Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 769893, 11 pages
doi:10.1155/2011/769893
Research Article
Necessary and Sufficient Conditions for
Positive Solutions of Second-Order Nonlinear
Dynamic Equations on Time Scales
Zhi-Qiang Zhu
Department of Computer Science, Guangdong Polytechnic Normal University, Guangzhou 510665, China
Correspondence should be addressed to Zhi-Qiang Zhu, z3825@163.com
Received 28 December 2010; Revised 22 March 2011; Accepted 13 April 2011
Academic Editor: Mihai Putinar
Copyright q 2011 Zhi-Qiang Zhu. 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.
This paper is concerned with the existence of nonoscillatory solutions for the nonlinear dynamic
Δ
equation ptψ ◦ xxσ Δ qtf ◦ xσ 0 on time scales. By making use of the generalized
Riccati transformation technique, we establish some necessary and sufficient criteria to guarantee
the existence. The last examples show that our results can be applied on the differential equations,
the difference equations, and the q-difference equations.
1. Introduction
In the recent decade there have been many literatures to study the oscillatory properties for
second-order dynamic equations on time scales; see, for example, 1–11
and the references
therein. In particular, the dynamic equation of the form
ptxΔ
Δ
qtf ◦ xσ 0
1.1
has been attracting one’s interesting; see, for example, 3, 5, 8
. Motivated by the papers
mentioned as above, in this paper we consider the existence of nonoscillatory solutions for
nonlinear dynamic equation
Δ
pt ψ ◦ x xσ Δ qtf ◦ xσ 0
on a time scale Ì, where xσ x ◦ σ.
1.2
2
International Journal of Mathematics and Mathematical Sciences
Referring to 12, 13
, a time scale Ì can be defined as an arbitrary nonempty subset
of the set Ê of real numbers, with the properties that every Cauchy sequence in Ì converges
to a point of Ì with the possible exception of Cauchy sequences which converge to a finite
infimum or finite supremum of Ì. On any time scale Ì, the forward and backward jump
operators are defined, respectively, by
σt : inf{s ∈ Ì : s > t},
ρt : sup{s ∈ Ì : s < t},
1.3
where inf ∅ : sup Ì and sup ∅ : inf Ì. A point t ∈ Ì is said to be right-scattered if σt > t,
right-dense if σt t, left-scattered if ρt < t, and left-dense if ρt t. A derived set from
Ì is defined as follows: Ì k Ì − {m} when Ì has a left-scattered maximum m, otherwise
Ì k Ì.
Definition 1.1. For a function f : Ì → Ê and t ∈ Ì k , we define the delta-derivative f Δ t of
ft to be the number provided it exists with the property that, for any ε > 0, there is a
neighborhood U of t i.e., U t − δ, t δ ∩ Ì for some δ such that
fσt − fs − f Δ tσt − s
≤ ε|σt − s| ∀s ∈ U.
1.4
We say that f is delta-differentiable or in short: differentiable on Ì k provided f Δ t exists
for all t ∈ Ì k .
For two differentiable functions f and g at t ∈ Ì k with gtgσt /
0, it holds that
fg
Δ
t f Δ tgt fσtg Δ t,
Δ
f
f Δ tgσt − fσtg Δ t
.
t g
gtgσt
1.5
1.6
Definition 1.2. A function F : Ì → Ê is called an antiderivative of f provided F Δ t ft
b
holds for all t ∈ Ì k . By the antiderivative, the Cauchy integral of f is defined as a fsΔs ∞
t
Fb − Fa, and a fsΔs limt → ∞ a fsΔs.
Definition 1.3. Let f : Ì → Ê be a function, where f is called right-dense continuous rdcontinuous if it is right continuous at right-dense points in Ì and its left-sided limits exist
finite at left-dense points in Ì.
To distinguish from the traditional interval in Ê , we define the interval in Ì by
a, ∞Ì : {t ∈ Ì : a ≤ t < ∞}.
1.7
Let Crd Ì or Crd Ì, Ê denote the set of all rd-continuous functions defined on Ì, and
1
1
Crd
Ì or Crd
Ì, Ê denote the set of all differentiable functions whose derivative is rdcontinuous.
