In this paper, we first give some bifurcation results near the origin for the $p$-Laplacian equation, then multiple solutions are obtained based on the combination of perturbation methods in critical groups and minimax methods.

Let $X$, $Y$, $S$ be closed connected surfaces and $\tau \times \beta$ a diagonal involution on $X \times Y$ where $\tau$ and $\beta$ are free involutions on $X$ and $Y$, respectively. In this work we study when the triple $(X \times Y, \tau \times \beta, S)$ satisfies the Borsuk-Ulam property. The problem is formulated in terms of an algebraic diagram, involving the 2-string braid group $B_{2}(S)$.

In this article, we investigate the existence of solutions for a coupled differential system of fractional differential equations at resonance. The existence results of solutions are obtained by using the coincidence degree theory. In addition, an example is presented to demonstrate the application of our main results.

In this article, we study the existence of weak solutions and of ground state solutions using the Nehari manifold approach, and existence of infinitely many solutions using the fountain theorem and the dual fountain theorem for a class of doubly nonlocal Kirchhoff-Choquard type equations involving the variable-order fractional $p(\cdot)$-Laplacian operator. Here the nonlinearity does not satisfy the well known Ambrosetti-Rabinowitz type condition.

In this paper we obtain a rate of convergence for the asymptotic behavior of some semilinar parabolic problems with Dirichlet boundary conditions relatively to smooth perturbations of the domain. We will obtain a rate of convergence dependent on convergence of domains for eigenvalues, eigenfunctions, invariant manifolds and continuity of attractors.

This paper deals with nonlinear elliptic boundary value problems with complicated geometry of nonlinearities. A new method for obtaining multiple solutions based on a recursive use of the nonlinear generalized Rayleigh quotients to the split Nehari manifold into subsets without degeneracies is introduced. The method is applied to prove the multiplicity result for nonnegative solutions, as well as to find a ground state of elliptic boundary value problems with nonlinearities of polynomial type.

J. Hadamard studied the geometric properties of geodesic flows on surfaces of negative curvature, thus initiating ``Symbolic Dynamics". In this article, we follow the same geometric approach to study the geodesic trajectories of billiards in ``rational polygons'' on the hyperbolic plane. We particularly show that the billiard dynamics resulting thus are just 'Subshifts of Finite Type' or their dense subsets. We further show that 'Subshifts of Finite Typ' play a central role in subshift dynamics and while discussing the topological structure of the space of all subshifts, we demonstrate that they approximate any shift dynamics.

We establish semi-convergence of a non-autonomous bi-spatial random attractor towards to an autonomous attractor under the topology of the regular space when time-parameter goes to infinity, where the criteria are given by forward compactness of the attractor in the terminal space as well as forward convergence of the random dynamical system in the initial space. We then apply to both non-autonomous and autonomous stochastic 2D Navier-Stokes equations with general delays (including variable and distribution delays). The forward-pullback asymptotic compactness in the space of continuous Sobolev-valued functions is proved by the method of spectrum decomposition.

In the present paper, we concentrate on the following critical Kirchhoff type problem\begin{equation*}-\bigg(a+b\int_{\mathbb{R}^N}|\nabla u|^{2}dx\bigg)\triangle u+u=|u|^{2^*-2}u+\mu|u|^{p-2}u,\quad \text{in } \mathbb{R}^N,\end{equation*}where $N\geq 3$, $a, b> 0$, $p\in (2,\ 2^*)$ and $\mu$ is an arbitrary positive parameter. With the help of an equivalent transformation, we first obtain at least one ground state nodal solution with precisely two nodal domains for $N=3$, all $b> 0$ and $N\geq4$, $b> 0$ small enough. Moreover, we give a convergence property of ground state nodal solutions as $b\searrow 0$. Besides, we attain infinitely many nodal solutions for $N=3$, $p\in(4, 6)$, all $b> 0$ and $N\geq4$, $p\in (2,\ 2^*)$, $b> 0$ sufficiently small, and also establish nonexistence results of nodal solutions for $N\geq 4$ and $b$ large enough.

In this paper we are interested in the following nonlocal elliptic equation$$\begin{cases}\displaystyle -\Delta u=\lambda_{k}u+\bigg[\frac{1}{|x|^{\mu}}\ast G(x,u)\bigg]g(x,u)&\mbox{in } \Omega,\\ u=0 &\mbox{on } \partial\Omega,\end{cases}$$where an open $\Omega\subset\mathbb{R}^{2}$ is bounded with smooth boundary. The nonlinearity $g(x,s)$ has the critical exponential growth in the sense of the Trudinger-Moser inequality and $\lambda_{k}$ denotes the $k$th eigenvalue of $(-\Delta,H_{0}^{1}(\Omega))$, $k\geq2$. Employing variational methods we prove the existence of a nontrivial solution for this nonlocal elliptic problem.

