... interval symmetrizes about the origin. Then, similar to the proof of (2.9), we
can verify the condition f
(α) = f
(β), and the other conditions in Lemma 2.2 are satisfied
all the way. Hence, it also ... contributions
QZ carried out the theoretical proof and drafted the manuscript. XH participated in the
design and coordination. Both ofthe two authors read and approved the final manuscript.
References
[1] ... [α, β], too. Then, by a similar
method to the proof of (2.3) together with Lemma 2.2, we can obtain (2.4) immediately.
For the other ordinary cases, i.e., a = 0, we only need to move the interval...
... and the existence results of
nontrivial solutions and positive solutions are given by means ofthe topological degree
theory. Motivated by the above works, we consider the singular third -order ... “Existence of solutions for a class of third -order nonlinear boundary value
problems,” Journal of Mathematical Analysis and Applications, vol. 294, no. 1, pp. 104–112, 2004.
14 D. Guo, Semi-Ordered ... Analysis and Applications, vol. 323, no. 1, pp. 413–425, 2006.
12 Y. Sun, “Positive solutions of singular third -order three-point boundary value problem,” Journal of
Mathematical Analysis and Applications,...
... while
all the regularly varying solutions of index 1 are asymptotic to a positive multiple of k.
On the other hand, in the remaining two cases, the study ofthe regular variation of the
solutions ... examined by means ofthe contraction mapping theorem
in the Banach space of sequences converging towards zero. The following properties of h
will play a crucial role in the proof. The first two are ... 1
α
α−1
.
5.2
Proof. The main idea ofthe proof is the analogous ofthe linear case, apart from some
additional technical problems. We omit all the details, pointing out only the main differences.
Necessity
Set...
... 1+hν
2
⎤
⎥
⎦
.
(2:7)
Further , let N be the number of positive roots ofthe function in (2.6), and W be the
number of sign changes in its coefficients. Because the radius of convergence of this
series is ∞, then ... verified that in the vicinity of zero, the function g(z) is oforder O (z
ν
).
By virtue of this asymptotic and because g(z) is an odd function, the integral along the
left-hand side ofthe contour ... δ >0.
Then, we can state the following theorem.
Theorem 3.2. Let the conditions of Theorem 2.1, (3.6) and (3.7) hold. If the operator-
value function q(t) has properties 1-3, then the following...
... 11
as an important area of investigation. For the general theory and applications of integer
order differential equations wit h deviat ed arguments, we refer the reader to the refer-
ences [39-45].
As ... fractional -order models. In
consequence, the subject of fractional differential equations is gaining much impor-
tance and attention. For details, see [18-36] and the references therein.
Differential equations ... equations.Toidentifyafew,wereferthereaderto
[2-13] and the references therein.
Fractional differential equations arise in many engineering and scientific disciplines
as the mathematical modeling of systems and processes in the fields of ph...
... completed the estimate ofthe error bounds for asymptotic solutions to second
order linear difference equations in the first case. For thesecond case, we leave it to the second
part of this paper: ... Asymptotic Solutions of Second- order Linear Difference
Equation II: thesecond case.
In the rest of this paper, we would like to give an example to show how to use the
results of this paper to obtain ... solutions to 1.2 is still open. The purpose of this and the next paper Error
bounds for asymptotic solutions of second- order linear difference equations II: the second
case is to estimate error...
... Michael J. Wallace.)
The second instruments are the two set of speaking tests. The first one is the final tests ofthe
three previous semesters (pre-tests) and the other is the final semester ...
V. The design ofthe study:
The study is divided into three parts, which are presented as follows.
Part A is the introduction, which states the background to the study, the statement ofthe ... foremost objective ofthe study is for the sake ofthe students. Besides, the study is expected
to serve as a source of reference for teachers of English on the teaching of speaking skills,...
... since the presence of
the weak degeneracy poses an obstacle to the improvement ofthe differential
properties ofthe functions forming the equation.
This monograph is not a survey ofthe theory of ... degenerate equations. We then establish local
estimates ofthe gradients of solutions of equations ofthe form (1) by combining the
use of certain methods characteristic ofthe technique of global ... depend on the behavior ofthe right side ofthe equation. It is shown in
[163] that these conditions are inherent in the nature ofthe problem.
§5. Estimates of IV ul on the boundary by means of local...
... statistical analysis ofthe capacity of N∗Nakagami-m
channels. Specifically, we have studied the influence ofthe severity of fading and the
number of hops on the PDF, CDF, LCR, and ADF ofthe channel capacity. ... Ψ
(1)
(·) is the first derivative ofthe Digamma function Ψ(·) [33, Equation (8.360)]. In
order to derive the PDF ofthe capacity of N∗Nakagami-m channels, we need to find the
PDF p
Ξ
2
∞
(z) ofthe squared ... levels r. Furthermore, the
variance ofthe channel capacity decreases by increasing the number of hops, while increase
in the severity of fading has an opposite influence on the variance ofthe channel...