It is proved that, in most cases, a scalar multiple of a linear-fractional generated composition operator $\lambda C_\varphi$ acting on a weighted Dirichlet space $S_\nu$ of holomorphic functions in the open unit disk is frequently hypercyclic if and only if it is hypercyclic. In fact, this holds for all triples $(\nu ,\lambda , \varphi )$ with the possible exception of those satisfying $\nu \in [1/4,1/2), \, |\lambda | = 1, \, \varphi =$ a parabolic automorphism.

In the present work, we investigate the simultaneous and converse approximation by trigonometric polynomials of the functions in the Orlicz spaces with weights satisfying so called Muckenhoupt's $A_{p}$ condition.

We show that every non-precompact topological group admits a fixed point-free continuous action by affine isometries on a suitable Banach space. Thus, precompact groups are defined by the fixed point property for affine isometric actions on Banach spaces. For separable topological groups, in the above statements it is enough to consider affine actions on one particular Banach space: the unique Banach space envelope $\langle\mathbb U\rangle$ of the universal Urysohn metric space $\mathbb U$, known as the Holmes space. At the same time, we show that Polish groups need not admit topologically proper (in particular, free) affine isometric actions on Banach spaces (nor even on complete metric spaces): this is the case for the unitary group $U(\ell^2)$ with strong operator topology, the infinite symmetric group $S_\infty$, etc.

Let $f:X\to Z$ and $g:Y\to Z$ be maps between connected pointed CW-complexes. Recall the definition of pairing with axes $f$ and $g$ due to N.Oda. In this paper, we introduce {\it (n)-pairing}, which is a generalization of {\it H(n)}-space due to Y.Félix and D.Tanré and define a family of subsets of the homotopy set of maps. We give some rational characterizations of it and illustrate some examples in Sullivan models. Also we consider about the $G(n)$-sequence of a fibration which is a generalization of $G$-sequence.

We derive an inequality for the $\mathbb Z_2$-cup-length of any smooth closed connected manifold unorientedly cobordant to zero. In relation to this, we introduce a new numerical invariant of a smooth closed connected manifold, called the characteristic rank. In particular, our inequality yields strong upper bounds for the cup-length of the oriented Grassmann manifolds $\tilde G_{n,k}\cong SO(n)/SO(k)\times SO(n-k)$ $(6\leq 2k\leq n)$ if $n$ is odd; if $n$ is even, we obtain new upper bounds in a different way. We also derive lower bounds for the cup-length of $\tilde G_{n,k}$. For $\tilde G_{2^t-1,3}$ $(t\geq 3)$ our upper and lower bounds coincide, giving that the $\mathbb Z_2$-cup-length is $2^t-3$ and the characteristic rank equals $2^t-5$. Some applications to the Lyusternik-Shnirel'man category are also presented.

We consider a nonlinear elliptic equation driven by the $p$-Laplacian and with a Carathéodory right hand side nonlinearity which exhibits an asymmetric asymptotic behaviour at $+\infty$ and at $-\infty$. These hypotheses imply that the Euler functional of the problem is noncoercive (indefinite). Using critical point theory, we prove the existence of at least two nontrivial smooth solutions. Also in the last section for the asymmetric functionals considered here, we compute the critical groups at infinity.

We study properties of the differences of weighted composition operators on the disk algebra and will see the equivalence of the compactness, the weak compactness and the complete continuity of them. Moreover, we characterize the differences of weighted composition operators acting from the space of bounded analytic functions to the disk algebra.

In this paper, we study the existence of periodic solutions for $n^{\text{th}}$ order functional differential equations $x^{ (n) } (t) +\sum\limits ^{n-1}_{i=0}b_{i}[x^{ (i) } (t) ]^{k}+ f (t, x (t-\tau) ) =p (t) $. Some new results on the existence of periodic solutions of the equations are obtained. Our approach is based on the coincidence degree theory of Mawhin.

The main aim of this paper is to introduce a modified Noor iterations scheme to provide a unified approach to the Mann and Ishikawa iteration processes. We study weak and strong convergence of the new iterations scheme of a nonself asymptotically nonexpansive map satisfying a new control condition in uniformly convex Banach spaces. Several recent results about Mann-type and Ishikawa-type iteration schemes for nonself(self) asymptotically nonexpansive maps follow directly and concurrently from our results.

In this paper we are interested in the existence of solutions for Dirichlet problem associated to the degenerate quasilinear elliptic equations $$-\,{\rm div}\, [v(x)\,{\cal A}(x, u, {\nabla}u)] + {\omega}(x){\cal A}_0(x,u(x))= f_0 - \sum_{j=1}^nD_jf_j, \ \ {\rm on } \ \ {\Omega}$$ in the setting of the weighted Sobolev spaces ${\rm W}_0^{1,p}(\Omega,\omega,v)$.

The class of PLS-spaces covers most of the natural spaces of analysis, e. g. the space of real analytic functions, spaces of distributions. We investigate the property of PLS-spaces called {\it dual interpolation estimate} and show that in many important and classical cases this property is inherited by tensor products of two PLS-spaces. We establish the inheritance if at least one of the spaces is a nuclear Fréchet space or a PLN-space. The latter includes the important, classical case when one of the spaces is the dual of a nuclear Fréchet space.

In this paper we shall be concerned with the existence of almost homoclinic solutions of the Hamiltonian system $\ddot{q}+V_q(t,q)=f(t)$, where $t\in\mathbb{R}$, $q\in\mathbb{R}^n$ and $V(t,q)=-\frac{1}{2}(L(t)q,q)+W(t,q)$. It is assumed that $L$ is a conti\-nuous matrix valued function such that $L(t)$ are symmetric and positive definite uniformly with respect to $t$. A map $W$ is $C^1$-smooth, $W_q(t,q)=o(|q|)$, as $q\to 0$ uniformly with respect to $t$ and $W(t,q)|q|^{-2}\to\infty$, as $|q|\to\infty$. Moreover, $f\neq 0$ is continuous and sufficiently small in $L^2(\R,\mathbb{R}^n)$. It is proved that this Hamiltonian system possesses a solution $q_{0}\colon\mathbb{R}\to\mathbb{R}^n$ such that $q_{0}(t)\to 0$, as $t\to\pm\infty$. Since $q\equiv 0$ is not a solution of our system, $q_{0}$ is not homoclinic in a classical sense. We are to call such a solution almost homoclinic. It is obtained as a weak limit of a sequence of almost critical points of an appropriate action functional $I$.

In quadratic form theory over fields, a much studied field invariant is the $u$-invariant, defined as the supremum of the dimensions of anisotropic quadratic forms over the field. We investigate the corresponding notions of $u$-invariant for hermitian and for skew-hermitian forms over a division algebra with involution, with a special focus on skew-hermitian forms over a quaternion algebra with canonical involution. Under certain conditions on the center of the quaternion algebra, we obtain sharp bounds for this invariant.