Since we are interested in the existence of nonoscillatory solutions of 1.2, we make
the blanket assumption that inf Ì t0 and sup Ì ∞. As defined in 1
, by a solution of 1.2
International Journal of Mathematics and Mathematical Sciences
3
1
t1 , ∞Ì and ptψxtxσ tΔ ∈
we mean a nontrivial real function xt, with xt ∈ Crd
1
Crd t1 , ∞Ì for some t1 ≥ t0 , which satisfies 1.2 on t1 , ∞Ì. A solution of 1.2 is said to be
nonoscillatory if it is eventually positive or eventually negative.
2. Preliminaries
Let us assume in 1.2 under consideration that
1
Ì, Ì with σ Δ t > 0 on Ì,
H1 σ ∈ Crd
H2 p, q ∈ Crd Ì, Ê and there
∞ exists a t1 ∈ Ì such that pt > 0 and qt ≥ 0 on t1 , ∞Ì
∞
and t1 Δs/ps ∞, t1 qsΔs < ∞; also, q is not identically zero on t1 , ∞Ì,
H3 ψ ∈ C1 Ê , Ê with ψx ≤ m0 on Ê for some constant m0 > 0 and, ψ is nonincreasing on 0, ∞ and nondecreasing on −∞, 0
,
H4 f ∈ C1 Ê , Ê satisfies that xfx > 0 for x / 0, f x ≥ m1 on Ê for some constant
m1 > 0, and f is nondecreasing on 0, ∞ and nonincreasing on −∞, 0
.
{xt ∈ Crd Ì, Ê : t ∈ Ì} is a time scale. By σ and ρ we denote the
Suppose that Ì
we denote the derivative
, respectively, and by Δ
forward and backward jump operators on Ì
. Let σ 2 σ ◦ σ . Then, f Δ by assumptions H3-H4.
on Ì
σ x/ψx is bounded below on Ì
Δ
2
Indeed, in case σ x σ x, we have f σ x/ψx f σ x/ψx and the assertion
holds. In case σ x < σ 2 x, by the definition of delta-derivative and the mean value theorem,
there exits a constant c ∈ σ x, σ 2 x
such that
σ x
f σ 2 x − f
f f c,
σ x σ
2 x − σ x
Δ
2.1
and then
σ x f c m1
f Δ ≥
.
ψx
ψx m0
2.2
Furthermore, f Δ σ x is nondecreasing on 0, ∞Ì
by virtue of H4. Indeed, for any
x σ 2 x and σ
y x, y ∈ 0, ∞Ì
with x ≤ y, there are four cases to consider. In case σ
σ 2 y, f Δ σ x f σ x ≤ f σ y f Δ σ y. In case σ x σ 2 x and σ y < σ 2 y,
Δ
we have f σ x f σ x and
2 f σ
y − f σ y
f
cy ,
f Δ σ y σ 2 y − σ y
2.3
σ y, σ 2 y
. Since cy ≥ σ x, it follows that f Δ σ x ≤ f Δ σ y. The other
where cy ∈ cases can be shown likewise.
σ x/ψx is nondecreasing
As a consequence, we see from assumption H3 that f Δ Δ
on 0, ∞Ì
σ x/ψx is nonincreasing on −∞, 0
Ì
. Similarly, we can show that f provided
σ x ∈ −∞, 0
Ì
.
4
International Journal of Mathematics and Mathematical Sciences
As thus, we can extract the essences as above and obtain a result as follows.
is a time scale and f Δ x/ψx is defined on it. Then it follows that
Lemma 2.1. Suppose that Ì
σ x/ψx is positively bounded below and
i f Δ σ x/ψx is nondecreasing on 0, ∞Ì
ii f Δ and nonincreasing on −∞, 0
Ì
.
with a /
as follows:
For a given a ∈ Ì
0, we introduce a function on Ì
Γa x x
a
ψ ρu
Δu
fu
with x /
for x ∈ Ì
0.
