Title and Author(s) 
Page 

Positive solutions for nonlinear nonhomogeneous parametric equations
Nikolaos S. Papageorgiou and George Smyrlis
ABSTRACT.
We consider a nonlinear parametric Dirichlet problem driven by a
nonhomogeneous differential operator which includes as special
cases the $p$Laplacian, the $(p,q)$Laplacian and the
generalized $p$mean curvature operator. Using variational
methods, we prove a bifurcationtype theorem describing the
dependence of positive solutions on the parameter.


1


An averaging formula for Reidemeister traces
Jiyou Liu and Xuezhi Zhao
ABSTRACT.
In this note, we shall give an averaging formula for Reidemeister traces, which is a simple relation among Reidemeister traces of a selfmap and those of its liftings with respect to a finitefold regular covering.


17


Analytic invariant manifolds for nonautonomous equations
Luis Barreira and Claudià Valls
ABSTRACT.
We construct real analytic stable invariant manifolds for sufficiently small perturbations of a linear equation $v'=A(t)v$ admitting a~nonuniform exponential dichotomy. As a byproduct of our
approach we obtain an exponential control not only of the
trajectories on the invariant manifolds, but also of all their
derivatives.


29


On solutions of semilinear elliptic equation with linear growth nonlinearity in $\mathbb{R}^{N}$
Rong Cheng and Jianhua Hu
ABSTRACT.
We study nontrivial solutions for a class of semilinear elliptic
equation which could be resonant at infinity. We establish the
existence of solutions for the equation by considering the modified
nonresonant problem associated with the original equation through
Morse theory. Moreover, only linear growth assumption is imposed on
the nonlinearity and condition on the potential is weaker than the
coercive assumption.


45


A partial positive solution to a conjecture of Ricceri
Francisco Javier GarciaPacheco and Justin R. Hill
ABSTRACT.
In this manuscript we introduce a new class of convex sets called quasiabsolutely convex
and show that a Hausdorff locally convex topological vector space satisfies the weak
antiproximinal property if and only if every totally antiproximinal quasiabsolutely
convex subset is not rare. This improves results from \cite{GPtop} and provides a partial
positive solution to a Ricceri's Conjectured posed in \cite{R} with many applications to
the theory of partial differential equations. We also study the intrinsic structure of
totally antiproximinal convex subsets proving, among other things, that the absolutely
convex hull of a linearly bounded totally antiproximinal convex set must be finitely open.
Finally, a new characterization of barrelledness in terms of comparison of norms is provided.


57


The blowup and global existence of solutions of Cauchy problems for a time fractional diffusion equation
QuanGuo Zhang and HongRui Sun
ABSTRACT.
In this paper, we investigate the blowup and global existence of
solutions to the following time fractional nonlinear diffusion
equations
{_0^C D_t^\alpha u}\triangle u=u^{p1}u, & x\in \Bbb{R}^N,\ t>0,
u(0,x)=u_0(x), & x\in \Bbb{R}^N,
where $0<\alpha<1$, $p>1$, $u_0\in C_0(\Bbb{R}^N)$ and
${_0^CD_t^\alpha u}=({\partial}/{\partial
t}){_0^{}I_t^{1\alpha}(u(t,x)u_0(x))}$, ${_0^{}I_t^{1\alpha}}$
denotes left RiemannLiouville fractional integrals of order
$1\alpha$. We prove that if $1


69


A further generalization of midpoint convexity of multimaps towards common fixed point theorems and applications
Irene Benedetti and Anna Martellotti
ABSTRACT.
We furtherly generalize midpoint convexity for multivalued maps and derive Fixed Point Theorems and Common Fixed Point Theorems without requiring strong compactness. As an application we obtain some Best Approximation results, and minimax and variational inequalities.


93


On properties of solutions for a functional equation
Zeqing Liu and Shin Min Kang
ABSTRACT.
his paper studies properties of solutions for a
functional equation arising in dynamic programming of multistage
decision processes. Using the Banach fixed point theorem and the
Mann iterative methods, we prove the existence and uniqueness of
solutions and convergence of sequences generated by the Mann
iterative methods for the functional equation in the Banach spaces
$BC(S)$ and $B(S)$ and the complete metric space $BB(S)$, and
discuss behaviors of solutions for the functional equation in the
complete metric space $BB(S)$. Four examples illustrating the
results presented in this paper are also provided.


