1. CMB 2018 (vol 61 pp. 768)
 Li, Liangliang; Tian, Jing; Chen, Goong

Chaotic Vibration of a Twodimensional Nonstrictly Hyperbolic Equation
The study of chaotic vibration for multidimensional PDEs due
to nonlinear boundary conditions is challenging. In this paper,
we mainly investigate the chaotic oscillation of a twodimensional
nonstrictly hyperbolic equation due to an energyinjecting
boundary condition and a distributed selfregulating boundary
condition. By using the method of characteristics, we give a
rigorous proof of the onset of the chaotic vibration phenomenon
of the 2D nonstrictly hyperbolic equation. We have also found
a regime of the parameters when the chaotic vibration phenomenon
occurs. Numerical simulations are also provided.
Keywords:chaotic vibration, reflection boundary condition, perioddoubling bifurcation, method of characteristics Categories:32H50, 34C28, 37K50, 54H20, 58J45 

2. CMB 2018 (vol 61 pp. 717)
 Bu, Shangquan; Cai, Gang

Periodic Solutions of Second Order Degenerate Differential Equations with Delay in Banach Spaces
We give necessary and sufficient
conditions of the $L^p$wellposedness (resp. $B_{p,q}^s$wellposedness) for the second order degenerate
differential equation with finite delays:
$(Mu)''(t)+Bu'(t)+Au(t)=Gu'_t+Fu_t+f(t),(t\in [0,2\pi])$ with periodic
boundary conditions $(Mu)(0)=(Mu)(2\pi)$, $(Mu)'(0)=(Mu)'(2\pi)$, where
$A, B, M$ are closed linear operators on a complex Banach space $X$ satisfying
$D(A)\cap D(B)\subset D(M)$, $F$ and $G$ are bounded linear operators from
$L^p([2\pi,0];X)$ (resp. $B_{p,q}^s([2\pi,0];X)$) into $X$.
Keywords:second order degenerate differential equation, Fourier multiplier theorem, wellposedness, LebesgueBochner space, Besov space Categories:34G10, 34K30, 43A15, 47D06 

3. CMB 2017 (vol 61 pp. 3)
 Alhasanat, Ahmad; Ou, Chunhua

Periodic steadystate solutions of a liquid film model via a classical method
In this paper, periodic steadystate of a liquid film flowing
over a periodic uneven wall is investigated via a classical method.
Specifically, we analyze a longwave model that is valid at
the nearcritical Reynolds number. For the periodic wall surface,
we construct an iteration scheme in terms of an integral form
of the original steadystate problem. The uniform convergence
of the scheme is proved so that we can derive the existence and
the uniqueness, as well as the asymptotic formula, of the periodic
solutions.
Keywords:film flow, classical methods, asymptotic analysis Categories:34E05, 34E10, 34E15 

4. CMB 2017 (vol 61 pp. 423)
 Zhang, GuoBao; Tian, Ge

Stability of Traveling Wavefronts for a TwoComponent Lattice Dynamical System Arising in Competition Models
In this paper, we study a twocomponent LotkaVolterra competition
system
on an onedimensional spatial lattice. By the method of the comparison
principle together with
the weighted energy, we prove that the traveling wavefronts with
large speed are exponentially asymptotically stable,
when the initial perturbation around the traveling wavefronts
decays
exponentially as $j+ct \rightarrow \infty$, where $j\in\mathbb{Z}$,
$t\gt 0$, but the initial perturbation
can be arbitrarily large on other locations. This partially answers
an open problem by J.S. Guo and C.H. Wu.
Keywords:lattice dynamical system, competition model, traveling wavefront, stability Categories:34A33, 34K20, 92D25 

