We consider the initial boundary value problem for a semi-linear partial functional differential equation of the first order on a cylindrical domain in $n+1$ dimensions. Projection of the domain onto the $n$-dimensional hyperplane is a connected set with boundary satisfying certain type of cone condition. Using the method of characteristics and the Banach contraction theorem, we prove the global existence, uniqueness and continuous dependence on data of Carathéodory solutions of the problem. This approach cover equations with deviating variables as well as differential integral equations.

Some new existence theorems are obtained for periodic solutions of nonautonomous second-order differential systems with $(q,p)$-Laplacian.

Let $G$ be a group and $L^\infty(G)$ be the $C^*$-algebra of bounded complex-valued functions on $G$. $G$ is called inner amenable if there exists a positive norm 1 functional $m$ on $L^\infty(G)$ such that $m(\rho(y)f) = m(f)$ for each $y \in G, f \in L^\infty(G)$ (where $\rho(y)f(x) = f(yxy^{-1})$); the functional $m$ is called an inner invariant mean. In this paper, among the other things, we prove a variety of characterizations of inner amenable groups. We also give sufficient conditions on an inner invariant mean to be a topologically inner invariant mean.

We determine the structure of finite groups whose cyclic subgroups, elementary abelian 2-subgroups and abelian subgroups of order at most $4p$, $p$ a prime, are TI-subgroups.

A $2p$ $(p\geq 1)$ times continuously differentiable complex-valued function $F=u+iv$ in a domain $D\subseteq \mathbb{C}$ is $p$-{\it harmonic} if $F$ satisfies the $p$-harmonic equation $\Delta^p F=\Delta(\Delta^{p-1})F=0$, where $\Delta$ represents the complex Laplacian operator $$\Delta=4\frac{\partial^{2}}{\partial z\partial\bar{z}}:=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}.$$ In this paper, the main aim is to investigate the subordination of $p$-harmonic mappings. First, the characterization for $p$-harmonic mappings to be subordinate are obtained. Second, we get two results on the relation of integral means of subordinate $p$-harmonic mappings. Finally, we discuss the existence of extreme points for subordination families of $p$-harmonic mappings. Two sufficient conditions for $p$-harmonic mappings to be extreme points of the closed convex hulls of the corresponding subordination families are established.

Consider a positive Banach lattice valued vector measure $\bf m:\Sigma \to X$, its space of 2-integrable functions $L^2(\bf m)$ and a sequence $S$ in it. We analyze the notion of weak $\bf m$-orthogonality for such an $S$ in these spaces and we prove a Menchoff-Rademacher Theorem on the almost everywhere convergence of series in them. In order to do this, we provide a criterion for determining when there is a functional $0 \le x' \in X'$ such that $S$ is orthogonal with respect to the scalar positive measure $\langle \bf m, x' \rangle$. As an application, we use the representation of $\ell-$sums of $L^2$-spaces as spaces $L^2 (\bf m)$ for a suitable vector measure $\bf m$ centering our attention in the case of $c_0$-sums.

Let $\mathcal{A}$ denote the family of all functions that are analytic in the unit disk $\Delta := \{ z \in \mathbb{C} : |z|<1 \}$ and normalized by $f(0)=f'(0)-1=0$. In this paper, we investigate the class $\mathcal{T}_G$ defined as follows \[\mathcal{T}_G:= \left\{ \sqrt{F(z)G(z)} : F \in \mathcal{T} \right\},\quad G \in \mathcal{T},\] where $\mathcal{T}$ denotes the class of all semi-typically real functions i.e. $\mathcal{T} := \{ F \in \mathcal{A} : F(z)>0 \iff z \in (0,1) \}$. We find the sets $\bigcup_{G \in \mathcal{T}} \mathcal{T}_G$ and $\bigcap_{G \in \mathcal{T}} \mathcal{T}_G$, the set of all extreme points of $\mathcal{T}_G$ and the set of all support points of $\mathcal{T}_G$. Moreover, for the fixed $G$, we determine the radii of local univalence, of starlikeness and of univalence of $\mathcal{T}_G$.

