Issue 2, Volume 41, June 2016

2016

1. Pullback Attractors for a Non-Autonomous Semilinear Degenerate Parabolic Equation on $\mathbb R^N$ Nguyen Dinh Binh , Nguyen Nhu Thang , Le Thi Thuy PDF

   ▸ Abstract

   In this paper, we prove the existence of pullback attractors for the following non-autonomous semilinear degenerate parabolic equation on $\mathbb R^N$: $$\frac{\partial u}{\partial t} - \textup{div} (\sigma (x)\nabla u) + \lambda u+ f(x,u) = g(x,t),$$ under a new condition concerning a variable non-negative diffusivity $\sigma(\cdot)$, an arbitrary polynomial growth order of the nonlinearity $f$ and an exponent growth of the non-autonomous external force $g$.

   183-199
2. Value Sharing by an Entire Function with Its Derivatives Indrajit Lahiri , Rajib Mukherjee PDF

   ▸ Abstract

   We prove a uniqueness theorem for an entire function, which shares certain values with its higher-order derivatives.

   201-207
3. Weighted BMO Type Spaces Associated to Admissible Functions and Applications Nguyen Ngoc Trong , Nguyen Thanh Tung PDF

   ▸ Abstract

   Let $\rho$ be an admissible function on an RD-space $X$. The main aim of this paper is to establish the theory of weighted bounded mean oscillation (BMO) type spaces $BMO^{\beta }(X,w), BMO^{\beta }_{\rho } (X,w)$ and their basic properties, including the John-Nirenberg inequality for these spaces. As applications, we also study the boundedness of some singular integrals such as the Hardy-Littlewood maximal operators, the radial maximal functions, the Poisson semigroup maximal functions, the Littlewood-Paley $g$-functions, and fractional integrals on these weighted BMO type spaces. Our findings extend well-known results in Bongioanni et al. (J. Math. Anal. Appl. 348, 12–27, 2008), Liu et al. (Math. Nachr. 357, 1–35, 2012), Yang and Zhou (Trans. Am. Math. Soc. 363, 1197–1239, 2011), Yang et al. (Commun. Pure Appl. Anal. 9, 779–812, 2010), and Yang et al. (Nagoya Math. J. 198, 77–119, 2010).

   209-241
4. Some Differential Estimates in Linear Programming Nguyen Nang Thieu PDF

   ▸ Abstract

   This paper proposes two sets of formulas for the sensitivity analysis of the optimal value function of a parametric linear programming problem. The first one evaluates the changes of the optimal value with respect to perturbations of the right-hand side vector in the constraint system and of the cost vector. The second one gives a lower bound and an upper bound for the Dini directional derivative of the optimal value function.

   243-249
5. A Modified Extragradient Method for Infinite-Dimensional Variational Inequalities Pham Duy Khanh PDF

   ▸ Abstract

   A modified form of the extragradient method for solving infinite-dimensional variational inequalities is considered. The weak convergence and the strong convergence for the iterative sequence generated by this method are studied. We also propose several examples to analyze the obtained results.

   251-263
6. Generalization of $C^{1,1}$ Property in Infinite Dimension Marie Dvorská , Karel Pastor PDF

   ▸ Abstract

   The class of functions with locally Lipschitz gradient, i.e., the class of $C^{1,1}$ functions, has been deeply studied and many optimization conditions has been stated for this class. A generalization of $C^{1,1}$ property leads to the class of $\ell$–stable functions. In this paper, we study two formally different definitions of $\ell$–stability in infinite dimension and show their equivalence.

   265-275
7. The Existence of Unbounded Closed Convex Sets with Trivial Recession Cone in Normed Spaces Huynh The Phung PDF

   ▸ Abstract

   It’s well known that a closed convex set in a finite-dimensional normed space is unbounded if and only if it has a nonzero recession direction. In this work, we shall prove that in every infinite-dimensional normed space there exists an unbounded closed convex set whose recession cone consists of the zero vector alone.

   277-282
8. On the Global Attractor for a Semilinear Strongly Degenerate Parabolic Equation Mai Xuan Thao PDF

   ▸ Abstract

   We prove the existence of a global attractor in $\mathcal {S}^{2}_{0}({\Omega })\cap L^{2p-2}({\Omega })$ for a semilinear strongly degenerate parabolic equation in a bounded domain with the homogeneous Dirichlet boundary condition, in which the nonlinearity satisfies a polynomial type condition of arbitrary order and the external force belongs to $L^2(\Omega)$. This global attractor is then shown to have a finite fractal dimension in $L^2(\Omega)$. We also study the existence and exponential stability of the unique stationary solution to the problem.

   283-297
9. An Efficient Class of Multipoint Root-Solvers With andWithout Memory for Nonlinear Equations Tahereh Eftekhari PDF

   ▸ Abstract

   A new class of optimal eighth-order iterative methods without memory for solving nonlinear equations is presented. Based on this new class of methods, we present an efficient class of multipoint methods with memory by suitable variation of a free parameter in each iterative step. Consequently, the order of convergence is increased without any additional function evaluations from 8 to 12. Numerical comparisons are made to show the performance of the presented methods.

   299-311
10. Determination of a Time-Dependent Term in the Right-Hand Side of Linear Parabolic Equations Nguyen Thi Ngoc Oanh , Bui Viet Huong PDF

    ▸ Abstract

    The problem of determining a time-dependent term from integral observations in the right-hand side of parabolic equations is studied. It is reformulated into a variational problem, and a formula for the gradient of the functional to be minimized is derived via an adjoint problem. The variational problem is discretized by the splitting method based on finite differences. A formula for the gradient of the discretized functional is given and the conjugate gradient method is suggested for numerically solving the problem. Several numerical examples are presented for illustrating the efficiency of the algorithm.

    313-335
11. On the Existence and Exponential Attractivity of a Unique Positive Almost Periodic Solution to an Impulsive Hematopoiesis Model with Delays Trinh Tuan Anh , Tran Van Nhung , Le Van Hien PDF

    ▸ Abstract

    In this paper, a generalized model of hematopoiesis with delays and impulses is considered. By employing the contraction mapping principle and a novel type of impulsive delay inequality, we prove the existence of a unique positive almost periodic solution of the model. It is also proved that, under the proposed conditions in this paper, the unique positive almost periodic solution is globally exponentially attractive. A numerical example is given to illustrate the effectiveness of the obtained results.

    337-354
12. Generalized Convolution for the Kontorovich-Lebedev, Fourier Transforms and Applications to Acoustic Fields Trinh Tuan , Nguyen Thanh Hong , Pham Van Hoang PDF

    ▸ Abstract

    To study the boundedness of the acoustic field in $L_{p}(\mathbb {R}_{+} ; dx)$ and its asymptotic behavior, we introduce a new generalized convolution for the Kontorovich-Lebedev ($\mathcal {K}\mathcal {L}$) transform and the Fourier transforms, and a new representation of the acoustic field via this generalized convolution. Some properties of the new generalized convolution are obtained. Moreover, an analog of Watson’s theorem is established, in which we obtain the necessary and sufficient conditions for a class of generalized convolution transforms to be isomorphism-isometric.

    355-367