African Diaspora Journal of Mathematics Articles (Project Euclid)

Intégrabilité algébrique : une introduction

Thu, 05 Aug 2010 15:41 EDT

Pol Vanhaecke

Source: Afr. Diaspora J. Math. (N.S.), Volume 9, Number 2, 1--16.

On Symplectomorphisms of the Symplectization of a Compact Contact Manifold

Thu, 05 Aug 2010 15:41 EDT

Augustin Banyaga

Source: Afr. Diaspora J. Math. (N.S.), Volume 9, Number 2, 66--73.

Abstract:
Let $(N,\alpha)$ be a compact contact manifold and $(N \times {\mathbb R}$, $d(e^t\alpha))$ its symplectization. We show that the group $G$ which is the identity component in the group of symplectic diffeomorphisms $\phi$ of $(N\times {\mathbb R}, d(e^t\alpha))$ that cover diffeomorphisms $\underline {\phi}$ of $ N\times S^1$ is simple, by showing that $G$ is isomorphic to the kernel of the Calabi homomorphism of the associated locally conformal symplectic structure.

Decomposability of a Poisson Tensor Could Be a Stable Phenomenon

Thu, 05 Aug 2010 15:41 EDT

Jan-Paul Dufour

Source: Afr. Diaspora J. Math. (N.S.), Volume 9, Number 2, 74--81.

Abstract:
In this paper, we develop one of the questions raised by the author in the mini-course he gave at the conference Geometry and Physics V held at the University Cheikh Anta Diop, Dakar in May 2007). Let $\Pi$ be a Poisson tensor on a manifold $M.$ We suppose that it is decomposable in a neighborhood $U$ of a point $m,$ i.e. we have $\Pi=X\wedge Y$ on $U$ where $X$ and $Y$ are two vector fields. We will exhibit examples where every Poisson tensor near enough $\Pi$ seems to be also decomposable in a neighborhood of a point which can be chosen arbitrarily near $m$; and this works even if $M$ has a big dimension. This idea is a consequence of a cohomology calculation which can be interesting by itself.

Equivalence for Differential Equations

Thu, 05 Aug 2010 15:41 EDT

Odinette Renée Abib

Source: Afr. Diaspora J. Math. (N.S.), Volume 9, Number 2, 82--97.

Abstract:
We shall study the equivalence problem for ordinary differential equations with respect to the affine transformation group $A( 2,{\mathbb R})$.

Volume and Energy of Reeb Vector Fields

Thu, 05 Aug 2010 15:41 EDT

Philippe Rukimbira

Source: Afr. Diaspora J. Math. (N.S.), Volume 9, Number 2, 98--111.

Abstract:
This paper contains a characterization of Reeb vector fields of K-contact forms in terms of J-holomorphic embeddings into the tangent unit sphere bundle. A consequence of this characterization is that these vector fields are critical points of a volume and an energy functionals defined on the set of unit vector fields. Reeb vector fields on closed, K-contact Einstein manifolds are absolute minimizers for the energy functional with a mean curvature correction. On odd-dimensional Einstein manifolds of positive sectional curvature, these unit vector fields are characterized by their minimizing property. It is also proved that any closed flat contact manifold admits a parallelization by three critical unit vector fields, one parallel (hence minimizing), the other two are Reeb vector fields of contact forms, not Killing and not minimizers of any of the volume or the energy functionals.

Observabilité des systèmes linéaires sur les groupes de Lie

Thu, 05 Aug 2010 15:41 EDT

Philippe Jouan

Source: Afr. Diaspora J. Math. (N.S.), Volume 9, Number 2, 112--119.

Introduction to the Group of Symplectomorphisms

Thu, 05 Aug 2010 15:41 EDT

Augustin Banyaga

Source: Afr. Diaspora J. Math. (N.S.), Volume 9, Number 2, 120--138.

Abstract:
In these Lecture Notes of a mini-course delivered in the " Séminaire Itinérant de Géometrie et Physique Mathématique, " Geometry and Physics V" at the University Cheikh Anta Diop, Dakar in May 2007, we introduce the group of symplectic diffeomorphisms, the main results on its algebraic structure and on some of its local and global properties. This survey culminates with the most recent results on Hofer geometry, the definitions of the groups of symplectic and hamiltonian homeomorphisms, and the introduction to the $C^0$ symplectic topology.

