We describe the asymptotic behavior of functions of the Royden $p$-algebra in terms of $p$-extremal length. We also prove that each bounded $\mathcal A$-harmonic function with finite energy on a complete Riemannian manifold is uniquely determined by the behavior of the function along $p$-almost every curve.

Asymptotic behavior of three-dimensional mixed, periodic and rotating magnetohydrodynamic system is investigated as the Rossby number goes to zero. The system presents the difficulty to be singular and mixed, that is hyperbolic in the vertical direction and parabolic in the horizontal one. The divergence free condition and the spectral properties of the penalization operator are crucial in the proofs. The main tools are the energy method, the Schochet's method and products laws in anisotropic Sobolev spaces.

It is well known that there is a residual subset $J$ of the space of $C^1$-diffeomorphisms on a compact Riemannian manifold $M$ such that the maps $f\mapsto \text{chain recurrent}$ $\text{set of}\, f$ and $f\mapsto$ number of chain components of $f$ are continuous on $J.$ In this paper we get the flow version of the above results on diffeomorphisms.

In this paper, we introduce an iterative method to study oscillatory properties of delay difference equations of the following form $$\nabla_{\alpha}\left[x\left(t\right)-r\left(t\right)x\left(t-\kappa\right)\right]+p\left(t\right)x\left(t-\tau\right)-q\left(t\right)x\left(t-\sigma\right)=0,\quad t\geq t_{0},$$ where $t_{0}\in\mathbb{R}$, $t$ varies in the real interval $\left[t_{0},\infty\right)$, $\alpha>0$, $\kappa,\tau,\sigma\geq0$, $r\in C\left(\left[t_{0}-\alpha,\infty\right),\mathbb{R}^{+}\right)$, $p,q\in C\left(\left[t_{0},\infty\right),\mathbb{R}^{+}\right)$ and $\nabla_{\alpha}x\left(t\right)=x\left(t\right)-x\left(t-\alpha\right)$ for $t\geq t_{0}$.

We obtain a result on complete convergence of weighted sums for arrays of rowwise independent Banach space valued random elements. No assumptions are given on the geometry of the underlying Banach space. The result generalizes the main results of Ahmed et al.~[1], Chen et al.~[2], and Volodin et al.~[14].

In this paper, a new class of general nonlinear variational inclusions involving $H$-monotone is introduced and studied in Hilbert spaces. By applying the resolvent operator associated with $H$-monotone, we prove the existence and uniqueness theorems of solution for the general nonlinear variational inclusion, construct an iterative algorithm for computing approximation solution of the general nonlinear variational inclusion and discuss the convergence of the iterative sequence generated by the algorithm. The results presented in this paper improve and extend many known results in recent literatures.

In this paper, we prove that if $f^nf'-P$ and $g^ng'-P$ share $0$ CM, where $f$ and $g$ are two distinct transcendental meromorphic functions, $n\geq 11$ is a positive integer, and $P$ is a nonzero polynomial such that its degree $\gamma_P \leq 11$, then either $f=c_1e^{cQ}$ and $g=c_2e^{-cQ},$ where $c_1,$ $c_2$ and $c$ are three nonzero complex numbers satisfying $(c_1c_2)^{n+1}c^2=-1,$ $Q$ is a polynomial such that $Q=\int_0^zP(\eta)d\eta,$ or $f=tg$ for a complex number $t$ such that $t^{n+1}=1.$ The results in this paper improve those given by M. L. Fang and H. L. Qiu, C. C. Yang and X. H. Hua, and other authors.

