Scholarly article on topic 'Oscillation on a Class of Differential Equations of Fractional Order'

Oscillation on a Class of Differential Equations of Fractional Order Academic research paper on "Mathematics"

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Academic research paper on topic "Oscillation on a Class of Differential Equations of Fractional Order"

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 830836,13 pages http://dx.doi.org/10.1155/2013/830836

Research Article

Oscillation on a Class of Differential Equations of Fractional Order

Tongbo Liu,1 Bin Zheng,1 and Fanwei Meng2

1 School of Science, Shandong University of Technology, Zhangzhou Road 12, Zibo, Shandong 255049, China

2 School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

Correspondence should be addressed to Bin Zheng; zhengbin2601@126.com Received 24 May 2013; Accepted 9 July 2013 Academic Editor: Yuji Liu

Copyright © 2013 Tongbo Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Based on Riccati transformation and certain inequality technique, some new oscillatory criteria are established for the solutions of a class of sequential differential equations with fractional order defined in the modified Riemann-Liouville derivative. The oscillatory criteria established are of new forms compared with the existing results so far in the literature. For illustrating the validity of the results established, we present some examples for them.

1. Introduction

Recently, research for oscillation of various equations including differential equations, difference equations, and dynamic equations on time scales, has been a hot topic in the literature, and much effort has been done to establish new oscillation criteria for these equations so far (e.g., see [1-21] and the references therein). In these investigations, we notice that very little attention is paid to oscillation of fractional differential equations. Recent results in this direction only include Chen's work [22, 23] and Zheng's work [24].

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

D? \a (t) (Dat (r (t) Datx (i)))rj + q(t)f (x (t)) = 0,

t>t0 > 0, 0 < a < 1,

where D"(■) denotes the modified Riemann-Liouville derivative [25] with respect to the variable t, y is the quotient of two odd positive numbers, the functions a e Ca([t0,rn),R+), r e C2a([i0, ot),.R+), and q e C([t0,rn),R+), Ca denotes continuous derivative of order a, and the function f is continuous satisfying f(x)/xY > K for some positive constant K and for all x = 0.

The definition and some important properties for Jumar-ie's modified Riemann-Liouville derivative of order a are listed as follows (see also in [26-28]):

DaJ(t)

r(1 -a) dt

kf(V- f(0))d^, 0 < a < 1,

(fn) (t)fa-n), n<a<n+l,n>l,

* r(l + r-a) Dat (f (t) g (t)) = g (t) D"f (t) + f (t) D"g (t), (3)

D"f [g (t)] = f'g [g (t)] Datg (t) = DaJ [g (t)] (g' (t))".

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

We organize this paper as follows. In Section 2, using Riccati transformation, inequality, and integration average

technique, we establish some new oscillatory criteria for (1), while we present some examples for them in Section 3.

2. Oscillatory Criteria for (1)

In the following, we denote $ = ta/T(1 + a),^ = t"/1(1 + a), i = 0,1,2,3,4,5, a(t) = a($), r(t) = r{t), q(t) = q(£),

R+ = (0r&Si) = J*(1/ct'v(s))ds, S1(t,t,) = r&Si), and S2(Z,Z,) = J* (i^1(s,^i)/f(s))ds,S2(t,ti) = S2(Z,Z,).

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

?c A1/r (s)

to r(1+a)r(t)

r i r\ 1 k ml q(s)ds

ds = m,

dt = m,

drdÇ = m. (7)

Then there exists a sufficiently large T such that D"(r(t)

-s n \T ,

D"x(t)) > 0 on [T,m) and either D"x(t) > 0 on [T,m) or

Dx(t) = 0.

Proof. Let x(t) = x(%), where % = ta/T(1 + a). Then by use of (2) we obtain D"£,(t) = 1, and furthermore by use of the first equality in (4), we have

D«x (t) = rfxß) = x' (0 D^ (t) = x' (0

Similarly we have Data(t) = a (Z),Datr(t) = r (!;). So (1) can be transformed into the following form:

ï(ï)((r(ï)x' ($)')] +q(Vf(x(Ç)) = 0, >0.

