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Oscillation behavior for a class of differential equation with fractional-order derivatives.

1. Introduction

Differential equations with fractional-order derivatives have gained importance due to their various applications in science and engineering such as rheology, dynamical processes in self-similar and porous structures, heat conduction, control theory, electroanalytical chemistry, chemical physics, and economics; for example, see [1-7]. It is well recognized that fractional calculus leads to better results than classical calculus.

Many articles have investigated some aspects of differential equation with fractional-order derivatives, such as the existence and uniqueness for p-type fractional neutral differential equations, smoothness and stability of the solutions, and the methods for explicit and numerical solutions; for example, see [8-16]. However, to the best of the author's knowledge very little is known regarding the oscillatory behavior of differential equation with fractional-order derivatives up to now except for [17-27].

Grace initiated the study of oscillatory theory of FDE, and he considered the equations of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where [D.sup.q.sub.a] denotes the Riemann-Liouville differential operator of order q with 0 < q < 1. In fact, the IVP is equivalent to the Volteria fractional integral equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

He made use of the conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where [p.sub.1], [p.sub.2] [member of] C([a, [infinity]), [R.sup.+]) and [beta], [gamma] > 0 are real numbers. He talked over the four cases of [f.sub.2] = 0; [beta] > 1 and [gamma] = 1; [beta] = 1 and [gamma] < 1; [beta] > 1 and [gamma] < 1. Besides that, he replaced 0 < q < 1 with m - 1 < q < m and got some results on the same cases by using an inequality; refer to [17].

Chen studied the oscillation of the differential equation with fractional-order derivatives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

where [D.sup.[alpha].sub.-]y denotes the Liouville right-sided fractional derivative of order [alpha] with the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

and he obtained four main results under the condition of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

by using a generalized Riccati transformation technique and an inequality; see [18].

Using the same method, in 2013, Chen [23] studied oscillatory behavior of the fractional differential equation with the form

([D.sup.1+[alpha].sub.-]y)(t) - p(t)([D.sup.[alpha].sub.-])(t)

+ q(t)f([[integral].sup.[infinity].sub.t][(v - t).sup.-[alpha]]y(v)dv) = 0 for t > 0,

where [D.sup.[alpha].sub.-]y is the Liouville right-sided fractional derivative of order [alpha] [member of] (0, 1) of y.

Zheng [24] considered the oscillation of the nonlinear fractional differential equation with damping term:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

where [D.sup.a.sub.-]x(t) denotes the Liouville right-sided fractional derivative of order a of x. Using a generalized Riccati function and inequality technique, he established some new oscillation criteria.

Han et al. [19] considered the oscillation for a class of fractional differential equation:

[r(t)g(([D.sup.[alpha].sub.-]y)(t))]' - p(t)f([[integral].sup.[infinity].sub.t]([s - t).sup.-[alpha]]y(s)ds) = 0,

for t > 0, (9)

where 0 < [alpha] < 1 is a real number and [D.sup.[alpha].sub.-]y is the Liouville right-sided fractional derivative of order [alpha] of y.

By generalized Riccati transformation technique, oscillation criteria for the nonlinear fractional differential equation are obtained.

Qi and Cheng [20] studied the oscillation behavior of the equation with the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

where [D.sup.[alpha].sub.-]x(t) also denotes the Liouville right-sided fractional derivative and some sufficient conditions for the oscillation of the equation were given.

The above works on the oscillation are all on fractional equations with Liouville right-sided fractional derivative by Riccati transformation technique.

We notice that very little attention is paid to oscillation of fractional differential equations with Riemann-Liouville derivative. For the relative works of study for oscillatory behavior of fractional differential equations Riemann-Liouville derivative we refer to [17, 21, 25,26].

Marian et al. [25] presented the oscillatory behavior of forced nonlinear fractional difference equation of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

where [[DELTA].sup.[alpha]] is a Riemann-Liouville like discrete fractional difference operator of order [alpha], and some oscillation criteria are established by the same method with 17].

In 2013, Chen et al. [21] improved and extended some work in [17] by considering the forced oscillation of fractional differential equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

with the conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)

where [D.sup.q.sub.a] denotes the Riemann-Liouville or Caputo differential operator of order q with m - 1 < q [less than or equal to] m, m [greater than or equal to] 1, and the operator [I.sup.m-q.sub.a] is the Riemann-Liouville fractional integral operator. The authors obtained some new oscillation criteria by the same method with [17].

