# Oscillation Criteria for Nonlinear Fractional Differential Equations

Authors: Run Xu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Several oscillation criteria are
established for nonlinear fractional differential equations of the format(rtD-αxt)′η′-Ft,∫t∞v-t-αxvdv=0, where D-αx is the Liouville right-side fractional
derivative of order α∈(0,1) of x and η is a quotient
of two odd positive integers. We also give some examples to
illustrate the main results. To the best of our knowledge, the
results are initiation for the oscillatory behavior of the equations.


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## Body

## 1. Introduction

In this paper, we are dealing with the oscillation problem of nonlinear fractional differential equations of the form(1){a(t)[(r(t)D-αx(t))′]η}′-F(t,∫t∞‍(v-t)-αx(v)dv)=0,t≥t0>0,
where α∈(0,1) is a constant, η is a ratio of two odd positive integers. D-αx(t) is the Liouville right-side fractional derivative of order α of x(t) defined by
(2)D-αx(t)=-1Γ(1-α)ddt×∫t∞‍(v-t)-αx(v)dv,t∈R+=(0,∞).Throughout this paper, we will suppose that the following conditions hold:(A1)
a(t)∈C1([t0,∞);R+), r(t)∈C2([t0,∞);R+), ∫t0∞‍(ds/a1/η(s))=∞, ∫t0∞‍(ds/r(s))=∞;(A2)
F(t,G)∈C([t0,∞)×R;R), there exists a function q(t)∈C([t0,∞);R+) such that F(t,G)/Gη≥q(t) for G≠0 and x≠0, t≥t0.By a solution of (1), we mean a function x(t)∈C(R+;R) such that ∫t∞‍(v-t)-αx(v)dv∈C1(R+;R), r(t)D-αx(t)∈C2(R+;R) and satisfies (1) on [Tx,∞). A nontrivial solution of (1) is called oscillatory if it has arbitrary large zero. Otherwise, it is called nonoscillatory. Equation (1) is called oscillatory if all of its solutions are oscillatory.The theory of fractional derivatives goes back to Leibniz’s note in [1], and this led to the appearance of the theory of derivatives and integrals of arbitrary order. The theory had taken a more or less finished form due primarily to Liouville, Grünwald, Letnikov, and Riemann by the end of nineteenth century. We can see some of the books such as [2, 3] on the subject of fractional derivatives and fractional integrals.Fractional differential equations are generalizations of classical differential equations of integer order. In the last few decades, many researchers found that fractional derivatives and fractional integrals were applied in widespread fields of science and engineering, especially in mathematical modeling and simulation of systems and processes, instead of simply being applied in pure theoretical fields of mathematics. Nowadays, many articles have investigated some aspects of fractional differential equations, such as the existence, the uniqueness and stability of solutions, the methods for explicit and numerical solutions, and the stability of solutions (we refer the reader to see [4–9] and the references cited in there). In very recent days, the research on oscillation of various fractional differential equations is being a hot topic; see [10–16].In [10], Grace et al. discussed the oscillation of the following question:
(3)Daqx+f1(t,x)=v(t)+f2(t,x),limt→a+Ja1-qx(t)=b1,
where Daq denotes the Riemann-Liouville differential operator of order q with 0<q≤1, and the functions f1, f2, and v are continuous functions.In [11], Chen et al. established several oscillation criteria for (3) with some additional initial conditions and m-1≤q≤m, m≥1 is an integer. They improved and extended some results of [10].In [12], Chen considered the oscillation of the fractional differential equation
(4)[r(t)(D-αγ)η(t)]′-q(t)f(∫t∞‍(v-t)-αγ(v)dv)=0,t>0
with 0<α≤1.In [13], Han et al. have established some oscillation criteria for the equation
(5)[r(t)g((D-αy)(t))]′-p(t)f(∫t∞‍(s-t)-αy(s)ds)=0,t>0.In [14], Qi and Cheng studied the following equation:
(6)(a(t)[r(t)D-αx(t)]′)′+p(t)[r(t)D-αx(t)]′-q(t)∫t∞‍(ξ-t)-αx(ξ)dξ=0,t>t0
with 0<α<1 and established some new interval oscillation criteria by using a generalized Riccati transformation and inequality technique.In [15], Feng and Meng paid attention to the oscillation of the fractional differential equation
(7)Dtα[r(t)ψ(x(t))Dtαx(t)]+q(t)f(x(t))=e(t),t>t0>0,0<α<1.In [16], Chen considered the oscillation of the fractional differential equation
(8)[r(t)(D-αy)η]′-q(t)f(∫t∞‍(s-t)-αx(s)ds)=0,t>0,0<α<1.The purpose of this paper is to establish some oscillation criteria for (1) by generalized Riccati function and present some applications for our results.In order to prove our theories, we use the general weighted functions from the classX. We say that a function H=H(t,s) belongs to the function class X, if H∈C(D;R+), where D={(t,s):t0≤s≤t<∞}, which satisfies H(t,t)=0, H(t,s)>0 for t>s, and has nonpositive continuous partial derivative Hs′(t,s) on D.