Transverse Geometry and Generalized Complex Structures

Thu, 05 Aug 2010 15:41 EDT

Aïssa Wade

Source: Afr. Diaspora J. Math. (N.S.), Volume 9, Number 2, 139--149.

Abstract:
Transversal generalized complex structures provide a framework unifying both transversely holomorphic foliations and generalized complex geometry. In this paper, we give characterizations of transversal generalized complex structures. Moreover, a natural extension of the basic Dolbeault cohomology is obtained.

Pseudo-Szabó Operators and Lightlike Szabó Hypersurfaces

Thu, 21 Apr 2011 09:19 EDT

C. Atindogbe, S. Lungiambudila, J. Tossa

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 1, 1--23.

Abstract:
In this paper, the pseudo-inversion of degenerate metric is considered. We extend Szabó operators associated to algebraic covariant derivative curvature maps (tensors) to lightlike hypersurfaces. Some examples are given with explicit determination of their Szabó operators. Finally, we introduce the notion of lightlike Szabó hypersurfaces and give some characterization results of locally symmetric lightlike hypersurfaces and semi-symmetric lightlike hypersurfaces from Szabó condition.

Left Multipliers and Jordan Ideals in Rings with Involution

Thu, 21 Apr 2011 09:19 EDT

Lahcen Oukhtite

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 1, 24--28.

Abstract:
The purpose of this paper is to study left multipliers satisfying certain identities on Jordan ideals of rings with involution. Some well known results characterizing commutativity of prime rings by left multipliers have also been extended to Jordan ideals.

Existence Results for Nonlinear Fractional Differential Equations with Four-Point Nonlocal Type Integral Boundary Conditions

Thu, 21 Apr 2011 09:19 EDT

Bashir Ahmad, Sotiris K. Ntouyas

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 1, 29--39.

Abstract:
In this paper, we investigate some new existence results for nonlinear fractional differential equations of order $q \in (1,2]$ with four-point nonlocal integral boundary conditions by applying standard fixed point theorems and Leray-Schauder degree theory. Our results are new in the sense that the nonlocal parameters in four-point integral boundary conditions for the problem appear in the integral part of the conditions in contrast to the available literature on four-point fractional boundary value problems which deals with the four-point boundary conditions restrictions on the solution or gradient of the solution of the problem. Some illustrative examples are presented.

On Armendariz-Like properties

Thu, 21 Apr 2011 09:19 EDT

Fuad Ali Ahmed Almahdi , Najib Mahdou

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 1, 40--47.

Abstract:
In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the transfer of the Armendariz-Like properties to trivial ring extensions to localization and direct product of rings, and then generate new families of rings with zero-divisors subject to some given Armendariz-like properties. The article includes a brief discussion of the scope and precision of our results.

Global Dynamics for a Relativistic Charged Fluid with Potential in Temporal Gauge in a Robertson-Walker Space-Time

Thu, 21 Apr 2011 09:19 EDT

Norbert Noutchegueme , Eric Magloire Zangue

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 1, 48--65.

Abstract:
Global existence of solutions to the coupled Einstein-Maxwell system which rules the dynamics of the considered relativistic charged fluid is proved and asymptotic behavior is investigated, in the case of positive cosmological constant and positive initial velocity of the cosmological expansion factor..

A First-Order Periodic Differential Equation at Resonance

Thu, 21 Apr 2011 09:19 EDT

Eric R. Kaufmann

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 1, 66--74.

Abstract:
We consider the existence of a periodic solution to the first-order nonlinear problem \begin{eqnarray*} &&x'(t) = -a(t)x(t)+ q ( t, x(t) ),\; \mbox{ a.e. on } (0, T),\\ &&x(0) = x(T), \end{eqnarray*} where the nonlinear term $q$ is Carathéodory with respect to $L^1[0, T]$. The coefficient function $a$ is such that the differential equation is non-invertible. The technique used to establish our existence result is Mahwin's coincidence degree theory.

Flow-Box Theorem and Beyond

Thu, 21 Apr 2011 09:19 EDT

Issa Amadou Tall

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 1, 75--102.