2.4
Then the function Γa x possesses the following properties.
i If a > 0, then Γa x is strictly increasing for x > 0 and Γa−1 y ≥ a is strictly increasing
for y ≥ 0.
ii If a < 0, then Γa x is strictly decreasing for x < 0 and Γa−1 y ≤ a is strictly
decreasing for y ≥ 0.
iii If a / 0, then FΓa−1 y is nondecreasing for y ≥ 0 by Lemma 2.1 ii. Here F is
defined as in 2.6.
For the sake of convenience, we let
Rt ptψxtxσ tΔ
,
fxσ t
Fx f Δ σ x
ψx
2.5
2.6
whenever they are defined.
Note that, if xt is an eventually negative solution of 1.2, then yt −xt satisfies
that
Δ Δ
pt ψ ◦ y y σ
qtf ◦ y σ 0,
2.7
ψ y ψ −y ,
2.8
where
f y −f −y
possess the same properties as ψ and f, respectively. Therefore, in what follows we will
restrict our attention to the eventually positive solutions of 1.2.
3. Main Results
Before entering our main discussions, we remark that xt1 , ∞Ì is a time scale when
1
t1 , ∞Ì and x is monotonic on t1 , ∞Ì, where t1 ∈ Ì. In the following discussions,
x ∈ Crd
International Journal of Mathematics and Mathematical Sciences
5
σ
the notations Δ,
and ρ will act upon the time scale xt1 , ∞Ì, while Δ, σ, and ρ will do on
Ì. Then σ ◦ x x ◦ σ when x is strictly increasing on t1 , ∞Ì.
Lemma 3.1. Suppose that t1 ∈
holds that
Rt Ì and xt is a solution of
∞
qsΔs ∞
t
t
1.2 with xt > 0 on t1 , ∞Ì. Then it
RsRσsFxs
Δs
ps
3.1
for all t ∈ t1 , ∞Ì and
Rt ∞
qsΔs t
ρs
∞ RsRσsF Γ −1σ
RuΔu/pu
x t1 t1
ps
t
Δs
3.2
for all t ∈ σt1 , ∞Ì.
Proof. Without loss of generality, let pt > 0 and qt ≥ 0 on t1 , ∞Ì. Then it follows from
1.2 that
ptψxtxσ tΔ
Δ
−qtfxσ t ≤ 0
∀t ∈ t1 , ∞Ì.
3.3
We assert that
ptψxtxσ tΔ > 0
∀t ∈ t1 , ∞Ì.
3.4
Otherwise, note from assumption H2 that q is not identically zero on t1 , ∞Ì, 3.3 implies
that there exits a t2 ∈ t1 , ∞Ì and a constant m > 0 such that
ptψxtxσ tΔ ≤ −m
∀t ∈ t2 , ∞Ì.
3.5
Consequently we find that
m
x t ≤ x t2 −
m0
σ
t
σ
t2
Δs
−→ −∞
ps
as t −→ ∞,
3.6
which contradicts xt > 0 on t1 , ∞Ì. Since 3.4 holds, it is clear that
xσ tΔ > 0,
Rt > 0
∀t ∈ t1 , ∞Ì.
3.7
On the other hand, we see from 3.3 that Rt is nonincreasing on t1 , ∞Ì, which,
associated with 3.7, means that limt → ∞ Rt exists as a finite number.
Next we will show that
∞
t1
RsRσsFxs
Δs < ∞.
ps
3.8
6
International Journal of Mathematics and Mathematical Sciences
Note that xσ is strictly increasing due to assumption H1 and 3.7; by 1.2, 1.5 and 1.6
and the chain rule 13, Theorem 1.93
we have
Δ 2 ptψxtxσ tΔ f xσ t
RΔ t fxσ tf xσ 2 t
2 Δ σ
f Δ xσ txσ tΔ
pσtψx t xσ t
−
fxσ tf xσ 2 t
2 Δ
f Δ xσ t p ψ ◦ x xσ Δ t
pσtψxσ t xσ t
·
−qt −
ptψxtfxσ t
f xσ 2 t
−qt −
3.9
RσtRtFxt
.
pt
Taking Δ-integral on 3.9 from t1 to t, we obtain that
Rt − Rt1 −
t
qsΔs −
t1
t
t1
RsRσsFxs
Δs.
ps
3.10
Now that the limit of Rt exists as t → ∞, by assumption H2 and 3.10 we see that
3.8 holds. Note that Fxt is positive bounded below see Lemma 2.1i, 3.8 infers that
limt → ∞ Rt 0. To sum up, it is easy to see that 3.10 yields 3.1.