## 2. Main Results

First, we set(9)G(t)=∫t∞‍(v-t)-αx(v)dv,
then it follows that
(10)G′(t)=-Γ(1-α)(D-αx)(t).We give the following lemmas for our results.Lemma 1 (see [17]).
LetX and Y be nonnegative; then
(11)mXYm-1-Xm≤(m-1)Ymform>1.Lemma 2.
Assume thatx(t) is an eventually positive solution of (1), and
(12)∫t0∞‍1r(ξ)∫ξ∞‍[1a(τ)∫τ∞‍q(s)ds]1/ηdτdξ=∞;
then there exists a sufficiently large T such that (r(t)D-αx(t))′<0 on [T,∞), and one of the following two conditions hold:(i)
D-αx(t)<0 on [T,∞),(ii)
D-αx(t)>0 on [T,∞) and limt→∞G(t)=0.Proof.
From the hypothesis, there exists at1 such that x(t)>0 on [t1,∞), so that G(t)>0 on [t1,∞), and we have
(13){a(t)[(r(t)D-αx(t))′]η}′=F(t,∫t∞‍(v-t)-αx(v)dv)≥q(t)Gη(t)>0,t>t1.
Then a(t)[(r(t)D-αx(t))′]η is strictly increasing on [t1,∞), and we can conclude that (r(t)D-αx(t))′ is eventually of one sign. We claim that (r(t)D-αx(t))′<0 on [t2,∞), where t2>t1 is sufficiently large. Otherwise, if there exists a t3>t2 such that (r(t)D-αx(t))′|t3>0, then we can get (r(t)D-αx(t))′>0 on [t3,∞); from those conditions we get
(14)a(t)[(r(t)D-αx(t))′]η≥a(t)[(r(t)D-αx(t))′]η|t3≐δ>0,t≥t3,
that is,
(15)(r(t)D-αx(t))′≥δ(a(t))1/η>0.
Integrating two sides of the previous inequality from t3 to t leads to
(16)r(t)D-αx(t)-r(t3)D-αx(t3)≥δ∫t3t‍ds(a(s))1/η.
Then from (A1) we have limt→∞r(t)D-αx(t)=+∞, which implies that for a certain constant t4>t3, D-αx(t)>0, t∈[t4,∞), then
(17)G(t)-G(t4)=∫t4t‍G′(s)ds=-Γ(1-α)∫t4t‍r(s)D-αx(s)r(s)ds≤-Γ(1-α)r(t4)D-αx(t4)∫t4t‍dsr(s);
by (A1) we obtain limt→∞G(t)=-∞, which contradicts to G(t)>0 on [t1,∞).
So we have(r(t)D-αx(t))′<0 on [t2,∞), and D-αx(t) is eventually of one sign. There are two possibilities: (i) D-αx(t)<0 on [T,∞), (ii) D-αx(t)>0 on [T,∞), where T is sufficiently large.
Assume thatD-αx(t)>0, t∈[t5,∞) for certain sufficiently large constant t5>t4; then G′(t)<0, t∈[t5,∞) and we have limt→∞G(t)=β≥0. We claim that β=0. Otherwise, let β>0; then G(t)≥β on [t5,∞), and by (13), we have
(18){a(t)[(r(t)D-αx(t))′]η}′≥q(t)Gη(t)≥q(t)βη,t∈[t5,∞).