Abstract:
For a given vector field $\nu(x)$ around a nonsingular point $x_0$, we provide explicit coordinates $z=\varphi(x)$ in which the vector field is straightened out, i. e., $\varphi_{*}(\nu)(z)=\frac{\partial}{\partial z_1}.$ The procedure is generalized to Frob\"{e}nius Theorem, namely, for an involutive distribution $\Delta={\rm span} \, \left \{\nu_1, \dots, \nu_m \right \}$ around a nonsingular point $x_0$, we give explicit coordinates $z=\varphi(x)$ in which $$ {\varphi_{*}\Delta= {\rm span} \left \{\frac{\partial}{\partial z_1}, \dots, \frac{\partial}{\partial z_m} \right \}.} $$ The method is illustrated by several examples and is applied to the linearization of control systems.

Affine Osserman Connections on 2-Dimensional Manifolds

Thu, 21 Apr 2011 09:19 EDT

Aboul Salam Diallo

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 1, 103--109.

Abstract:
This paper deals with affine Osserman connections on 2-dimensional manifolds. We give in an explicit form, a sufficient condition for an affine connection to be Osserman. As applications, examples of affine Osserman connections are given.

Novel Exponential Estimate for Nonlinear Systems with Mixed Interval Time-Varying Nondifferentiable Delays

Thu, 21 Apr 2011 09:19 EDT

Mai Viet Thuan

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 1, 110--123.

Abstract:
This paper addresses exponential stability problem for a class of nonlinear systems with mixed time-varying delays. The time delays are non-differentiable functions belonging to a given interval, in which the lower bound of delay is not restricted to zero. By constructing a suitable augmented Lyapunov’s functional, new criteria for the exponential stability of the system are established in terms of linear matrix inequalities. The result has been applied to robust stability problem of uncertain systems with interval time-varying delays. Numerical examples are given to show the effectiveness of the conditions.

Exact Controllability of Semilinear Stochastic Evolution Equation

Thu, 21 Apr 2011 09:19 EDT

D. Barraez, H. Leiva, Nelson Merentes, Miguel Narváez

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 1, 124--139.

Abstract:
In this paper we study the exact controllability of the following semilinear stochastic evolution equation in a Hilbert space $X$ $$ dx(t)=\{Ax(t)+Bu(t)+f(t,\omega,x(t),u(t)) \}dt + \{\Sigma(t) +\sigma(t,\omega,x(t),u(t)) \}dw(t), $$ where the control $u$ is a stochastic process in the Hilbert space $U$, $A:D(A)\subset X\rightarrow X,$ is the infinitesimal generator of a strongly continuous semigroup $\left\{S(t)\right\}_{t\geq 0}$ on $X$ and $B\in L(U,X)$. To this end, we give necessary and sufficient conditions for the exact controllability of the linear part of this system $$ dx(t)=Ax(t)dt+Bu(t)dt+\Sigma(t)dw(t). $$ Then, under a Lipschitzian condition on the non linear terms $f$ and $\sigma$ we prove that the exact controllability of this linear system is preserved by the semilinear stochastic system. Moreover, we obtain explicit formulas for a control steering the system from the initial state $\xi_0$ to a final state $\xi_1$ on time $T >0$, for both system, the linear and the nonlinear one. Finally, we apply this result to a semilinear damped stochastic wave equation.

Asymptotic Behaviour of Solutions for Some Differential Equations in a Banach Space

Thu, 11 Aug 2011 16:00 EDT

E. Ait Dads, K. Ezzinbi, S. Fatajou

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 1--18.

Abstract:
We present some results of Eberlein-weakly almost periodic functions with values in a Banach space. Then, we apply these results to investigate the Eberlein-weakly almost periodic solutions of some differential equations in a Banach space.

Positive Pseudo Almost Automorphic Solutions for Some Nonlinear Infinite Delay Integral Equations

Thu, 11 Aug 2011 16:00 EDT

P. Cieutat, K. Ezzinbi

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 19--33.

Abstract:
We state sufficient conditions for the existence of positive pseudo almost automorphic solutions of the following nonlinear infinite delay integral equation: $$ x(t)= \int^t_{-\infty}a(t,t-s)f(s,x(s))\: ds . $$ We deduce some corollaries on a finite delay integral equation and on a delay differential equation.

Local Existence and Global Continuation for Some Partial Functional Integrodifferential Equations

Thu, 11 Aug 2011 16:00 EDT

K. Ezzinbi, S. Ghnimi

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 34--45.

Abstract:
In this work, we study the local existence of mild solutions for some partial functional integrodifferential equations. We suppose that the linear part has a resolvent operator in the sense of Grimmer [13]. The nonlinear part is just assumed to be continuous. We have also that the solution may blow up in finite time. An application is provided to illustration.

