## Abstract

In this study, we give two sequences *{L*
^{+}
_{n}}_{n≥}_{1} and *{L ^{−}_{n}}_{n≥}*

_{1}derived by altering the Lucas numbers with {

*±*1,

*±*3}, terms of which are called as altered Lucas numbers. We give relations connected with the Fibonacci

*F*and Lucas

_{n}*L*numbers, and construct recurrence relations and Binet’s like formulas of the

_{n}*L*

^{+}

*and*

_{n}*L*numbers. It is seen that the altered Lucas numbers have two distinct factors from the Fibonacci and Lucas sequences. Thus, we work out the greatest common divisor (

^{−}_{n}*GCD*) of

*r*-consecutive altered Lucas numbers. We obtain

*r*-consecutive

*GCD*sequences according to the altered Lucas numbers, and show that their

*GCD*sequences are unbounded or periodic in terms of values

*r*.

## Abstract

A general fixed point theorem for two pairs of absorbing mappings satisfying a new type of implicit relation ([37]), without weak compatibility in *G _{p}*-metric spaces is proved. As applications, new results for mappings satisfying contractive conditions of integral type and for

*ϕ*-contractive mappings are obtained.

## Abstract

In this article, we will define Padovan’s hybrid numbers, based on the new noncommutative numbering system studied by Özdemir ([7]). Such a system that is a set involving complex, hyperbolic and dual numbers. In addition, Padovan’s hybrid numbers are created by combining this set, satisfying the relation *ih* = *−hi* = *ɛ* + *i*. Given this, some properties and identities are shown for these numbers, such as Binet’s formula, generating matrix, characteristic equation, norm, and generating function. In addition, these numbers are extended to the integer field and some identities are made.

## Abstract

In this paper we consider the hypo-*q*-operator norm and hypo-*q-*numerical radius on a Cartesian product of algebras of bounded linear operators on Banach spaces. A representation of these norms in terms of semi-inner products, the equivalence with the *q*-norms on a Cartesian product and some reverse inequalities obtained via the scalar reverses of Cauchy-Buniakowski-Schwarz inequality are also given.

## Abstract

In this research we introduce the concept of *m*-convex function of higher order by means of the so called *m*-divided difference; elementary properties of this type of functions are exhibited and some examples are provided.

## Abstract

Let *{r _{n}}_{n}*

_{∈}be a strictly increasing sequence of nonnegative real numbers satisfying the asymptotic formula

*r*, where

_{n}~ αβ^{n}*α, β*are real numbers with

*α >*0 and

*β >*1. In this note we prove some limits that connect this sequence to the number

*e*. We also establish some asymptotic formulae and limits for the counting function of this sequence. All of the results are applied to some well-known sequences in mathematics.

## Abstract

This work presents a numerical comparison between two efficient methods namely the fractional natural variational iteration method (FNVIM) and the fractional natural homotopy perturbation method (FNHPM) to solve a certain type of nonlinear Caputo time-fractional partial differential equations in particular, nonlinear Caputo time-fractional wave-like equations with variable coefficients. These two methods provided an accurate and efficient tool for solving this type of equations. To show the efficiency and capability of the proposed methods we have solved some numerical examples. The results show that there is an excellent agreement between the series solutions obtained by these two methods. However, the FNVIM has an advantage over FNHPM because it takes less time to solve this type of nonlinear problems without using He’s polynomials. In addition, the FNVIM enables us to overcome the diffi-culties arising in identifying the general Lagrange multiplier and it may be considered as an added advantage of this technique compared to the FNHPM.

## Abstract

Topological indices play a very important role in the mathematical chemistry. The topological indices are numerical parameters of a graph. The degree sequence is obtained by considering the set of vertex degree of a graph. Graph operators are the ones which are used to obtain another broader graphs. This paper attempts to find degree sequence of vertex–*F* join operation of graphs for some standard graphs.

## Abstract

Three main tools to study graphs mathematically are to make use of the vertex degrees, distances and matrices. The classical graph energy was defined by means of the adjacency matrix in 1978 by Gutman and has a large number of applications in chemistry, physics and related areas. As a result of its importance and numerous applications, several modifications of the notion of energy have been introduced since then. Most of them are defined by means of graph matrices constructed by vertex degrees. In this paper we define another type of energy called *q*-distance energy by means of distances and matrices. We study some fundamental properties and also establish some upper and lower bounds for this new energy type.