# Volume 6 (2014), Number 2

## Hahn’s Problem with Respect to a Third-order Differential Operator

### Author(s): BAGHDADI ALOUI

**Abstract:** In the present work, we are interested to the lowering operator
${\mathrm{\u0161\x9d\x92\u0156}}_{c;1,3,2}$, given by a Linear combination of three successive
Laguerre derivatives $${\mathrm{\u0161\x9d\x92\u0156}}_{c;1,3,2}:={\mathcal{\u0101\x84\x92}}_{2,c}+3{\mathcal{\u0101\x84\x92}}_{1,c}+2{\mathcal{\u0101\x84\x92}}_{0,c},$$ where ${\mathcal{\u0101\x84\x92}}_{i,c}:=D\left(x-c\right)D\dots \left(x-c\right)D,\phantom{\rule{1em}{0ex}}0\le i\le 2$, Ā (i.e. containing
$\left(i+1\right)$ ordinary derivatives with respect to the
$x$ variable),
and $c$
is an arbitrary complex number. Then, we establish an intertwining relation between the
operators ${\mathrm{\u0161\x9d\x92\u0156}}_{c;1,3,2}$ and the standard derivative
$D$.
Besides, an analogue to the Hahn problem for the operator
${\mathrm{\u0161\x9d\x92\u0156}}_{c;1,3,2}$ is studied.
As a consequence, some integral relations between the corresponding polynomials are deduced.
Finally, some expansions in series of Laguerre polynomials are presented.

## Integrable Solutions for Implicit Fractional Order Differential Equations

### Author(s): MOUFFAK BENCHOHRA and MOHAMMED SAID SOUID

**Abstract:** In this paper we study the existence of integrable solutions for initial
value problem for fractional order implicit differential equations. Our results are based on
Schauder’s ļ¬xed point theorem and the Banach contraction principle ļ¬xed point theorem.

## Some Integral Operators of Analytic Functions

### Author(s): RADU DIACONU

**Abstract:** In the present paper we deļ¬ne two integral operators
${F}_{{\gamma}_{1},\dots ,{\lambda}_{l}}^{m,n}$
and ${G}_{{\gamma}_{1},\dots ,{\lambda}_{l}}^{m,n}$, deļ¬ned using the differential
operator $S{R}^{m,n}$.
We introduce some classes deļ¬ned by these operators and we investigate properties of the integral
operators on these classes. Also, are obtained subordination results for functions
$f\in \mathcal{\u0161\x9d\x92\AE}$ associated with
the differential operator $S{R}^{m,n}$.

## Space and Time Quantization. Particles with Spin 0

### Author(s): LAUREAN HOMORODEAN

**Abstract:** The dynamical state of a particle with the spin 0 is described by a (4-dimensional)
scalar wave function in the momentum representation. The time and the coordinates are associated with
Hermitian linear operators acting on the wave functions. The eigenvalues and the eigenfunctions of the
time and of the coordinates are determined by the wave equation. With its aid, we deļ¬ne the time-coordinate
4-tensor and the ācurrent densityā 4-vector, which satisfy some āconservation lawsā and allow to write
the time, the coordinates and the āchargeā as integrals of the components of these quantities in the
momentum space. The expression of wave function in the second quantization leads to two kinds of
particles: particles proper and antiparticles. We express the time, the coordinates and the āchargeā
by creation and annihilation operators and determine their eigenvalues. Because the momentum is limited
in value, the space and the time become discrete. Each position is populated with particles and
antiparticles living a ļ¬nite time. The changes in numbers of particles and antiparticles in different
positions by creation and annihilation lead to displacement of the center of the particle system.
This is interpreted as the macroscopic motion of a particle. Particularly, the strictly neutral
particles are discussed.