Since x(t) is an eventually positive solution of (1), then x(Ç) is an eventually positive solution of (9), and there exists ^ > Ç0 such that x(Ç) > 0 on œ). Furthermore, we have

5(Ç)((r(Ç)x? ($)') = -q(Ç)f(x(ï))

such that (r(Ç)x'(Ç)) < 0 on [^3, œ). Then r(Ç)x (Ç) is strictly decreasing on [£3, œ), and we have

r(t)x' (Q-rfo)* fe)

? a1/y (s)(r(s)x (s))' ä1/y (s)

)* &))' l

1/y (s)

By (5),wehavelim^TOr(£);r' (£,) = -<m. So there exists a sufficiently large with £4 > such that x(^) < 0, ^ e [£4,ot). Furthermore,

:(0-x(t4)= J? x? (s)ds= J?

r(s)x (s)

?4 r (s)

?4 r(S)'

t4 r(1 + a)r(t)

By (6)wededucethat lim^ mx(Ç) = -œ,whichcontradicts

the fact that x(%) is an eventually positive solution of (6). So (r(Ç)x(Ç))' > 0 on [£2, œ), and Dat (r(t)Dat x(t)) > 0 on [t2, œ). Thus D"x(t) = x'(^) is eventually of one sign. Now we assume that x'(^) < 0, Ç e [£5, œ), for some sufficiently large Ç5 > £4. Since x(Ç) > 0, furthermore we have lim?^œx(Ç) = ¡3 > 0. We claim that p = 0. Otherwise,

assume that p > 0. Then x(Ç) > p on [£5, œ), and for Ç e [^5, œ) by (9) we have

a(t)((r(t)x' (!■))') ] <-Kq(ï)xV (Ç) < -Kq(%)ßY.

Substituting % with s in the inequality previously, an integration with respect to s from % to >x> yields

■a(ï)((r(ï)x' ({))'f <- lim â(ï)((r(ï)x(ï))')r

Îœ çœ

q(s)ds<-Kßy J q(s)ds,

Then a(<;)((r(i;)x (<;)) )r is strictly decreasing on [^1, œ), and thus (r(Ç)x'(Ç)) is eventually of one sign. We claim (r(Ç)x(O) > 0 on [<;2, œ), where <;2 > is sufficiently large. Otherwise, assume that there exists a sufficiently large Ç3 > Ç2

which means

f T 1 fTO

(r(ï)x' ($) >K1/"ß^—J^ q(s) ds

Substituting % with t in (15), an integration for (15) with respect to t from % to >x> yields

-r(Z)x' (!■)>- lim r(Z)x' (S)

fœ r I rœ

+ K1lyß\ —I q(s)ds Jç la(T) )r

fœ f 1 ftt > K1/Yß \ — \ q(s) ds Jç la(r) Jr

that is,

1 rœ r I rœ

Ï (t)<-K1lyß—\ q(s)ds

r (S) Jç ,a (t) Jr

Substituting % with C in (17), an integration for (17) with respect to % from %5 to % yields

x(Ç)-x(Çs)

rÇ 1 rœ r i rœ

<-K1lyß\ J- \ q(s)ds

r (0 k , a (t) jr

drdÇ.

By (7), one can see limt^œ%(£) = -œ, which causes a cont-

radiction. So the proof is complete.

Datx (t) >

x(t)>82 (t,t1 )a1ly (t)Dat (r(t)Datx(t)). (21)

Proof. By (10), we obtain that â(Ç)((r(Ç)x'(Ç))')y is strictly decreasing on [^ œ). So

r(t)x (ï)>r(ï)x' (ï)-r(ïi)x' (Si)

Ç a1ly (s)(r(s)x (s))'

a1/y (s)

>a1ly (Ç) (r(t)x' ($)'

hi à1ly (s) = ä1ly (Ç) (f(t)x' ($)% (HJ;

that is

r (t) Datx (t) > S1 (t, t1 )a1ly (t) D? (r (t) D?x (t)),

which admits (20). On the other hand, we have

Ç 81 (s,Ï1)a1ly (s) (r(s)x' (s))'

■Çi r (s)

k r (s)

Jçi p M

= 02 (ï,ï1)â1ly (ï)(r(t)x' (Ç))',

dr (17) which can be rewritten as (21). So the proof is complete. □

Lemma 3 (see [29, Theorem 41]). Assume that A and B are nonnegative real numbers. Then

XABX-1 -AX < (X - 1) BX

Lemma 2. Assume that x is an eventually positive solution of (1) such that

D? (r (t) Dtx (t)) > 0, Dtx (t) > 0 (19)

on [t1, œ)T, where t1 > t0 is sufficiently large. Then one has S1 (t,t1)a1ly (t)Dt;t (r (t)Ü?x(t))

for all X > l.