Integrating two sides of above inequality fromt to ∞ leads to
(19)-a(t)[(r(t)D-αx(t))′]η≥∫t∞‍q(s)βηds-limt→∞a(t)[(r(t)D-αx(t))′]η≥βη∫t∞‍q(s)ds,
which means
(20)(r(t)D-αx(t))′≤-β[1a(t)∫t∞‍q(s)ds]1/η,t∈[t5,∞).
Integrating two sides of (20) from t to ∞ yields
(21)-r(t)D-αx(t)≤-limt→∞r(t)D-αx(t)-β∫t∞‍[1a(τ)∫τ∞‍q(s)ds]1/ηdτ≤-β∫t∞‍[1a(τ)∫τ∞‍q(s)ds]1/ηdτ,t∈[t5,∞).
That is,(22)G′(t)≤-βΓ(1-α)1r(t)∫t∞‍[1a(τ)∫τ∞‍q(s)ds]1/ηdτ.
Integrating two sides of (22) from t5 to t, we have
(23)G(t)-G(t5)≤-βΓ(1-α)×∫t5t‍1r(ξ)×{∫ξ∞‍[1a(τ)∫τ∞‍q(s)ds]1/ηdτ}dξ.
By (12), we have limt→∞G(t)=-∞, which contradicts to the fact that G(t)>0. Then we get that β=0, which is limt→∞G(t)=0. The proof is complete.Lemma 3.
Assume thatx(t) is an eventually positive solution of (1) such that (r(t)D-αx(t))′<0, D-αx(t)<0 on [t1,∞), where t1 is sufficiently large and t1>t0. Then one has
(24)G′(t)≥-Γ(1-α)1r(t)A(t1,t)a1/η(t)(r(t)D-αx(t))′,
where A(t1,t)=∫t1t‍a-(1/η)(s)ds.Proof.
From (13), we get that a(t)[(r(t)D-αx(t))′]η is strictly increasing on [t1,∞), so we get
(25)r(t)D-αx(t)≤r(t)D-αx(t)-r(t1)D-αx(t1)=∫t1t‍a1/η(s)(r(s)D-αx(s))′a1/η(s)ds≤a1/η(t)(r(t)D-αx(t))′∫t1t‍dsa1/η(s)=A(t)a1/η(t)(r(t)D-αx(t))′.
Then we get
(26)G′(t)≥-Γ(1-α)1r(t)A(t)a1/η(t)(r(t)D-αx(t))′.Theorem 4.
Assume that(A1), (A2), and (12) hold; if there exists a function ρ∈C1([t0,∞);R+) such that
(27)∫T∞‍[ρ(s)q(s)-1(η+1)η+1(ρ+′(s)ρ(s))η+1×ρ(s)rη(s)[Γ(1-α)A(t2,s)]η1(η+1)η+1(ρ′+(s)ρ(s))η+1]ds=∞
for all sufficiently large constants t2 and T, where ρ+′(s)=max{ρ′(s),0}, A(t2,t) is defined in Lemma 3, t≥T≥t2≥t0>0, then every solution of (1) is oscillatory or satisfies limt→∞G(t)=0.Proof.
Suppose to the contrary that (1) has a nonoscillatory solution x(t) on [t0,∞); without loss of generality, we assume that x(t)>0 on [t1,∞), where t1 is sufficiently large. By Lemma 2 we have (r(t)D-αx(t))′<0 on [t2,∞), where t2 is sufficiently large and t2>t1, and either D-αx(t)<0 on [t2,∞) or limt→∞G(t)=0. (In the case x(t) is eventually negative, it can be proved similarly.)