Next we prove 3.2. Again note that xσ is strictly increasing on t1 , ∞Ì, by the
substitution theorem 13, Theorem 1.98
, it follows that
t
t1
RtΔs
pt
t
t1
ψxsxσ sΔ Δs
fxσ s
xσ t
xσ t1 ψ ρu
Δu
,
fu
3.11
which, together with the definition of Γa , induces
xt Γ−1
xσ t1 ρt
t1
RsΔs
ps
∀t ∈ σt1 , ∞Ì,
3.12
where we have used ρσt t see assumption H1. Now let us substitute xt in 3.12
into 3.1 and then we obtain 3.2. The proof is complete.
Theorem 3.2. Suppose that t1 ∈ Ì. Then xt is a solution of 1.2 with xt > 0 on t1 , ∞Ì (or, on
σt1 , ∞Ì) if and only if there exists a constant α > 0 and a function ϕ ∈ Crd t1 , ∞Ì, Ê such
that
∞
∞
ρs
ϕsϕσs
ϕuΔu
−1
F Γα
Δs
3.13
qsΔs ϕt ≥
ps
pu
t
t
t1
for all t ∈ σt1 , ∞Ì.
International Journal of Mathematics and Mathematical Sciences
7
Proof. Suppose that xt is a solution of 1.2 with xt > 0 on t1 , ∞Ì. Then, by Lemma 3.1,
R defined as in 2.5 satisfies R ∈ Crd t1 , ∞Ì, Ê and 3.13 holds, where α xσ t1 .
Conversely, suppose that there exists a constant α > 0 and a function ϕ ∈ Crd Ì, Ê such that 3.13 holds. Let y0 t ≡ 0 and define a sequence of functions {yn t}∞
n0 on
σt1 , ∞Ì as follows:
yn1 t ∞
qsΔs t
∞ yn syn σsF Γ −1 ρs yn uΔu/pu
α
t1
ps
t
Δs.
3.14
It is clear that 0 ≤ y1 t ≤ ϕt on σt1 , ∞Ì. Suppose that yn−1 t ≤ yn t ≤ ϕt on
σt1 , ∞Ì. Then, by the monotone of FΓa−1 y for y ≥ 0, we learn that
yn1 t ≥
∞
qsΔs t
∞ yn−1 syn−1 σsF Γ −1 ρs yn−1 uΔu/pu
α
t1
ps
t
Δs yn t
3.15
as well as
yn1 ≤ ϕt.
3.16
So by the mathematical induction we obtain that
0 ≤ yn t ≤ yn1 t ≤ ϕt
∀σt1 , ∞Ì,
3.17
lim yn t yt ≤ ϕt ∀t ∈ σt1 , ∞Ì.
3.18
which means that there exists a function y such that
n→∞
Now by Lebesgue’s domination convergence theorem on time scales 14, Chapter 5
,
we may deduce from 3.14 that
yt ∞
t
qsΔs ∞ ysyσsF Γ −1 ρs yuΔu/pu
α
t1
ps
t
Δs.
3.19
Let
xt Γα−1
ρt
t1
ysΔs
ps
.
3.20
8
International Journal of Mathematics and Mathematical Sciences
Then xt ≥ α on σt1 , ∞Ì and xσ t is strictly increasing on t1 , ∞Ì. Moreover,
xσt1 , ∞Ì is another time scale with σ ◦ x x ◦ σ. Therefore, by the theorem on derivative
of the inverse 13, Theorem 1.97
we learn that
xσ tΔ fxσ tyt
ψxtpt
for t ∈ t1 , ∞Ì.