Interior Controllability of the Thermoelastic Plate Equation

Thu, 11 Aug 2011 16:00 EDT

H. Leiva, N. Merentes

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 46--59.

Abstract:
In this paper we prove the interior controllability of the Thermoelastic Plate Equation $$ \left\{ \begin{array}{ll} w_{tt}+\Delta^2w+\alpha\Delta w=1_{\omega}u_{1}(t,x),& \mbox{in} \quad (0, \tau) \times \Omega,\\ \theta_t-\beta\Delta\theta-\alpha\Delta w_t=1_{\omega}u_{2}(t,x), & \mbox{in} \quad (0, \tau) \times \Omega,\\ \theta=w=\Delta w=0, & \mbox{on} \quad (0, \tau) \times \partial \Omega, \end{array} \right.$$ where $\alpha\neq 0$, $\beta>0$, $\Omega$ is a sufficiently regular bounded domain in $\R^{N}$ $(N\geq 1)$, $\omega$ is an open nonempty subset of $\Omega$, $1_{\omega}$ denotes the characteristic function of the set $\omega$ and the distributed control $u_{i}\in L^{2}([0,\tau]; L^{2}(\Omega)), i=1,2.$ Specifically, we prove the following statement: For all $\tau >0$ the system is approximately controllable on $[0, \tau]$. Moreover, we exhibit a sequence of controls steering the system from an initial state to a final state in a prefixed time $\tau >0$ .

Global Dynamics for a Relativistic Charged Fluid with Potential in Temporal Gauge in a Robertson-Walker Space-Time

Thu, 11 Aug 2011 16:00 EDT

P. H. Bezandry, T. Diagana

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 60--79.

Abstract:
In this paper we first introduce and study the concepts of $p$-th mean pseudo almost automorphy and that of $p$-th mean pseudo almost periodicity for $p \geq 2$. Next, we make extensive use of the well-known Schauder fixed point principle to obtain the existence of $p$-th mean pseudo almost automorphic mild solutions to some nonautonomous stochastic differential equations.

Iterated Operator Inequalities on Ordered Linear Spaces

Thu, 11 Aug 2011 16:00 EDT

M. Turinici

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 80--88.

Abstract:
An operator version of the Young's result [Proc. Amer. Math. Soc., 94 (1985), 636-640] is obtained, via non-differential techniques.

Weighted Ostrowski--Grüss Inequalities on Time Scales

Thu, 11 Aug 2011 16:00 EDT

M. Bohner, T. Matthews, A. Tuna

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 89--99.

Abstract:
In this paper, we study Ostrowski--Grüss and Ostrowski-like inequalities on time scales and thus unify and extend corresponding continuous and discrete versions from the literature. We present corresponding inequalities by using the time scales $L^\infty$-norm and also by using the time scales $L^p$-norm. Several interesting inequalities representing special cases of our general results are supplied.

Gevrey Regularity for a Class of Solutions of the Linearized Spatially Homogeneous Boltzmann Equation Without Angular Cutoff

Thu, 11 Aug 2011 16:00 EDT

S. Y. Lin

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 100--112.

Abstract:
In this paper, we study the Gevrey smoothing property for the non-negative solution of the linearized spatially homogeneous Boltzmann equation. Using pseudo-differential calculus and some techniques of mathematical analysis, we show that in the non-cutoff and non-Maxwellian case with the inverse power law potential, if the solution is Lipschitz continuous on the velocity variable, then it has the local Gevrey regularity.

Consensus Analysis of Multi-agent System with a Varying-velocity Leader and Time-varying Delays

Thu, 11 Aug 2011 16:00 EDT

J. L. Yao

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 113--120.

Abstract:
In this paper, consensus problem of the leader-following multi-agent system with a varying-velocity leader is analyzed. The system is considered with both time-varying input-delay and directed dynamic topologies, where the system delay is unknown and time-varying with a pre-specified upper bounded derivative. The stability analysis is performed with a proposed Lyapunov-Krosovskii functional. Sufficient delay-dependent condition in the form of Linear Matrix Inequalities (LMIs) is given to guarantee system consensus. Finally, numerical simulation verifies the theoretical results.

Doubly-Weighted Pseudo Almost Periodic Functions

Thu, 11 Aug 2011 16:00 EDT

T. Diagana

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 121--136.

