufficiently large ξ₃ > ξ₂ such that (r̃(ξ)x̃′(ξ))′ < 0 on [ξ₃, ∞). Thus, (r̃(ξ)x̃′(ξ))′ is strictly decreasing on [ξ₃, ∞), and we get that

r~ξx~′ξ−r~ξ3x~′ξ3=∫ξ3ξAsa~sr~sx~′s′Asa~sds ≤Aξ3a~ξ3r~ξ3x~′ξ3′∫ξ3ξ1Asa~sds.

Therefore, we get

(15)r~ξx~′ξ≤r~ξ3x~′ξ3+Aξ3a~ξ3r~ξ3x~′ξ3′∫ξ3ξ1Asa~sds.

By (9), we have lim_ξ→∞r̃(ξ) x̃′(ξ) = −∞. So there exists a sufficiently large ξ₄ > ξ₃ such that x̃′(ξ) < 0, ξ ∈ [ξ₄, ∞). Then, we have

x~ξ−x~ξ4=∫ξ4ξx~′sds=∫ξ4ξr~sr~sx~′sds≤r~ξ4x~′ξ4∫ξ4ξ1r~sds

and so

(16)x~ξ≤r~ξ4x~′ξ4∫ξ4ξαtα−1Γ(1+α)rtdt.

By (10), we deduce that lim_ξ→∞x̃(ξ) = −∞, which contradicts the fact that x̃ (ξ) is an eventually positive solution of (13). Thus, r̃(ξ)x̃(ξ)) > 0 on [ξ₂, ∞) , and then Dtα(r(t)Dtαx(t))>0 on [t₂, ∞]. So Dtαx(t)=x˜′(ξ) is eventually of one sign. Now we assume x̃′ (ξ) < 0 on [ξ₅, ∞) where ξ₅ > ξ₄ is sufficiently large. Since x̃(ξ) > 0, we have lim_ξ→∞x̃(ξ) = β ≥ 0. We claim β = 0. Otherwise, assume β > 0. Then x̃ (ξ) ≥ β on [ξ₅, ∞), f (x (ξ))≥ k.x(ξ) > M for M ∈ ℝ₊ and by (14) we have

(17)[A(ξ)a˜(ξ)(r˜(ξ)x˜′(ξ))′]′=−A(ξ)q˜(ξ)f(x˜(ξ))≤−A(ξ)q˜(ξ)M.

substitutingξ with in (17), and integrating it with respect to s from ξ to ∞ yields

(18)∫ξ∞Asa~sr~sx~′s′′ds≤−M∫ξ∞Asq~sds. −Aξa~ξr~ξx~′ξ′ ≤−limξ→∞Aξa~ξr~ξx~′ξ′−M∫ξ∞Asq~sds <−M∫ξ∞Asq~sds

which means

(19)r~ξx~′ξ′>M1Aξa~ξ∫ξ∞Asq~sds

substituting ξ with τ in (19), and integrating it with respect to τ from ξ to ∞ yields

(20)∫ξ∞r~sx~′s′ds>M∫ξ∞1ϰ∫ξ∞Asq~sdsdτlimξ→∞r~ξx~′ξ−r~ξx~′ξ>M∫ξ∞1ϰ∫ξ∞Asq~sdsdτ−r~ξx~′ξ>−limξ→∞r~ξx~′ξ+M∫ξ∞1ϰ∫ξ∞Asq~sdsdτ−r~ξx~′ξ>M∫ξ∞1Aτa~τ∫ξ∞Asq~s,dsdτ

where ϰ = A(τ) ã(τ). That is,

(21)x~′ξ<−M1r~ξ∫ξ∞1ϰ∫ξ∞Asq~sdsdτ

substituting ξ with ζ in (21), and integrating it with respect to ζ from ξ₅ to ξ yields

∫ξ5ξx~′sds<−M∫ξ5ξ1r~ζ∫ξ∞1ϰ∫ξ∞Asq~sdsdτdζx~ξ−x~ξ5<−M∫ξ5ξ1r~ζ∫ξ∞1ϰ∫ξ∞Asq~sdsdτdζx~ξ<x~ξ5−M∫ξ5ξ1r~ζ∫ξ∞1ϰ∫ξ∞Asq~sdsdτdζ.

