Issue 2, Volume 37, June 2012

2012 · Open access

1. GLAESKE-KILBAS-SAIGO FRACTIONAL INTEGRATION AND FRACTIONAL DIXMIER TRACE EL-NABULSI AHMAD RAMI PDF

   ▸ Abstract

   The Dixmier trace was introduced by Jacques Dixmier in 1966 and its key role in noncommutative geometry was discovered by Connes around 1990 during his development of non-commutative infinitesimal calculus. Remarkably, the Dixmier trace is used to define dimension, integration and has been used along with heat kernel type expansions, to define ‘spectral actions’ for noncommutative quantum field theories. This work concerns a generalization of the Dixmier trace to its fractional counterpart. Some new properties are raised and explored in some details.

   149-160
2. AN IMPROVED DESCENT CONJUGATE GRADIENT METHOD AND ITS CONVERGENCE MIN SUN, JING LIU PDF

   ▸ Abstract

   Recently, a new family of conjugate gradient methods with a Grippo-Lucidi type step length rule is proposed by Shi and Guo [A new family of conjugate gradient methods. Journal of Computational and Applied Mathematics, 2009, 224:444-457]. In this paper, we improve Shi and Guo’s method by adopting an improved Grippo-Lucidi type step length rule, and the improvement is twofold: (1) We drop the local Lipschitz constant in the step length, which is beyond the problem data and has to be further estimated; (2) The search direction $d_k$ only needs to satisfy the descent property instead of the sufficient descent property. The global convergence result of the modified method is established under some mild conditions. Preliminary numerical results are also reported to show the efficiency of the improved method.

   161-169
3. ON THE TOPOLOGY OF RATIONAL FUNCTIONS IN TWO COMPLEX VARIABLES NGUYEN TAT THANG PDF

   ▸ Abstract

   We give some characterizations for the critical values at infinity of a rational function in two complex variables in terms of the Euler characteristic, the Malgrange condition and the M-tameness.

   171-187
4. STABILITY OF FUNCTIONAL EQUATIONS OF SEVERAL VARIABLES WHICH ARE ADDITIVE OR QUADRATIC IN EACH VARIABLES IN FRÉCHET SPACES MADJID ESHAGHI GORDJI, ALI EBADIAN, RASOUL AGHALARY PDF

   ▸ Abstract

   In this paper, we establish the Hyers–Ulam–Rassias stability of the system of functional equations $$\begin{cases} f(x_1, x_2, \dots, x_{i_r−1}, a + b, x_{i_r+1}, \dots, x_n) = f(x_1, x_2, \dots, x_{i_r−1}, a, x_{i_r+1}, \dots, x_n) \\ +f(x_1, x_2, \dots, x_{i_r−1}, b, x_{i_r+1}, \dots, x_n), \\ f(x_1, x_2, \dots, x_{j_s−1}, a + b, x_{j_s+1}, \dots, x_n) + f(x_1, x_2, \dots, x_{j_s−1}, a − b, x_{j_s+1}, \dots, x_n)\\ = 2f(x_1, x_2, \dots, x_{j_s−1}, a, x_{j_s+1}, \dots, xn) + 2f(x_1, x_2, \dots, x_{j_s−1}, b, x_{j_s+1}, \dots, x_n)\end{cases}$$ in Fréchet spaces, where $1 \leq i_1 < i_2 < \dots < i_k < n,$ $1 < j_1 < j_2 < \dots < j_{n−k},$ ${1, 2, \dots, n} = \{i_1, i_2, \dots, i_k\} \cup \{j_1, j_2, \dots, j_{n−k}\}$.

   189-200
5. A NEW DECOMPOSITION ALGORITHM FOR GLOBALLY SOLVING MATHEMATICAL PROGRAMS WITH AFFINE EQUILIBRIUM CONSTRAINTS L. D. MUU, Q. TRAN DINH, L. T. H. AN, P. D. TAO PDF

   ▸ Abstract

   This paper proposes a new decomposition method for globally solving mathematical programming problems with affine equilibrium constraints (AMPEC). First, we view AMPEC as a bilevel programming problem where the lower level problem is a parametric affine variational inequality. Then we use a regularization technique to formulate the resulting problem as a mathematical program with an additional constraint defined by the difference of two convex functions (DC function). A main feature of this DC decomposition is that the second component depends upon only the parameter in the lower level problem. This property allows us to develop branch-and-bound algorithms for globally solving AMPEC where the adaptive rectangular bisection takes place only in the space of the parameters. As an example, we use the proposed algorithm to solve a bilevel Nash-Cournot equilibrium market model. Computational results show the efficiency of the proposed algorithm.