113


Eigenvalue, bifurcation and convex solutions for MongeAmpère equations
Guowei Dai, Ruyun Ma and Xiaoling Han
ABSTRACT.
In this paper we study the following eigenvalue boundary value problem for MongeAmp\`{e}re equations
\det(D^2u)=\lambda^N f(u)& \text{in } \Omega,
u=0 &\text{on } \partial \Omega.
We establish global bifurcation results for the problem with $f(u)=u^N+g(u)$ and $\Omega$ being the unit ball of $\mathbb{R}^N$.
More precisely, under some natural hypotheses on the perturbation function $g\colon[0,+\infty)\rightarrow[0,+\infty)$, we show that
$(\lambda_1,0)$ is a bifurcation point of the problem and there exists an unbounded continuum of convex solutions,
where $\lambda_1$ is the first eigenvalue of the problem with $f(u)=u^N$. As the applications of the above results, we consider
with determining interval of $\lambda$,
in which there exist convex solutions for this problem in unit ball.
Moreover, we also get some results about the existence and nonexistence of convex solutions for this problem on general domain by domain comparison method.


135


Existence of Anosov diffeomorphisms on infranilmanifolds modeled on free nilpotent Lie groups
Karel Dekimpe and Jonas Deré
ABSTRACT.
An infranilmanifold is a manifold which is constructed as a~quotient space $\Gamma\setminus G$ of a simply connected nilpotent Lie group $G$, where
$\Gamma$ is a discrete group acting properly discontinuously and cocompactly on~$G$ via so called affine maps. The manifold $\Gamma\setminus G$
is said to be modeled on the Lie group~$G$. This class of manifolds is conjectured to be the only class of closed manifolds allowing an Anosov diffeomorphism.
However, it is far from obvious which of these infranilmanifolds actually do admit an Anosov diffeomorphism. In this paper we completely solve this
question for infranilmanifolds modeled on a free $c$step nilpotent Lie group.


165


Partially symmetric solutions of the generalized Hénon equation in symmetric domains
Ryuji Kajikiya
ABSTRACT.
We study the generalized H\'{e}non equation in a symmetric domain $\Omega$.
Let $H$ and $G$ be closed subgroups of the orthogonal group such that
$H \varsubsetneq G$ and $\Omega$ is $G$ invariant.
Then we show the existence of a positive solution which is $H$ invariant but $G$ noninvariant
under suitable assumptions of $H$, $G$ and the coefficient function of the equation.


191


Index 1 fixed points of orientation reversing planar homeomorphisms
Francisco R. Ruiz del Portal and José M. Salazar
ABSTRACT.
Let \(U \subset {\mathbb R}^2\) be an open subset, \(f\colon U
\rightarrow f(U) \subset {\mathbb R}^2\) be an orientation
reversing homeomorphism and let \(0 \in U\) be an isolated, as a~periodic orbit, fixed point. The main theorem of this paper says
that if the fixed point indices \(i_{{\mathbb
R}^2}(f,0)=i_{{\mathbb R}^2}(f^2,0)=1\) then there exists an
orientation preserving dissipative homeomorphism $\varphi\colon
{\mathbb R}^2 \rightarrow {\mathbb R}^2$ such that \(f^2=\varphi\)
in a~small neighbourhood of \(0\) and \(\{0\}\) is a~global
attractor for \(\varphi\). As a corollary we have that for
orientation reversing planar homeomorphisms a~fixed point, which
is an isolated fixed point for \(f^2\), is asymptotically stable
if and only if it is stable. We also present an application to
periodic differential equations with symmetries where orientation
reversing homeomorphisms appear naturally.


223


A predatorprey model of Hollingtype II with state dependent impulsive effects
Changming Ding and Zhongxin Zhang
ABSTRACT.
We investigate a predatorprey model with
state dependent impulsive effects, which is based on a modified
version of the LeslieGower scheme and on the Hollingtype II
scheme. Using topological methods, we present some sufficient
conditions to guarantee the existence and asymptotical stability of
semitrivial periodic solutions and positive periodic solutions,
respectively.


247


Harmonic perturbations with delay of periodic separated variables differential equations
Luca Bisconti and Marco Spadini
ABSTRACT.
We study the structure of the set of harmonic solutions to perturbed, nonautonomous,
$T$periodic, separated variables ODEs on manifolds. The perturbing term, supposed to
be $T$periodic in time, is allowed to contain a finite delay.
Our main result extends those of \cite{FS09} and \cite{spaSepVar} but it cannot be
simply deduced from them: It emerges from of a combination of the techniques
exposed in those two papers.


261


Pregular nonlinear dynamics
Beata Medak and Alexey A. Tret'yakov
ABSTRACT.
In this paper we generalize the notion of $p$factor operator
which is the basic notion of the socalled $p$regularity theory for
nonlinear and degenerated operators. We prove a theorem related to
a new construction of $p$factor operator. The obtained results
are illustrated by an example concerning nonlinear dynamical
system.


283


On the stability of new impulsive differential equations
Jinrong Wang, Zeng Lin and Yong Zhou
ABSTRACT.
In this paper, we study new impulsive ordinary differential
equations and apply fixed point approach to establish existence and
uniqueness theorem and derive an interesting stability result in the
sense of generalized $\beta$UlamHyersRassias. At last, two
examples are given to demonstrate the applicability of our result.