By (11), we have lim_t→∞x̃ (ξ) = −∞, which causes a contradiction. So, the proof is complete. □

Lemma 2

Assume that x(t)is an eventually positive solution of (1) such that

(22)Dtα(r(t)Dtαx(t))>0, Dtαx(t)>0

on [t₁, ∞), where t₁ > t₀is sufficiently large. Then, for t ≥ t₁, we have

(23)Dtαx(t)≥A(ξ)δ1(t,t1)a(t)Dtα(r(t)Dtαx(t))r(t)

(24)x(t)≥A(ξ)δ2(t,t1)a(t)Dtα(r(t)Dtαx(t)).

Proof

Assume that x is an eventually positive solution of (1). So, by (14), we obtain that A(ξ) ã (ξ) r̃ (ξ) x̃′ (ξ)) is strictly decreasing on [ξ₁, ∞). Then,

(25)r~ξx~′ξ≥r~ξx~′ξ−r~ξ1x~′ξ1=∫ξ1ξAsa~sr~sx~′s′Asa~sds≥Aξa~ξr~ξx~′ξ′∫ξ1ξ1Asa~sds=Aξa~ξ(r~ξx~′ξ)′δ~1ξ,ξ1

and so

(26)r(t)Dtαx(t)≥A(ξ)δ1(t,t1)a(t)Dtα(r(t)Dtαx(t))

multiplying both sides of (26) by 1/r(t), we obtain

Dtαx(t)≥A(ξ)δ1(t,t1)a(t)Dtα(r(t)Dtαx(t))r(t).

On the other hand, we have

x~ξ≥x~ξ−x~ξ1=∫ξ1ξx~′sds=∫ξ1ξr~sx′sr~sds.

Using (26), we obtain

(27)x~ξ≥∫ξ1ξAsa~sr~sx~′s′δ~1s,ξ1r~sds,x~ξ≥Aξa~ξr~ξx~′ξ′∫ξ1ξδ~1s,ξ1r~sds=Aξδ~2ξ,ξ1a~ξr~ξx~′ξ′.

That is

x(t)≥A(ξ)δ2(t,t1)a(t)Dtα(r(t)Dtαx(t)).

So, the proof is complete. □

Lemma 3

[30]: Assume that A and B are nonnegative real numbers. Then,

(28)λABλ−1−Aλ≤(λ−1)Bλ

Theorem 4

Assume that (9)-(11) hold and f(x)/x ≥ k > 0 for all x ≠ = 0. If there exists ϕ ∈ C^α ([t₀, ∞) , R₊) such that for any sufficiently large T ≥ ξ₀, there exist a, b, c with T ≤ a < c < b satisfying

(29)1H(b,c)∫cbH(b,s)kA(s)ϕ˜(s)q˜(s)ds+1H(c,a)∫acH(s,a)kA(s)ϕ˜(s)q˜(s)ds>1H(b,c)∫cbr˜(s)ϕ˜(s)4δ1(s,ξ2)Q22(b,s)ds+1H(c,a)∫acr˜(s)ϕ˜(s)4δ1(s,ξ2)Q12(s,a)ds,

where k ∈ ℝ₊, ϕ̃(ξ)Q1(s,ξ)=h1(s,ξ)−(ϕ˜′(s)/ϕ˜(s))H(s,ξ), Q2(ξ,s)=h2(ξ,s)−(ϕ˜′(s)/ϕ˜(s))H(ξ,s); then, (1) is oscillatory or satisfies lim_t→∞x(t) = 0.

Proof.

Suppose the contrary that x(t) is non-oscillatory solution of (1). Then without loss of generality, we may assume that there is a solution x(t) of (1) such that x(t) > 0 on [t₁, ∞), where t₁ is sufficiently large. By Lemma 1, we have Dtα(r(t)Dtαx(t))>0, wheret₂ > t₁ is sufficiently large, and either Dtαx(t)>0 on [t₂, ∞) or lim_t→∞x(t) = 0. If we take Dtαx(t)>0 on [t₂, ∞). Define the following generalized Riccati function:

(30)ω(t)=ϕ(t)A(ξ)a(t)Dtα(r(t)Dtαx(t))x(t).

For t ∈ [t₂, ∞) , we have

Dtαω(t)=Dtαϕ(t)A(ξ)a(t)Dtα(r(t)Dtαx(t))x(t)+ϕ(t)Dtα{A(ξ)a(t)Dtα(r(t)Dtαx(t))x(t)}.

