Very recently, Ceng and Yao [L. C. Ceng, J. C. Yao, A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem, Nonlinear Anal., 72 (2009), 1922-1937, suggested and analyzed a relaxed extragradient-like method for finding a common solution of a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem of a strict pseudocontractive mapping in a Hilbert space. In this paper, based on the authors' iterative method, we introduce a modification of the relaxed extragradient-like method for finding a common solution of a generalized mixed equilibrium problem with perturbed mapping, a general system of generalized nonlinear mixed composite-type equilibria and a fixed point problem of a strict pseudocontractive mapping in a Hilbert space, and then obtain a strong convergence theorem. Utilizing this theorem, we establish some new strong convergence results in fixed point problems, variational inequalities, mixed equilibrium problems and systems of generalized nonlinear mixed composite-type equilibria in Hilbert spaces.

Our purpose in this paper is first to prove a weak convergence theorem by Mann's iteration for positively homogeneous nonexpansive mappings in a Banach space. Further, using the shrinking projection method defined by Takahashi, Takeuchi and Kubota, we prove a strong convergence theorem for such mappings. From two results, we obtain weak and strong convergence theorems for linear contractive mappings in a Banach space. These results are new even if the mappings are linear and contractive.

In this paper, we prove the existence of solutions for impulsive differential equations of fractional order $q \in (1,2]$ with anti-periodic boundary conditions in a Banach space. Our study is based on the contraction mapping principle and Krasnoselskii's fixed point theorem.

We study the blow-up behaviors of two parabolic problems on a bounded domain. One is the heat equation with nonlinear memory and the other is a parabolic system with power nonlinearity in which the coefficients of the reaction terms (potentials) are nonnegative and spatially inhomogeneous. Our aim is to show that any zero of the potential, where there is no reaction, is not a blow-up point, if the solution is monotone in time. We also give sufficient conditions for the time monotonicity of solutions.

It is shown that the set of fixed-points of some asymptotically regular mappings in uniformly convex Banach spaces is not only connected but even a retract of a domain. The results presented in this paper improve and extend some results in [11, 13].

A new notion of the higher-order generalized adjacent derivative for a set-valued map is defined. By virtue of the derivative, a higher-order Mond- Weir type dual problem is introduced for a constrained set-valued optimization problem. The weak duality, strong duality and converse duality theorems are established.

By utilizing Nevanlinna's value distribution theory, we study the existence or solvability of meromorphic solutions of functional equations of the type $P(f) f'P(g) g' = 1$, where $P(z)$ is a polynomial with two distinct zeros at least. We show that such type of equations have no meromorphic solutions $f$ and $g$ when $P(z)$ has at least three distinct zeros. Moreover, for some polynomials $P(z)$ with two distinct zeros only, such type of equations possess transcendental meromorphic solutions which can be expressed by Weierstrass $\wp$ function.

In this paper we consider the Cauchy problem for the 3D generalized magnetohydrodynamic (MHD) equations and the quasi-geostrophic (QG) equations. We prove some new regularity criteria for weak solutions.

Let $A$ be a Banach algebra. In this paper, among the other things, we present a few results in the theory of homomorphisms on $A^*$. We want to find out when the equality $T(af) = aT(f)$ for every $a \in A$ and $f \in A^*$ implies the equality $T(Ff) = FT(f)$ for every $f \in A^*$ and $F \in A^{**}$. One of the main results of this paper is to introduce and study the notion of a weakly almost periodic operator.

The bounded (resp. bounded compact) approximation property is well known in the theory of Banach spaces. This paper is concerned with some bounded approximation properties in the more general setting. We establish various new criteria of bounded approximation properties.

The feedback number of a graph $G$ is the minimum number of vertices whose removal from $G$ results in an acyclic subgraph. We use $f(d,n)$ to denote the feedback number of the de Bruijn graph $UB(d,n)$. R. Královic and P. Ruzicka [Minimum feedback vertex sets in shuffle-based interconnection networks. Information Processing Letters, 86 (4) (2003), 191-196] proved that $f(2,n) = \lceil \frac{2^{n}-2}{3} \rceil$. This paper gives the upper bound on $f(d,n)$ for $d \ge 3$, that is, $f(d,n) \leq d^n \left(1 - \left( \frac{d}{1+d}\right)^{d-1} \right) + \binom{n+d-2}{d-2}$.

