Let $S$ and $T$ be continuous linear operators on an ultraprime Banach algebra $A$. We show that if $S$, $T$, and $ST$ are close to satisfy the derivation identity on $A$, then either $S$ or $T$ approaches to zero. If $T$ is close to satisfy the derivation identity and $[T(a),a]$ is near the centre of $A$ for each $a \in A$, then either $T$ approaches to zero or $A$ is nearly commutative. Further, we give quantitative estimates of these phenomena.

Let $\mathcal{N}$ be a non-trivial and complete nest on a Hilbert space $H$. Suppose $d = \{d_n: n \in N\}$ is a group of linear mappings from $Alg\mathcal{N}$ into itself. We say that $d = \{d_n: n \in N\}$ is a Jordan higher derivable mapping at a given point $G$ if $d_{n}(ST+TS) = \sum\limits_{i+j=n} \{d_{i}(S) d_{j}(T) + d_{j}(T) d_{i}(S)\}$ for any $S,T \in Alg \mathcal{N}$ with $ST = G$. An element $G \in Alg \mathcal{N}$ is called a Jordan higher all-derivable point if every Jordan higher derivable mapping at $G$ is a higher derivation. In this paper, we mainly prove that any given point $G$ of $Alg\mathcal{N}$ is a Jordan higher all-derivable point. This extends some results in [1] to the case of higher derivations.

In this paper, we study monotone hybrid method for finding a commonfixed point of a family of generalized nonexpansive mappings and then prove astrong convergence theorem for a family of generalized nonexpansive mappings inBanach spaces. Using this theorem, we obtain some new results for a generalizednonexpansive mapping and two generalized nonexpansive mappings in Banachspaces. Moreover, we apply our main result to obtain a strong convergence theoremfor a family of nonexpansive mappings in a Hilbert space.

This paper studies the observability and null controllability for a class of one-dimensional fourth order parabolic equation. By means of establishing the global Carleman estimates, we derive the observability inequalities for one-dimensional linear fourth order parabolic equation with potentials. The null controllability results for one-dimensional fourth order semilinear equation are also presented.

In this paper, we consider the long time behavior of solutions of the initial value problem for the viscoelastic wave equation under boundary damping \begin{eqnarray*} u_{tt} - \Delta u + \int_0^t g(t-\tau) \text{div}(a(x)\nabla u(\tau)) d\tau + u_t = 0 &\text{in}\, \Omega \times (0,\infty). \end{eqnarray*} For the low initial energy case, which is the non-positive initial energy, based on concavity argument we prove the blow up result. As for the high initial energy case, we give out sufficient conditions of the initial datum such that the solution blows up in finite time.

In this paper, we study the maximal dissipative singular Sturm-Liouville operators (in Weyl's limit-circle case at singular point $b$) acting in the Hilbert space $L_{w}^{2}\left[ a,b\right)$ ($-\infty \lt a\lt b\leq \infty$ ). In fact, we consider all extensions of a minimal symmetric operator and we investigate two classes of maximal dissipative operators with separated boundary conditions, called 'dissipative at $a$' and 'dissipative at $b$'. In both cases, we construct a selfadjoint dilation of the maximal dissipative operator and determine its incoming and outgoing spectral representations. This representations make it possible to determine the scattering matrix ofthe dilation in terms of the Titchmarsh-Weyl function of a selfadjoint Sturm-Liouville operator. We also construct a functional model of the maximal dissipative operator and determine its characteristic function in terms of the scattering matrix of the dilation (or of the Titchmarsh-Weyl function). Finally we prove theorems on the completeness of the eigenfunctions and associated functions of the maximal dissipative Sturm-Liouville operators.

A structure on an almost contact metric manifold is defined as a generalization of well-known cases: Sasakian, quasi-Sasakian, Kenmotsu and cosymplectic. This was suggested by a local formula of Eum [9]. Then we consider a semi-invariant $\xi^{\bot}$-submanifold of a manifold endowed with such a structure and two topics are studied: the integrability of distributions defined by this submanifold and characterizations for the totally umbilical case. In particular we recover results of Kenmotsu [11], Eum [9,10] and Papaghiuc [16].

Let $m$, $j$ and $k$ be positive integers with $j \ge k$. An $m$-circular-$L(j,k)$-edge-labeling of a graph $G$ is an assignment $f$ from $\{0,1,\dots,m-1\}$ to the edges of $G$ such that, for any two edges $e_1$ and $e_2$, $|f(e_1)-f(e_2)|_m\geq j$ if $e_1$ and $e_2$ are adjacent, and $|f(e_1)-f(e_2)|_m \geq k$ if $e_1$ and $e_2$ are at distance $2$, where $|a|_m = \min \{a,m-a\}$. The minimum $m$ such that $G$ has an $m$-circular-$L(j,k)$-edge-labeling is defined as the circular-$L(j,k)$-edge-labeling number of $G$, denoted by $\sigma_{j,k}'(G)$. This paper determines the circular-$L(2,1)$-edge-labeling numbers of the infinite $\Delta$-regular tree for $\Delta \ge 2$ and the $n$-dimensional cube for $n \in \{2,3,4,5\}$.