So,

Dtαω(t)=Dtαϕ(t)ω(t)ϕ(t)+ϕ(t)x(t)Dtα(A(ξ)a(t)Dtα(r(t)Dtαx(t)))x2(t)−ϕ(t)Dtαx(t)A(ξ)a(t)Dtα(r(t)Dtαx(t))x2(t)=Dtαϕ(t)ω(t)ϕ(t)+ϕ(t)[DtαA(ξ)a(t)Dtα(r(t)Dtαx(t))]x(t)+ϕ(t)[A(ξ)Dtα(a(t)Dtα(r(t)Dtαx(t)))]x(t)−ϕ(t)Dtαx(t)A(ξ)a(t)Dtα(r(t)Dtαx(t))x2(t)

(31)Dtαω(t)=Dtαϕ(t)ω(t)ϕ(t)+ϕ(t)[A′(ξ)Dtαξa(t)Dtα(r(t)Dtαx(t))]x(t)+ϕ(t)[A(ξ)Dtα(a(t)Dtα(r(t)Dtαx(t)))]x(t)−ϕ(t)Dtαx(t)A(ξ)a(t)Dtα(r(t)Dtαx(t))x2(t).

If we use Dtαξ=1 and (23), we obtain

Dtαω(t)≤Dtαϕ(t)ω(t)ϕ(t)+ϕ(t)[A(ξ)p(t)a(t)a(t)Dtα(r(t)Dtαx(t))]x(t)+ϕ(t)[A(ξ)Dtα(a(t)Dtα(r(t)Dtαx(t)))]x(t)−ϕ(t)δ1(t,t2)ϕ2(t)r(t)ω2(t)

and so,

Dtαω(t)≤Dtαϕ(t)ω(t)ϕ(t)+ϕ(t)A(ξ)[p(t)Dtα(r(t)Dtαx(t))]x(t)+ϕ(t)A(ξ)[Dtα(a(t)Dtα(r(t)Dtαx(t)))]x(t)−ϕ(t)δ1(t,t2)ϕ2(t)r(t)ω2(t) =Dtαϕ(t)ω(t)ϕ(t)−A(ξ)q(t)f(x(t))ϕ(t)x(t)−δ1(t,t2)ϕ(t)r(t)ω2(t).

Using f (x(t)) /x(t) ≥ k,

(32)Dtαω(t)≤Dtαϕ(t)ω(t)ϕ(t)−kA(ξ)q(t)ϕ(t)−δ1(t,t2)ϕ(t)r(t)ω2(t).

Now, let ω (t) = ω̃ (ξ). Then we have Dtαω(t)=ω˜′(ξ) and Dtαϕ(t)=ϕ˜′(ξ).. Thus (32) is transformed into

(33)ω˜′(ξ)≤ϕ′(ξ)ϕ(ξ)ω˜(ξ)−kA(ξ)q˜(ξ)ϕ˜(ξ)−δ˜1(ξ,ξ2)ϕ˜(ξ)r˜(ξ)ω˜2(ξ),ξ≥ξ2.

We can choose a, b, c arbitrary in [ξ₂, ∞) with b >c > a. Substituting ξ with s, we multiply both sides of (33) by H (ξ, s ) and integrating it with respect to s from c to ξ for ξ ∈ [c, b), then we get that

∫cξH(ξ,s)kA(s)q˜(s)ϕ˜(s)ds≤−∫cξH(ξ,s)ω˜′(s)ds+∫cξH(ξ,s)ϕ˜′(s)ϕ˜(s)ω˜(s)ds−∫cξH(ξ,s)δ˜1(s,ξ2)ϕ˜(s)r˜(s)ω˜2(s)

using the method of integration by parts

∫cξH(ξ,s)kA(s)q˜(s)ϕ˜(s)ds≤H(ξ,c)ω˜(c)−∫cξ[(H(ξ,s)ϑ)1/2ω˜(s)+12(ϑ)1/2Q2(ξ,s)]2ds+∫cξϑ4Q22(ξ,s)ds,

where ϑ=r˜(s)ϕ˜(s)δ˜1(s,ξ2) and therefore,

(34)∫cξH(ξ,s)kA(s)q˜(s)ϕ˜(s)ds≤H(ξ,c)ω˜(c)+∫cξϑ4Q22(ξ,s)ds.

Letting ξ → b⁻ in (34) and dividing both sides by H (ξ, c), we obtain,

(35)1H(b,c)∫cbH(b,s)kA(s)q˜(s)ϕ˜(s)ds≤ω˜(c)+1H(b,c)∫cbϑ4Q22(b,s)ds.