Theorem 4. Leth1,h2,H e C([^0,rn), R) satisfying= 0, H(%, s) > 0, ^ > s > and H has continuous partial derivatives H(£,, s) and H'S(E,, s) on [£0, <m). Assume that (5)-(7) hold, and for any sufficiently large T > %0, there exist $ e C1([i0, rn), R+) and f e C1([i0)ro),[0,ro)) and a, b, and c with T < a < c < b satisfying

1 (b ~ — \ H(b,s)

b,c) jc

H(b,c) Jc

K(f> (s) q(s)-<f> (s) (p (s) $(s)01 (s,Qf1+{1ly) (s)

- \ H(s,a)

c,a) Ja

H (c, a)

K<p(s)cp(s)-<p(s)(p (s)

P(s)S1 (s,Ï2)(p1+(1ly) (s) + p(s)

H(b,c) Jc

1-) £ H(b,s){(\(Y+ 1) (p1ly (s) <p (s) 81 (s, $2) + r(s)4>' (s)

+ p(s)<(s)H's (b,s)]r+1 )

x((y+1)y+1 x\<(s)81 (^fp^y1)^

T H(s,a)\([(Y+l) p1'* (s) < (s) 81 (s, Q H(c,a) Ja l v

+ r(s)fi (s)

+r(s);f(s)H's (s,«)]r+1)

x((Y+iy+1

x[$(s)81 ^^fr^y^ds, (26)

where <(£,) = <(t), ¿p(^) = p(t). Then every solution of (1) is oscillatory or satisfies limt^œx(i) = 0.

Proof. Assume that (1) has a nonoscillatory solution x on [i0, œ). Without loss of generality, we may assume that x(t) > 0 on [t1, œ), where t1 is sufficiently large. By Lemma 1 we have D^(r(t)D"x(t)) > 0, t e [t2, œ), where t2 > t1 is sufficiently large and either D"x(t) > 0 on [i2, œ) or limt^œx(i) = 0. Now we assume that D"x(t) > 0 on [t2 ,œ).

Define the generalized Riccati function

a(t) = $(t)

a(t) (D" (r(t)DÇx(t)))

XV (t)

+ p(t)

Then for t e [t2, m), we have

D"w (t) = (t)

a(t)(DÇ (r(t)DÇx(t)))

+ $(t)D\

xr (t)

a(t)(DÇ (r(t)DÇx(t))Y

XY (t)

+ D^(t)p(t) + $(t)Dtp(t) $(t)[(xy (t)D« (a(t)(D« (r(t)D«x(t))Y) -yxr-1 (t) D"x (t) a (t) x(Dt (r(t)D«x(t))Y)

x(x2' W)-1}

(t) a + u(t)+<Ht)BÇp(t) <p(t)

t(t)q(t)f(x(t)) XT (t)

y^(t)Datx(t)a(t) (Dat (r(t)Datx(t))Y

XY+1 (t)

+ tr\''w (t) + $ (t) Datp (t).