3.21
for t ∈ t1 , ∞Ì,
3.22
Now from 3.20 we have
Γα xσ t t
t1
ysΔs
ps
which implies that
ptψxtxσ tΔ ytfxσ t,
3.23
y σ tf Δ xσ txσ tΔ fxσ tyΔ t
ytyσtFxt
y σ tf Δ xσ txσ tΔ fxσ t −qt −
pt
Δ
σ
y σ tFxt
ptψxtx
t
Δ
Δ
σ
σ
σ
σ
y tf x tx t − qtfx t −
pt
3.24
and this results in
ptψxtxσ tΔ
Δ
−qtfxσ t,
where we have imposed formulas 3.19 and 3.20 for the second equal sign and 3.23 for
the third, respectively.
Now we see from 3.24 that xt defined by 3.20 is a positive solution of 1.2. The
proof is complete.
In the remainder of this section, we define formally a sequence of functions {Qn t}∞
n0
as follows. Let t1 ∈ Ì, α > 0, and
Q0 t ∞
3.25
qsΔs,
t
Q1 t ∞ Q0 sQ0 σsF Γ −1 ρs Q0 uΔu/pu
α
t1
t
ps
Δs
3.26
International Journal of Mathematics and Mathematical Sciences
9
as well as
Qn1 t
∞ Q0 s Qn s
Q0 σs Qn σs
F Γ −1 ρs Q0 u Qn u/puΔu
α
t1
ps
t
Δs.
3.27
Then Q1 t ≤ Q2 t. By induction, it follows that
0 < Qn t ≤ Qn1 t for n 1, 2, 3, . . . .
3.28
If xt is a solution of 1.2 with xt > 0 on t1 , ∞Ì for some t1 ∈
by Lemma 3.1 and hence
Ì, then 3.1 holds
3.29
Q0 t ≤ Rt ∀t ∈ σt1 , ∞Ì,
which, together with 3.26, results in
Q1 t ≤
∞ RsRσsF Γ −1 ρs RuΔu/pu
a
t1
ps
t
Δs ∀t ∈ σt1 , ∞Ì.
3.30
Δs,
3.31
Let us now define R0 t by
R0 t ∞ RsRσsF Γ −1 ρs RuΔu/pu
a
t1
ps
t
t ∈ σt1 , ∞Ì.
Then, in view of 3.2 and 3.30, we have
∀t ∈ σt1 , ∞Ì.
3.32
Q0 t Qn t ≤ Rt ∀t ∈ σt1 , ∞Ì, n 1, 2, 3, . . . .
3.33
Q0 t Q1 t ≤ Q0 t R0 t Rt
By the mathematical induction again we educe
Now we learn from 3.28 and 3.33 that {Qn t}∞
0 is well defined and converges to
some function Qt when xt is a solution of 1.2 with xt > 0 on t1 , ∞Ì.
Conversely, suppose that {Qn t}∞
n0 is well defined and
lim Qn t Qt < ∞
n→∞
on σt1 , ∞Ì.
3.34
10
International Journal of Mathematics and Mathematical Sciences
Then, by 3.28, we have Qn t ≤ Qt for all n ≥ 1. Hence, by Lebesgue’s domination convergence theorem on time scales, we may obtain from 3.27 that
Qt ∞ Q0 s Qs
Q0 σs Qσs
F Γ −1 ρs Q0 u Qu/puΔu
α
t1
ps
t
Δs.
3.35
Let ϕt Q0 t Qt. Then, from 3.35 we see that 3.13 holds. Furthermore, by
means of Theorem 3.2, 1.2 has a solution xt > 0 on σt1 , ∞Ì.
To sum up, we obtain our last result as follows.
Corollary 3.3. Suppose that t1 ∈ Ì. Then xt is a solution of 1.2 with xt > 0 on t1 , ∞Ì (or, on
σt1 , ∞Ì) if and only if there exists a constant α > 0 such that the sequence of functions {Qn t}∞
n0
defined as in 3.25–3.27 is well defined and
lim Qn t Qt < ∞
n→∞
on σt1 , ∞Ì.
3.36
Example 3.4. Let Ì 0, ∞. Suppose in 1.2 that pt t1, qt 1/4t2 , ψx 1/21|x|,
and fx x. Let ϕt 1/2t. It is easy to verify that
Γ1−1
s
1
ϕsds
ps
s,
s
ϕudu
1
F Γ1−1
21 |s|
1 pu
for s ≥ 1
3.37
as well as
∞
qsds t
∞
t
s
ϕ2 s
ϕudu
3
−1
F Γ1
≤ ϕt,
ds ps
pu
8t
1
t ≥ 1.