On the other hand, substituting ξ with s, multiplying both sides of (33) by H (s, ξ) and integrating it with respect to s from ξ to c for ξ ∈ (a, c ], we deduce that

(36)∫ξcH(s,ξ)kA(s)q˜(s)ϕ˜(s)ds≤−H(c,ξ)ω˜(c)+∫ξcr˜(s)ϕ˜(s)4δ˜1(s,ξ2)Q12(s,ξ)ds.

Letting ξ → a⁺ in (36) and dividing both sides of it by H(c, ξ) and we obtain

(37)1H(c,a)∫acH(s,a)kA(s)q˜(s)ϕ˜(s)ds≤−ω˜(c)+1H(c,a)∫acr˜(s)ϕ˜(s)4δ˜1(s,ξ2)Q12(s,a)ds.

A combination of (35) and (37) yields the inequality

(38)1H(b,c)∫cbH(b,s)kA(s)q˜(s)ϕ˜(s)ds+1H(c,a)∫acH(s,a)kA(s)q˜(s)ϕ˜(s)ds≤1H(b,c)∫cbr˜(s)ϕ˜(s)4δ˜1(s,ξ2)Q22(b,s)ds+1H(c,a)∫acr˜(s)ϕ˜(s)4δ˜1(s,ξ2)Q12(s,a)

which contradicts (29). Thus, the proof is complete. □

Theorem 5.

Under the conditions of Theorem 4, if for any sufficiently large l ξ₀,

(39)limξ→∞sup∫lξ[H(s,l)kA(s)q˜(s)ϕ˜(s)−ϑ4Q12(s,l)]ds>0,

(40)limξ→∞sup∫lξ[H(ξ,s)kA(s)q˜(s)ϕ˜(s)−ϑ4Q22(ξ,s)]ds>0,

then (1) is oscillatory.

Proof.

For any sufficiently large T ≥ ξ₀, let a = T. If we choose l = a in (39), then there exists c > a such that

(41)∫ac[H(s,a)kA(s)q˜(s)ϕ˜(s)−ϑ4Q12(s,a)]ds>0.

If we choose l = c > a in (40), then there exists b > c such that

(42)∫cb[H(b,s)kA(s)q˜(s)ϕ˜(s)−ϑ4Q22(b,s)]ds>0.

Finally, we combine (41) and (42), to obtain (29). Thus, the proof is complete from Theorem 4. □

If we choose H(ξ, s) = (ξ − s), ξ s ξ₀, where λ > 1 is a constant in Theorem 4 and Theorem 5, then we obtain the following corollaries.

Corollary 1

Under the conditions of Theorem 4, if for any sufficiently large T ≥ ξ₀, there exist a, b, c with T ≤ a < c < b satisfying

(43)1(c−a)λ∫ac(s−a)λkA(s)q˜(s)ϕ˜(s)ds+1(b−c)λ∫cb(b−s)λkA(s)q˜(s)ϕ˜(s)ds>1(c−a)λ∫acϑ4(s−a)λ−2(λ+ϕ˜′(s)ϕ˜(s)(s−a))2ds+1(b−c)λ∫cbϑ4(b−s)λ−2(λ−ϕ˜′(s)ϕ˜(s)(b−s))2ds

then (1) is oscillator

Corollary 2

Under the conditions of Theorem 5, if for any sufficiently large l ≥ ξ₀,

(44)limξ→∞sup∫lξ[(s−l)λkA(s)q˜(s)ϕ˜(s)−ϑ4(s−l)λ−2(λ+ϕ˜′(s)ϕ˜(s)(s−l))2]ds>0,limξ→∞sup∫lξ[(ξ−s)λkA(s)q˜(s)ϕ˜(s)−ϑ4(ξ−s)λ−2(λ−ϕ˜′(s)ϕ˜(s)(ξ−s))2]ds>0

then (1) is oscillatory.

Theorem 6.

If (9)-(11) hold, ffi is defined as in Teorem 4 and

(45)∫ξ0∞[kA(s)q˜(s)ϕ˜(s)−r˜(s)[ϕ˜′(s)]24δ˜1(s,ξ2)ϕ˜(s)]ds=∞.

Then every solution of (1) is oscillatory or satisfies lim_t→∞x(t) = 0.

