Assume that a cubic polynomial differential system in the plane has four invariant straight lines in generic position, i.e., they are not parallel and no more than two straight lines intersect in a point. Then such a differential system only can have $0$, $1$ or $3$ centers.

In this paper, the existence and exponential stability of anti-periodic solutions for fuzzy BAM neural networks with inertial terms and time-varying delays is investigated. Firstly, some sufficient conditions ensuring the existence of anti-periodic solutions of the system are obtained by employing a new continuation theorem of coincidence degree theory. Secondly, by constructing an appropriate Lyapunov function, some sufficient conditions are derived to guarantee the global exponential stability of anti-periodic solutions of the system. Our results of this paper are completely new. Finally, two numerical examples are given to show the effectiveness of the obtained results.

In the paper we study a class of semilinear differential variational systems with nonlocal boundary conditions, which are obtained by mixing evolution equations and generalized variational inequalities. Firstly, we show the properties of the solution set for generalized variational inequalities. Then, the existence results are established and proved mainly by the topological degree theory and the Yosida approximations of the generator of $C_0$-semigroup.

In this paper we study the existence and orbital stability of the Choquard equation with partial confinement. This type equation originates from Fröhlich and Pekar's model of the polaron, where free electrons in an ionic lattice interact with phonons associated with deformations of the lattice or with the polarisation that it creates on the medium (interaction of an electron with its own hole). On the one hand, we prove the existence of global minimizer of the associate energy functional subject to the $L^2$-constraint. On the other hand, we discuss the orbital stability and asymptotic behavior of the global minimizer.

In this paper notions of strong specification property and quasi-weak specification property for non-autonomous discrete systems are introduced and studied. It is shown that these properties are dynamical properties and are preserved under finite product. It is proved that a $k$-periodic non-autonomous system on intervals having weak specification is Devaney chaotic. Moreover, it is shown that if the system has strong specification then the result is true in general. Specification properties of induced systems on hyperspaces and probability measures spaces are also studied. Examples/counterexamples are provided wherever necessary to support results obtained.

This paper investigates the existence of mild solutions for fractional stochastic evolution hemivariational inequalities and optimal controls. An existence theorem concerned with the mild solution for the presented system is proved by means of the fractional calculation, stochastic analysis theory, Bohnenblust-Karlin fixed point theorem and some properties of the Clarke subdifferential. Moreover, an existence result of optimal control pair that governed by a fractional stochastic evolution hemivariational inequality is also obtained. Finally, an example is given for demonstration.

Using the homological link theorem and iteration inequalities of Maslov-type index, we prove the multiplicity of subharmonic solutions for some variant subquadratic non-autonomous Hamiltonian systems. Moreover, the minimal period problem has also been considered for the variant subquadratic autonomous Hamiltonian systems.

We incorporate the notion of a distal system into the continuum theory [14] through the notion of the continuum-wise distal homeomorphism. Results concerning distal homeomorphisms will be generalized to the case of cw-distal homeomorphisms. Notions of cw-distality for measures will be studied. We also analyze the variation of distality for flows obtained by making the proximal cell [1] to depend on a given subset of the full set of reparametrizations. Some properties of these reparametrized distality will be obtained.

We give a different and possibly more accessible proof of a general Borsuk-Ulam theorem for a product of spheres, originally due to Ramos. That is, we show the non-existence of certain $(\mathbb Z/2)^k$-equivariant maps from a product of $k$ spheres to the unit sphere in a real $(\mathbb Z/2)^k$-representation of the same dimension. Our proof method allows us to derive Borsuk-Ulam theorems for certain equivariant maps from Stiefel manifolds, from the corresponding results about products of spheres, leading to alternative proofs and extensions of some results of Fadell and Husseini.

In this paper we prove the existence of multiple periodic (harmonic and subharmonic) solutions for a class of planar Hamiltonian systems which includes the case of the second order scalar ODE $x'' + a(t)g(x) = 0$ with $g$ satisfying a one-sided condition of sublinear type. We consider the classical approach based on the Poincaré-Birkhoff fixed point theorem as well as some refinements on the side of the theory topological horseshoes. A Duffing-type equation and an exponential nonlinearity case are studied as applications.

We consider the elliptic partial differential equation in the divergence form \[ -{\rm div}(\nabla G(\nabla u(x)))+ F_u(x,u(x))=0, \] where $G$ is a convex, anisotropic function satisfying certain growth and ellipticity conditions. We prove that weak solutions in $W^{1,G}$ are in fact of class $W^{2,2}_{\rm loc}\cap W^{1,\infty}_{\rm loc}$.

We provide a counter-example to Theorem 1.4(a) in [3], Topol. Methods Nonlinear Anal. 49 (2017), no. 1, 105-132, by showing that the closure of the $\Gamma$-orbit of a point $x$ in the pointwise basin of Ls-attraction of a quasi-attractor $A$ is not compact. In order to fix this gap, we modified the definition of Ls-basin of attraction. In addiction, we propose a better place to play the chaos game and as a consequence we get some additional results on strongly-fibred quasi-attractors and Conley attractors.

We consider a smooth submanifold $N$ with a smooth boundary in an ambient closed manifold $M$ and assign a spectral invariant $c(\alpha,H)$ to every singular homological class $\alpha\in H_*(N)$ and a Hamiltonian $H$ defined on the cotangent bundle $T^*M$. We also derive certain properties of spectral numbers, for example we prove that spectral invariants $c_\pm(H,N)$ associated to the whole Floer homology $HF_*(H,N:M)$ of the submanifold $N$, are the limits of decreasing nested family of open sets.

We provide some existence results for Sturm-Liouville boundary value problems associated with the planar differential system $Jz'=g(t,z) + r(t,z)$ where $g$ is suitably controlled by the gradient of two positively homogeneous functions of degree 2 and $r$ is sublinear with respect to the variable $z$ at infinity. We study the existence of solutions when a double resonance phenomenon occurs by the introduction of Landesman-Lazer type conditions. Applications to scalar second order differential equations are given.

In this paper, we study the interplay between the Orlicz-Sobolev spaces $L^{M}$ and $W^{1,M}$ and the fractional Sobolev spaces $W^{s,p}$. More precisely, we give some qualitative properties of a new fractional Orlicz-Sobolev space $W^{s,M}$, where $s\in (0,1)$ and $M$ is a Young function. We also study a related non-local operator, which is a fractional version of the nonhomogeneous $M$-Laplace operator. As an application, we prove existence of a weak solution for a non-local problem involving the new fractional $M$-Laplacian operator.

In this paper we study ergodic optimization problems for subadditive sequences of functions on a topological dynamical system. We prove that for $t\rightarrow \infty$ any accumulation point of a family of equilibrium states is a maximizing measure. We show that the Lyapunov exponent and entropy of equilibrium states converge in the limit $t\rightarrow \infty$ to the maximum Lyapunov exponent and entropy of maximizing measures. In the particular case of matrix cocycles we prove that the maximal Lyapunov exponent can be approximated by Lyapunov exponents of periodic trajectories under certain assumptions.