A feedback vertex set of a graph is a set of vertices whose removal results an acyclic graph. This paper shows that for every planar graph the minimum cardinality of a feedback vertex set is at most three times the maximum number of vertex disjoint cycles in the graph.

In this paper a unified approach for finding soliton solutions is applied to the modified Fornberg-Whitham equation. Variety of peakon, periodic, and solitary new exact solutions are constructed.

We consider the boundedness and compactness of the differences of two weighted composition operators acting from the Banach space of bounded analytic functions on the open unit disk to the Bloch space. In the sequel, we will present some explicit examples to bring our reseach into forcus.

By an equivalent characterization of Morrey space associated with the fractional heat semigroup, we establish a relation between the generalized $Q-$type spaces and Morrey spaces. By this relation, in this paper, we prove the boundedness of the singular integral operatoes on the Q-type spaces $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$. As an application, we get the well-posedness and regularity of the quasi-geostrophic equation with initial data in $Q_{\alpha}^{\beta, -1}(\mathbb{R}^{n})$.

In this paper we prove the global existence and the global exponential stability of weak solutions to a class of semilinear parabolic equations with discrete and distributed time-varying delays. Moreover, the exponential stability of stationary solutions of the equations is also studied. The obtained results can be applied to some models in biology and physics.

Let $\{\varphi_n^{\alpha}\}_{n \in \mathbb{N}}$ be the Laguerre functions of Hermite type with index $\alpha$. These are eigenfunctions of the Laguerre differential operator $L_\alpha = \dfrac{1}{2}(-\frac{d^2}{dy^2} + y^2 + \frac{1}{y^2}(\alpha^2-\frac{1}{4}))$. We define and study a $BMO$-type space $BMO_{L_{\alpha}}$, which is identified as the dual space of the Hardy-type space associated with $L_{\alpha}$. We characterize $BMO_{L_{\alpha}}$ by Carlesson measures related to appropriate square functions. Finally, we prove the boundedness on this space of the fractional integral operator and the Riesz transform related to $L_{\alpha}$.

Let $s \in (0,\,1/2)$, $M_{0,\,s}$ be the local maximal operator of John and Strömberg, and ${\cal M}_{0,\,s}$ the multi(sub)linear local maximal operator. In this paper, the authors give some characterizations of the weights $w_1,\,...,\,w_{\ell}$ for which the operator ${\cal M}_{0,\,s}$ is bounded from $L^{p_1}(\mathbb{R}^n,\,w_1) \times ... \times L^{p_{\ell}}(\mathbb{R}^n,\,w_{\ell})$ to $L^{p}(\mathbb{R}^n,\,\nu_{\vec{w}})$ with $\nu_{\vec{w}} = \prod_{k=1}^{\ell} w_k^{p/p_k}$, $p_1,\,...,\,p_{\ell} \in (0,\,\infty)$ and $1/p = \sum_{1 \leq k \leq \ell} 1/p_k$. A new characterization of $A_{\infty}(\mathbb{R}^n)$ weights and a characterization of weights $w$ which satisfies $w^{\theta} \in A_{\infty}(\mathbb{R}^n)$ for some $\theta \in (0,\,\infty)$, are also obtained.

We give an elementary proof of the well-known congruence $$ \binom{\frac{p-1}{2}}{\frac{p-1}{4}} \equiv \frac{2^{p-1}+1}{2} \bigg(2a-\frac{p}{2a}\bigg) \pmod{p^2}, $$ where $p \equiv 1 \pmod{4}$ is prime and $p = a^2 + b^2$ with $a \equiv 1 \pmod{4}$.

Let $({\mathcal X},d,\mu)$ be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions. In this paper, the authors prove that multilinear commutators of Calderón-Zygmund operators with RBMO($\mu$) functions are bounded on Orlicz spaces, especially, on $L^p(\mu)$ with $p \in (1,\infty)$. The weak type endpoint estimate of multilinear commutators of Calderón-Zygmund operators with Orlicz type functions in $Osc_{\exp L^r}(\mu)$ for $r \in [1,\infty)$ is also presented.

In this paper, we introduce weak Hardy spaces $H^{p,\infty}$ on spaces of homogeneous type. We establish an atomic decomposition characterization of these spaces, show the boundedness of fractional integral operators and provide an $H^{p,\infty}$ interpolation theorem. Applications to the Nagel-Stein's singular integral operators and fractional integral operators are also discussed.

In this paper, we study a sufficient condition for the existence of a periodic solution for the $n$th order dynamic equations on time scales. The results are shown by the use of coincidence degree theory. The necessary a priori bounds are based on Wirtinger type inequality established as Lemma B.

In this paper, we showed some results of similarity degrees of von Neumann algebras satisfying co-amenability. We also obtain some results of Christensen's property $D_k$ for such von Neumann algebras.

Using a relation between domination number and incidence chromatic number, we obtain necessary and sufficient conditions for $r$-regular graphs to be $(r+1)$-incidence colorable. Also, we determine the optimal Nordhaus-Gaddum inequality for the incidence chromatic number.