Proof

Suppose the contrary that x(t) is a non-oscillatory solution of (1). Then without loss of generality, we may assume that there is a solution x(t) of (1) such that x(t) > 0 on [t₁, 1) , where t₁ is suffciently large. By Lemma 1, we have Dtα(r(t)Dtαx(t))>0,, t ∈ [t₂, ∞) , where t₂ > t₁ is suffciently large, and either Dtαx(t)>0 on [t₂, ∞) or lim_t→∞x(t) = 0. Now we assume that Dtαx(t)>0 on [t₂, ∞).Let ω(t), ω̃(ξ) be defined as in Theorem 1. Thus, we obtain (33). So,

w˜′(ξ)≤−kA(ξ)q˜(ξ)ϕ˜(ξ)−δ˜1(ξ,ξ2)r˜(ξ)ϕ˜(ξ)w˜2(ξ)+ϕ˜′(ξ)ϕ˜(ξ)w˜(ξ)=−kA(ξ)q˜(ξ)ϕ˜(ξ)+14r˜(ξ)[ϕ˜′(ξ)]2δ˜1(ξ,ξ2)ϕ˜(ξ)−[(δ˜1(ξ,ξ2)r˜(ξ)ϕ˜(ξ))1/2w˜(ξ)−12(r˜(ξ)ϕ˜(ξ)δ˜1(ξ,ξ2))1/2ϕ˜′(ξ)ϕ˜(ξ)]2≤−kA(ξ)q˜(ξ)ϕ˜(ξ)+14r˜(ξ)[ϕ˜′(ξ)]2δ˜1(ξ,ξ2)ϕ˜(ξ), ξ≥ξ2

and thus,

(46)kA(ξ)q˜(ξ)ϕ˜(ξ)−14r˜(ξ)[ϕ˜′(ξ)]2δ˜1(ξ,ξ2)ϕ˜(ξ)≤−w˜′(ξ).

Substituting ξ with s in (46) and integrating it with respect to s from ξ₂ to ξ, then we get that

(47)∫ξ2ξ[kA(s)q˜(s)ϕ˜(s)−14r˜(s)[ϕ˜′(s)]2δ˜1(s,ξ2)ϕ˜(s)]ds≤w˜(ξ2)−w˜(ξ)≤w˜(ξ2)<∞

which contradicts (45). So, the proof is complete. ⎕

Theorem 7

Assume (9)-(11) hold, and there exists a function G ∈ C([ξ₀, ∞),ℝ) such that G(ξ,ξ) = 0, for ξ ≥ ξ₀, G(ξ, s) ≥ 0, for ξ > sξ₀, and G has a non-positive continuous partial derivative G_s (ξ, s).Ifϕ̃ is defined as in Theorem 4 and

(48)limξ→∞sup1G(ξ,ξ0){∫ξ0ξG(ξ,s){ϖ−14ϱ}ds}=∞,

where ϱ=r˜(s)[ϕ˜′(s)]2δ˜1(s,ξ2)ϕ˜(s) and ω̄ = kA (s) q̃ (s) ϕ̃ (s) Then every solution of (1) is oscillatory or satisfies lim_t→∞x (t) = 0.

Proof

Suppose the contrary that x(t) is a non-oscillatory solution of (1). Then without loss of generality, we may assume that there is a solution x(t) of (1) such that x(t) > 0 on [t₁, ∞), where t₁ is suffciently large. By Lemma 1, we have Dtα(r(t)Dtαx(t))>0, t ∈ [t₂, ∞), where t₂ > t₁ is suffciently large, and either Dtαx(t)>0 on [t₂, ∞) or lim_t→∞x(t) = 0. Now we assume that Dtαx(t)>0 on [t₂, ∞). Let ω(t), ω̃(ξ) be defined as in Theorem 4. Thus we have (46). So,

(49)ϖ−ϱ4≤−w˜′(ξ), ξ≥ξ2.

Substituting ξ with s in (49), multiplying both sides by G(ξ, s) and then integrating it with respect to s from ξ₂ to ξ, we get that

(50)∫ξ2ξG(ξ,s){ϖ−ϱ4}ds≤−∫ξ2ξG(ξ,s)w˜′(s)ds

and thus,

∫ξ2ξG(ξ,s){ϖ−ϱ4}ds≤−G(ξ,ξ)w˜(ξ)+G(ξ,ξ2)w˜(ξ2)+∫ξ2ξGs′(ξ,s)w˜(s)Δs≤G(ξ,ξ2)w˜(ξ2).

Then,

(51)∫ξ2ξG(ξ,s){ϖ−ϱ4}ds≤G(ξ,ξ0)w˜(ξ2)

and

∫ξ0ξG(ξ,s){ϖ−ϱ4}ds=∫ξ0ξ2G(ξ,s){ϖ−ϱ4}ds+∫ξ2ξG(ξ,s){ϖ−ϱ4}ds≤G(ξ,ξ0)w˜(ξ2)+G(ξ,ξ0)∫ξ0ξ2|ϖ−ϱ4|ds.