## Space and Time Quantization. Particles with Spin 1/2

### Author(s): LAUREAN HOMORODEAN

**Abstract:** The dynamical state of a particle with the spin 1/2 is described by a (4-dimensional)
bispinor wave function in the momentum representation. The time and the coordinates are associated
with Hermitian linear operators acting on the wave functions. The eigenvalues and the eigenfunctions
of the time and of the coordinates are determined by the wave equation expressed in diverse forms.
With its aid, we deļ¬ne the time, the coordinates and the āchargeā of the particle ļ¬eld. The expression
of wave function in the second quantization leads to two kinds of particles: particles proper and
antiparticles. We express the time, the coordinates and the āchargeā by creation and annihilation
operators and determine their eigenvalues. Because the momentum is limited in value, the space and
the time become discrete. Each position is populated with particles and antiparticles living a ļ¬nite
time. The changes in numbers of particles and antiparticles in diļ¬erent positions by creation and
annihilation lead to displacement of the center of the particle system. This is interpreted as the
macroscopic motion of a particle.

## New Ostrowski Type Inequalities for Co-ordinated $\left(\alpha ,m\right)$ -Convex Functions

### Author(s): MUHAMMAD AMER LATIF

**Abstract:** In this paper, we establish some new Ostrowski type inequalities for functions
of two variables whose derivatives in absolute value are co-ordinated
$\left(\alpha ,m\right)$-convex.

## Positive Periodic Solutions for First-Order Nonlinear Neutral Functional Differential Equations with Periodic Delay

### Author(s): MOUATAZ BILLAH MESMOULI, ABDELOUAHEB ARDJOUNI and AHCENE DJOUDI

**Abstract:** In this paper, we study the existence of positive periodic solutions
of two classes for first-order nonlinear neutral functional differential equations with
periodic delay. The main tool employed here is the Krasnoselskii’s hybrid ļ¬xed point
theorem dealing with a sum of two mappings, one is a contraction and the other is compact.
Two examples are included to illustrate our results. The results obtained here generalize
the work [12].

## New Class of Integral Operators Preserving Subordination and Superordination for Analytic Meromorphic Functions

### Author(s): AABED MOHAMMED and MASLINA DARUS

**Abstract:** A certain class of integral operators
${\mathcal{\u0161\x9d\x92\u201e}}_{\beta ,\gamma}\left(f\right)$, $\gamma ,\beta \in \mathrm{\u0101\x84\x82}$, $\beta \in \mathrm{\u0101\x84\x82}\u0101\x88\x96\left\{0\right\}$ and $\Re \left(\gamma -\beta \right)>1\u0101\x88\x952$, of meromorphic functions in the punctured open unit disk is introduced.
The main object of this paper is to investigate some subordination and superordination
preserving properties of these integral operators with the sandwich type theorem.

## Hermite-Hadamard Inequalities for Modiļ¬ed $h$ -convex Functions

### Author(s): MUHAMMAD ASLAM NOOR, KHALIDA INAYAT NOOR and MUHAMMAD UZAIR AWAN

**Abstract:** In this paper, we consider the class of modiļ¬ed
$h$
-convex functions, which was introduced by Toader [14]. We derive Hermite-Hadamard
type inequalities for the modiļ¬ed $h$-convex functions. Some special cases are also discussed. We try
to show that this class enjoys some nice properties which the convex functions have.

## A New Technique to Derive Many Explicit Thermoelastic Green’s Functions

### Author(s): VICTOR ČEREMET

**Abstract:** This study is devoted to a new technique to derive main
thermoelastic Green’s functions (MTGFs), based on their new integral
representations via Green’s functions for Poison’s equation (GFPE).
The derived new integral representations for MTGFs permitted to prove a theorem
about their constructive formulas expressed in terms of respective GFPE and some
simplest integrals. The inļ¬uence functions of thermoelastic displacements are
generated by a unitary heat source. This source is applied in an arbitrary inner
point of a generalized thermoelastic octant at the different homogeneous mechanical
and thermal boundary conditions, prescribed on its marginal quadrants. According to
the proved theorem many MTGFs for a group of two-and three dimensional BVPs of
thermoelasticity for a plane, a half-plane, a quadrant, a space, a quarter-space
and an octant may be obtained by changing the respective well-known GFPE and
calculating some simplest integrals. Some concrete new MTGFs for octant, quarter-space
and half-space are presented. The graphical and numerical computer evaluation of
MTGFs for a thermoelastic octant by using Maple 15 software also is included.
All MTGFs are obtained in terms of elementary functions that are very important
for their numerical implementation. The analytical checking of MTGFs is given
for a new BVP for thermoelastic octant. Using the proposed technique it is possible
to extend all obtained results for any domain of Cartesian system of coordinates.