An approximation method for strictly pseudocontractive mappings

Let K$K$ be a closed convex subset of a q$q$-uniformly smooth separable Banach space, T:K→K$T:K\to K$ a strictly pseudocontractive mapping, and f:K→K$f:K\to K$ an L$L$-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1)$t\in (0,1)$, let _xt$x_t$ be the unique fixed point of tf+(1-t)T$\mathit{tf}+(1-t)T$. We prove that if T$T$ has a fixed point, then {_xt}$\{x_t\}$ converges to a fixed point of T$T$ as t$t$ approaches to 0.     

Some Liouville-type theorems for harmonic functions on Finsler manifolds

In this paper, we give some Liouville-type theorems for L^p$L^p$(p∈R)$(p\in \mathbb{R})$ harmonic (resp. subharmonic, superharmonic) functions on forward complete Finsler manifolds. Moreover, we derive a gradient estimate for harmonic functions on a closed Finsler manifold. As an application, one obtains that any harmonic function on a closed Finsler manifold with nonnegative weighted Ricci curvature _RicN${\mathit{Ric}}_N$(N∈(n,∞))$(N\in (n,\infty ))$ must be constant.     

Kadec–Klee property and fixed points

Let X be a reflexive Banach space which does not have the Kadec–Klee property. Then there exists a compact mapping f   from the unit ball _BX$B_X$ of X   to the dual space X^?$X^{?}$ such that _infx∈BX‖f(x)‖>0${\inf }_{x\in B_X}\Vert f(x)\Vert >0$ and 〈f(x),x〉<‖f(x)‖$〈f(x),x〉<\Vert f(x)\Vert$ for every x∈_BX$x\in B_X$.     

Rigidity of symmetric products

Given a metric continuum X, we consider the following hyperspaces of X  : 2^X$2^X$, _Cn(X)$C_n(X)$ and _Fn(X)$F_n(X)$ (n∈N$n\in \mathbb{N}$). Let _F1(X)={{x}:x∈X}$F_1(X)=\{\{x\}:x\in X\}$. A hyperspace K(X)$K(X)$ of X   is said to be rigid provided that for every homeomorphism h:K(X)→K(X)$h:K(X)\to K(X)$ we have that h(_F1(X))=_F1(X)$h(F_1(X))=F_1(X)$. In this paper we study under which conditions a continuum X   has a rigid hyperspace _Fn(X)$F_n(X)$.     

Unbounded order convergence in dual spaces

A net (_xα)$(x_{\alpha })$ in a vector lattice X   is said to be unbounded order convergent (or uo-convergent, for short) to x∈X$x\in X$ if the net (|_xα−x|∧y)$(|x_{\alpha }-x|\wedge y)$ converges to 0 in order for all y∈_X+$y\in X_{+}$. In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let X   be a Banach lattice. We prove that every norm bounded uo-convergent net in X^?$X^{?}$ is w^?$w^{?}$-convergent iff X   has order continuous norm, and that every w^?$w^{?}$-convergent net in X^?$X^{?}$ is uo-convergent iff X is atomic with order continuous norm. We also characterize among σ  -order complete Banach lattices the spaces in whose dual space every simultaneously uo- and w^?$w^{?}$-convergent sequence converges weakly/in norm.     

Spectral problems of a class of non-self-adjoint one-dimensional Schrodinger operators

In this paper we investigate the one-dimensional Schrodinger operator L(q)$L(q)$ with complex-valued periodic potential q   when q∈_L1[0,1]$q\in L_1[0,1]$ and _qn=0$q_n=0$ for n=0,−1,−2,...$n=0,-1,-2,...$, where _qn$q_n$ are the Fourier coefficients of q   with respect to the system {e^i2πnx}$\{e^{i2\pi nx}\}$. We prove that the Bloch eigenvalues are (2πn+t)²${(2\pi n+t)}^2$ for n∈Z$n\in \mathbb{Z}$, t∈C$t\in \mathbb{C}$ and find explicit formulas for the Bloch functions. Then we consider the inverse problem for this operator.     

On rigidity of Roe algebras

Roe algebras are C^?$C^{?}$-algebras built using large scale (or ‘coarse’) aspects of a metric space (X,d)$(X,d)$. In the special case that X=Γ$X=\Gamma$ is a finitely generated group and d   is a word metric, the simplest Roe algebra associated to (Γ,d)$(\Gamma ,d)$ is isomorphic to the crossed product C^?$C^{?}$-algebra l^∞(Γ)_?rΓ$l^{\infty }(\Gamma ){?}_r\Gamma$.     