In 2014, Wang et al. [26] extended some oscillation results from integer differential equation to the fractional differential equation:

[D.sup.a.sub.a]x(t) + q(t)f(x(t)) = 0, t [member of] [a, +[infinity]), a > 0, (14)

where [D.sup.[alpha].sub.a] denotes the standard Riemann-Liouville differential operator of order a with 0 < [alpha] [less than or equal to] 1, q is a positive real-valued function, f is a continuous functional defined on [0, +[infinity]) [right arrow] [0, +[infinity]) satisfying that

f(x)/[I.sup.2-[alpha]]x [greater than or equal to] K > 0, (15)

and [I.sup.2-a] denotes Riemann-Liouville integral operator. The authors obtained some new oscillation criteria by the method of Riccati transformation technique.

The main purpose of this paper is giving several oscillation theorems for the fractional differential equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)

where [alpha] [member of] (0, 1) is a constant, [gamma] > 0 is a quotient of odd positive integers, and ([D.sup.[alpha].sub.-]x) is the Liouville right-sided fractional derivative of order a of x defined by

([D.sup.a.sub.-]x)(t) := [-1/[GAMMA](1 - [alpha])][d/dt][[integral].sup.[infinity].sub.t][(s - t).sup.-[alpha]]x(s)ds (17)

for t [member of] [R.sub.+] := (0, [infinity]); here [GAMMA] is the gamma function defined by [GAMMA](f) := [[integral].sup.[infinity].sub.0] [s.sup.t-1][e.sup.-s]ds for t [member of] [R.sub.+], and the following conditions are assumed to hold.

(A) f: R [right arrow] R is a continuous function such that f([eta])/[[eta].sup.[gamma]] [much greater than] K for a certain constant K > 0 and for all [eta] [not equal to] 0. a, b, and q are positive continuous functions on [[t.sub.0], [infinity]) for a certain [t.sub.0] > 0, and p is a nonnegative continuous function on [[t.sub.0], [infinity]) for a certain [t.sub.0] > 0. There exists M > 0, q(t) [much less than] M, for t [member of] [[t.sub.0], [infinity]). And [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(B) (p(t)/q(t))' [not equal to] 0, for t [member of] [[t.sub.0], [infinity]).

By a solution x of (16) we mean a nontrivial function x [member of] C([R.sub.+], R) such that [[integral].sup.[infinity].sub.t][(s - t).sup.-[alpha]]x(s)ds [member of] [C.sup.1]([R.sub.+], R) and a(t)(p(t) + q(t)[([D.sup.[alpha].sub.-]x)(t)).sup.[gamma]] [member of] [C.sup.1]([R.sub.+], R),satisfying (16)for t [greater than or equal to] [t.sub.0] > 0. We consider only those solutions of (16) that satisfy sup{[absolute value of x(t): t > [t.sub.x]]} > 0 for any [t.sub.x] [greater than or equal to] [t.sub.0]. A solution x of (16) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory. Equation (16) is said to be oscillatory if all its solutions are oscillatory.

Our results obtained here improve and extend the main results of [18]. In [18], the author studied the oscillation of (16), where p(t) = 0 and q(t) = 1. We are dealing with the oscillation theorems for (16).

For the sake of convenience, we remember

z(t) = p(t) + q(t)([D.sup.[alpha].sub.-]x)(t). (18)

2. Preliminaries and Lemmas

In this section, we present some useful preliminaries and lemmas, which will be used in the proof of our main results.

Definition 1 (see [28]). The Liouville right-sided fractional integral of order [sigma] > 0 of a function y: [R.sub.+] [right arrow] R on the half-axis [R.sub.+] is given by

([I.sup.[sigma].sub.-])(t) := [1/[GAMMA]([sigma])][[integral].sup.[infinity].sub.t][(s - t).sup.[sigma]-1]y(s)ds (19)

for t > 0, provided that the right-hand side is pointwise defined on [R.sup.+], where [GAMMA] is the gamma function.

