Publications

2019
Merahi W, Guedjiba S. SOME PROPERTIES OF COMMON HERMITIAN SOLUTIONS OF MATRIX EQUATIONS A1XA*1= B1 AND A2XA2 = B2. MATEMATICKI VESNIK ˇ MATEMATIQKI VESNIK. 2019;71 (3) :214–229.
Guelfen H, Kittaneh F. On Numerical Radius Inequalities for Operator Matrices. Numerical Functional Analysis and Optimization. 2019;40 (11) :1231-1241.
Ameur S. SELFADJOINT OPERATORS, NORMAL OPERATORS, AND CHARACTERIZATIONS. Operators and Matrices. 2019;13 (3) :835–842.Abstract

Let B(H) be the C -algebra of all bounded linear operators acting on a complex separable Hilbert space H . We shall show that:

1. The class of all selfadjoint operators in B(H) multiplied by scalars is characterized by ∀X ∈ B(H), S2X +XS2 =>2||SXS||, (S ∈ B(H)).

2. The class of all normal operators in B(H) is characterized by each of the three following properties (where DS = SS−SS , for S ∈ B(H)),

(i) ∀X ∈ B(H), S2X + XS2 =>2||SXS||,(S ∈ B(H)),

(ii) S∗DSS = 0 = SDSS∗,(S ∈ B(H)),

(iii) SDSS=> 0 =>SDSS,(S ∈ B(H)).

 

Lombarkia F, Boussaid A. Operator equations and inner inverses of elementary operators. Linear and Multilinear Algebra . 2019.Abstract

Let E,F,G,D be infinite complex Banach spaces and B(F,E) the Banach space of all bounded linear operators from F into E. Consider A1,A2∈B(F,E), B1,B2∈B(D,G)B1,B2∈B(D,G). Let MA1,B1:X→A1XB1 be the multiplication operator on B(G,F) induced by A1,B1. In particular, LA1=MA1,I and RB1=MI,B1, where I is the identity operator are the left and the right multiplication operators, respectively. The elementary operator Ψ defined on B(G,F)B(G,F) is the sum of two multiplication operators Ψ=MA1,B1+MA2,B2. This paper gives necessary and sufficient conditions for the existence of a common solution of the operator equations MA1,B1(X)=C1 and MA2,B2(X)=C2 and derive a new representation of the general common solution via the inner inverse of the elementary operator Ψ; we apply this result to determine new necessary and sufficient conditions for the existence of a Hermitian solution and a representation of the general Hermitian solution to the operator equation MA,B(X)=C. As a consequence, we obtain well-known results of Daji and Koliha.

2018
Bouraya C, Ameur S. On the characterizations of some distinguished subclasses of Hilbert space operators. Acta Scientiarum Mathematicarum. 2018;84 (34) :611-627.Abstract

In this note, we present several characterizations for some distinguished classes of bounded Hilbert space operators (self-adjoint operators, normal operators, unitary operators, and isometry operators) in terms of operator inequalities.

Mennouni A, Ramdani N-E, Zennir K. A New Class of Fredholm Integral Equations of the Second Kind with Non Symmetric Kernel: Solving by Wavelets Method. Boletim da Sociedade Paranaense de Matem´atica. 2018.Abstract

In this paper, we introduce an efficient modification of the wavelets method to solve a new class of Fredholm integral equations of the second kind with non symmetric kernel. This method based on orthonormal wavelet basis, as a consequence three systems are obtained, a Toeplitz system and two systems with condition number close to 1. Since the preconditioned conjugate gradient normal equation residual (CGNR) and preconditioned conjugate gradient normal equation error (CGNE) methods are applicable, we can solve the systems in O(2n log(n)) operations, by using the fast wavelet transform and the fast Fourier transform.

CHEBBAH H, Mennouni A, Ramdani N-E. Numerical Solution of Generalized Logarithmic Integral Equations of the Second Kind by Projections. Malaysian Journal of Mathematical Sciences. 2018;12 (3) :349–367.Abstract

In this work, we present a new techniques to solve the integral equations of the second kind with logarithmic kernel. First, we show the existence and uniqueness of the solution for the given problem in a Hilbert space. Next, we discuss a projection method for solving integral equations with logarithmic kernel of the second kind; the present method based on the shifted Legendre polynomials. We examine the existence of the solution for the approximate equation, and we provide a new error estimate for the numerical solutions. At the end, numerical examples are provided to illustrate the theoretical results.

2017
Aksas B, Rebiai S-eddine. Stabilization of the Fourth Order Schrödinger Equation. New Trends in Analysis and Interdisciplinary Applications. 2017 : 529-535.Abstract

We study both boundary and internal stabilization problems for the fourth order Schrödinger equation in a smooth bounded domain Ω of Rn. We first consider the boundary stabilization problem. By introducing suitable dissipative boundary conditions, we prove that the solution decays exponentially in an appropriate energy space. In the internal stabilization problem, by assuming that the damping term is effective on the neighborhood of the boundary, we prove the exponential decay of the L2(Ω)-energy of the solution. Both results are established by using multipliers technique and compactness/uniqueness arguments.

Mennouni A, Zaouia S. Discrete septic spline quasi-interpolants for solving generalized Fredholm integral equation of the second kind via three degenerate kernel methods. Mathematical Sciences. 2017;11 :345–357.Abstract

Three main contributions are presented in this paper. First, the septic quasi-interpolants are calculated with all their coefficients. Second, we explore the results to solve a generalized and broad class of Fredholm integral equations of the second kind. Finally, we present three degenerate kernel methods; the latter is a combination of the two previously established methods in the literature. Moreover, we provide a convergence analysis and we give new error bounds. Finally, we exhibit some numerical examples and compare them with previous results in the literature.

Mennouni A. Piecewise constant Galerkin method for a class of Cauchy singular integral equations of the second kind in L2. Journal of Computational and Applied Mathematics. 2017;326 :268-272.Abstract

In this work, we present piecewise constant Galerkin method for a class of Cauchy singular integral equations of the second kind with constant coefficients in L2([0,1],C), using a sequence of orthogonal finite rank projections. We prove the existence and uniqueness theorems for the Cauchy integral equation and the approximate equation, respectively. We perform the error analysis for which we give new and improved estimates for the rates of convergence. Numerical example illustrates the theoretical results.

Mennouni A, Youkana A. FINITE TIME BLOW-UP OF SOLUTIONS FOR A NONLINEAR SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS. Electronic Journal of Differential Equations. 2017;2017 (152) :1–15.
Lombarkia F, Amouch M. Asymmetric Fuglede Putnam's Theorem for operators reduced by their eigenspaces. FILOMAT. 2017.Abstract

Fuglede-Putnam Theorem have been proved for a considerably large number of class of operators. In this paper by using the spectral theory, we obtain a theoretical and general framework from which Fuglede-Putnam theorem may be promptly established for many classes of operators.

2016
Rebiai S-eddine, Sidi-Ali FZ. Uniform exponential stability of the transmission wave equation with a delay term in the boundary feedback. IMA Journal of Mathematical Control and Information. 2016;33 (1) :1 - 20.Abstract

We consider a system of transmission of the wave equation with Neumann feedback control that contains a delay term and acts on the exterior boundary. First, we prove under some assumptions that the closed-loop system generates a C 0 -semigroup of contractions on an appropriate Hilbert space. Then, under further assumptions, we show that the closed-loop system is exponentially stable. To establish this result, we introduce a suitable energy function and use multiplier method together with an estimate taken from Lasiecka & Triggiani (1992, Appl. Math. Optim., 25, 189–244.) (Lemma 7.2) and compactness-uniqueness argument.