Abstract:
We introduce and study a new concept called doubly-weighted pseudo-almost periodicity, which generalizes the notion of weighted pseudo-almost periodicity due to Diagana. Properties of such a new concept such as the stability of the convolution, translation-invariance, existence of a doubly-weighted mean for almost periodic functions, and a composition result will be studied.

Interior Controllability of the $nD$ Semilinear Heat Equation

Thu, 13 Oct 2011 15:48 EDT

H. Leiva, N. Merentes, J. L. Sanchez

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 1--12.

Abstract:
In this paper we prove the interior approximate controllability of the following Semilinear Heat Equation $$ \left\{ \begin{array}{lr} z_{t}(t,x) = \Delta z(t,x) + 1_{\omega}u(t,x)+f(t,z,u(t,x)) & \mbox{in} \quad (0, \tau] \times \Omega,\\ z = 0, & \quad \mbox{on} \quad (0, \tau) \times \partial \Omega, \\ z(0,x) = z_{0}(x), & x \in\Omega, \end{array} \right. $$ where $\Omega$ is a bounded domain in $\mathbb{R}^{N}(N\geq1)$, $z_0 \in L^{2}(\Omega)$, $\omega$ is an open nonempty subset of $\Omega$, $1_{\omega}$ denotes the characteristic function of the set $\omega$,the distributed control $u$ belong to $\in L^{2}([0,\tau]; L^{2}(\Omega;))$ and the nonlinear function $f:[0, \tau] \times \R \times \R \rightarrow \R$ is smooth enough and there are $a,b, c \in \R$, with $c \neq -1$, such that $$ \sup_{(t,z,u) \in Q_{\tau}} |f(t,z,u) -az-cu-b | < \infty, $$ where $Q_{\tau}= [0, \tau] \times \R \times \R$. Under this condition we prove the following statement: For all open nonempty subset $\omega$ of $\Omega$ the system is approximately controllable on $[0, \tau]$. Moreover, we could exhibit a sequence of controls steering the nonlinear system (\ref{eq1}) from an initial state $z_0$ to an $\epsilon$ neighborhood of the final state $z_1$ at time $\tau >0$, which is very important from a practical and numerical point of view..

Fractional Integro-differential Equations with State-Dependent Delay on an Unbounded Domain

Thu, 13 Oct 2011 15:48 EDT

M. Benchohra, S. Litimein

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 13--25.

Abstract:
We are concerned with the existence of solutions for fractional integro-differential equations with state-dependent delay on an infinite interval. Our results are based on Schauder's fixed point theorem combined with the diagonalization process.

Some PPF Dependent Random Fixed Point Theorems and Periodic Boundary Value Problems of Random Differential Equation

Thu, 13 Oct 2011 15:48 EDT

B. C. Dhage, S. K. Ntouyas, V. S. Patil

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 26--42.

Abstract:
In this paper some random fixed point theorems with PPF dependence are proved for random operators in separable Banach spaces with different domain and range spaces. The obtained abstract results are applied to prove PPF dependence existence results for first order periodic boundary value problems for functional random differential equations.

$AP_{r}$-Almost Periodic Solutions to the Equation $\dot{x}(t)= Ax(t)+(k\ast x)(t)+f(t)$

Thu, 13 Oct 2011 15:48 EDT

M. Mahdavi

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 43--47.

Abstract:
This note is dedicated to the existence of almost periodic solutions of a certain class of functional equations, of the form (1) in the text, in spaces like $AP_r(R, {\cal C}^n)$, $1\leq r\leq 2$. Frequency domain conditions are involved in this study.

Existence of Positive Almost Automorphic Solutions to a Class of Integral Equations

Thu, 13 Oct 2011 15:48 EDT

W. Long

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 48--56.

Abstract:
This paper is concerned with positive almost automorphic solutions to a class of nonlinear infinite delay integral equation. By using a fixed point theorem in partially ordered Banach spaces, we establish an existence theorem about positive almost automorphic solutions to the addressed integral equation. Our theorem extend some earlier results to a more general class of integral equations.

Existence and Uniqueness of the Weak Solutions for the Steady Incompressible Navier-Stokes Equations with Damping

Thu, 13 Oct 2011 15:48 EDT

W. Li, X. Wang, Q. Jiu

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 57--72.