5. CMB 2017 (vol 61 pp. 240)
 Bu, Shangquan; Cai, Gang

HÃ¶lder Continuous Solutions of Degenerate Differential Equations with Finite Delay
Using known operatorvalued Fourier multiplier results on vectorvalued
HÃ¶lder continuous function spaces $C^\alpha (\mathbb R; X)$, we completely
characterize the $C^\alpha$wellposedness of the first order
degenerate differential equations with finite delay $(Mu)'(t)
= Au(t) + Fu_t + f(t)$ for $t\in\mathbb R$
by the boundedness of the $(M, F)$resolvent of $A$ under suitable
assumption on the delay operator $F$, where $A, M$ are closed
linear
operators on a Banach space $X$ satisfying $D(A)\cap D(M) \not=\{0\}$,
the delay operator $F$ is a bounded linear operator
from $C([r, 0]; X)$ to $X$ and $r \gt 0$ is fixed.
Keywords:wellposedness, degenerate differential equation, $\dot{C}^\alpha$multiplier, HÃ¶lder continuous function space Categories:34N05, 34G10, 47D06, 47A10, 34K30 

6. CMB 2017 (vol 60 pp. 747)
 Huang, Yanhe; Sottile, Frank; Zelenko, Igor

Injectivity of Generalized Wronski Maps
We study linear projections on PlÃ¼cker space whose restriction
to the Grassmannian is a nontrivial branched
cover.
When an automorphism of the Grassmannian preserves the fibers,
we show that the Grassmannian is necessarily
of $m$dimensional linear subspaces in a symplectic vector
space of dimension $2m$, and the linear map is
the Lagrangian involution.
The Wronski map for a selfadjoint linear differential operator
and pole placement map for
symmetric linear systems are natural examples.
Keywords:Wronski map, PlÃ¼cker embedding, curves in Lagrangian Grassmannian, selfadjoint linear differential operator, symmetric linear control system, pole placement map Categories:14M15, 34A30, 93B55 

7. CMB 2017 (vol 60 pp. 381)
 Rousseau, C.

The Bifurcation Diagram of Cubic Polynomial Vector Fields on $\mathbb{C}\mathbb{P}^1$
In this paper we give the bifurcation diagram
of the family of cubic vector fields $\dot z=z^3+ \epsilon_1z+\epsilon_0$
for $z\in \mathbb{C}\mathbb{P}^1$, depending on the values of
$\epsilon_1,\epsilon_0\in\mathbb{C}$.
The bifurcation diagram is in $\mathbb{R}^4$, but its conic structure
allows describing it for parameter values in $\mathbb{S}^3$. There are
two open simply connected regions of structurally stable vector
fields separated by surfaces corresponding to bifurcations of
homoclinic connections between two separatrices of the pole at
infinity. These branch from the codimension 2 curve of double
singular points. We also explain the bifurcation of homoclinic
connection in terms of the description of Douady and Sentenac
of polynomial vector fields.
Keywords:complex polynomial vector field, bifurcation diagram, DouadySentenac invariant Categories:34M45, 32G34 

8. CMB 2017 (vol 60 pp. 364)
 Preda, Ciprian

On the Roughness of Quasinilpotency Property of Oneâparameter Semigroups
Let $\mathbf{S}:=\{S(t)\}_{t\geq0}$ be a C$_0$semigroup of quasinilpotent
operators
(i.e. $\sigma(S(t))=\{0\}$ for each $t\gt 0$).
In the dynamical systems theory the above quasinilpotency property
is equivalent
to a very strong concept of stability for the solutions of autonomous
systems.
This concept is frequently called superstability and weakens
the classical finite time extinction property
(roughly speaking, disappearing solutions).
We show that under some assumptions, the quasinilpotency, or
equivalently, the superstability property
of a C$_0$semigroup is preserved under the perturbations of
its infinitesimal generator.
Keywords:oneparameter semigroups, quasinilpotency, superstability, essential spectrum Categories:34D05, 34D10, 34E10 

