We consider a nonlinear parametric Dirichlet problem driven by anonhomogeneous differential operator which includes as specialcases the $p$-Laplacian, the $(p,q)$-Laplacian and thegeneralized $p$-mean curvature operator. Using variationalmethods, we prove a bifurcation-type theorem describing thedependence of positive solutions on the parameter.

In this note, we shall give an averaging formula for Reidemeister traces, which is a simple relation among Reidemeister traces of a self-map and those of its liftings with respect to a finite-fold regular covering.

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 ourapproach we obtain an exponential control not only of thetrajectories on the invariant manifolds, but also of all theirderivatives.

We study nontrivial solutions for a class of semilinear ellipticequation which could be resonant at infinity. We establish theexistence of solutions for the equation by considering the modifiednon-resonant problem associated with the original equation throughMorse theory. Moreover, only linear growth assumption is imposed onthe nonlinearity and condition on the potential is weaker than thecoercive assumption.

In this manuscript we introduce a new class of convex sets called quasi-absolutely convex and show that a Hausdorff locally convex topological vector space satisfies the weak anti-proximinal property if and only if every totally anti-proximinal quasi-absolutely convex subset is not rare. This improves results from [F.J. García-Pacheco, An approach to a Ricceri's conjecture, Top. Appl. 159 (2012), 3307-3313] and provides a partial positive solution to a Ricceri's Conjecture posed in [B. Ricceri, Topological problems in nonlinear and functional analysis, Open Problemsin Topology II (E. Pearl, ed.), Elsevier, 2007, 585-593] with many applications to the theory of partial differential equations. We also study the intrinsic structure of totally anti-proximinal convex subsets proving, among other things, that the absolutely convex hull of a linearly bounded totally anti-proximinal convex set must be finitely open. Finally, a new characterization of barrelledness in terms of comparison of norms is provided.

In this paper, we investigate the blow-up and global existence ofsolutions to the following time fractional nonlinear diffusionequations\begin{equation*}\begin{cases}{_0^C D_t^\alpha u}-\triangle u=|u|^{p-1}u, & x\in \mathbb{R}^N,\ t > 0,\\u(0,x)=u_0(x), & x\in \mathbb{R}^N,\end{cases}\end{equation*}where $0 < \alpha < 1$, $p > 1$, $u_0\in C_0(\mathbb{R}^N)$ and${_0^CD_t^\alpha u}=({\partial}/{\partialt}){_0^{}I_t^{1-\alpha}(u(t,x)-u_0(x))}$, ${_0^{}I_t^{1-\alpha}}$denotes left Riemann-Liouville fractional integrals of order$1-\alpha$. We prove that if $1 < p < 1+2/{N}$, then everynontrivial nonnegative solution blow-up in finite time, and if$p\geq 1+2/{N}$ and $\|u_0\|_{L^{q_c}(\mathbb{R}^N)}$,$q_c=N(p-1)/{2}$ is sufficiently small, then the problem hasglobal solution.

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.

This paper studies properties of solutions for afunctional equation arising in dynamic programming of multistagedecision processes. Using the Banach fixed point theorem and theMann iterative methods, we prove the existence and uniqueness ofsolutions and convergence of sequences generated by the Manniterative methods for the functional equation in the Banach spaces$BC(S)$ and $B(S)$ and the complete metric space $BB(S)$, anddiscuss behaviors of solutions for the functional equation in thecomplete metric space $BB(S)$. Four examples illustrating theresults presented in this paper are also provided.

In this paper we study the following eigenvalue boundary value problem for Monge-Ampère equations\begin{equation*}\begin{cases}\det(D^2u)=\lambda^N f(-u)& \text{in } \Omega,\\u=0 &\text{on } \partial \Omega.\end{cases}\end{equation*}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.

An infra-nilmanifold 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 infra-nilmanifolds actually do admit an Anosov diffeomorphism. In this paper we completely solve this question for infra-nilmanifolds modeled on a free $c$-step nilpotent Lie group.