On the norming constants for normal maxima

Given n   independent standard normal random variables, it is well known that their maxima _Mn$M_n$ can be normalized such that their distribution converges to the Gumbel law. In a remarkable study, Hall proved that the Kolmogorov distance _dn$d_n$ between the normalized _Mn$M_n$ and its associated limit distribution is less than 3/log?n$3/\log ?n$. In the present study, we propose a different set of norming constants that allow this upper bound to be decreased with _dn≤C(m)/log?n$d_n\le C(m)/\log ?n$ for n≥m≥5$n\ge m\ge 5$. Furthermore, the function C(m)$C(m)$ is computed explicitly, which satisfies C(m)≤1$C(m)\le 1$ and _limm→∞?C(m)=1/3${\lim }_{m\to \infty }?C(m)=1/3$. As a consequence, some new and effective norming constants are provided using the asymptotic expansion of a Lambert W type function.     

Parameter space for families of parabolic-like mappings

In this paper we study families of degree 2 parabolic-like mappings _(fλ)λ∈Λ${(f_{\lambda })}_{\lambda \in \Lambda }$ (as defined in [4]). We prove that the hybrid conjugacies between a nice analytic family of degree 2 parabolic-like mappings and members of the family _Per1(1)${\mathit{Per}}_1(1)$ induce a continuous map χ:Λ→C$\chi :\Lambda \to \mathbb{C}$, which under suitable conditions restricts to a ramified covering from the connectedness locus of _(fλ)λ∈Λ${(f_{\lambda })}_{\lambda \in \Lambda }$ to the connectedness locus _M1?{1}$M_1?\{1\}$ of _Per1(1)${\mathit{Per}}_1(1)$. As an application, we prove that the connectedness locus of the family _Ca(z)=z+az²+z³$C_a(z)=z+a{z}^2+z^3$, a∈C$a\in \mathbb{C}$ presents baby _M1$M_1$.     

Characterization of soliton solutions in 2D nonlinear Schrödinger lattices by using the spatial disorder

In this paper, the pattern of the soliton solutions to the discrete nonlinear Schrödinger (DNLS) equations in a 2D   lattice is studied by the construction of horseshoes in _l∞$l_{\infty }$-spaces. The spatial disorder of the DNLS equations is the result of the strong amplitudes and stiffness of the nonlinearities. The complexity of this disorder is log(N+1)$\log (N+1)$ where N   is the number of turning points of the nonlinearities. For the case N=1$N=1$, there exist disjoint intervals _I0$I_0$ and _I1$I_1$, for which the state _um,n$u_{m,n}$ at site (m,n)$(m,n)$ can be either dark (_um,n∈_I0$u_{m,n}\in I_0$) or bright (_um,n∈_I1$u_{m,n}\in I_1$) that depends on the configuration _km,n=0$k_{m,n}=0$ or 1, respectively. Bright soliton solutions of the DNLS equations with a cubic nonlinearity are also discussed.     

On the least distance eigenvalue of a graph

Denote by D(G)=_(di,j)n×n$D(G)={(d_{i,j})}_{n\times n}$ the distance matrix of a connected graph G with n   vertices, where _dij$d_{ij}$ is equal to the distance between vertices _vi$v_i$ and _vj$v_j$ in G  . The least eigenvalue of D(G)$D(G)$ is called the least distance eigenvalue of G  , denoted by _λn${\lambda }_n$. In this paper, we determine all the graphs with _λn∈[−2.383,0]${\lambda }_n\in [-2.383,0]$.     

A sharp upper bound on the signless Laplacian spectral radius of graphs

Let G be a simple connected graph of order n   with degree sequence _d1,_d2,…,_dn$d_1,d_2,\dots ,d_n$ in non-increasing order. The signless Laplacian spectral radius ρ(Q(G))$\rho (Q(G))$ of G   is the largest eigenvalue of its signless Laplacian matrix Q(G)$Q(G)$. In this paper, we give a sharp upper bound on the signless Laplacian spectral radius ρ(Q(G))$\rho (Q(G))$ in terms of _di$d_i$, which improves and generalizes some known results.