An eigenvalue problem on a bounded domain for the Laplacian with a nonlinear Robin-like boundary condition is investigated. We prove the existence, isolation and simplicity of the first two eigenvalues.

An integer $g \geq 2$ is said to be a genus of a finite group $G$ if there is a compact Riemann surface of genus $g$ on which $G$ acts as a group of automorphisms. In this paper finite abelian $p$-groups of arbitrarily large rank, where $p$ is an odd prime, are investigated. For certain classes of abelian $p$-groups the minimum reduced stable genus $\sigma_0$ of $G$ is calculated and consequently the genus spectrum of $G$ is completely determined for certain "extremal" abelian $p$-groups. Moreover for the case of $Z_p^{r_1} \oplus Z_{p^2}^{r_2}$ we will see that the genus spectrum determines the isomorphism class of the group uniquely.

We study the ruled surfaces in Minkowski 3-space with pointwise 1-type Gauss map. As a result, we introduce some new examples of the ruled surfaces with pointwise 1-type Gauss map in Minkowski 3-space.

In this paper we consider the following Neumann problem $$ \left\{ \begin{array}{lll} -\Delta u = \alpha(x)(F_u(u,v)-u) + \lambda G_u(x,u,v) \ in \; \Omega \\ -\Delta v = \alpha(x) (F_v(u,v)-v) + \lambda G_v(x,u,v) \ in \; \Omega \\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0 \ on \ \partial \Omega \end{array} \right. $$ In particular, by means of a multiplicity theorem obtained by Ricceri, we establish that if the set of all global minima of the function $\mathbb{R}^2 \ni y \longmapsto \frac{|y|^2}{2} - F(y)$ (where $F \in C^1(\mathbb{R}^2)$ and it satisfies the condition $F(0,0) = 0$) has at least $k \geq 2$ connected components, then the above Neumann problem admits at least $k+1$ weak solutions, $k$ of which are lying in a given set.

A graph $G$ is said to be $H$-regular if for each vertex $v \in V(G)$, the graph induced by $N_G(v)$ is isomorphic to $H$. A graph $H$ is a feasible neighbor-subgraph if there exists an H-regular graph, otherwise $H$ is a forbidden neighbor-subgraph. In this paper, we obtain some classes of graphs $H$ which are forbidden and then we focus on searching $H$-regular graphs especially those graphs of smaller order.

Intersection bodies define a continuous and $GL(n)$ contravariant valuation which plays a crucial role in the solution of the Busemann-Petty problem. In this paper, we introduce the concept of $L_{p}$ radial Minkowski homomorphisms and consider the Busemann-Petty type problem whether $\Phi_{p} K \subseteq \Phi_{p} L$ implies $V(K) \leq V(L)$, where $\Phi_{p}$ is a homogeneous of degree $\displaystyle \left(\frac{n}{p}-1\right)$, continuous operator on star bodies which is an $SO(n)$ equivariant valuation. Previous results by Schuster are generalized to a large class of $L_{p}$ radial valuations.

We study the blowup behavior of non-negative solutions of the following problem: \begin{equation*} \begin{array}{rll} u_t & \hspace{-0.2cm} \displaystyle = u^{p}(\Delta u + u^q) & \quad {\rm in}\quad \Omega \times (0,T),\\ u(x,t) & \hspace{-0.2cm} \displaystyle = 0 & \quad {\rm whenever} \quad x \in \partial \Omega, \end{array} \end{equation*} with $p \gt 0$ and $q \gt 1$. We will show that it is possible to have solutions blowing up at only one point, and $$\limsup_{t \to T^-} \left( (T-t)^{1/(p+q-1)} \max_\Omega u(x,t) \right) = \infty.$$

In this paper, we prove strong convergence theorems of modified Halpern’s iteration for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping in a Banach space by using two hybrid methods. Using these results, we obtain new convergence results for resolvents of maximal monotone operators and relatively nonexpansive mappings in Banach spaces.

In this paper, we use the results in [4, 5, 11] to prove several interesting local or global density results of minimal, ergodic or reducible quasi-periodic linear systems on $SU(2)$ or $SO(3,\mathbb{R})$. In particular, we give a positive answer to a open problem of H. Eliasson in [3, 4].

In the present paper, we generalize the concept of well-posedness to a hemivariational inequality, give some metric characterizations of the wellposed hemivariational inequality, and derive some conditions under which the hemivariational inequality is strongly well-posed in the generalized sense. Also, we show that the well-posedness of the hemivariational inequality is equivalent to the well-posedness of the corresponding inclusion problem.

