Issue 1, Volume 36, April 2011

2011 · Open access

1. COMMON FIXED POINT THEOREMS FOR OCCASIONALLY WEAKLY COMPATIBLE MAPS HAKIMA BOUHADJERA, CHRISTIANE GODET-THOBIE PDF

   ▸ Abstract

   In this paper, we establish some common fixed point theorems for two pairs of occasionally weakly compatible single and set-valued maps satisfying a strict contractive condition in a metric space. Our results unify and extend many results existing in the literature including those of Aliouche [3], Bouhadjera [4] and some results of Popa [19]-[23]. Also, we establish another common fixed point theorem for four occasionally weakly compatible single and set-valued maps of Greguš type which improves the results of Djoudi and Nisse [5], Pathak et al. [17] and others and we end our work by giving another theorem which generalizes the results given by Elamrani and Mehdaoui [6], Mbarki [14] and references therein.

   1-17
2. UNIFORM ATTRACTORS FOR A NON-AUTONOMOUS PARABOLIC EQUATION INVOLVING GRUSHIN OPERATOR CUNG THE ANH, NGUYEN VAN QUANG PDF

   ▸ Abstract

   In this paper, we consider the first initial boundary value problem for a non-autonomous semilinear degenerate parabolic equation involving the Grushin operator, in which the nonlinearity satisfies the polynomial condition of arbitrary order and the external force is normal. Using the asymptotic a priori estimate method, we prove the existence of uniform attractors for this problem. The obtained results, in particular, improve some recent ones for the non-autonomous Laplacian equation.

   19-33
3. LYAPUNOV EXPONENTS AND CENTRAL EXPONENTS OF LINEAR ITO STOCHATIC DIFFERENTIAL EQUATIONS NGUYEN DINH CONG, NGUYEN THI THUY QUYNH PDF

   ▸ Abstract

   We study Lyapunov, central and auxiliary exponents of linear Ito stochastic equations. We show that the central exponents are nonrandom like Lyapunov exponents, the nonrandomness of which was proved in [8]. We prove that under a nondegeneracy condition the central exponents $\Theta_k$ of a linear Ito stochastic differential equation coincide with its auxiliary exponents $\gamma_k$, and, moreover, all the first exponents coincide: $\Theta_1 = \lambda_1 = \Omega_1 = \gamma_1$.

   35-53
4. RELATION BETWEEN THE HARDNESS OF A PROBLEM AND THE NUMBER OF ITS SOLUTIONS THAN QUANG KHOAT PDF

   ▸ Abstract

   In this note, we survey the effect of the number of solutions on solving problems. Theoretically, the number of solutions to a problem cannot help the problem to be easier in the sense of computation. That is, a problem having few solutions may be as easy as the one having many solutions. For the aim of giving some evidences, we show that the Subset sum problem, Knapsack problem and Bounded Integer Programming problem under the assumption that the problem has either no solution or exponentially many solutions are NP-hard. On the other hand, the Knapsack Optimization problem having at most one solution is also NP-hard.

   55-60
5. A NOTE ON WEAKLY KOSZUL MODULES YUAN PAN PDF

   ▸ Abstract

   Let $M$ be a weakly Koszul module. Then Martínez-Villa and Zacharia proved that $M$ admitted a filtration of submodules $0 = U_0 \subseteq U_1 \subseteq U_2 \subseteq \dots · · · \subseteq U_p = M$, such that all $U_{i+1}/U_i$ are Koszul modules (see [11]). Now further, let $\mathcal P_∗^i \to U_i/U_{i−1} \to 0$ and $\mathcal P_∗ \to M \to 0$ be the corresponding minimal graded projective resolutions. We prove that, for all $n \geqslant 0$, $\mathcal P_n \cong \oplus \mathcal P_n^i$ . Moreover, we also give a new characterization for a module $M$ to be weakly Koszul in terms of the filtration of the complex $\mathcal P_∗$, where $\mathcal P_∗ \to M \to 0$ is a minimal graded projective resolution.

   61-71
6. ON THE RATIONAL RECURSIVE SEQUENCE $x_{n+1}=\frac{A+\alpha_0x_n+\alpha_1x_{n-\sigma}}{B+\beta_0x_n+\beta_1x_{n-\tau}}$ E. M. E. ZAYED, M. A. EL-MONEAM PDF

   ▸ Abstract

   The main objective of this paper is to study the boundedness, the periodicity, the convergence and the global stability of the positive solutions of the difference equation $$x_{n+1}=\frac{A+\alpha_0x_n+\alpha_1x_{n-\sigma}}{B+\beta_0x_n+\beta_1x_{n-\tau}}, \ \ n=0,1,2,\dots$$ where the coefficients $A, B, \alpha_0, \alpha_1, \beta_0, \beta_1 \in  (0,\infty)$ and $\sigma, \tau \in N$. The initial conditions $x_{−\omega},\dots,x_{−1}, x_0$ are arbitrary positive real numbers and $\omega = \max\{\tau, \sigma\}$. Some numerical examples are presented.

   73-87
7. CONVERGENCE TO COMMON FIXED POINT OF MULTI-STEP ITERATION WITH ERRORS FOR ASYMPTOTICALLY QUASI-NONEXPANSIVE MAPPINGS GURUCHARAN SINGH SALUJA PDF

   ▸ Abstract

   In this paper, strong convergence theorems for modified multistep iteration scheme with errors for a finite family of uniformly $\phi$ - continuous and asymptotically quasi-nonexpansive mappings are established in the framework of real uniformly convex Banach spaces. The results presented in this paper extend and improve the corresponding results of Khan and Fukhar-uddin [7], Khan and Takahashi [8], Shahzad and Udomene [18], Xu and Noor [21], Cho et al. [3], Rhoades [14], Schu [15] and some others.

   89-103
8. AN EXTENSION OF MICHAEL’S SELECTION THEOREM LIANG-JU CHU, CHIEN-HAO HUANG PDF

   ▸ Abstract

   In this paper, we extend earlier Michael’s selection theorem in the general nonconvex setting. Indeed, we deal with the case of almost lower semicontinuous multifunctions $T$ on a zero-dimensional paracompact topological space $X$. Based on a kind of equicontinuous property, we prove that any a.l.s.c. ECP multifunction from $X$ into a complete metric space $Y$ still has a continuous selection.

   105-112
9. ON LOCALLY NILPOTENT MAXIMAL SUBGROUPS OF THE MULTIPLICATIVE GROUP OF A DIVISION RING BUI XUAN HAI PDF

   ▸ Abstract

   Let $D$ be a division ring with the center $F$ and $D^∗$ be the multiplicative group of $D$. In this paper we study locally nilpotent maximal subgroups of $D^∗$. We give some conditions that influence the existence of locally nilpotent maximal subgroups in division ring with infinite center. Also, it is shown that if $M$ is a locally nilpotent maximal subgroup that is algebraic over $F$, then either it is the multiplicative group of some maximal subfield of $D$ or it is center-by-locally finite. If, in addition we assume that $F$ is finite and $M$ is nilpotent, then the second case cannot occur, i.e. $M$ is the multiplicative group of some maximal subfield of $D$.

   113-118
10. A METHOD OF EXTENDING BY PARAMETER FOR APPROXIMATE SOLUTIONS OF OPERATOR EQUATIONS KHUAT VAN NINH PDF

    ▸ Abstract

    A method of extending by parameter has been researched in the works of V. A. Trenoghin [4]-[6], A. A. Fonarov [1] and Y. L. Gaponenco [2]. In this paper, the author presents an application of this method for approximating solutions of operator equations with any growth. Some results on the existence of solutions of operator equations are derived.

    119-127