We study the generalized Hénon 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 $G$ non-invariant under suitable assumptions of $H,G$ and the coefficient function of the equation.

Let \(U \subset {\mathbb R}^2\) be an open subset, \(f\colon U\rightarrow f(U) \subset {\mathbb R}^2\) be an orientationreversing homeomorphism and let \(0 \in U\) be an isolated, as a periodic orbit, fixed point. The main theorem of this paper saysthat if the fixed point indices \(i_{{\mathbbR}^2}(f,0)=i_{{\mathbb R}^2}(f^2,0)=1\) then there exists anorientation 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 globalattractor for \(\varphi\). As a corollary we have that fororientation reversing planar homeomorphisms a fixed point, whichis an isolated fixed point for \(f^2\), is asymptotically stableif and only if it is stable. We also present an application toperiodic differential equations with symmetries where orientationreversing homeomorphisms appear naturally.

We investigate a predator-prey model withstate dependent impulsive effects, which is based on a modifiedversion of the Leslie-Gower scheme and on the Holling-type IIscheme. Using topological methods, we present some sufficientconditions to guarantee the existence and asymptotical stability ofsemi-trivial periodic solutions and positive periodic solutions,respectively.

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 [M. Furi and M. Spadini, Periodic perturbations with delay of autonomous differential equations on manifolds, Adv. Nonlinear Stud. 9 (2009), No. 2, 263-276] and [M. Spadini, Harmonic solutions to perturbations of periodic separated variables ODEs on manifolds, Electron. J. Differential Equations, 2003 (2003), No. 88, 1-11] but it cannot be simply deduced from them: It emerges from of a combination of the techniques exposed in those two papers.

In this paper we generalize the notion of $p$-factor operatorwhich is the basic notion of the so-called $p$-regularity theory fornonlinear and degenerated operators. We prove a theorem related toa new construction of $p$-factor operator. The obtained resultsare illustrated by an example concerning nonlinear dynamicalsystem.

In this paper, we study new impulsive ordinary differentialequations and apply fixed point approach to establish existence anduniqueness theorem and derive an interesting stability result in thesense of generalized $\beta$-Ulam-Hyers-Rassias. At last, twoexamples are given to demonstrate the applicability of our result.

In this paper, we prove a weak convergence theorem of Ishikawa'stype for $m$-generalized hybrid mappings in a Hilbert space. Further,by using a new modification of Ishikawa iteration, weprove a strong convergence theorem for $m$-generalized hybridmappings in a Hilbert space.

We study the existence of solutions for a class of nonlinear Schrödinger equations involving a magnetic field with mixed Dirichlet-Neumann boundary conditions. We use Lusternik-Shnirelman 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.

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.

In a previous paper, the first and third author developed a degree theory for oriented locally compact perturbations of $C^1$ Fredholm maps of index zero between real Banach spaces. In the spirit of a celebrated Amann-Weiss 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 Leray-Schauder degree.

We prove new results on the existence ofpositive solutions of one-dimensional $p$-Laplacian equationsunder 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 criteriainvolving the first eigenvalue for one-dimensional $p$-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 modelarising in mathematical biology.

In a separable Hilbert space, we consider an evolution inclusion involving time-dependent subdifferential of a proper convex lower semicontinuous function with a set-valued 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.

This paper is devoted to the study of the dynamical behavior for the 3D viscous Magneto-hydrodynamics 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 $p\in(3,\infty)$.

This paper is devoted to the study of standing waves for so-called $N=2$ supersymmetric Chern-Simons-Schrödinger equations, a coupled system of Schrödinger equations in which Chern-Simons gauge fields are incorporated.We show that there is no nontrivial standing wave to the $N=2$ supersymmetric Chern-Simons-Schrödinger equations when the coupling constants are less than critical numbers.We also prove that static $N=2$ supersymmetric Chern-Simons-Schrödinger equations are equivalent to their first order self-dual system when the coupling constants are critical.