Let $\sigma$ be an endomorphism and $I$ a $\sigma$-ideal of a ring $R$. Pearson and Stephenson called $I$ a {\it $\sigma$-semiprime ideal} if whenever $A$ is an ideal of $R$ and $m$ is an integer such that $A\sigma^t(A) \subseteq I$ for all $t\geq m$, then $A \subseteq I$, where $\sigma$ is an automorphism, and Hong et al.~called $I$ a {\it $\sigma$-rigid ideal} if $a\sigma(a)\in I$ implies $a\in I$ for $a\in R$. Notice that $R$ is called a $\sigma$-semiprime ring (resp., a $\sigma$-rigid ring) if the zero ideal of $R$ is a $\sigma$-semiprime ideal (resp., a $\sigma$-rigid ideal). Every $\sigma$-rigid ideal is a $\sigma$-semiprime ideal for an automorphism $\sigma$, but the converse does not hold, in general. We, in this paper, introduce the quasi $\sigma$-rigidness of ideals and rings for an automorphism $\sigma$ which is in between the $\sigma$-rigidness and the $\sigma$-semiprimeness, and study their related properties. A number of connections between the quasi $\sigma$-rigidness of a ring $R$ and one of the Ore extension $R[x;\sigma,\delta]$ of $R$ are also investigated. In particular, $R$ is a (principally) quasi-Baer ring if and only if $R[x;\sigma,\delta]$ is a (principally) quasi-Baer ring, when $R$ is a quasi $\sigma$-rigid ring.

In this paper, the existence of periodic solutions is obtained for a kind of $p$-Laplacian systems by the minimax methods in critical point theory. Moreover, the existence of infinite periodic solutions is also obtained.

By using the fixed point index theory, we investigate the existence of at least two positive solutions for $p$-Laplace equation with sign-changing nonlinear terms $ (\varphi_{p}(u'))'+a(t)f(t,u(t),u'(t))=0, $ subject to some boundary conditions. As an application, we also give an example to illustrate our results.

{$\tilde{G}(P,Q)$}, a new generalization of the set of gap numbers of a pair of points, was described in \cite{P.B-N.T}. Here we study gap numbers of local subring $R=\mathcal{K}+C$ of algebraic function field over a finite field and we give a formula for the number of elements of {$\tilde{G}(P,Q)$} depending on pure gaps and {$R$}.

In this paper, we study the geometry of screen conformal lightlike real hypersurfaces of an indefinite Kaehler manifold. The main result is a characterization theorem for screen conformal lightlike real hypersurfaces of an indefinite complex space form.

The algebraic structure of the natural integral cohomology operations is explored by means of examples. We decompose the generators of the groups $H^m(\mathbb{Z},n)$ with $2 \leq n \leq 7$ and $2 \leq m \leq 13$ into the operations of cup products, cross-cap products and compositions. Examination of these decompositions and comparison with other possible generators demonstrates the existence of relations between integral operations that have withheld formulation. The calculated groups and generators are collected in a table for practical reference.

Some new \v{C}eby\v{s}ev type inequalities have been developed by working on functions whose first derivatives are absolutely continuous and the second derivatives belong to the usual Lebesgue space $L_{_{\infty }}\left[ a,b\right].$ A unified treatment of the special cases is also given.

This paper deals with a predator-prey model with Holling type III response function and cross-diffusion subject to the homogeneous Neumann boundary condition. We first give a priori estimates (positive upper and lower bounds) of positive steady states. Then the non-existence and existence results of non-constant positive steady states are given as the cross-diffusion coefficient is varied, which means that stationary patterns arise from cross-diffusion.

In this paper, we show that the modified Mann iteration with errors converges strongly to fixed point for uniformly L-Lipschitzian asymptotically $\Phi$-pseudocontractive mappings in real Banach spaces. Meanwhile, it is proved that the convergence of Mann and Ishikawa iterations is equivalent for uniformly L-Lipschitzian asymptotically $\Phi$-pseudocontractive mappings in real Banach spaces. Finally, we obtain the convergence theorems of Ishikawa iterative sequence and the modified Ishikawa iterative process with errors.

In this paper we find necessary and sufficient conditions for weak amenability of the convolution Banach algebras $A(K)$ and $L^2(K)$ for a compact hypergroup $K$, together with their applications to convolution Banach algebras $L^p(K)$ $(2\leq p<\infty)$. It will further be shown that the convolution Banach algebra $A(G)$ for a compact group $G$ is weakly amenable if and only if $G$ has a closed abelian subgroup of finite index.