9. CMB 2016 (vol 59 pp. 849)
 Nah, Kyeongah; Röst, Gergely

Stability Threshold for Scalar Linear Periodic Delay Differential Equations
We prove that for the linear scalar delay differential
equation
$$ \dot{x}(t) =  a(t)x(t) + b(t)x(t1) $$
with nonnegative periodic coefficients of period $P\gt 0$, the
stability threshold for the trivial solution is
$r:=\int_{0}^{P}
\left(b(t)a(t)
\right)\mathrm{d}t=0,$
assuming that $b(t+1)a(t)$ does not change its sign. By constructing
a class of explicit examples, we show the counterintuitive result
that in general, $r=0$ is not a stability threshold.
Keywords:delay differential equation, stability, periodic system Categories:34K20, 34K06 

10. CMB 2016 (vol 59 pp. 363)
 Li, Dan; Ma, Wanbiao

Dynamical Analysis of a StageStructured Model for Lyme Disease with Two Delays
In this paper, a
nonlinear stagestructured model for Lyme disease is considered.
The model is a system of differential equations with two time
delays. The basic reproductive rate, $R_0(\tau_1,\tau_2)$, is
derived. If $R_0(\tau_1,\tau_2)\lt 1$, then the boundary equilibrium
is globally asymptotically stable. If $R_0(\tau_1,\tau_2)\gt 1$,
then there exists
a unique positive equilibrium whose local asymptotical stability
and the existence of
Hopf bifurcations are established by analyzing the distribution
of the characteristic values.
An explicit algorithm for determining the direction of Hopf bifurcations
and the
stability of the bifurcating periodic solutions is derived by
using the normal form and
the center manifold theory. Some numerical simulations are performed
to confirm the correctness
of theoretical analysis. At last, some conclusions are given.
Keywords:Lyme disease, stagestructure, time delay, Lyapunov functional stability Hopf bifurcation. Category:34D20 

11. CMB 2015 (vol 58 pp. 818)
 Llibre, Jaume; Zhang, Xiang

On the Limit Cycles of Linear Differential Systems with Homogeneous Nonlinearities
We consider the class of polynomial differential systems of the
form
$\dot x= \lambda xy+P_n(x,y)$, $\dot y=x+\lambda y+ Q_n(x,y),$ where
$P_n$ and $Q_n$ are homogeneous polynomials of degree $n$. For
this
class of differential systems we summarize the known results
for the
existence of limit cycles, and we provide new results for their
nonexistence and existence.
Keywords:polynomial differential system, limit cycles, differential equations on the cylinder Categories:34C35, 34D30 

12. CMB 2014 (vol 58 pp. 44)
13. CMB 2011 (vol 56 pp. 39)
 Ben Amara, Jamel

Comparison Theorem for Conjugate Points of a Fourthorder Linear Differential Equation
In 1961, J. Barrett showed that if the first conjugate point
$\eta_1(a)$ exists for the differential equation $(r(x)y'')''=
p(x)y,$ where $r(x)\gt 0$ and $p(x)\gt 0$, then so does the first
systemsconjugate point $\widehat\eta_1(a)$. The aim of this note is to
extend this result to the general equation with middle term
$(q(x)y')'$ without further restriction on $q(x)$, other than
continuity.
Keywords:fourthorder linear differential equation, conjugate points, systemconjugate points, subwronskians Categories:47E05, 34B05, 34C10 

14. CMB 2011 (vol 56 pp. 388)
 Mursaleen, M.

Application of Measure of Noncompactness to Infinite Systems of Differential Equations
In this paper we determine the Hausdorff measure of noncompactness on
the sequence space $n(\phi)$ of W. L. C. Sargent.
Further we apply
the technique of measures of noncompactness to the theory of infinite
systems of differential equations in the Banach sequence spaces
$n(\phi)$ and $m(\phi)$. Our aim is to present some existence results
for infinite systems of differential equations formulated with the help
of measures of noncompactness.
Keywords:sequence spaces, BK spaces, measure of noncompactness, infinite system of differential equations Categories:46B15, 46B45, 46B50, 34A34, 34G20 