So,

limξ→∞sup1G(ξ,ξ0){∫ξ0ξG(ξ,s){ϖ−14ϱ}ds}≤w˜(ξ2)+∫ξ0ξ2|ϖ−14ϱ|ds<∞

which contradicts (48). So the proof is complete.

3 Applications of the results

Example 1

Consider the nonlinear fractional differential equation with damping term

(52)Dt1/2[t1/4Dt1/2Dt1/2x(t)]+Γ(3/2)tDt1/2Dt1/2x(t)+t−1x(t)(1+sin2(x(t)))=0, t≥2.

This corresponds to (1) with t₀ = 2; α=12;a(t)=t1/4;r(t)=1;p(t)=Γ(3/2)/t;q(t)=t−1andf(x) = x + x sin²x. So, f(x)/x = x(1+sin²x)/x ≥ 1 = k; ξ₀ = 2^1/2/Γ(3/2); a˜(ξ)=ξΓ(3/2);p˜(ξ)=ξ−1;q˜(ξ)=(ξΓ(3/2))−2. Furthermore, A(ξ)=exp((Γ(3/2))−1/2∫ξ0ξs−3/2ds)=exp((Γ(3/2))−1/2[2ξ0−1/2−2ξ−1/2])which implies 1<A(ξ)≤(exp(2[Γ(3/2)]−1/2)ξ0−1/2). On the other hand,

δ˜1(ξ,ξ2)=∫ξ2ξ(1/A(s)a˜(s))ds≥[Γ(3/2)]−1/2exp(−2[Γ(3/2)]−1/2ξ0−1/2)∫ξ2ξ1sds=2[Γ(3/2)]−1/2exp(−2[Γ(3/2)]−1/2ξ0−1/2)×(ξ−ξ2)

which implies lim_ξ→∞δ̃ (ξ, ξ₂) = ∞, and so, (9) holds. Then, there exists a suffciently large T > ξ₂such thatδ̃₁(ξ, ξ₂) > 1 on [T, ∞). In (10),

(53)∫t0∞αtα−1Γ(1+α)r(t)dt=∫ξ0∞1r(s)ds=∫ξ0∞ds=∞.

In (11),

(54)∫ξ0∞1r˜(ζ)∫ζ∞1A(τ)a˜(τ)%∫τ∞A(s)q˜(s)dsdτdζ≥[Γ(3/2)]−5/2exp(−2[Γ(3/2)]−1/2ξ0−1/2)×∫ξ0∞∫ζ∞1τ%∫τ∞s−2dsdτdζ=∞.

Letting ff (ξ) =ξ in (45),

(55)∫ξ0∞[kA(s)q˜(s)ϕ˜(s)−r˜(s)[ϕ˜′(s)]24δ˜1(s,ξ2)ϕ˜(s)]ds=∫ξ0∞[A(s)s[sΓ(32)]−2−14δ˜1(s,ξ2)s]ds=∫ξ0T[A(s)[Γ(32)]−2−14δ˜1(s,ξ2)]1sds+∫T∞[A(s)[Γ(32)]−2−14δ˜1(s,ξ2)]1sds≥∫u0T[A(s)[Γ(32)]−2−14δ˜1(s,ξ2)]1sds+∫T∞([Γ(32)]−2−14)1sds=∞.

So, (52) is oscillatory by Theorem 6.

Example 2

Consider the nonlinear fractional di˙erential equation with damping term

(56)Dt2/3[t2/9Dt2/3Dt2/3x(t)]+t−2/3Dt2/3Dt2/3x(t)+x(t)+x5(t)=0, t≥2.

This corresponds to (1) with t₀ = 2; α=23;a(t)=t2/9;r(t)=1;p(t)=t−2/3;q(t)=1andf(x) = x + x⁵x. So, f(x)/x=x(1+x4)/x≥1=k;ξ0=22/3/Γ(5/3);a˜(ξ)=(ξΓ(5/3))1/3;r˜(ξ)=1;p˜(ξ)=(ξΓ(5/3))−1;q˜(ξ)=1. Furthermore, A(ξ)=exp((Γ(5/3))−4/3∫ξ0ξs−4/3ds)=exp((Γ(5/3))−4/3[3ξ0−1/3−3ξ−1/3])which implies1<A(ξ)≤(exp(3[Γ(5/3)]−4/3)ξ0−1/3). On the other hand,