IfD-αx(t)<0 on [t2,∞), then the conclusion of Lemma 3 holds. We define
(28)w(t)=-ρ(t)a(t)[(r(t)D-αx(t))′]ηGη(t),
then w(t)>0 on [t2,∞), and from (1), (A2), and Lemma 3, we have
(29)w′(t)=ρ′(t)ρ(t)w(t)-ρ(t)(({a(t)[(r(t)D-αx(t))′]η}′Gη(t)-a(t)[(r(t)D-αx(t))′]ηηGη-1(t)G′(t))×(G2η(t))-1({a(t)[(r(t)D-αx(t))′]η}′Gη(t))≤ρ+′(t)ρ(t)w(t)-ρ(t)F(t,G(t))Gη(t)+ηρ(t)((A(t2,t)r(t)a(t)[(r(t)D-αx(t))′]η×[-Γ(1-α)A(t2,t)r(t)a1/η(t)×(r(t)D-αx(t))′A(t2,t)r(t)])×(Gη+1(t))-1)≤ρ+′(t)ρ(t)w(t)-ρ(t)q(t)-ηΓ(1-α)ρ(t)×A(t2,t)r(t)a(η+1)/η(t)[(r(t)D-αx(t))′]η+1Gη+1(t)=ρ+′(t)ρ(t)w(t)-ρ(t)q(t)-ηΓ(1-α)A(t2,t)r(t)1ρ1/η(t)w(η+1)/η(t).
Letm=1+(1/η), X=[ηΓ(1-α)A(t2,t)/ρ1/η(t)r(t)]1/mw(t), Ym-1=(η/(η+1))(ρ+′(t)/ρ(t))[ρ1/η(t)r(t)/ηΓ(1-α)A(t)]η/(η+1); apply Lemma 1 to (29), we get
(30)w′(t)≤-ρ(t)q(t)+1(η+1)η+1×(ρ+′(t)ρ(t))η+1ρ(t)rη(t)[Γ(1-α)A(t2,t)]η.
Integrating two sides of (30) from t2 to t, we have
(31)∫t2t‍[ρ(s)q(s)-1(η+1)η+1(ρ+′(s)ρ(s))η+1×ρ(s)rη(s)[Γ(1-α)A(t2,s)]η(ρ+′(s)ρ(s))η+1]≤w(t2)-w(t)≤w(t2),
which contradicts to (27).
IfD-αx(t)>0 on [t2,∞), then from Lemma 2, we get that limt→∞G(t)=0. This completes the proof.Corollary 5.
Assume that(A1), (A2), and (12) hold; if there exists a function ρ∈C1([t0,∞);R+) such that
(32)∫T∞‍ρ(s)q(s)ds=∞,∫T∞‍(ρ+′(s)ρ(s))η+1ρ(s)rη(s)Aη(t2,s)ds<∞
for a sufficiently large constant T, where ρ+′(s)=max{ρ′(s),0}, A(t2,t) is defined in Lemma 3, then every solution of (1) is oscillatory or satisfies limt→∞G(t)=0.Theorem 6.
Assume that(A1), (A2), and (12) hold; if there exist functions ρ∈C1([t0,∞);R+) and H(t,s)∈X such that
(33)limsupt→∞1H(t,t0)×∫t2t‍ρ(s){rη(s)h+η+1(t,s)(η+1)η+1[Γ(1-α)A(t2,s)H(t,s)]ηH(t,s)q(s)-rη(s)h+η+1(t,s)(η+1)η+1[Γ(1-α)A(t2,s)H(t,s)]η}ds=∞,
where t2 is sufficiently large, t≥t2≥t0,A(t2,t) is defined in Lemma 3, h(t,s)=H(t,s)(ρ+′(s)/ρ(s))+Hs′(t,s),h+(t,s)=max{0,h(t,s)}, then every solution of (1) is oscillatory or satisfies limt→∞G(t)=0.Proof.
Suppose thatx(t) is a nonoscillatory solution of (1); without loss of generality, we let x(t) be an eventually positive solution of (1). According to the proof of Theorem 4, if D-αx(t)<0 on [t2,∞), then (29) holds. Multiplying two sides of (29) by H(t,s) and integrating it from t2 to t, we get
(34)∫t2t‍ρ(s)q(s)H(t,s)ds≤-∫t2t‍H(t,s)w′(s)ds+∫t2t‍H(t,s)ρ+′(s)ρ(s)w(s)ds-ηΓ(1-α)∫t2t‍H(t,s)A(t2,s)ρ1/η(s)r(s)w(η+1)/η(s)ds≤H(t,t2)w(t2)+∫t2t‍[(H(t,s)ρ+′(s)ρ(s)+Hs′(t,s))w(s)-ηΓ(1-α)A(t2,s)H(t,s)ρ1/η(s)r(s)w(η+1)/η(s)(H(t,s)ρ+′(s)ρ(s)+Hs′(t,s))]ds≤H(t,t2)w(t2)+∫t2t‍[ηΓ(1-α)A(t2,s)H(t,s)ρ1/η(s)r(s)h+(t,s)w(s)-ηΓ(1-α)A(t2,s)H(t,s)ρ1/η(s)r(s)w(η+1)/η(s)]ds.