   201-217
6. THE GENERALIZED GAMMA FUNCTIONS TRAN GIA LOC, TRINH DUC TAI PDF

   ▸ Abstract

   In this paper, we introduce a way to generalize the Euler’s gamma function as well as some related special functions. With a given polynomial in one variable $f(t) \geq 0$, we can associate a function, so-called “gamma function associated with $f$”, defined by $\Gamma_f (s) := \int_0^{\infty}f^{s-1}e^{-t} dt$. This function has many features similar to the Euler’s gamma function. We also present some initial results on the gamma-type functional equation for $\Gamma_f (s)$ in some special cases.

   219-230
7. REMARKS ON EKELAND’S VARIATIONAL PRINCIPLE FOR POLYNOMIAL FUNCTIONS NGUYEN THI THAO PDF

   ▸ Abstract

   In this paper, we provide a method to determine points $v \in \mathbb R^n$ which satisfy Ekeland’s variational principle, and also to choose Palais-Smale minimizing sequences that satisfy the second order condition for polynomial functions bounded from below on $\mathbb R^n$ .

   231-242
8. THE COMPARISON OF THE CONVERGENCE SPEED BETWEEN PICARD, MANN, ISHIKAWA AND TWO-STEP ITERATIONS IN BANACH SPACES DUONG VIET THONG PDF

   ▸ Abstract

   The purpose of this paper is to compare convergence speed of the Picard and two-step iterations, Mann and two-step iterations, Ishikawa and two-step iterations for the class of Zamfirescu operators.

   243-249
9. ON THE RATIONAL RECURSIVE SEQUENCE $$x_{n+1}=Ax_n+Bx_{n-k}+\dfrac{1+x_nx_{n-k}}{px_n+x_{n-k}}$$ ELSAYED M. E. ZAYED PDF

   ▸ Abstract

   In this article, we study the global stability and the asymptotic properties of the positive solutions of the nonlinear difference equation $$x_{n+1}=Ax_n+Bx_{n-k}+\dfrac{1+x_nx_{n-k}}{px_n+x_{n-k}}, \quad n=0,1,2,\dots $$ where the parameters $A, B, p$ and the initial conditions $x_{−k}, \dots, x_{−1}, x_0$ are arbitrary positive real numbers, while $k$ is a positive integer number. Some numerical examples will be given to illustrate our results.

   251-266
10. ON THE PERMANENCE OF PREDATOR-PREY MODEL WITH THE BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE IN PERIODIC ENVIRONMENT TRINH TUAN ANH, NGUYEN HUU DU, TONG THANH TRUNG PDF

    ▸ Abstract

    This paper studies a predator-prey population system described by a differential equation with the Beddington-DeAngelis functional response in the periodic case. We establish a sufficient criterion for the permanence of the system and for the existence of a positive periodic solution.

    267-280
11. $\lambda$-CENTRAL BMO ESTIMATES FOR MULTILINEAR COMMUTATORS OF MARCINKIEWICZ OPERATOR ON CENTRAL MORREY SPACES XIAOMING HUANG, LANZHE LIU PDF

    ▸ Abstract

    In this paper, we establish $\lambda$-central BMO estimates for the multilinear commutator related to the Marcinkiewicz operator in central Morrey spaces.

    281-292
12. PRODUCT OF COMPOSITION AND MULTIPLICATION OPERATORS GOPAL DATT PDF

    ▸ Abstract

    The paper discusses some properties for the operator $W_{(u,T )} = C_TM_u$ given by $f\mapsto u \circ T \cdot f \circ T$, on Orlicz spaces using the conditional expectation operator.

    293-300