Issue 1, Volume 41, March 2016

2016

1. Asymptotically Stable Solutions for a Nonlinear Functional Integral Equation L. T. P. Ngoc , N. T. Long PDF

   ▸ Abstract

   In this paper, the solvability and the existence of asymptotically stable solutions for a nonlinear functional integral equation are studied. The main tools are a fixed point theorem of Krasnosel’skii type, the dominated convergence theorem, and other results from functional analysis. In order to illustrate the result obtained here, an example is given.

   1-24
2. Uniqueness of Meromorphic Functions Whose Certain Differential Polynomials Share a Small Function Pulak Sahoo , Biswajit Saha PDF

   ▸ Abstract

   n this paper, we study the uniqueness of meromorphic functions whose certain nonlinear differential polynomials share a small function with finite weight. Our result generalizes and improves the recent results due to A. Banerjee and the present first author Sarajevo (Sarajevo J. Math. 88(20), 69–89, (2012)).

   25-36
3. Existence and Long-Time Behavior of Variational Solutions to a Class of Nonclassical Diffusion Equations in Noncylindrical Domains Nguyen Duong Toan PDF

   ▸ Abstract

   We prove the existence and uniqueness of variational solutions to the following non-autonomous nonclassical diffusion equation $$ u_t - \Delta u_t - \Delta u + f(u)= g(x,t) $$ in a non-cylindrical domain with the homogeneous Dirichlet boundary condition, under assumptions that the spatial domains are bounded and increase with time, the nonlinearity $f$ satisfies growth and dissipativity conditions of Sobolev type, and the external force $g$ is time-dependent. Moreover, the non-autonomous dynamical system generated by this class of solutions is shown to have a pullback attractor $\hat{\mathcal A}=\{A(t): t\in \mathbb{R}\}$.

   37-53
4. Regularization Methods and Iterative Methods for Variational Inequality with Accretive Operator Nguyen Thi Thu Thuy PDF

   ▸ Abstract

   We consider a regularization method based on the Browder–Tikhonov and iterative regularization method that can be used to find a solution of a class of accretive variational inequalities in a $q$-uniformly smooth Banach space, and the solutions are sought in the set of common fixed points of a countable family of nonexpansive mappings. We also introduce an iteration method, which is implicit and converges strongly, based on the steepest descent method with a strongly accretive and strictly pseudocontractive mapping. Our results generalize and extend some well-known regularization methods in the literature.

   55-68
5. Strongly Koszul Edge Rings Takayuki Hibi , Kazunori Matsuda , Hidefumi Ohsugi PDF

   ▸ Abstract

   We classify the finite connected simple graphs whose edge rings are strongly Koszul. From the classification, it follows that if the edge ring is strongly Koszul, then its toric ideal possesses a quadratic Gröbner basis.

   69-76
6. Lyapunov Functionals that Lead to Exponential Stability and Instability in Finite Delay Volterra Difference Equations Catherine Kublik , Youssef Raffoul PDF

   ▸ Abstract

   We use Lyapunov functionals to obtain sufficient conditions that guarantee exponential stability of the zero solution of the finite delay Volterra difference equation $$ x(t+1) = a(t)x(t)+\sum^{t-1}_{s=t-r}b(t,s)x(s). $$ Also, by displaying a slightly different Lyapunov functional we obtain conditions that guarantee the instability of the zero solution. The highlight of the paper is the relaxing the condition $|a(t)| <1.$ Moreover we provide examples in which we show that our theorems provide an improvement of some recent results.

   77-89
7. Some Geometric Properties of Generalized Cesàro-Musielak-Orlicz Spaces Equipped with the Amemiya Norm Atanu Manna , P. D. Srivastava PDF

   ▸ Abstract

   A generalized Ces\`{a}ro-Musielak-Orlicz sequence space $Ces_\Phi(q)$ endowed with the Amemiya norm is introduced. Criteria for the coordinatewise uniformly Kadec-Klee property and the Uniform Opial property of the spaces $Ces_\Phi(q)$ with respect to the Amemiya norm are obtained.