By Lemma 2 and the definition of f we get that Datœ(t)

<-K<(t)q(t)

- ( (y< (t) 81 (t, t2) a1/v (t) D" (r (t) Datx (t)) a (t)

x(D« (r(t)D"x(t)))r)

x(r(t)xY+1 (t))-1)

-w(t) + $(t)Datp(t)

= -K$(t)q(t)-

V$(t)Ô1 (t,t2 r(t)

D^(t) $(t)

w(t) + $(t)Datp(t)

Using the following inequality (see [2, Equation (2.17)])

(u - v)1+(1W > u1+{1'y) + - v1^ -(l + -) v1'yu, (30) v V v/

we obtain

1+(1/Y) a1+(1/y) (f) 1

> " ^ + 1-p1+(1/Y) (t) f+^M (t) yr v y

1\p1/y (t)a(t)

-(1 + -

A combination of (29) and (31) yields:

]-w(t) + <(t)Datp(t)

Data (t) < (t) q (t) + Dt $(t) a (+) ■ $ (+)

Y$(t)$1 (t,t2 r(t)

1+(1/y)

+ 1 1+(1/y)

_ f+(1/y) (t) Y

1\p1/y (t)œ(t)

-( 1 +

= -K$(t)q(t) + $(t)D«p(t)

$(t)81 (t,t2)p1+(1W (t) r(t)

Y$(t)$1 (t,h) a1+(1/y) (t) r (t) $1+(1/Y) (t)

(Y+1)p1/y (t)4(t)81 (t,t2) + r(t)D^(t)

r(t)$(t)

xœ(t), t>t2

Let u(t) = titf). Then D"w(t) = w($), D"$(t) = fitf), and Setting

D"f(t) = <p'(£), and (32) is transformed into the following

tWSi (t)

r(t)<j>Uy (t)

(y + i)<p1ly (Z)$(Z)$i (S,S2) + r(t)f (t)

+-=-Z (t),

s(s)s(?)

Select a, b, and c arbitrarily in [^2, >x) with b > c > a. Substituting % with s, multiplying both sides of (33)by H(£,, s), and integrating it with respect to s from c to % for % e [c, b), we get that

K<t>(s)q(s)-<f>(s)<p' (s)

$(s)8i (s,t2)<pl+(lly) (s)

■ ds

j H(t,s)w (s) ds j^H^s)

YSi M -WM (s)

7(s)<ftlly (s)

■(((y+l)<p1lv (s)j(s)8i (s,t2) +f(s)j (s))

x(r(s)j(s))-1 )z(s)

= H(t,c)w(c) + j?H(t,s)

- ^ (s, t2) Zl+(1lr) (s)

r(s) <ff1ly (s)

(((y+l)<p1ly (s)jf(s)Si (s,t2) +r(s)j (s) + r(s)ff(s)H's (t,s))

x(r(s)ff(s))-1 )z(s)

\=1 + 1, AX = lSl (,t2) z1+(1ly) (s), 1 r (s) f1^ (s)

B*-1 = y1l(y+1)

x(((Y+l)f1ly (s)$(s)8i №2)

+ r(s)$ (s)+r(s)P(s)H's ($,s))

x((y+l)[4>(s)8i {s^f^r11^1 (s))-1),

by a combination of Lemma 3 and (33), we get that

[SH(s,s)

K<p(s)q(s) -<r(s) < (s)

j(s)Si (s,s2)<p1+(1M (s) + r(s)

< H (s, c) w (c) + j H(s,s)

x{([(Y+l)<p1ly (s) j (s) Si (s,t2) + r(s)j' (s)

r(s)j(s)H's (is)]**"

x((Y+l)y+1[jt(s)Sl (s^f7(s)]'1}ds.

Dividing both sides of inequality (36) by H(£,, c) and letting % ^ b-,we obtain

H(b,c) Jc

j H(b,s)

KS(S)S(S)

- j (s) < (s) <S(s) Si (s,t2)<Pmly) (s)

+ r(s)

< w(c)

1 (h ~

—- I H(b,s)

H(b,c) Jc

x{([(y+l)<p1ly (s)j(s)8i (s, t2)

+ r(s)p' (s) +p(s)P(s)H's (b,s)]r+1

*((Y+1)

x((y+iY+1

x[p(s)p ^^fr^y^ds.

x[4,(s)S1 (s^)]^))" }ds

On the other hand, substituting % with s, multiplying both sides of (33) by H(s, %), and integrating it with respect to s from % to c for % e (a, c], similar to (36)-(37), we get that

A combination of (37) and (39) yields

H(b,c) Jc

j H(b,s)

\CH(s,V

Kp (s)q(s)-4>(s)<p' (s)

P(s)P (s)

+ »(s)

■ ds

< -H(c, Z)w(c) + j" H?(s,^){([(r+ 1) <lly (s) P(s) r (s, &

+r(s)p' (s) + r(s)4>(s)H's (a)]

"((Y+1)

x[4>(s)8i (^yfr^V1}^.