In order to unify various concepts of distance-two labelings, we consider a general setting of distance-two labelings as follows. Given a graph $H$, an $L(2,1)$-$H$-labeling of a graph $G$ is a mapping $f$ from $V(G)$ to $V(H)$ such that $d_H(f(u),f(v)) \ge 2$ if $d_G(u,v) = 1$ and $d_H(f(u),f(v)) \ge 1$ if $d_G(u,v) = 2$. Suppose $\cal F$ is a family of graphs. The $L(2,1)$-${\cal F}$-labeling problem is to determine the $L(2,1)$-${\cal F}$-labeling number $\lambda_{\cal F}(G)$ of a graph $G$ which is the smallest number $|E(H)|$ such that $G$ has an $L(2,1)$-$H$-labeling for some $H \in {\cal F}$. Notice that the $L(2,1)$-${\cal F}$-labeling is the $L(2,1)$-labeling (respectively, the circular distance-two labeling) if ${\cal F}$ is the family of all paths (respectively, cycles). The purpose of this paper is to study the $L(2,1)$-${\cal F}$-labeling problem.

In the note we investigate the main duality properties of quantum (or local operator) spaces involving quantum bornology. Namely, we prove that each finite complete bornology admits precisely one quantization and each complete quantum space is a matrix bornology quotient of a local trace class algebra.

For $p\!\gt \!3$ an odd prime, let $\Gamma$ be a congruence subgroup between $\Gamma_1(p)$ and $\Gamma_0(p)$. In this article, we give an explicit basis for the group of modular units on $X(\Gamma)$ that have divisors defined over $\mathbb{Q}$. As an application, we determine the order of the cuspidal $\mathbb{Q}$-rational torsion subgroup of $J(\Gamma)$ generated by the divisor classes of cuspidal divisors of degree $0$ defined over $\mathbb{Q}$.

In the present paper, we estimate the covering number for some reproducing kernel Hilbert spaces on the unit sphere. Both the upper bounds and the lower bounds are provided.

Very recently, Plubtieng and Kumam [S. Plubtieng, P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings, J. Comput. Appl. Math. 224 (2009) 614-621] proposed an iterative algorithm for finding a common solution of a variational inequality problem for an inverse-strongly monotone mapping and a fixed point problem of a countable family of nonexpansive mappings, and obtained a weak convergence theorem. In this paper, based on Plubtieng-Kumam's iterative algorithm we introduce a new iterative algorithm for finding a common solution of a generalized mixed equilibrium problem with perturbation and a fixed point problem of a countable family of nonexpansive mappings in a Hilbert space. We first derive a strong convergence theorem for this new algorithm under appropriate assumptions and then consider a special case of this new algorithm. Moreover, we establish a weak convergence theorem for this special case under some weaker assumptions. Such a weak convergence theorem unifies, improves and extends Plubtieng-Kumam's weak convergence theorem. It is worth pointing out that the proof method of strong convergence theorem is very different from the one of weak convergence theorem.

In this paper, some existence theorems are obtained for multiple solutions of an ordinary $p$-Laplacian system by using the minimax principle. Our results generalize and improve those in the literatures.

Let $X, Y$ be linear topological subspaces of the space of all measurable functions over a $\sigma$-finite, atomless and complete measure space $\Omega$. The question of existence of a nonzero, linear continuous operator $T: X \to Y$ is a natural extension of the question "does X admit a nonzero, linear continuous functional?". In case of $X = L^\Phi(\Omega)$, $Y = L^\Psi(\Omega)$ being Orlicz spaces Ph.Turpin ([11]) gave a criterion telling when there is no nonzero, linear and continuous operator between $L^\Phi(\Omega)$ and $L^\Psi(\Omega)$. That result was extended to the case of Musielak-Orlicz spaces by A.K.Kalindé, R.Płuciennik and M.Wisła ([2, 8]) but only necessary conditions have been presented by them - in fact conditions assuring that there is no nonzero, linear and continuous operator between $L^\Phi(\Omega)$ and $L^\Psi(\Omega)$. In this paper we give inverse theorems to the Kalindé and Płuciennik-Wisła theorems. Generally speaking we state that there exists a nonzero, linear and continuous operator if and only if it there exists a set $A \subset \Omega$ of positive and finite measure such that the inclusion operator $i: L^\Phi(A) \to L^\Psi(A)$ is nonzero and continuous.