Definition 2 (see [28]). The Liouville right-sided fractional derivative of order [sigma] > 0 of a function y: [R.sub.+] [right arrow] R on the half-axis [R.sub.+] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)

provided that the right-hand side is pointwise defined on [R.sub.+], where [[sigma]] := min{z [member of] Z: z [greater than or equal to] [sigma]] is the ceiling function.

Lemma 3 (see [29]). If A and B are nonnegative constants, then

[A.sup.[beta]] + ([beta] - 1)[B.sup.[beta]] - [beta][AB.sup.[beta]-1] [greater than or equal to] 0, [beta] > 1, (21)

where the equality holds if and only if A = B.

Lemma 4 (see [18]). Let x be a solution of (16) and

G(t) := [[integral].sup.[infinity].sub.t][(s - t).sup.-[alpha]]x(s)ds, for [alpha] [member of] (0, 1), t > 0. (22)

Then

G'(t) = -[GAMMA](1 - [alpha])([D.sup.[alpha].sub.-]x) (t), for [alpha] [member of] (0, 1), t > 0. (23)

The proof of Lemma 4 is the same as the proof of Lemma 2.1 in [18].

3. Main Results

In this section, we establish some new oscillation criteria for (16).

Theorem 5. Assume that (A) holds, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (24)

Furthermore, assume that there exists a positive function r [member of] [C.sup.1][[t.sub.0], [infinity]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (25)

where [r'.sub.+](s) := max{r'(s), 0}. Then every solution of (16) is oscillatory.

Proof. Suppose that x(t) is a nonoscillatory solution of (16). Without loss of generality, we may assume that x(t) is an eventually positive solution of (16). Then there exists [t.sub.1] [member of] [[t.sub.0], [infinity]) such that

x(t) > 0, G(t) > 0, for t [member of] [[t.sub.1], [infinity]), (26)

where G(t) is defined as in (22). Therefore, it follows from (16) that

[a(t)[z.sup.[gamma]](t)]' = b(t)f(G(t)) > 0, for t [member of] [[t.sub.1], [infinity]). (27)

Thus, a(t)[z.sup.[gamma]](t) is strictly increasing on [[t.sub.1], [infinity]) and is eventually of one sign. Since a(t) > 0 for t [member of] [[t.sub.0], [infinity]) and [gamma] > 0 is a quotient of odd positive integers, we see that z(t) is eventually of one sign. We first show

z(t) < 0, for t [member of] [[t.sub.1], [infinity]). (28)

Otherwise, there exists [t.sub.2] [greater than or equal to] [t.sub.1] such that z([t.sub.2]) > 0, and since a(t)[z.sup.[gamma]](t) is strictly increasing on [[t.sub.1], [infinity]), it is clear that a(t)[z.sup.[gamma]](t) [greater than or equal to] a([t.sub.2])[z.sup.[gamma]]([t.sub.2]) := [c.sub.1] > 0 for t [member of] [[t.sub.2], [infinity]). Therefore, we have

z(t) [greater than or equal to] [c.sup.1/[gamma].sub.1][a.sup.-1/[gamma]](t). (29)

Due to q(t) > 0, from 18), we get

p(T/q(t) + ([D.sup.[alpha].sub.-]x)(t) [greater than or equal to] [c.sup.1/[gamma].sub.1][[alpha].sup.- 1/[gamma]](t)/q(t) [greater than or equal to] [c.sup.1/[gamma].sub.1][a.sup.-1/[gamma]](t)/M. (30)

Integrating both sides of last inequality from [t.sub.2] to t, from (23), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (31)

So, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (32)

and this contradicts (26). Hence, we have that (28) holds.

From (A), (18), and (23), we have

z(t) = p(t) + q(T)([D.sup.[alpha].sub.-]x)(t) = p(t) + q(t)(-G'(t)/[GAMMA](1 - [alpha])). (33)

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

Define the function w(t) by a generalized Riccati transformation

w(t) = r(t) - a(t)[z.sup.[gamma]](t)/[G.sup.[gamma]](t), for t [member of] [[t.sub.1], [infinity]). (35)

Then, we have w(t) > 0 for t [member of] [[t.sub.0], [infinity]), and from (16), (34), (35), and (A), it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (36)

where [r'.sub.+](t) is defined as in Theorem 5. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (37)

From (21) and 36), we derive

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (38)

Integrating both sides of (38) from [t.sub.1] to t, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

Letting t [right arrow] [infinity], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (40)

which contradicts (25). The proof is complete.