Taking(35)m=1+1η,X=[ηΓ(1-α)A(t2,s)H(t,s)ρ1/η(s)r(s)]η/(η+1)w(s),Ym-1=ηη+1×ρ(s)rη/(η+1)(s)[ηΓ(1-α)A(t2,s)H(t,s)]η/(η+1)h+(t,s),
applying Lemma 1 to (34), we obtain that for t≥t2,
(36)∫t2t‍ρ(s)q(s)H(t,s)ds≤H(t,t2)w(t2)+1(η+1)η+1Γη(1-α)×∫t2t‍h+η+1(t,s)ρ(s)rη(s)(A(t2,s)H(t,s))ηds.
Then we have
(37)∫t2t‍[ρ(s)rη(s)h+η+1(t,s)(η+1)η+1[Γ(1-α)A(s)H(t,s)]ηρ(s)q(s)H(t,s)-ρ(s)rη(s)h+η+1(t,s)(η+1)η+1[Γ(1-α)A(s)H(t,s)]η]ds≤H(t,t2)w(t1)≤H(t,t0)w(t2).
From above inequality we have
(38)1H(t,t0)×∫t2t‍[rη(s)h+η+1(t,s)(η+1)η+1[Γ(1-α)A(t2,s)H(t,s)]ηρ(s)q(s)H(t,s)-rη(s)h+η+1(t,s)(η+1)η+1[Γ(1-α)A(t2,s)H(t,s)]η]ds≤w(t2).
Letting t→∞ we have
(39)limsupt→∞1H(t,t0)×∫t2t‍{rη(s)h+η+1(t,s)(η+1)η+1[Γ(1-α)A(t2,s)H(t,s)]ηρ(s)q(s)H(t,s)-rη(s)h+η+1(t,s)(η+1)η+1[Γ(1-α)A(t2,s)H(t,s)]η}ds<∞,
which contradicts to (33). If D-αx(t)>0 on [t2,∞), from Lemma 2, we have limt→∞G(t)=0. So the proof is complete.Corollary 7.
Assume that(A1), (A2), and (12) hold; if there exist functions ρ∈C1([t0,∞);R+) and H(t,s)∈X such that
(40)limsupt→∞1H(t,t0)∫t2t‍ρ(s)q(s)H(t,s)ds=∞,limsupt→∞1H(t,t0)∫t2t‍ρ(s)rη(s)h+η+1(t,s)[A(t2,s)H(t,s)]ηds<∞,
where t2 is sufficiently large, t≥t2≥t0, A(t2,t) is defined in Lemma 3, then every solution of (1) is oscillatory or satisfies limt→∞G(t)=0.With an appropriate choice of the functionsH and ρ, one can derive from Theorem 6 a number of oscillation criteria for (1). For example, let H(t,s)=ln(t/s), (t,s)∈D, and ρ(t)=tμ; then h(t,s)=(1/s)(μln(t/s)-1), h+(t,s)=max{h(t,s),0}. Based on the above results we obtain the following corollary.Corollary 8.
Assume that(A1), (A2), and (12) hold; if
(41)limsupt→∞1lnt×∫t2t‍sμ[rη(s)h+(t,s)η+1(η+1)η+1Γη(1-α)(A(t2,s)ln(t/s))ηq(s)lnts-rη(s)h+(t,s)η+1(η+1)η+1Γη(1-α)(A(t2,s)ln(t/s))η]ds=∞,
where t2 is sufficiently large, t≥t2≥t0,A(t2,t) is defined in Lemma 3, then every solution of (1) is oscillatory or satisfies limt→∞G(t)=0.Corollary 9.
Assume that(A1), (A2), and (12) hold; if
(42)limsupt→∞1lnt∫t2t‍sμq(s)lntsds=∞,limsupt→∞1lnt∫t2t‍sμrη(s)[(1/s)(μln(t/s)-1)]η+1(A(t2,s)ln(t/s))ηds<∞,
where t2 is sufficiently large, t≥t2≥t0, A(t2,t) is defined in Lemma 3, then every solution of (1) is oscillatory or satisfies limt→∞G(t)=0.