3.38
By Theorem 3.2, 1.2 has a solution xt > 0 on 1, ∞.
Example 3.5. Let h > 0 and q0 > 1, and let Æ 0 be a set of nonnegative integers. Suppose in 1.2
that ψx 1 and fx x.
In case Ì hÆ 0 , let
pt 1,
qt 1
.
2t 2h2t h
3.39
Then, for a given ϕt 1/2t h and any given α > 0, we have
∞
t
∞
t
qsΔs ≤
∞
t
1
Δt
,
σ2s h2s h 22t h
ρs
∞
ϕsϕσs
ϕuΔu
Δs
1
−1
F Γα
Δs ≤
ps
pu
22t
h
h2σs
h
2s
0
t
3.40
International Journal of Mathematics and Mathematical Sciences
11
and hence 3.13 holds. By means of Theorem 3.2, 1.2 has a solution xt > 0 on Ì hÆ 0 .
In case Ì {q0k : k ∈ Æ 0 }, let
pt 2,
ϕt 1
,
t
qt 1
.
2σtt
3.41
Then, for any given α > 0, we have
∞
t
qsΔs ∞
t
ρs
ϕsϕσs
ϕuΔu
−1
F Γα
Δs ϕt,
ps
pu
1
3.42
and hence Theorem 3.2 implies that 1.2 has a solution xt > 0 on Ì {q0k : k ∈ Æ 0 }.
References
1
R. P. Agarwal, D. O’Regan, and S. H. Saker, “Oscillation criteria for nonlinear perturbed dynamic
equations of second-order on time scales,” Journal of Applied Mathematics & Computing, vol. 20, no. 1-2,
pp. 133–147, 2006.
2
H. A. Agwo, “Oscillation of nonlinear second order neutral delay dynamic equations on time scales,”
Bulletin of the Korean Mathematical Society, vol. 45, no. 2, pp. 299–312, 2008.
3
D. R. Anderson, “Interval criteria for oscillation of nonlinear second-order dynamic equations on time
scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 12, pp. 4614–4623, 2008.
4
D. R. Andersona and A. Zafer, “Nonlinear oscillation of second-order dynamic equations on time
scales,” Applied Mathematics Letters, vol. 22, no. 10, pp. 1591–1597, 2009.
5
M. Bohner and C. C. Tisdell, “Oscillation and nonoscillation of forced second order dynamic
equations,” Pacific Journal of Mathematics, vol. 230, no. 1, pp. 59–71, 2007.
6
L. Erbe, A. Peterson, and S. H. Saker, “Kamenev-type oscillation criteria for second-order linear delay
dynamic equations,” Dynamic Systems and Applications, vol. 15, no. 1, pp. 65–78, 2006.
7
T. S. Hassan, “Kamenev-type oscillation criteria for second order nonlinear dynamic equations on
time scales,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5285–5297, 2011.
8
J. Baoguo, L. Erbe, and A. Peterson, “Oscillation of a family of q-difference equations,” Applied
Mathematics Letters, vol. 22, no. 6, pp. 871–875, 2009.
9
S. H. Saker, “Oscillation of nonlinear dynamic equations on time scales,” Applied Mathematics and
Computation, vol. 148, no. 1, pp. 81–91, 2004.
10
S. H. Saker, “Boundedness of solutions of second-order forced nonlinear dynamic equations,” The
Rocky Mountain Journal of Mathematics, vol. 36, no. 6, pp. 2027–2039, 2006.
11
S. H. Saker, “Oscillation of second-order forced nonlinear dynamic equations on time scales,”
Electronic Journal of Qualitative Theory of Differential Equations, no. 23, pp. 1–17, 2005.
12
C. D. Ahlbrandt, M. Bohner, and J. Ridenhour, “Hamiltonian systems on time scales,” Journal of
Mathematical Analysis and Applications, vol. 250, no. 2, pp. 561–578, 2000.
13
M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications,
Birkhäuser, Boston, Mass, USA, 1st edition, 2001.
14
M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass,
USA, 2003.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014