# Volume 12 (2020), Number 1

## Some New Simpson-Like Type Inequalities Via Preqausiinvexity

### Author(s): T. CHIHEB, N. BOUMAZZA and B. MEFTAH

**Abstract:** In this paper, we first establish a new integral identity which represent a partielle result, by using this identity we derive some new Simpson like type integral inequalities for functions whose second derivatives are prequasiinvex functions. we also discuss some special cases where the second derivatives are monotonous functions. At the end we give some applications to special means.

## An Application of the Admissibility Types in $b$-Metric Spaces

### Author(s): MARIA DOBRIȚOIU

**Abstract:** In this paper we present a fixed point theorem for the solution of the nonlinear
Fredholm integral equation
$$x\left(t\right)={\displaystyle {\int}_{a}^{b}K\left(t,r,x\left(r\right),x\left(g\left(r\right)\right)\right)dr}+f\left(t\right),\text{\hspace{0.17em}}t\in \left[a,b\right],$$
as an application of a result of S. Radenović et al., in [14]. In order to obtain this existence result,
we used also, the notions of admissibility types defined on a
$b$-metric space, in [17].

## Modified $S$-Metric Spaces and Some Fixed Point Results

### Author(s): ABDULLAH KARAMI, SHABAN SEDGHI, NABI SHOBE and ZORAN D. MITROVIĆ

**Abstract:** In this paper, a structure of modified
$S$-metric spaces is introduced which can be viewed a
generalization of both $S$-metric and
${S}_{b}$-metric spaces. Also, the
notions of $\tilde{S}$
-contractive mappings in the modified $S$-metric spaces is given.
We also investigate the existence of fixed point for such mappings under various contractive conditions.
We provide example and an application to illustrate the results presented herein.

## Chebyshev Type Inequalities Involving Generalized Proportional Fractional Integral Operators

### Author(s): İLKER MUMCU, ERHAN SET and ARTION KASHURI

**Abstract:** A number of definition of fractional integral operators have, recently, been
presented. In [11], Jarad et al. introduced the proportional generalized fractional integrals. In this
paper, we motivated essentially by the earlier works and established some Chebyshev type inequalities for
synchronous functions involving generalized proportional fractional integral operators. Also we have
results containing confluent hypergeometric functions and incomplete gamma functions.

## A Problem With Second Kind’s Nonlocal Condition for Pseudohyperbolic Equation

### Author(s): AICHA SAKHRI and AHCENE MERAD

**Abstract:** In this article, the Faeod-Galerkin method is proposed for solving a pseudohyperbolic
type equation with an integral condition. We construct a discrete numerical solution of the approximate
problem.

## Three-Dimensional Influence Functions and Integration Formulas for Many Boundary Value Problems Within a Thermoelastic Half-Layer

### Author(s): VICTOR ȘEREMET and ION CREȚU

**Abstract:** The aim of this paper is the constructing of the main thermoelastic displacements
Green’s functions (MTDGFs) for a generalized 3D BVP of uncoupled thermoelasticity for a half-layer. To
reach this aim are derived structural formulas for MTDGFs expressed via respective Green’s functions for
Poisson’s equation (GFPE) by using harmonic integral representations method (HIRM). These structural
formulas are validated by the checking the equations of thermoelasticity with respect to point of
application the heat source and the nonhomogeneous Poisson’s equation with respect to point of response
in which the thermoelastic displacements appeared. In addition, they satisfy boundary conditions for
temperature Green’s function with respect to point application the displacements and to mechanical
boundary conditions with respect to point of application the heat source. The thermoelastic volume
dilatation (TVD) derived separately from respective integral representations have to be equal to the TVD
derived by using structural formulas for MTDGFs. The final analytical expressions for MTDGFs obtained on
the base of mentioned above structural formulas for eight new 3D BVPs of thermoelasticity within
half-layer contain Bessel functions of the zero-order and of the second type. Finally, the integration
formulas for thermal displacements and stresses, created by inner heat source and by the thermal data
given of the surface of the half-layer are presented also.

## The Direct Problem and the Inverse Problem of the Normal Distribution. Mathematical Models-Algorithms-Program

### Author(s): CONSTANTIN ZĂVOIANU and FELICIA ZĂVOIANU

**Abstract:** In this article we present two mathematical models, the algorithms that implement them, and the software that solves the direct problem and the inverse problem of the normal distribution. Tandem solving of the two issues provides both the possibility of forming an overview of the normal distribution as well as a set of useful and strictly necessary information to perform an informative analysis of the effectiveness of the used mathematical models and algorithms. The efficiency of mathematical models and, implicitly, of their algorithms is ultimately expressed through the accuracy and precision of the results obtained by using the software encoding these algorithms. The accuracy of the results reflects the extent to which the calculated value approaches the real value and the precision of the results refers to the exact number of digits in the representation of a double-precision mobile floating-point solution. Here, we have also demonstrated, using the concepts of absolute error and relative error, that the scientific approach in this article is rigorously substantiated both from a mathematical point of view and from the point of view of implementing algorithms in a high-level programming language.