Abstract:
This paper is concerned with the boundary-value problem for the steady incompressible Navier-Stokes equations with damping. Two cases are considered here: 1) the Dirichlet's boundary condition; 2) the nonhomogeneous boundary condition. we obtain the existence and uniqueness of the weak solutions for the steady incompressible Navier-Stokes equations with damping using different methods for the above cases.

Adomian Method for Under Determined Systems of Differential Equations

Thu, 13 Oct 2011 15:48 EDT

M. N. Hounkonnou, P. A. Dkengne Sielenou

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 73--103.

Abstract:
In this paper, the Adomian decomposition method for solving nonlinear partial differential equations (NPDEs) is revisited. Then we show how this method can be extended and used to solve under-determined systems of NPDEs. The examples of Kompaneets, Novikov and Ginzburg-Landau equations are considered as illustration.

Asymptotically Almost Automorphic Solutions to Nonautonomous Semilinear Evolution Equations

Thu, 13 Oct 2011 15:48 EDT

X. J. Zheng, C. Z. Ye, H. S. Ding

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 104--112.

Abstract:
This paper is concerned with a class of nonautonomous semilinear evolution equation in a Banach space. We establish an existence theorem about asymptotically almost automorphic mild solution to the addressed evolution equation. An example is given to illustrate our abstract result.

S-Asymptotically $\omega$-Periodic Functions and Applications to Evolution Equations

Thu, 13 Oct 2011 15:48 EDT

J. Blot, P. Cieutat, G. M. N'Guérékata

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 113--121.

Abstract:
In this paper, we first study further properties of S-asymptotically $\omega$-periodic functions taking values in Banach spaces including a theorem of composition. Then we apply the results obtained to study the existence and uniqueness of S-asymptotically $\omega$-periodic mild solutions to a nonautonomous semilinear differential equation.

Bumps of Potentials and Almost Periodic Oscillations

Thu, 13 Oct 2011 15:48 EDT

J. Blot, D. Lassoued

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 122--133.

Abstract:
We establish the existence of a Besicovitch almost periodic solution of a second-order differential equation, $u''(t)+ D_1V(u(t),t) = 0$, in a Hilbert space, when the potential $V(.,t)$ possesses a bump surrounded with a hollow. We use a variational method on a Hilbert space of Besicovitch almost periodic functions.

Optimization and Flow Invariance via First and Second Order Tangent Cones

Thu, 13 Oct 2011 15:48 EDT

E. Constantin, N. H. Pavel

Source: Afr. Diaspora J. Math. (N.S.), Volume 12, Number 1, 134--148.

Abstract:
This is a survey paper on optimization problems via the technique of first and second order tangent cones to a nonempty subset of a Banach space X. Such a technique is also used in the study of the flow invariance of a closed set with respect to a second order differential equation (motion on a given orbit in a force field). Many of the known results in these areas are included here.

Bounded and Compact Operators on the Bergman Space $L^{1}_{a}$ in the Unit Disk of $\mathbb{C}$

Tue, 06 Dec 2011 09:04 EST

D. Agbor, D. Békollé, E. Tchoundja

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 2, 1--29.

Abstract:
We characterize boundedness and compactness of the Toeplitz operator $T_{\mu}$, on the Bergman space $L_{a}^{1}(\Delta)$, where the symbols, $\mu$, are complex Borel measures on the unit disk of the complex plane, $\Delta$. The case of Toeplitz operators whose symbols are anti-analytic integrable functions is settled. Our results are related to the reproducing kernel thesis. We also study the case of symbols which are positive measures and the case of radial symbols. Moreover, we give a characterization of compactness for general bounded operators on $L^1_a.$

The Corona Problem in Carleman Algebras on Non-Stein Domains in $\mathbb{C}^n$

Tue, 06 Dec 2011 09:04 EST

P. W. Darko

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 2, 30--35.

Abstract:
New estimates are obtained for the $\overline{\partial}$ on non-Stein domains in $\mathbb{C}^n$and the results are applied to the Corona problem in Carleman algebras on those domains.

Well-Posedness Result For a Nonlinear Elliptic Problem Involving Variable Exponent and Robin Type Boundary Condition

Tue, 06 Dec 2011 09:04 EST

S. Ouaro, A. Tchousso

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 2, 36--64.