303


Weak and strong convergence theorems for $m$generalized hybrid mappings in Hilbert spaces
Sattar Alizadeh and Fridoun Moradlou
ABSTRACT.
In this paper, we prove a weak convergence theorem of Ishikawa's
type for $m$generalized hybrid mappings in a Hilbert space. Further,
by using a new modification of Ishikawa iteration, we
prove a strong convergence theorem for $m$generalized hybrid
mappings in a Hilbert space.


315


Nontrivial solutions for a mixed boundary problem for Schrödinger equations with an external magnetic field
Claudianor O. Alves, Rodrigo C. M. Nemer and Sergio H. Monari Soares
ABSTRACT.
We study the existence of solutions for a class of nonlinear Schr\"{o}dinger equations involving a magnetic field with mixed
DirichletNeumann boundary conditions. We use LusternikShnirelman category and the Morse theory to estimate the number of nontrivial solutions in terms of the topology of the part of the boundary where the Neumann condition is prescribed.


329


Geometric proof of strong stable/unstable manifolds with application to the restricted three body problem
Maciej J. Capiński and Anna WasieczkoZając
ABSTRACT.
We present a method for establishing strong stable/unstable manifolds of fixed points for maps and ODEs. The method is based on cone conditions, suitably formulated to allow for application in computer assisted proofs. In the case of ODEs, assumptions follow from estimates on the vector field, and it is not necessary to integrate the system.
We apply our method to the restricted three body problem and show that for a given choice of the mass parameter, there exists a homoclinic orbit along matching strong stable/unstable manifolds of one of the libration points.


363


On the degree for oriented quasiFredholm maps: its uniqueness and its effective extension of the LeraySchauder degree
Alessandro Calamai, Pierlugi Benevieri and Massimo Furi
ABSTRACT.
In a previous paper, the first and third author developed a~degree theory for oriented locally compact perturbations of $C\sp{1}$ Fredholm maps of index zero between real Banach spaces.
In the spirit of a celebrated AmannWeiss paper, we prove that this degree is unique if it is assumed to satisfy three axioms: Normalization, Additivity and Homotopy invariance.
Taking into account that any compact vector field has a canonical orientation, from our uniqueness result we shall deduce that the above degree provides an effective extension of the LeraySchauder degree.


401


Positive solutions of onedimensional $p$Laplacian equations and applications to population models of one species
Kunquan Lan, Xiaojing Yang and Guangchong Yang
ABSTRACT.
We prove new results on the existence of
positive solutions of onedimensional $p$Laplacian equations
under sublinear conditions involving the first eigenvalues of the corresponding homogeneous Dirichlet boundary value problems.
To the best of our knowledge, this is the first paper to use fixed point index theory of compact maps to give criteria
involving the first eigenvalue for onedimensional $p$Laplacian equations with $p\ne 2$.
Our results generalize some previous results where either $p$ is required to be greater than $2$ or
the nonlinearities satisfy stronger conditions.
We shall apply our results to tackle a logistic population model
arising in mathematical biology.


431


Setvalued perturbation for time dependent subdifferential operator
Soumia Saïdi and Mustapha Fateh Yarou
ABSTRACT.
In a separable Hilbert space, we consider an evolution inclusion involving timedependent subdifferential of a proper convex lower semicontinuous function with a setvalued perturbation depending on both time and state variable. We prove, under a compactness condition on the perturbation, that there exists at least one absolutely continuous solution.


447


Hopfbifurcation theorem and stability for the magnetohydrodynamics equations
Weiping Yan
ABSTRACT.
This paper is devoted to the study of the dynamical behavior for the 3D viscous Magnetohydrodynamics equations. We first prove that this system under smooth external forces possesses time dependent
periodic solutions, bifurcating from a steady solution. If the time periodic solution is smooth, then the linear stability of the time periodic solution implies nonlinear stability is obtained in $\textbf{L}^p$ for all $p\in(3,\infty)$.


471


Equivalence and nonexistence of standing waves for coupled Schrodinger equations with ChernSimons gauge fields
Hyungjin Huh and Jinmyoung Seok
ABSTRACT.
This paper is devoted to the study of standing waves for socalled $N=2$ supersymmetric ChernSimonsSchrodinger equations, a coupled system of Schr\"odinger equations in which ChernSimons gauge fields are incorporated.
We show that there is no nontrivial standing wave to the $N=2$ supersymmetric ChernSimonsSchrodinger equations when the coupling constants are less than critical numbers.
We also prove that static $N=2$ supersymmetric ChernSimonsSchrodinger equations are equivalent to their first order selfdual system when the coupling constants are critical.


495