## 3. Examples

Example 1.
Consider the fractional differential equation(43){tη/2[(D-αx(t))′]η}′-F(t,∫t∞‍(v-t)-αx(v)dv)=0,t>t0,
where α∈(0,1) and η is a quotient of two odd positive integers.Here,a(t)=tη/2, r(t)=1, F(t,G)/Gη≥q(t) for G≠0 and x≠0, t≥t0, q(t)≥t-(η+1).We setρ(t)=tμ; if t is sufficiently large, then A(t1,t)=∫t1t‍(ds/s1/2)=2(t-t1)≥t, and ∫t0∞‍(ds/a1/η(s))=∫t0∞‍(ds/s1/2)=∞,∫t0∞‍(ds/r(s))=∞. Consider
(44)∫t0∞‍1r(ξ)∫ξ∞‍[1a(τ)∫τ∞‍q(s)ds]1/ηdτdξ≥∫t0∞‍∫ξ∞‍[1τη/2∫τ∞‍s-(η+1)ds]1/ηdτdξ=∞,
so (A1),(A2), and (12) hold.Furthermore, for a sufficiently large constantT and η<μ<(3/2)η, we have
(45)∫T∞‍ρ(s)q(s)≥∫T∞‍sμs-(η+1)ds=∞,∫T∞‍(ρ+′(s)ρ(s))η+1ρ(s)rη(s)A(t1,s)ηds≤∫T∞‍(μs)η+1sμsη/2ds=μη+1sμ-(3/2)ημ-(3/2)η|T∞<∞.So the conditions of Corollary5 hold, and we deduce that every solution of (43) is oscillatory or satisfies limt→∞G(t)=0.Example 2.
Consider the fractional differential equation(46){tη[(lnttD-αx(t))′]η}′-F(t,∫t∞‍(v-t)-αx(v)dv)=0,t>t0,
where α∈(0,1) and η is a quotient of two odd positive integers.Here,a(t)=tη, r(t)=lnt/t, F(t,G)/Gη≥q(t) for G≠0 and x≠0, t≥t0, q(t)≥t-(η+1).We setρ(t)=1, H(t,s)=ln(t/s); then h(t,s)=(1/s)(μln(t/s)-1), and ∫t0∞‍(ds/a1/η(s))=∫t0∞‍(ds/s)=∞,∫t0∞‍(ds/r(s))=∫t0∞‍(sds/lns)=∞.May as well we assume thatt2<t; then we have A(t2,t)=ln(t/t2)>(1/2)lnt. By calculation we have
(47)∫t0∞‍1r(ξ)∫ξ∞‍[1a(τ)∫τ∞‍q(s)ds]1/ηdτdξ≥∫t0∞‍ξlnξ∫ξ∞‍[1τη∫τ∞‍s-(η+1)ds]1/ηdτdξ=∞,
so (A1),(A2), and (12) hold.Furthermore,(48)limsupt→∞1lnt∫t0t‍sμlntsq(s)ds≥limsupt→∞1lnt∫t0t‍sμ-η-1lntsds=∞,
and for η<μ<2η,
(49)limsupt→∞1lnt×∫t2t‍sμrη(s)[(1/s)(μln(t/s)-1)](η+1)(A(t2,s)ln(t/s))ηds≤limsupt→∞1lnt×∫t2t‍sμlnηs[(1/s)(μln(t/s)-1)](η+1)sη((lns/2)ln(t/s))ηds≤2(1/η)limsupt→∞1lnt∫t0t‍sμ-2η-1[μln(t/s)]η+1(ln(t/s))ηds=2(1/η)limsupt→∞μη+1lnt∫t0t‍sμ-2η-1lntsds=2(1/η)limsupt→∞μη+1lnt1μ-2η×(-lntt0t0μ-2η+1μ-2η(tμ-2η-t0μ-2η))<∞,
the conditions of Corollary 9 hold, and we get that every solution of (46) is oscillatory or satisfies limt→∞G(t)=0.



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*Source: 971357-2013-12-16.xml*