Dividing both sides of inequality (38) by H(c, %) and letting

£ ^ fl+,weobtain

„ - j H(s,a)

H(c,a) Ja

K<r(s)q(s)

-p(s)< (s)

<T(s)Si (s,^2)<P1+(1IY) (s)

< -w (c) _J

H(c,a) Ja

+ —- j H(s,a)

c,a) Ja

x{([(Y+l)<>1ly (s)<p(s)si (s,q

+ r(s)p (s)

(s)<P(s)H's (s, a)]

K<p(s)>(s)

- <r (s) < (s)

P(S)S1 (S,^2)v1+(1ly) (S)

+ —- j H(s,a)

H (c, a)

r(s) K<(S)P(S)

- P (s) < (s)

P(S)P (s,^)^1™ (S) + P(S)

■ ds

H&7) jc H (b, *){([(Y+1) <p1Iy (s) p (s) S1 (s, O

+ p(s)p' (s)

+P(s)P(s)H's (b,s)]r+1) x((Y+1)n1 x[P(s)81 (s,$2 ^r^y^ds

+ —- j H(s,a)

H (c, a)

x{([(Y+1)<1ly (s)P(s)81 (s,Q + p(s)< (s)

+ p(s)p(s)h's (5,fl)]r+1) x((Y+1)n1

x [P(s)81 ^^fr^y^ds, which contradicts (26). So the proof is complete.

Theorem 5. Under the conditions of Theorem 4, if for any sufficiently large I > %0,

lim sup | H (s, I) {K< (s) q(s) - < (s) f> (s)

p(s)81 (s,$2)f1+(1/y) (s) + r(s)

+ l)f1/y (s)$(s)$1 (s,l-2)

+ r(s)<' (s)

+r(s)<(s)H's (s,l)]n1 V+1

x( (V+1)

x\<(s)81 (s^rb))-1) )ds

lim sup \ H (Ç, s) -

Ç^œ Jl

K<p(s)q(s)-<p(s) ( (s)

<P(s)S1 (s,Ï2)(p1+(1ly) (S) + 7(s)

-((\(1ly (s)P(s)81 (s,Ï2 ) + r(s)< (s) +r(s)$(s)H's (^p x((y+1)v+1

x\<p(s)81 M2)]V(s))-1)

then (1) is oscillatory.

Proof. For any T > , let a = T. In (41) we choose I = a. Then there exists c > a such that

^H^^^^cr^-tr^f' (s)

<P(s)81 (s,^2)v1+(1ly) (s)

((\(v+l)(p1|y (s)t(s)81 (s,^)

>, Niy+1

+r(s)t' (s)+r(s)t(s)H's (s.fl)]

x((y+1)

x\t(s)81 Oa2)]yr(s))-1)}ds>0.

In (42) we choose I = c > a. Then there exists b > c such that

£H(b,s){K4>(s)cl(s)-4>(s)f'(s)

$(3)0! (s,Qfl+{lly) (S)

+ r(s)

(y+l)<p1lr (s)$(s)8i (s.Q

+r(s)$ (s) + r(s)p(s)% (b,s)

x({Y+l)y+1

x [$(s)8i (s,y]rr(s))-1)}ds>0.

Combining (43) and (44) we obtain (26). The conclusion thus comes from Theorem 4, and the proof is complete. □

In Theorems 4 and 5, if we choose H(s) = (£,- s) , ^ > s > £0,where X > l is a constant, then we obtain the following two corollaries.