15. CMB 2011 (vol 56 pp. 366)
 Kyritsi, Sophia Th.; Papageorgiou, Nikolaos S.

Multiple Solutions for Nonlinear Periodic Problems
We consider a nonlinear periodic problem driven by a
nonlinear nonhomogeneous differential operator and a
CarathÃ©odory reaction term $f(t,x)$ that exhibits a
$(p1)$superlinear growth in $x \in \mathbb{R}$
near $\pm\infty$ and near zero.
A special case of the differential operator is the scalar
$p$Laplacian. Using a combination of variational methods based on
the critical point theory with Morse theory (critical groups), we
show that the problem has three nontrivial solutions, two of which
have constant sign (one positive, the other negative).
Keywords:$C$condition, mountain pass theorem, critical groups, strong deformation retract, contractible space, homotopy invariance Categories:34B15, 34B18, 34C25, 58E05 

16. CMB 2011 (vol 56 pp. 102)
 Kong, Qingkai; Wang, Min

Eigenvalue Approach to Even Order System Periodic Boundary Value Problems
We study an even order system boundary value problem with
periodic boundary conditions. By establishing
the existence of a positive eigenvalue of an associated linear system
SturmLiouville problem, we obtain new conditions for the boundary
value problem to have a positive solution. Our major tools are the
KreinRutman theorem for linear spectra and the fixed point index theory
for compact operators.
Keywords:Green's function, high order system boundary value problems, positive solutions, SturmLiouville problem Categories:34B18, 34B24 

17. CMB 2011 (vol 55 pp. 214)
 Wang, DaBin

Positive Solutions of Impulsive Dynamic System on Time Scales
In this paper, some criteria for the existence of positive solutions of a class
of systems of impulsive dynamic equations on time scales are obtained by
using a fixed point theorem in cones.
Keywords:time scale, positive solution, fixed point, impulsive dynamic equation Categories:39A10, 34B15 

18. CMB 2011 (vol 55 pp. 285)
 Eloe, Paul W.; Henderson, Johnny; Khan, Rahmat Ali

Uniqueness Implies Existence and Uniqueness Conditions for a Class of $(k+j)$Point Boundary Value Problems for $n$th Order Differential Equations
For the $n$th order nonlinear differential equation, $y^{(n)} = f(x, y, y',
\dots, y^{(n1)})$, we consider uniqueness implies uniqueness and existence
results for solutions satisfying certain $(k+j)$point
boundary conditions for $1\le j \le n1$ and $1\leq k \leq nj$. We
define $(k;j)$point unique solvability in analogy to $k$point
disconjugacy and we show that $(nj_{0};j_{0})$point
unique solvability implies $(k;j)$point unique solvability for $1\le j \le
j_{0}$, and $1\leq k \leq nj$. This result is
analogous to
$n$point disconjugacy implies $k$point disconjugacy for $2\le k\le
n1$.
Keywords:boundary value problem, uniqueness, existence, unique solvability, nonlinear interpolation Categories:34B15, 34B10, 65D05 

19. CMB 2011 (vol 55 pp. 736)
20. CMB 2011 (vol 55 pp. 400)
 Sebbar, Abdellah; Sebbar, Ahmed

Eisenstein Series and Modular Differential Equations
The purpose of this paper is to solve various differential
equations having Eisenstein series as coefficients using various tools and techniques. The solutions
are given in terms of modular forms, modular functions, and
equivariant forms.
Keywords:differential equations, modular forms, Schwarz derivative, equivariant forms Categories:11F11, 34M05 