δ˜1(ξ,ξ2)=∫ξ2ξ(1/A(s)a˜(s))ds≥[Γ(3/2)]−1/2exp(−2[Γ(3/2)]−1/2ξ0−1/2)∫ξ2ξ1sds=2[Γ(3/2)]−1/2exp(−2[Γ(3/2)]−1/2ξ0−1/2)×(ξ−ξ2)

which implies lim_ξ→∞δ̃₁(ξ, ξ₂) = ∞ and so (9) holds. Then, there exists a sufficiently large T > ξ₂such thatδ̃₁ (ξ, ξ₂) > 1

on [T ∞). In(10),

(57)∫t0∞αtα−1Γ(1+α)r(t)dt=∫ξ0∞1r(s)ds=∫ξ0∞ds=∞.

In (11),

(58)∫ξ0∞1r˜(ζ)∫ζ∞1A(τ)a˜(τ)%∫τ∞A(s)q˜(s)dsdτdζ≥[Γ(5/3)]−1/3exp(−3[Γ(5/3)]−4/3ξ0−1/3)×∫ξ0∞∫ζ∞τ−1/3∫τ∞dsdτdζ.=∞

Letting ϕ(ξ) = 1 andλ = 2 in (44), for any suffciently large l, we have

limξ→∞sup∫lξ[(s−l)λϖ−ϑ4(s−l)λ−2(λ+ϕ˜′(s)ϕ˜(s)(s−l))2]ds≥limξ→∞sup∫lξ[(s−l)2−14(2)2]ds=limξ→∞sup∫lξ[(s−l)2−1]ds=∞limξ→∞sup∫lξ[(ξ−s)λϖ−ϑ4(ξ−s)λ−2(λ−ϕ˜′(s)ϕ˜(s)(ξ−s))2]ds≥limξ→∞sup∫lξ[(ξ−s)2−1]ds=∞.

So (44) holds, and then we deduce that (56) is oscillatory by Corollary 2.

4 Conclusion

In this paper, we are concerned with the oscillation of solutions to nonlinear fractional di˙erential equations with a damping term. Based on the variable transformation used in ξ, the fractional di˙erential equations are converted into another di˙erential equation of integer order. Then, some new oscillation criteria for the equations are established by using inequalities, the integration average technique and the Riccati transformation. Consequently, it can be seen that this approach can also be applied to the oscillation of other fractional di˙erential equations involving the modified Riemann-Liouville derivative.

References

[1] Das S., Functional Fractional Calculus for System and Controls, Springer, New York, 2008.Search in Google Scholar

[2] Diethelm K., Freed A., On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity, In: F., Mackens W., Vob H., Werther J. (Eds.) Scientific Computing in Engineering II: Computational Fluid Dynamics, Reaction Engineering Molecular Properties, Springer, Heidelberg, 1999, 217-224.10.1007/978-3-642-60185-9_24Search in Google Scholar

[3] Gaul L., Klein P., Kempfle S., Damping description involving fractional operators, Mech. Syst. Signal Process., 1991, 5, 81-88.10.1016/0888-3270(91)90016-XSearch in Google Scholar

[4] Glöckle W., Nonnenmacher T., A fractional calculus approach to self-similar protein dynamics, Biophys. J., 1995, 68, 46-53.10.1016/S0006-3495(95)80157-8Search in Google Scholar

[5] Mainardi F., Fractional calculus: some basic problems in and statistical mechanics, In: Carpinteri A., Mainardi F. (Eds.) and Fractional Calculus in Continuum Mechanics, Springer, Vienna, 1997, 291-348.10.1007/978-3-7091-2664-6_7Search in Google Scholar

[6] Metzler R., Schick W., Kilian H., Nonnenmacher T., Relaxation in filled polymers: a fractional calculus approach, J. Chem. Phys., 1995, 103, 7180-7186.10.1063/1.470346Search in Google Scholar

[7] Diethelm K., The Analysis of Fractional Differential, Springer, Berlin, 2010.10.1007/978-3-642-14574-2Search in Google Scholar

[8] Miller K., Ross B., An Introduction to the Calculus and Fractional Differential Equations, Wiley, New York, 1993.Search in Google Scholar

[9] Podlubny I., Fractional Differential Equations, Press, San Diego, 1999.Search in Google Scholar

[10] Kilbas A., Srivastava H., Trujillo J., Theory and of Fractional Differential Equations, Elsevier, Amsterdam, 2006.10.3182/20060719-3-PT-4902.00008Search in Google Scholar

