Title and Author(s) 
Page 

Existence of multiple solutions of some second order impulsive differential equations
Jing Xiao, Juan J. Nieto and Zhiguo Luo
ABSTRACT.
This paper uses critical point
theory and variational methods to investigate the multiple solutions
of a boundary value problem for second order impulsive differential
equations. The conditions for the existence of multiple solutions
are established.


287


Existence of a solution to a nonmonotone dynamic model in poroplasticity with mixed boundary conditions
Sebastian Owczarek
ABSTRACT.
In this note, we investigate a nonmonotone and noncoercive dynamic model of poroplasticity with mixed boundary conditions. The existence of the solution to this nonmonotone model, where the inelastic constitutive equation is satisfied in the sense of Young measures, is proved using the coercive and monotone approximations.


297


The existence of nontrivial critical point for a class of strongly indefinite asymptotically
quadratic functional without compactness
Guanggang Liu, Shaoyun Shi and Yucheng Wei
ABSTRACT.
In this paper, we show the existence of nontrivial
critical point for a class of
strongly indefinite asymptotically quadratic functional without
compactness,
by using the technique of penalized functionals and an infinite dimensional Morse theory developed by Kryszewski and
Szulkin. Two applications are given to Hamiltonian systems and
elliptic systems.


323


Approximate controllability of fractional functional equations with infinite delay
Rathinasamy Sakthivel, Ramakrishnan Ganesh and Nazim I. Mahmudov
ABSTRACT.
Fractional differential equations have been used for constructing
many mathematical models in science and engineering. In this paper,
we study the approximate controllability results for a class of
impulsive fractional differential equations with infinite delay.
A new set of sufficient conditions are formulated and proved for
achieving the required result. In particular, the results are
established under the natural assumptions that the corresponding
linear system is approximately controllable. The results are
obtained by using the fractional calculus, solution operators and
fixed point technique. An example is also provided to illustrate the
theory. Further, as a corollary, exact controllability result is
discussed without assuming compactness of characteristic solution
operators.


345


Nonconvex retracts and computation for fixed point index in cones
Guowei Zhang and Pengcheng Li
ABSTRACT.
In this paper we construct two retracts in a cone by nonnegative functionals of convex and concave types, and an example is given to illustrate that the retracts are nonconvex. Then the nonconvex retracts are used to compute the fixed point index for the completely continuous operator in the domains $D_1\cap D_2$ and $D_1\cup D_2$, where $D_1$ and $D_2$ are bounded open sets in the cone. The computation for fixed point index can be applied to the existence and the more precise location of positive fixed points.


365


Signchanging critical points for noncoercive functionals
Yaotian Shen, Zhouxin Li and Youjun Wang
ABSTRACT.
We study the existence of infinitely many signchanging critical points and nonexistence of critical points to a class of noncoercive functionals.


273


Existence of solutions of singularly perturbed Hamiltonian systems with nonlocal nonlinearities
Minbo Yang and Yuanhong Wei
ABSTRACT.
In the present paper we study singularly perturbed Hamiltonian elliptic systems with nonlocal nonlinearities
\vr^2\Delta u +V(x)u =\bigg(\int_{\R^N} \frac{z^{p}}{xy^{\mu}}\,dy\bigg)z^{p2}u,
\vr^2\Delta v +V(x)v =\bigg(\int_{\R^N} \frac{z^{p}}{xy^{\mu}}\,dy\bigg)z^{p2}v,
where $z=(u,v)\in H^1(\R^N,\R^2)$, $V(x)$ is a continuous real function on $\R^N$, $0<\mu2$ and $2+({2\mu})/({N2})


385


Periodic solutions for nonlinear differential systems: the second order bifurcation function
Adriana Buica, Jaume Gine and Jaume Llibre
ABSTRACT.
We are concerned here with the classical problem of Poincar\'{e} of
persistence of periodic solutions under small perturbations. The
main contribution of this work is to give the expression of the
second order bifurcation function in more general hypotheses than
the ones already existing in the literature. We illustrate our main
result constructing a second order bifurcation function for the
perturbed symmetric Euler top.