Theorem 6. Suppose that (A) and (24) hold. Furthermore, suppose that there exists a positive function r [member of] [C.sup.1][[t.sub.0], [infinity]), and a function H [member of] C(1, R), where I := {(s, t): s [greater than or equal to] t [greater than or equal to] [t.sub.0]}, such that

H(t, t) = 0, for t [greater than or equal to] [t.sub.0],

H(s, t) > 0, for (s, t) [member of] [I.sub.0], (41)

where [I.sub.0] := {(s, t): s > t [greater than or equal to] [t.sub.0]}, and H has a nonpositive continuous partial derivate [H'.sub.t](s, t) = [partial derivative]H(s, t)/[partial derivative]t on [I.sub.0], with respect to the second variable, and satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (42)

where [h.sub.+](s,t) = max{0, [H'.sub.t](s,t) + ([r'.sub.+](t)/r(t))H(s, t)} for (s, t) [member of] [I.sub.0]; here [r'.sub.+](t) is defined as in Theorem 5. Then all solutions of (16) are oscillatory.

Proof. Suppose that x(t) is a nonoscillatory solution of (16). Without loss of generality, we may assume that x(t) is an eventually positive solution of (16). We proceed as in proof of Theorem 5 to get that (36) holds. Multiplying 36) by H(s, t) and integrating from [t.sub.1] to s - 1, for s [member of] [(1 + 1, [infinity]), we derive

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (43)

From

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (44)

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (45)

where [h.sub.+](t) is defined as in Theorem 6. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (46)

From (21) and 45), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (47)

From [H'.sub.t](s, t) < 0, for s > t [greater than or equal to] [t.sub.0], we have

0 < H (s, [t.sub.1]) [less than or equal to] H(s, [t.sub.0]), for s > [t.sub.1] [greater than or equal to] [t.sub.0], (48)

and 0 < H(s, t) [less than or equal to] H(s, [t.sub.0]), for s > t [greater than or equal to] [t.sub.0]; then H(s,t)

0 < H(s, t)/H(s, [t.sub.0]) [less than or equal to] 1, for s > t [greater than or equal to] [t.sub.0]. (49)

Therefore, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (50)

Letting s [right arrow] [infinity], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (51)

which is a contradiction to (42). The proof is complete.

Next, we consider the condition of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (52)

which yields that (24) does not hold. Under this condition, we have the following results.

Theorem 7. Suppose that (A), (B), and (52) hold, and there exists a positive function r [member of] [C.sup.1][[t.sub.0], [infinity]) such that (25) holds. Furthermore, assume that, for every constant T [greater than or equal to] [t.sub.0],

[[integral].sup.[infinity].sub.T][[[1/a(t)][[integral].sup.t.sub.T]b(s)ds].sup.1/[gamma]]dt = [infinity]. (53)

Then every solution x of (16) is oscillatory or satisfies [lim.sub.t[right arrow][infinity]]G'(t) = 0 or [lim.sub.t[right arrow][infinity]]G(t) = 0, where G(t) is defined as Lemma 4.

Proof. Assume that x(t) is a nonoscillatory solution of (16). Without loss of generality, assume that x(t) is an eventually positive solution of (16). Proceeding as in the proof of Theorem 5, we get that (26) and (27) hold. Then there are two cases for the sign of z(t).

When z(t) is eventually negative, from the proof of Theorem 5, we get that every solution x(t) of (16) is oscillatory.

Next, assume that z(t) is eventually positive; then there exists [t.sub.21] [greater than or equal to] [t.sub.0], such that z(t) > 0, for t [greater than or equal to] [t.sub.21]. From (18) and (23), we get

z(t) = p(t) + q(t)([D.sup.[alpha].sub.-]x)(t) = p(t)

+ q(t)(-G'(t)/[GAMMA](1 - [alpha])) > 0. (54)

Therefore,

G'(t) < [GAMMA](1 - [alpha])p(t)/q(t). (55)

Since (B) holds and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (56)

Letting t [right arrow] [infinity] in (55), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (57)

If [lim.sub.t[right arrow][infinity]]G'(t) < 0, then there exists [t.sub.22] [greater than or equal to] [t.sub.1] such that G'(t) < 0, for t [greater than or equal to] [t.sub.22]. We set [t.sub.2] := max{[t.sub.21], [t.sub.22]}. Thus, we get [lim.sub.t[right arrow][infinity]]G(t) = C, and G(t) [greater than or equal to] C, t [greater than or equal to] [t.sub.2].