[11] Delbosco D., Rodino L., Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., 1996, 204, 609-625.10.1006/jmaa.1996.0456Search in Google Scholar

[12] Bai Z., Lü H., Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 2005, 311, 495-505.10.1016/j.jmaa.2005.02.052Search in Google Scholar

[13] Jafari H., Daftardar-Gejji V., Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Appl. Math. Comput., 2006, 180, 700-706.10.1016/j.amc.2006.01.007Search in Google Scholar

[14] Sun S., Zhao Y., Han Z., Li Y., The existence of solutions for boundary value problem of fractional hybrid differential equations, Commun. Nonlinear Sci. Numer. Simul., 2012, 17, 4961-4967.10.1016/j.cnsns.2012.06.001Search in Google Scholar

[15] Muslim M., Existence and approximation of solutions to fractional differential equations, Math. Comput. Model., 2009, 49, 1164-1172.10.1016/j.mcm.2008.07.013Search in Google Scholar

[16] Saadatmandi A., Dehghan M., A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl., 2010, 59, 1326-1336.10.1016/j.camwa.2009.07.006Search in Google Scholar

[17] Ghoreishi F., Yazdani S., An extension of the spectral Tau method for numerical solution of multi-order fractional differential equations with convergence analysis, Comput. Math. Appl., 2011, 61, 30-43.10.1016/j.camwa.2010.10.027Search in Google Scholar

[18] Edwards J., Ford N., Simpson A., The numerical solution of linear multi-term fractional differential equations: systems of equations, J. Comput. Appl. Math., 2002, 148, 401-418.10.1016/S0377-0427(02)00558-7Search in Google Scholar

[19] Galeone L., Garrappa R., Explicit methods for fractional differential equations and their stability properties, J. Comput. Appl. Math., 2009, 228, 548-560.10.1016/j.cam.2008.03.025Search in Google Scholar

[20] Trigeassou J., Maamri N., Sabatier J., Oustaloup A., A Lyapunov approach to the stability of fractional differential equations, Signal Process., 2011, 91, 437-445.10.1016/j.sigpro.2010.04.024Search in Google Scholar

[21] Deng W., Smoothness and stability of the solutions for nonlinear fractional differential equations, Nonlinear Anal., 2010, 72, 1768-1777.10.1016/j.na.2009.09.018Search in Google Scholar

[22] Agarwal R.P., Grace S.R., O'Regan D., Theory for Second Order Linear, Half Linear, Super Linear and Sub Dynamic Equations, Kluwer Academic Publishers, The Netherlands, 2002.10.1007/978-94-017-2515-6Search in Google Scholar

[23] Agarwal R.P., Bohner M., Wan-Tong L., and Os-\newpage{app0} cillation: Theory for Functional Differential Equations, Dekker Inc., New York, 2004.10.1201/9780203025741Search in Google Scholar

[24] Jumarie G., Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results, Comput. Math. Appl., 2006, 51, 1367-1376.10.1016/j.camwa.2006.02.001Search in Google Scholar

[25] Jumarie G., Table of some basic fractional calculus formulae derived froma modified Riemann--Liouville derivative for nondifferentiable functions, Appl. Math. Lett., 2009, 22, 378-385.10.1016/j.aml.2008.06.003Search in Google Scholar

[26] Lu B., The first integral method for some time fractional differential equations, J. Math. Anal. Appl., 2012, 395, 684-693.10.1016/j.jmaa.2012.05.066Search in Google Scholar

[27] Lu B., Backlund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations, Phys. Lett. A, 2012, 376, 2045-2048.10.1016/j.physleta.2012.05.013Search in Google Scholar

[28] Faraz N., Khan Y., Jafari H., Yildirim A., Madani M., Fractional variational iteration method via modified Riemann-Liouville derivative, J. King. Saud. Univ., 2011, 23, 413-417.10.1016/j.jksus.2010.07.025Search in Google Scholar

[29] Qin H., Zheng B., Oscillation of a class of fractional differential equations with damping term, Sci. World J., 2013, 2013, Article ID 685621.10.1155/2013/685621Search in Google Scholar PubMed PubMed Central

[30] Hardy G.H., Littlewood J.E., Polya G., Inequalities, 2nd Edtn., Cambridge University Press, Cambridge, 1988.Search in Google Scholar

Received: 2015-11-7

Accepted: 2016-3-3

Published Online: 2016-4-28

Published in Print: 2016-1-1

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