403


Nodal solutions for nonlinear nonhomogeneous Neumann equations
Sergiu Aizicovici, Nikolaos S. Papageorgiou and Vasile Staicu
ABSTRACT.
We consider a nonlinear Neumann problem driven by a nonhomogeneous
differential operator with a Caratheodory reaction which is $(p1)$sublinear near $\pm\infty$. Using variational tools we show
that the problem has at least three nontrivial smooth solutions (one positive,
one negative and a third nodal). Our formulation unifies problems driven by
the $p$Laplacian, the $(p,q) $ Laplacian and the $p$generalized mean curvature operator.


421


Quasilinear elliptic equations with singular potentials and bounded discontinuous nonlinearities
Hongrui Cai, Anran Li and Jiabao Su
ABSTRACT.
In this paper we study the quasilinear equation
 \text{div}(\nabla u^{p2} \nabla u)+V(x)u^{p2} u=
Q(x)f(u), & x\in \mathbb{R}^N,
u(x)\rightarrow 0,\quad x\rightarrow \infty.
with singular radial potentials $V$, $Q$ and bounded measurable function $f$.
The approaches used here are based on a compact embedding from
the space $W^{1,p}_r(\mathbb{R}^N; V)$ into $L^1 (\mathbb{R}^N; Q)$ and a new multiple critical point theorem for locally Lipschitz continuous functionals.


439


Nonlinear parabolic boundary value problems of infinite order
Moussa Chrif, Mohamed Housseine Abdou and Said El Manouni
ABSTRACT.
In this paper an existence result is presented for solution of a parabolic boundary value problem under Dirichlet null boundary
conditions for a class of general equations of infinite order with
strongly nonlinear perturbation terms.


451


Pseudodifferential parabolic equations; two examples
Tomasz Dlotko and Maria B. Kania and Chunyou Sun
ABSTRACT.
The paper is devoted to local and global solvability and existence
of a global attractor for an exemplary 'parabolic' problem
containing fractional powers of the minus Laplace operator. We want
to compare, which properties of the similar semilinear heat equation
are preserved when we replace the pure minus Laplace operator by a
fractional power of that operator. Several useful technical tools
and estimates are collected in that paper.


463


Multiplicity results to a class of variationalhemivariational inequalities
Gabriele Bonanno and Patrick Winkert
ABSTRACT.
This paper deals with variationalhemivariational inequalities involving the $p$Laplace operator and a nonlinear Neumann boundary condition. Based on an abstract critical point result, which is developed at the beginning of the paper, it is shown the existence of at least three solutions to such inequalities whereby the cases $p>N$ and $p \leq N$ are treated separately. The applicability of these results is emphasized with suitable examples.


493


The effect of diffusion on critical quasilinear elliptic problems
Renato Jose de Moura and Marcos Montenegro
ABSTRACT.
We discuss the role of the diffusion coefficient $a(x)$ on the existence of
a positive solution for the quasilinear elliptic problem involving critical exponent
 \text{div}( a(x) \nabla u^{p2} \nabla u) = u^{p^*  1} + \lambda u^{p1} & \text{in } \Omega,
u = 0 & \text{on } \partial\Omega,
where $\Omega$ is a smooth bounded domain in $\R^n$, $n \geq 2$, $1 < p < n$, $p^* = np/(np)$ is the critical exponent from the
viewpoint of Sobolev embedding, $\lambda$ is a real parameter and $a\colon \overline{\Omega} \rightarrow \R$ is a positive continuous function. We prove that if the function $a(x)$ has an interior global minimum point $x_0$ of order $\sigma$, then the range of values $\lambda$ for which the problem above has a positive solution relies strongly on $\sigma$. We discover in particular that the picture changes drastically from $\sigma > p$ to $\sigma \leq p$. Some sharp answers are also provided.


517


Existence of solutions for a fractional hybrid boundary value problem via measures of noncompactness in Banach algebras
Josefa Caballero, Mohamed Abdalla Darwish and Kishin Sadarangani
ABSTRACT.
We study the existence of solutions for the following fractional hybrid boundary value problem
D_{0^+}^{\alpha}\bigg[\frac{x(t)}{f(t,x(t))}\bigg]+g(t,x(t))=0, &0
x(0)=x(1)=0,
where $1<\alpha\leq 2$ and $D_{0^+}^{\alpha}$ denotes the RiemannLiouville fractional derivative. The main tool is our study is
the technique of measures of noncompactness in the Banach algebras. Some examples are presented to illustrate our results. Finally, we compare the results of paper with the ones obtained by other authors.


535