Corollary 6. Under the conditions of Theorem 4, if for any sufficiently large T > %0, there exist a, b, c with T < a < c < b satisfying

(c - a)

(s - a)

Kt(s)q(s)

- t (s) ( (s) $(s)81 (s,^2)p1+(1ly) (s)

x]Kt(s)q(s)

-<r(s)v' (s)

t(s)81 (s,^2)y1+(1ly) (S)

' ?(s)

(c - a)

(s - a)

x{([(Y+1)p1/y (s)<r(s)Ô1 (s,^) + r(s)4>' (s) +\(s-a)X-1r(s)j>(s)]Y+1)

x((Y+1r

X [$(s)81 (s,y]rT(s)) 1}ds

xJ (b - s)x {([(Y+1)p1/y (s) <r (s) 81 (s,Ï2 ) + T(s)(' (s)

-A(b-s)X-1r(s)<(s)]y+1)

x((Y+1)r+1 x[<(s)81 (s,y]r r&Y'lds,

then (1) is oscillatory.

Corollary 7. Under the conditions of Theorem 5, if for any sufficiently large I > %0,

lim sup J (s -1)

i — m Jl

(s)q(s) - < (s) p (s)

<(3)81 (s,Ï2)p1+(1/y) (s) + r(s)

Y+1)p1/r (s) <8(s) 81 (s,^)

+ r(s)< (s)

lim sup J (S, - s)X {k<T(s)ît(s)

-<T(s) p t (s)

T(S)81 (s,Ï2)p1+(1/y) (s)

+ T(s)

Y+1)p>1/y (s) (f (s) 81 (s,^) + T(s)(' (s)

-\(t-s)X-1r(s)<T(s)]nl) x({Y+1f1

x[<T(s)81 (s^Ym)- )}ds

then (1) is oscillatory.

Theorem 8. Assume (5)-(7) hold, and there exists two functions p e C1 ([i0, ot), R+) and < e C1([i0, ot), [0, ot)) such that

K<f>(s)T(s)-<t>(s)p> (s) T(3)81 (s,Ï2)p1+(1/y) (s)

[(Y+1)p1/y (s)T(s)81 (s,Ï2) + T(s)<T' (5)] (Y+1)y+1 [<8(3)81 (U2)]V(S)

where p, <p are defined as in Theorem 4. Then every solution of (1) is oscillatory or satisfies limt^mx(t) = 0.

+X(s-l)X-1T(s)T(s)] x((Y+1)y+1

Proof. Assume that (1) has a nonoscillatory solution x on [i0, œ). Without loss of generality, we may assume that x(t) > 0 on [t1, œ), where t1 is sufficiently large. By Lemma 1 we have D^(r(t)D"x(t)) > 0, t e [t2, œ), where t2 > t1 is sufficiently large and either D"x(t) > 0 on [t2, œ) or x [<p(s) 81 (s Ç2)]Yr(s)yL)} ds limt^mx(t) = 0. Now we assume that D"x(t) > 0 on [t2, œ).

Let w(t), w($) be defined as in Theorem 4. Then we obtain (32).

Setting

, , 1 ,A Yt(t)S1 № ^1+{1ly) (t)

X = 1 +—, A = -—--, „ , .-,

V r(t) <1+(1ly) (t)

B^-1 = y1l(y+1)

(Y+1)y1ly (t)t(t)S1 (t,t2) + r(t)Ptt(t) (y+1)[r(t)S1 (t,t2)]yl{y+1)r1l(y+1) (t)

Using Lemma 3 in (32) we get that

D"w (t)

< -K< (t) q(t)+< (t) D"<p (t)

t(t)8i (t,t2)cp1+{1ly) (t) r(t)

[(y+l)<p1ly (t)4(t)8i (t,t2) + r(t)DÇt (t)] (y+l)r+1 [<(t)81 (t,t2)]yr(t)

which is rewritten in the following form: W (S)<-K<(S)q(S)+<(S)y' (S)

$(t)81 (ÎÏ2)v1H1ly) (S) r(S)

+ [(y+i)v1ly (S)$(S)$1 (S,Ï2) + r(S)$' (S)] (y+i)r+1[<P(S)81 (S,S2)]r r(S)

Substituting % with s in (50), an integration for (50) with respect to s from to % yields

Kt(s)q(s)

- t (s) ( (s) +

t(s)81 (s,Ï2)(p1+{1ly) (s) r(s)

[(r+l)y1ly (s)<p(s)81 (s,Ï2) + r(s)<p' (s)] (y+l)y+1[;p(s)81 (s,S2)]rr(s)

<w(S2)-w(S)<w(S2) < OT, which contradicts (47). So the proof is complete.