   91-102
8. A Predictor-Corrector Algorithm for $P_∗(κ)$-Linear Complementarity Problems Based on a Specific Self-Regular Proximity Function B. Kheirfam , K. Ahmadi PDF

   ▸ Abstract

   First-order predictor-corrector methods working in a large neighborhood of the central path are among the most efficient interior point methods. In Peng et al. (SIAM J. Optim. 15(4):1105–1127, 2005), based on a specific proximity function, a wide neighborhood of the central path is defined which matches the standard large neighborhood defined by the infinity norm. In this paper, we extend the predictor-corrector algorithm proposed for linear optimization in Peng et al. (SIAM J. Optim. 15(4):1105–1127, 2005) to $P_∗(κ)$-linear complementarity problems. Our algorithm performs two kinds of steps. In corrector steps, we use the specific self-regular proximity function to compute the search directions. The predictor step is the same as the predictor step of standard predictor-corrector method in the interior point method literature. We prove that our predictor-corrector algorithm has an ${\mathcal {O}}\left ((1+2\kappa )\sqrt {n}\log n\log \frac {(x^{0})^{T} s^{0}}{\epsilon }\right )$ iteration bound, which is the best known iteration complexity for problems of this type.

   103-120
9. Power Values of Derivations on Multilinear Polynomials in Prime Rings Basudeb Dhara , Sukhendu Kar , Sachhidananda Mondal PDF

   ▸ Abstract

   Let $R$ be a prime ring with center $Z(R)$ and with extended centroid $C, d$ a derivation of $R$ and $f(x_1,\dots,x_n )$ a nonzero multilinear polynomial over $C$, $m \geq 1$ and $p \geq 1$ two integers. In the present paper, we study the situations (i) $((d(f(x_1,\dots,x_n )))^m − f(x_1,\dots,x_n ))^p = 0$; (ii) $((d(f(x_1,\dots,x_n )))^m − f(x_1,\dots,x_n ))^p \in Z(R)$ for all $x_1,\dots,x_n$ in some subsets of $R$.

   121-131
10. Some Properties, Explicit Evaluations, and Applications of Ramanujan’s Remarkable Product of Theta Functions Nipen Saikia PDF

    ▸ Abstract

    We study several properties of Ramanujan’s remarkable product of theta functions $a_{m,n}$ and establish its connections with some other parameters involving theta-functions. We also show that the explicit values of $a_{m,n}$ can be used to evaluate Ramanujan’s class invariants and some other parameters of theta functions connected with Ramanujan’s continued fractions.

    133-142
11. The Backward Problem for Ginzurg-Landau-Type Equation Dang Duc Trong , Bui Thanh Duy , Nguyen Dang Minh PDF

    ▸ Abstract

    Let $\Omega$ be a bounded domain of $\mathbb R^n$. In this paper, we consider a final value problem for the nonlinear parabolic equation $$\begin{array}{@{}rcl@{}} u_{t} &=& {\Delta} u + au-bu^{3}+h(x,t),\quad x \in {\Omega},\quad t \in \left( {0,T} \right), \\ u&=&0 ,\quad x \in \partial {\Omega},\quad t \in \left( {0,T} \right),\\ u(T) &=& g,\quad x \in {\Omega}, \end{array}$$ where $g, h$ are given functions and the numbers $a, b$ $( b > 0)$ are modeling parameters. The problem does not fulfill Hadamard’s postulates of well posedness: it might not have a solution in the strict sense; its solutions might not be unique or might not depend continuously on the data. Hence, its mathematical analysis is subtle. However, it has many applications in physics and other fields. For this reason, a regularization for the problem is proposed. In our problem, the function $f(u) = a u − b u^3$ is not globally Lipschitz. So, we cannot apply directly recent methods that have been used in Trong et al. (Zeitschrift Analysis und ihre Anwendungen 26(2), 231–245, 2007), Trong et al. (Electron. J. Differ. Equ. 2009(109), 1–16, 2009), and Trong and Tuan (Electron. J. Differ. Equ. 2009(77), 1–13, 2009). We have to approximate the function f by a globally Lipschitz function and use an approximate equation to find the regularization solution of the problem. Error estimations between the exact solution and the approximate solution, established from noise data $g_{\epsilon}, a_{\delta} , b_{\delta}$, are given.

    143-169
12. Linear Difference Equations and Periodic Sequences over Finite Fields Dang Vu Giang PDF

    ▸ Abstract

    First, we study linear equations over finite fields in general. An explicit formula for a common period is found for every solution of a linear difference equation over a finite field. It will help to estimate the $p$-adic modulus of polynomial roots. Second, we focus our attention on periodic sequences over finite fields and Hamiltonian cycles in de Bruijn directed graph.

    171-181