We now prove C = 0. If not, that is, C > 0, then from (27), we derive

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (58)

Integrating both sides of (58) from [t.sub.2] to t, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (59)

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (60)

Hence, from (18), (A), and (23), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (61)

Integrating both sides of (61) from [t.sub.2] to t, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (62)

Then, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (63)

This contradicts (26). Therefore, we have C = 0; that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (64)

The proof is complete.

Theorem 8. Suppose that (A), (B), and (52) hold. Let r(t) and H(s, t) be defined as in Theorem 6 such that (42) holds. Furthermore, assume that, for every T [greater than or equal to] [t.sub.0], (53) holds. Then every solution x of (16) is oscillatory or satisfies [lim.sub.t[right arrow][infinity]]G'(t) = 0 or [lim.sub.t[right arrow][infinity]]G'(t) = 0, where G(t) is defined as in Lemma 4.

Proof. Assume that x is a nonoscillatory solution of (16). Without loss of generality, assume that x is an eventually positive solution of (16). Proceeding as in the proof of Theorem 5, we get that (26) and (27) hold. Then there are two cases for the sign of z(t).

When z(t) is eventually negative, the proof is similar to that of Theorem 6. When z(t) is eventually positive, the proof is similar to that of Theorem 7. Here we omitted it.

Remark 9. From Theorems 5-8, we can get many different sufficient conditions for the oscillation of (16) with different choices of the functions r and H.

4. Examples

Example 10. Consider the differential equation with fractional-order derivatives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (65)

where [alpha] [member of] (0, 1), and [gamma] > 0 is a quotient of odd positive integers.

Here, a(t) = [t.sup.[gamma]-2], b(t) = 1/[t.sup.2], p(t) = [e.sup.-t], and q(t) = 1/t. Take [t.sub.0] = 1, K = 1, and M = 1. From

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (66)

we see that (A) and (24) hold. Letting r(s) = s, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (67)

which satisfies condition (25). Therefore, by Theorem 5, every solution of (65) is oscillatory.

Example 11. Consider the differential equation with fractional-order derivatives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (68)

where [alpha] [member of] (0, 1), and [gamma] > 0 is a quotient of odd positive integers.

Here, a(t) = [t.sup.2+[gamma]], b(t) = 6t, p(t) = 1/[t.sup.3], q(t) = 1/t, and f(t) = (2 + [t.sup.2])[t.sup.[gamma]]. Take K = 2, M = 1 and [t.sub.0] = 2, f(u) = u. From

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (69)

we find that (A), (B), and (52) hold.

Take r(s) [equivalent to] 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (70)

which satisfies condition (25). For every constant T [greater than or equal to] [t.sub.0], t [member of] [2T, [infinity]), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (71)

which implies that (53) holds. Therefore, by Theorem 7, every solution x of (68) is oscillatory or satisfies [lim.sub.t[right arrow][infinity]]G'(t) = 0 or [lim.sub.t[right arrow][infinity]]G(t) = 0.

http://dx.doi.org/10.1155/2014/419597

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. This research is supported by the Natural Science Foundation of China (61374074), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119), and Shandong Provincial Natural Science Foundation (ZR2012AM009 and ZR2013AL003).

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Shouxian Xiang, (1) Zhenlai Han, (1) Ping Zhao, (2) and Ying Sun (1)

(1) School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China

(2) School of Electrical Engineering, University of Jinan, Jinan, Shandong 250022, China

Correspondence should be addressed to Zhenlai Han; hanzhenlai@163.com

Received 26 April 2014; Accepted 21 July 2014; Published 6 August 2014

Academic Editor: Shurong Sun
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Title Annotation:Research Article
Author:Xiang, Shouxian; Han, Zhenlai; Zhao, Ping; Sun, Ying
Publication:Abstract and Applied Analysis
Article Type:Report
Date:Jan 1, 2014
Words:4636
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