Given non empty open subsets $\Omega$ of $\mathbb{R}^r$ and $\Omega'$ of $\mathbb{R}^s$, and sequences $\mathscr{M}$ and $\mathscr{M}'$, we recall the definition of the space ${\mathscr{D}^{\{\mathscr{M},\mathscr{M}'\}}{(\Omega \times \Omega')}}$. Given $p \in [1,+\infty[$, we also introduce the space $\mathscr{D}_{(L^p)}^{\{\mathscr{M},\mathscr{M}'\}}{(\Omega \times \Omega')}$. By use of a basic idea due to Valdivia, we obtain a global representation of the corresponding ultradistributions, i.e. of the elements of the topological duals of these spaces.

We prove a necessary and sufficient conditions for discreteness of the set of all eigenvalues (with the usual Sturm--Liouville properties) of half--linear eigenvalue problem with locally integrable weights. Our conditions appear to be equivalent to the compact embedding of certain weighted Sobolev and Lebesgue spaces. Every eigenvalue allows the variational characterization of Ljusternik--Schnirelmann type.

By using minimax methods in critical point theory, some new existence theorems of infinitely many periodic solutions are obtained for a second-order ordinary $p$-Laplacian system. The results obtained generalize many known works in the literature.

In previous work the author constructed a convolution algebra and an isomorphic multiplication algebra of one-dimensional associated homogeneous distributions with support in $R$. In this paper we investigate the various algebraic substructures that can be identified in these algebras. Besides identifying ideals and giving polynomial representations for six subalgebras, it is also shown that both algebras contain an interesting Abelian subgroup, which can be used to construct generalized integration/derivation operators of complex degree on the whole line $R$.

The purpose of this paper is to prove a general fixed point theorem by altering distances for two owc pairs of mappings and to reduce the study of fixed points for pairs of mappings satisfying a contractive condition of integral type to the study of fixed points in a metric space by altering distances satisfying a new type of implicit relation generalizing a main result obtained by Pathak and Shahzad in a recent paper.

Let ${\cal B}$ be a certain Banach space consisting of analytic functions defined on a bounded domain $G$ in the complex plane. Let $\varphi$ be an analytic multiplier of ${\cal B}$ we denote by $M_{\varphi}$ and $\{M_{\varphi}\}^{'} $ respectively, the operator of multiplication by $\varphi$ and the commutant of $M_{\varphi}$. In this article under certain conditions on $\varphi$ and $G$ we characterize the commutant of $M_{\varphi}$. In particular, when $\varphi$ is a rational function with poles off $\overline G$, under certain conditions on $\varphi$ we show that $\{M_{\varphi}\}^{'} =\{M_{z}\}^{'} $. We extend some results obtained in [4] and [6] about the commutant of the operator $M_{\varphi}$.

Let $M(X,\mathbb{K})$ be the non-archimedean Banach space of all additive and bounded $\mathbb{K}$-valued measures on the ring of all clopen subsets of a zero-dimensional compact space $X$, where $\mathbb{K}$ is a non-archimedean non-trivially valued complete field. It is known that $M(X,\mathbb{K})$ is isometrically isomorphic to the dual of the Banach space $C(X,\mathbb{K})$ of all continuous $\mathbb{K}$-valued maps on $X$ with the sup-norm topology. Does the non-archimedean Lebesgue Dominated Convergence Theorem hold for the space $M(X,\mathbb{K})$? Only in the trivial case! We show (Theorem 2) that for every sequence $(f_{n})_n$ in $C(X,\mathbb{K})$ such that $f_{n}(x)\rightarrow 0$ for all $x\in X$ and $\| f_n \| \leq 1$ for all $n\in\mathbb{N}$, one has $\int_{X}f_{n}d\mu\rightarrow 0$ for each $\mu\in M(X, \mathbb{K})$ iff $X$ is finite. In the second part we characterize weakly Lindelöf non-archimedean Banach spaces $E$ with a base as well as Corson $\sigma(E',E)$-compact unit balls in their duals $E'$. We also look at the Kunen space from the non-archimedean point of view.

In this paper we supply a new proof that the Browder spectrum is the largest part of the spectrum that remains unchanged under compact perturbations from the commutant.