21. CMB 2011 (vol 54 pp. 506)
 Neamaty, A.; Mosazadeh, S.

On the Canonical Solution of the SturmLiouville Problem with Singularity and Turning Point of Even Order
In this paper, we are going to investigate the canonical property of solutions of
systems of differential equations having a singularity and turning
point of even order. First, by a replacement, we transform the system
to the SturmLiouville equation with turning point. Using of the
asymptotic estimates provided by Eberhard, Freiling, and Schneider
for a special fundamental system of solutions of the SturmLiouville
equation, we study the infinite product representation of solutions of the systems. Then we
transform the SturmLiouville equation with
turning point to the
equation with singularity, then we study the asymptotic behavior of its solutions. Such
representations are relevant to the inverse spectral problem.
Keywords:turning point, singularity, SturmLiouville, infinite products, Hadamard's theorem, eigenvalues Categories:34B05, 34Lxx, 47E05 

22. CMB 2011 (vol 55 pp. 882)
 Xueli, Song; Jigen, Peng

Equivalence of $L_p$ Stability and Exponential Stability of Nonlinear Lipschitzian Semigroups
$L_p$ stability and exponential stability are two important concepts
for nonlinear dynamic systems. In this paper, we prove that a
nonlinear exponentially bounded Lipschitzian semigroup is
exponentially stable if and only if the semigroup is $L_p$ stable
for some $p>0$. Based on the equivalence, we derive two sufficient
conditions for exponential stability of the nonlinear semigroup. The
results obtained extend and improve some existing ones.
Keywords:exponentially stable, $L_p$ stable, nonlinear Lipschitzian semigroups Categories:34D05, 47H20 

23. CMB 2011 (vol 55 pp. 3)
24. CMB 2011 (vol 54 pp. 580)
 Baoguo, Jia; Erbe, Lynn; Peterson, Allan

Kiguradzetype Oscillation Theorems for Second Order Superlinear Dynamic Equations on Time Scales
Consider the second order superlinear dynamic equation
\begin{equation*}
(*)\qquad
x^{\Delta\Delta}(t)+p(t)f(x(\sigma(t)))=0\tag{$*$}
\end{equation*}
where $p\in C(\mathbb{T},\mathbb{R})$, $\mathbb{T}$ is a time scale,
$f\colon\mathbb{R}\rightarrow\mathbb{R}$ is
continuously differentiable and satisfies $f'(x)>0$, and $xf(x)>0$ for
$x\neq 0$. Furthermore, $f(x)$ also satisfies a superlinear condition, which
includes the nonlinear function $f(x)=x^\alpha$ with $\alpha>1$, commonly
known as the EmdenFowler case. Here the coefficient function $p(t)$ is
allowed to be negative for arbitrarily large values of $t$. In addition to
extending the result of Kiguradze for \eqref{star1} in the real case $\mathbb{T}=\mathbb{R}$, we
obtain analogues in the difference equation and $q$difference equation cases.
Keywords:Oscillation, EmdenFowler equation, superlinear Categories:34K11, 39A10, 39A99 

25. CMB 2010 (vol 54 pp. 527)
 Preda, Ciprian; Sipos, Ciprian

On the Dichotomy of the Evolution Families: A DiscreteArgument Approach
We establish a discretetime criteria guaranteeing the existence of an
exponential dichotomy in the continuoustime
behavior of an abstract evolution family. We prove that an evolution
family ${\cal U}=\{U(t,s)\}_{t
\geq s\geq 0}$ acting on a Banach space $X$ is uniformly
exponentially dichotomic (with respect to its continuoustime
behavior) if and only if the
corresponding difference equation with the inhomogeneous term from
a vectorvalued Orlicz sequence space $l^\Phi(\mathbb{N}, X)$
admits
a solution in the same $l^\Phi(\mathbb{N},X)$. The technique of
proof effectively eliminates the continuity hypothesis on the
evolution family (\emph{i.e.,} we do not assume that $U(\,\cdot\,,s)x$
or $U(t,\,\cdot\,)x$ is continuous on $[s,\infty)$, and respectively
$[0,t]$). Thus, some known results given by
Coffman and Schaffer, Perron, and Ta Li are extended.
Keywords:evolution families, exponential dichotomy, Orlicz sequence spaces, admissibility Categories:34D05, 47D06, 93D20 