Motivated by Lazer-Leach type results, we study the existence of periodic solutions for systems of functional-differential equations at resonance with an arbitrary even-dimensional kernel and linear deviating terms involving a general delay of the form $\int_0^{2\pi}u(t+s)d\lambda(s)$, where $\lambda$ is a finite regular signed measure. Our main technique shall be the Coincidence Degree Theorem due to Mawhin.

In this paper we deal with the Banach space $C_b^m[0, + \infty)$ of all $m$-times continuously derivable, bounded with all derivatives up to the order $m$, real functions defined on $[0,+ \infty)$. We prove, for any $\epsilon > 0$, the existence of a new proper $k$-ball-contractive retraction with $k < 1+ \epsilon$ of the closed unit ball of the space onto its boundary, so that the Wośko constant $W_\gamma (C_b^m[0, + \infty))$ is equal to $1$.

It is natural to ask whether the weak fixed point property for nonexpansive mappings in Banach spaces is inherited by other generalized nonexpansive mappings without using weak normal structure or close-to normal structure (also called quasi-normal structure) (see C.S. Wong, Close-to-normal structure and its applications, J. Func. Anal. 16 (1974), no. 4, 353-358). In this paper, we give an affirmative answer to this question for Suzuki mappings and other generalized nonexpansive mappings in the setting of various Banach spaces. In addition, we prove the existence of common fixed points for commuting affine $(c)$-mappings and Suzuki mappings acting on convex bounded $L^{0}$-closed subsets in the Banach space $L^{1}(\Omega, \Sigma, \mu)$.

The existence of the solutions with prescribed $L^{p}$-norm for a fractional $p$-Laplacian equation is investigated in this paper. The obtained result is suitable for all the order of the derivative $0< s< 1$ and $p> 1$, which extends the previous results for $s=1$ or $p=2$. In particular, to the best of our knowledge, as the $L^{p}$-subcritical or $L^{p}$-critical constrained minimization problem for fractional $p$-Laplacian equation, the critical exponent $({pN+p^{2}s})/{N}$ is properly established for the first time. On one hand, using Lions Vanishing Lemma and Brézis-Lieb Lemma, the compactness of minimizing sequences for the related constrained minimization problem is derived, then based on which the existence of constrained minimizers is achieved. On the other hand, the existence of weak solution and the nonexistence result are also provided.

An evolution inclusion driven by a time-dependent maximal monotone operator and a Lipschitz closed valued perturbation, in a separable Hilbert space is considered. The inclusion with a convexified perturbation term is also studied. Then, the existence of solutions and the relaxation property between these evolution inclusions are proved. Applications to dynamical systems governed by a couple of a fractional equation and an evolution inclusion involving time-dependent maximal monotone operators with a Lipschitz perturbation are presented.

In this partly expository paper we discuss conditions for the global injectivity of $C^2$ semi-algebraic local diffeomorphisms $f\colon \mathbb{R}^n \to \mathbb{R}^n$. In case $n> 2$, we consider the foliations of $\mathbb{R}^n$ defined by the level sets of each $n-2$ projections of $f$, i.e. the maps $\mathbb{R}^n \to \mathbb{R}^{n-2}$ obtained by deleting two coordinate functions of $f$. It is known that if the set of non-proper points of $f$ has codimension greater than or equal to $2$ and the leaves of the above-defined foliations are simply connected, then $f$ is bijective. In this work we relate this simply connectedness with the notion of locally trivial fibrations. Then some computable regularity conditions at infinity ensuring such simply connectedness are presented. Further, we provide an equivalent statement of the Jacobian conjecture by using fibrations. By means of examples we prove that the results presented here are different from a previous result based on a spectral hypothesis. Our considerations are also applied to discuss the behaviour of some conditions when $f$ is composed with linear isomorphisms: this is relevant due to some misunderstandings appearing in the literature.

For $p \in (1, \infty)$, for an integer $N \geq 2$ and for a bounded Lipschitz domain $\Omega \subset \mathbb R^N$, we consider the following nonlinear Steklov bifurcation problem$$-\Delta_p \phi = 0 \quad \text{in } \Omega, \qquad |\nabla \phi|^{p-2} \frac{\partial \phi}{\partial \nu} = \lambda \big( g |\phi|^{p-2}\phi + f r(\phi) \big) \quad \text{on } \partial \Omega,$$where $\Delta_p$ is the $p$-Laplace operator, $g,f \in L^1(\partial \Omega)$ are indefinite weight functions and $r \in C(\mathbb R)$ satisfies $r(0)=0$ and certain growth conditions near zero and at infinity. For $f$, $g$ in some appropriate Lorentz-Zygmund spaces, we establish the existence of a continuum that bifurcates from $(\lambda_1,0)$, where $\lambda_1$ is the first eigenvalue of the following nonlinear Steklov eigenvalue problem$$-\Delta_p \phi = 0 \quad \text{in } \Omega, \qquad |\nabla \phi|^{p-2} \frac{\partial \phi}{\partial \nu} = \lambda g |\phi|^{p-2}\phi \quad \text{on } \partial \Omega.$$