Abstract:
In this work we study the following nonlinear elliptic boundary value problem, $b(u)-div \; a(x,\nabla u)=f \hbox{ in }\Omega$, $a(x,\nabla u).\eta=-\left|u\right|^{p(x)-2}u \hbox{ on }\partial \Omega$, where $\Omega$ is a smooth bounded open domain in $\mathbb{R}^{N}$, $N \geq 1$ with smooth boundary $\partial\Omega$. We prove the existence and uniqueness of a weak solution for $f \in L^{\infty}(\Omega)$, the existence and uniqueness of an entropy solution for $L^{1}$-data $f$. The functional setting involves Lebesgue and Sobolev spaces with variable exponent.

Gerbes for the Chow

Tue, 06 Dec 2011 09:04 EST

A. Tsemo

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 2, 65--77.

Abstract:
The definition of the coherence relations in non-abelian cohomology is a difficult problem studied by many authors. The purpose of this paper is to simplify the solution provided by the author which uses the notion of sequences of fibred categories and to apply the resulting theory to higher divisors and Chow theory.

Feuilletages sans feuille compacte sur les fibrés en surfaces sur le cercle

Tue, 06 Dec 2011 09:04 EST

H. Dathe, A. Saidou

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 2, 78--89.

Positive Solutions for a System of Periodic Neutral Delay Difference Equations

Tue, 06 Dec 2011 09:04 EST

E. Yankson

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 2, 90--97.

Abstract:
In this article we consider the existence of positive solutions of a system of periodic neutral difference equations. The main tool employed is the Krasnosel'skii's fixed point theorem for the sum of a completely continuous operator and a contraction.

The Polar Decomposition in Banach Spaces

Tue, 06 Dec 2011 09:04 EST

T. L. Gill, V. Steadman, W. W. Zachary

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 2, 98--131.

Abstract:
In this paper we survey research progress related to the existence of an adjoint for linear operators on Banach spaces. We introduce a new pair separable Banach spaces which are required for the general theory. We then discuss a number ways one can explicitly construct an adjoint and then prove that one always exists for bounded linear operators. However, this is not true for the class of closed densely defined linear operators. In this case, we can only show that one exists for operators of Baire class one. The existence of an adjoint allows us to construct the polar decomposition. As applications, we extend the Poincaré inequality and the Stone-von Neumann version of the spectral theorem to all operators of Baire class one on a separable Banach space. Our results even show that the spectral theorem is natural for Hilbert spaces (in a certain well-defined sense). As a final application, we provide the natural Banach space version of the Schatten class of compact operators.

Some Discrete Fractional Inequalities of Chebyshev Type

Tue, 06 Dec 2011 09:04 EST

M. Bohner, R. A. C. Ferreira

Source: Afr. Diaspora J. Math. (N.S.), Volume 11, Number 2, 132--137.

Abstract:
Using the discrete fractional sum operator, we establish some inequalities of Chebyshev type.

Interview with Prof. Yves Meyer

Tue, 24 Jan 2012 23:30 EST

M. Mboup

Source: Afr. Diaspora J. Math. (N.S.), Volume 13, Number 1.

Quasicrystals, Almost Periodic Patterns, Mean-periodic Functions and Irregular Sampling

Tue, 24 Jan 2012 23:30 EST

Y. Meyer

Source: Afr. Diaspora J. Math. (N.S.), Volume 13, Number 1, 1--45.

Abstract:
Three properties of quasicrystals will be proved in this essay. Quasicrystals are almost periodic patterns (such patterns are carefully defined below). Every mean-periodic function whose spectrum is contained in a quasicrystal is almost periodic. Finally simple quasicrystals are universal sampling sets .

Multivalued Stochastic Partial Differential-Integral Equations Via Backward Doubly Stochastic Differential Equations Driven by a Lévy Process

Fri, 21 Dec 2012 15:45 EST

Y. Ren, A. Aman

Source: Afr. Diaspora J. Math. (N.S.), Volume 13, Number 2, 1--22.

Abstract:
In this paper, we deal with a class of backward doubly stochastic differential equations (BDSDEs, in short) involving subdifferential operator of a convex function and driven by Teugels martingales associated with a Lévy process. We show the existence and uniqueness result by means of Yosida approximation. As an application, we give the existence of stochastic viscosity solution for a class of multivalued stochastic partial differential-integral equations (MSPIDEs, in short).

