Issue 3, Volume 43, September 2018

2018

1. Discrete Embeddings for Lagrangian and Hamiltonian Systems Jacky Cresson , Isabelle Greff , Charles Pierre PDF

   ▸ Abstract

   The topic of this paper is to study the conservation of variational properties for a given problem when discretising it. Precisely, we are interested in Lagrangian or Hamiltonian structures and thus with variational problems attached to a least action principle. Consider a partial differential equation (PDE) deriving from a variational principle. A natural question is to know whether this structure is preserved at the discrete level when discretising the PDE. To address this question, a concept of coherence is introduced. Both the differential equation (the PDE translating the least action principle) and the variational structure can be embedded at the discrete level. This provides two discrete embeddings for the original problem. If these procedures finally provide the same discrete problem, we will say that the discretisation is coherent. Our purpose is illustrated with the Poisson problem. Coherence for discrete embeddings of Lagrangian structures is studied for various classical discretisations. For Hamiltonian structures, we show the coherence between a discrete Hamiltonian and the discretisation of the mixed formulation of the Poisson problem.

   391-413
2. Solutions to Partial Functional Differential Equations with Infinite Delays: Periodicity and Admissibility Thieu Huy Nguyen , Quy Dang Ngo PDF

   ▸ Abstract

   Under some appropriate conditions, we prove the existence and uniqueness of periodic solutions to partial functional differential equations with infinite delay of the form $\dot {u}=A(t)u+g(t,u_{t})$ on a Banach space $X$ where $A(t)$ is 1-periodic, and the nonlinear term $g(t, \phi)$ is 1-periodic with respect to $t$ for each fixed $\phi$ in fading memory phase spaces, and is $\varphi(t)$-Lipschitz for $\varphi$ belonging to an admissible function space. We then apply the attained results to study the existence, uniqueness, and conditional stability of periodic solutions to the above equation in the case that the family $(A(t))_{t\geq  0}$ generates an evolution family having an exponential dichotomy. We also prove the existence of a local stable manifold near the periodic solution in that case.

   415-432
3. On the Structure of $\mathcal{N}_{p}$-Spaces in the Ball Bingyang Hu , Le Hai Khoi , Trieu Le PDF

   ▸ Abstract

   We study the structure of $\mathcal{N}_{p}$-spaces in the ball. In particular, we show that any such space is Moebius-invariant and for $0 < p \leq n$, all $\mathcal{N}_{p}$-spaces are different. Our results will be used in the study of operator theory on $\mathcal{N}_{p}$-spaces.

   4330448
4. The Backward Problem for a Nonlinear Riesz-Feller Diffusion Equation Dinh Nguyen Duy Hai , Dang Duc Trong PDF

   ▸ Abstract

   In this paper, we reconstruct the solution $u(x,t)$ of the backward space-fractional diffusion problem with a locally Lipschitzian nonlinear source This problem is severely ill-posed in the Hadamard sense, hence, a regularization is in order. In the paper, we introduce one spectral regularization method and establish stability error estimates with optimal order under an a priori choice of regularization parameter. Finally, numerical implementations are given to show the effectiveness of the proposed regularization methods.

   449-470
5. Properties of Harmonic Functions Defined by Shear Construction Rasoul Aghalary , Mahdi Jahani Rad PDF

   ▸ Abstract

   In this paper, we introduce a new subclass of harmonic functions defined by shear construction. Among other properties of this subclass, we study the convolution of its elements of it with some special subclasses of harmonic functions. Also, we provide coefficient conditions leading to harmonic mappings which are starlike of order $\alpha$. Some interesting applications of the general results are also presented.

   471-483
6. New Stability Criteria for Nonlinear Volterra Integro-Differential Equations Pham Huu Anh Ngoc , Tran The Anh PDF

   ▸ Abstract

   Using a novel approach, we present some new explicit criteria for the uniform asymptotic stability and the exponential stability of nonlinear Volterra integro-differential equations. Some examples are given to illustrate the obtained results.

   485-501
7. A van Leer-Type Numerical Scheme for the Model of a General Fluid Flow in a Nozzle with Variable Cross Section Mai Duc Thanh , Dao Huy Cuong PDF

   ▸ Abstract

   A van Leer-type numerical scheme for the model of a general fluid in a nozzle with variable cross section is presented. The model is nonconservative, making it hard for standard numerical schemes. Exact solutions of local Riemann problems are incorporated in the construction of this scheme. The scheme can work well in regions of resonance, where multiple waves are colliding. Numerical tests are conducted, where we compare the errors and orders of accuracy for approximating exact solutions between this scheme and a Godunov-type scheme. Results from numerical tests show that this van Leer-type scheme has a much better accuracy than the Godunov-type scheme.

   503-547
8. Approximation Properties of Generalized Szász-Type Operators Arun Kajla PDF

   ▸ Abstract

   In the present paper, we study some approximation properties of the generalized Szász type operators introduced by V. Miheşan (Creat. Math. Inf. 17:466–472, 2008). We present a quantitative Voronovskaya-type theorem, local approximation theorem by means of second-order modulus of continuity and weighted approximation for these operators. The rate of convergence for differential functions whose derivatives are of bounded variation is also obtained.

   549-563
9. Cofiniteness of Local Cohomology Modules over Homomorphic Image of Cohen-Macaulay Rings Asghar Farokhi , Alireza Nazari PDF

   ▸ Abstract

   Let $(R,\mathfrak{m})$ be a Noetherian local ring, $M$ a non-zero finitely generated $R$-module, and let $I$ be an ideal of $R$. In this paper, we establish some new properties of local cohomology modules $\mathrm{H}^{i}_{I}(M)$, $i\geq 0$. In particular, we show that if $R$ is catenary, $M$ an equidimensional $R$-module of dimension $d$, and $x_{1},x_{2},\dots ,x_{t}$ is an $I$-filter regular sequence on $M$, then $(0:_{\mathrm{H}^{d-j}_{I}(\frac{M}{\langle x_{1},x_{2},\dots ,x_{i-1}\rangle M})} x_{i})$ is $I$-cofinite for all $i = 1,2,\dots ,t$ and all $i \leq j \leq t$ if and only if $\mathrm{H}^{d-j}_{I}(\frac{M}{\langle x_{1},x_{2},\dots ,x_{i-1}\rangle M})$ is $I$-cofinite for all $i = 1,2,\dots ,t$ and all $i \leq j \leq t$. Also we study the cofiniteness of local cohomology modules over homomorphic image of Cohen-Macaulay rings and we show that $\frac {\mathrm{H}^{\mathcal{W}(I,M)}_{I}(M)}{I\mathrm{H}^{\mathcal{W}(I,M)}_{I}(M)}$ has finite support, where $$\mathcal{W}(I,M) := \text{Max} \{i : \mathrm{H}^{i}_{I}(M) \text{~is not weakly Laskerian}\}.$$

   565-574
10. A Generalization of Strongly Preserver Problems of Drazin Invertibility Mourad Oudghiri , Khalid Souilah PDF

    ▸ Abstract

    Let $\phi$ be an additive map between unital complex Banach algebras such that $\phi(1)$ is invertible. We show that $\phi$ satisfies $\phi(a^D)\phi(b)^D = \phi(a)^D\phi(b^D)$ for every Drazin invertible elements $a, b$ if and only if $\phi(1)^{− 1}\phi$ is a Jordan homomorphism and $\phi(1)$ commutes with the range of $\phi$. A similar result is established for group invertible elements, and more explicit forms of such maps are given in the context of the algebra of all bounded linear operators on a complex Banach space.

    575-583