Theorem 9. Assume that (5)-(7) hold, and there exists a function H e C([£0,ot), R) such that H(%,%) = 0, for £ >

S0, H(S,s) > 0, for S> s> So, and H has a nonpositive continuous partial derivative H's(S, s). If

lim sup —7—TT

£ H (Ç, s)

x\Kt(s)q(s)

- t (s) ( (s)

P(s)Ô1 (s,^2)y1+(1ly) (s) r(s)

(Y+1)(1ly (s)t(s)81 (s,Ï2

7 , Niy+1N

+r(s)<r' (s)] x((v+1)y+1

x \t(s)81 (S,^2)]yr(s))-1)}ds

where q> are defined as in Theorem 4, then every solution of (1) is oscillatory or satisfies limt^mx(t) = 0.

Proof. Assume that (1) has a nonoscillatory solution x on [t0, ot). Without loss of generality, we may assume that x(t) > 0 on [t1, ot), where t1 is sufficiently large. By Lemma 1 we have Dat(r(t)Datx(t)) > 0, t e [t2, ot), where t2 > t1 is sufficiently large and either D"x(t) > 0 on [t2, ot) or limt^œx(t) = 0. Now we assume that D*x(t) > 0 on [t2, ot).

Let w(t), w(S) be defined as in Theorem 4. By (50) we have

K<(S)q(S)- <(S)y (S)

<(S)81 (S,S2)y1+(1ly) (S) + 7(S)

[(y + l) r1ly (S) <(S) 81 (S, S2) + r(S) < (S)]r+1

(y + i)r+1[<r(S)81 (U2)f r(S)

<w(S), S>Ï2-

Substituting S with s in (53), multiplying both sides by H(S, s), and then integrating with respect to s from S2 to S yield

£ H(S,s){Kt>(s)q(s)-t>(s)( (s)

$(s)81 (s,^2)v1+(1ly) (s)

Y+ 1)<P1Iy (S)P(S)81 (S,$2) (S)P' «D

x((Y+1Y+1[p(s)81 (s,^ j H($,s)p' (s)ds

+ j H ($,s)p(s)As

\ioH(^,s){k<P(s)cP(s)-<P(s)P' (s)

p(s)81 (s,^2)<1+(1lv) (s) + P(S)

-(([(Y+1)^ (s)p(s)81 (s,^) +P(s)P' (s)]n1) x((Y+1f+1[p(s)81 (s^)]^))"1)^

j2 H(Z,S){KP(S)P(S)-P(S)P' (S)

P(S)P (S,^2)<P1+(1IY) (S)

+ P(S)

Y+1)p1lr (s)P(s)P (s,^) 1 , Nir+1N

+r (s)p'(s)]

"((Y+1)

x[p(s)81 (s^)]^))"1)^ jH(Z,s){kP(s)P(s)-P(s)P'(s)

P(S)P (S,^2)p1+(1ly) (S)

-(([(Y+1)^ (s)p(s)81 (s,S2) +P(S)P' (5)]r+1) "((Y+1)-1

x [P(S)P (s^)]^))"1)^

f?2 _ _ h(z,zo)\ KP(s)p(s)-p(s)p' (s)

p(s)81 (s,^2)<P1+(1Iy) (s) P(s)

(([(Y+1)p1ly (S)P(S)P (S,^2 )

+p(S)P' (s)]r+1N x((Y+1)n1

x [p(s)81 (s^P^))'1)

ds. (55)

lim sup

i™ rH(i,(o)

X {K<P(S)P(S)-<P(S)<P' (S)

P(s)P (s,^2)p1+(1h) (S) + P(S)

-(([(Y+1)p1/y (S)P(S)P (S,S2)

+P(S)P>' (s)]r+1 nV+1

X ( (Y+ 1)

X [P(S)P (s^)]^))"1)^

\ 2 KP(S)P(S)-P(S)P' (S)

P(s)P (s,^2)p1+(1ly) (s) + P(S)

(([(y+l)^ (8)4,(8)8! (s,S2)

x[4(s)8i (s,Ofr^y1

which contradicts (52). So the proofis complete.