Entropy Solution for Some $p(x)$-Quasilinear Problem with Right-Hand Side Measure

Fri, 21 Dec 2012 15:45 EST

E. Azroul, M. B. Benboubker, M. Rhoudaf

Source: Afr. Diaspora J. Math. (N.S.), Volume 13, Number 2, 23--44.

Abstract:
In this paper we study the existence of entropy solution for the following $p(x)$-quasilinear elliptic problem $$ \mbox{div}(a(x,u,\nabla u))+ g(x,u,\nabla u) = \mu$$ where the right-hand side $\mu$ is a measure, which admits a decomposition in $L^{1}(\Omega)+W^{-1,p'(x)}(\Omega)$ and $g(x,s,\xi)$ is a nonlinear term which has a growth condition with respect to $\xi$ and has no growth with respect to $s$ while satisfying a sign condition on $s$.

Fractional Complexified Field Theory from Saxena-Kumbhat Fractional Integral, Fractional Derivative of Order ($\alpha, \beta$) and Dynamical Fractional Integral Exponent

Fri, 21 Dec 2012 15:45 EST

A. R. El-Nabulsi, C. G. Wu

Source: Afr. Diaspora J. Math. (N.S.), Volume 13, Number 2, 45--61.

Abstract:
Fractional complexified field theory based on Saxena-Kumbhat fractional integrals with the presence of fractional derivative of order $(\alpha, \beta )$and dynamical fractional exponent is considered. Some interesting results are explored and discussed in some details.

A Regularization Proximal Point Algorithm for Zeros of Accretive Operators in Banach Spaces

Fri, 21 Dec 2012 15:45 EST

T. M. Tuyen

Source: Afr. Diaspora J. Math. (N.S.), Volume 13, Number 2, 62--73.

Abstract:
In this paper, we study the strong convergence of a regularization proximal point algorithm for the problem of finding a zero of $m-$accretive operators in a uniformly smooth Banach space $E$, and the stability of the regularization algorithms are considered.

Projective Motion in Special Finsler Spaces

Fri, 21 Dec 2012 15:45 EST

P. N. Pandey, S. K. Shukla

Source: Afr. Diaspora J. Math. (N.S.), Volume 13, Number 2, 74--80.

Abstract:
The present paper deals with the differential geometry of a Finsler space whose projective deviation tensor satisfies certain conditions. It discusses the projective motion in such Finsler space. A sufficient condition has been obtained for the projective motion to be an affine motion in a Finsler space whose projective deviation tensor is recurrent. Similar problems have been discussed for recurrent and projective recurrent Finsler spaces. It is established that the projective motion, in a Finsler space for which the covariant derivative of the projective deviation tensor is symmetric, is an affine motion or the space is of scalar curvature .

Parameters Identification in Population Dynamics Problem

Fri, 21 Dec 2012 15:45 EST

S. Somdouda

Source: Afr. Diaspora J. Math. (N.S.), Volume 13, Number 2, 81--99.

Abstract:
We are interested in the identification of parameters in a problem of pollution modeled by a population dynamics problem. We use the notion of sentinel introduced by O.Nakoulima in [13]. We prove the existence of such sentinels by solving a problem of null-controllability with constraint on the control. The key of our results is an observability inequality of Carleman type adapted to the constraint .

Sur les déformations d'un feuilletage de codimension 1 en structures de contact

Fri, 21 Dec 2012 15:45 EST

H. Dathe, C. Khoulé

Source: Afr. Diaspora J. Math. (N.S.), Volume 13, Number 2, 100--107.

Abstract:
Dans cet article on étudie quelques déformations particulières d'un feuilletage de codimension 1 en structures de contact. Ces déformations dites affines sont un peu plus générales que les déformations linéaires (voir [3]) au sens de Dathe-Rukimbira. Elles permettent aussi de donner dans une variété de contact compacte de dimension $2n+1$, une condition nécéssaire et suffisante de déformabilité d'une 1-forme intégrable quelconque en structures de contact.

Finite and Infinite Time Interval of BDSDEs Driven by Lévy Processes

Fri, 21 Dec 2012 15:45 EST

I. Faye, A. B. Sow

Source: Afr. Diaspora J. Math. (N.S.), Volume 13, Number 2, 108--126.

Abstract:
In this work we deal with a backward doubly stochastic differential equation (BDSDE) associated to a Poisson random measure. We establish existence and uniqueness of solution in the case of non-Lipschitz coefficients. The novelty of our result lies in the fact that we allow the time interval to be infinite.