(56) □

3. Applications of the Results

Example 10. Consider the following fractional differential equation:

St(Dl'2 (D\/2x(t)))

r3 (3/2)

[x5/3 (t) + x11/3 (t)]=0, t>2.

In (1), if we set t0 = 2, a =1/2, y = 5/3, a(t)=^t, r(t) = 1, q(t) = t3l2/T3(3/2), then we obtain (57). So P(£) = a(t) = jt = T(3/2)$, f($) = 1, and ftf) = and f(x) = x5/3 + x11/3, which implies that f(x)/x513 = 1 + x2 > 1 = K. Furthermore, SA^,^) = J\ (1/P1/y(s))ds = [r(3/2)]-3/5

J\ s-3/5ds = (5/2)[I(3/2)]-315(?l5-&5), which implies that

lim^^S^, %2) = ot. So there exists a sufficiently large T >

such that 81(£,,£,2) > 1 on [T,ot). Furthermore, one can easily see that (5)-(7) all hold. On the other hand, in (46), after letting <({,) = 1, p($) = 0, and A = 2, for any sufficiently large I (we may let I > T without loss of generality), it holds that

lim sup \ (s - l)X {Kp (s) P(s) - p (s) <p (s)

<(3)81 (s,^2)p1+(1ly) (s) + p(s)

Y+1)p1lr (s)<(s)81 (s,^) + p(s)<p (s)

+A (s-l)X-1 r(s)(t>(s)Y+1

x((r+iy

x [4(s)81 (s,^2)]rr(s))-1)|ds

n /3\ 8/3

> lim sup I (s - I)2 s3 - (-) (s- if

Z^rn )l I \4/

lim sup I (£, - s)X {K< (s) q(s) - < (s) fr (s)

ds = x,

p (s) 81 (s,^2)p1+(1/y) (s) + P(s)

-(([(Y+1)p1ly (s)<(s)81 (s,^) + P(s)p>' (s)

-A ($-s)X-1f(s)$(s)]y+1)

x((Y+1)n1

X [<(s)81 0a2)]V(s))-1)}ds

C? r /-\ 8/3

> lim sup \ (^ - s)2 s3 - ( - ) (£, - s)8/3 ds = ot. J1 [ \4J

So according to Corollary 7 we deduce that (57) is oscillatory.

Example 11. Consider the following fractional differential equation:

t-1/6{D\/2 {D\/2x(t)))

+qmx1/3 <t)[i+/>-> ] (59)

= 0, t>2.

In (1), if we set t0 = 2, a = 1/2, y = 1/3, a(t) = t-1/6, r(t) = 1, q(t) = T1/3(3/2)/t1/6, then we obtain (57). So a(^) = a(t) = t-1/6 = [T(-/2)]-1/3^-1/3, ?(£,) = 1, and q(<;) = $-1/3,

and f(x) = x 1 [1 + ex ], which implies that f(x)/x 1+e*2 > 1 = K. Furthermore, f&U = jl (1/P1 ly(s))ds =

jl T(3/2)sds = (1/2)T(3/2)(? - g), which implies that

lim^^TOp(^, ^2) = ot. So there exists a sufficiently large

T > such that ^2) > 1 on [T,ot). Furthermore, after some computation one can see that (5)-(7) all hold, and in (47), letting p>($) = £ p($) = 0, we obtain

K<p (s) q(s) - < (s) (f (s) +

<r (s) 81 (s,Qfl+{1/y) (s)

[(y+l)f1/y (s)<r(s)81 (s,l-2) + r(s)<r' (s)] (y+l)y+1[<P(s)81 (s,y]V(s)

2l3 S ' -

2l3 S ' -

2l3 S ' -

(4/3)4l3\4 M)] 1

(4/3)il3\sSl (5,0] 1

(4/3)^3\4 M)]

2l3 S ' -

ds = œ.

Therefore, (59) is oscillatoryby Theorem 8.

Acknowledgments

This work is partially supported by National Natural Science Foundation of China (11171178) and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province. The authors would like to thank the reviewers very much for their valuable suggestions on the paper.

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