Asian Research Journal of Mathematics

The goal of this article is to consider an Ishikawa type iteration process with errors to approximate the fixed point of -asymptotically quasi non-expansive mapping in convex cone metric spaces. Our results extend and generalize many known results from complete generalized convex metric spaces to cone metric spaces.

The ultimate goal of this work is to provide in a concise manner old and new results relating to the simplicial polytopic numbers.

Collatz conjecture (stated in 1937 by Collatz and also named Thwaites conjecture, or Syracuse, 3n+1 or oneness problem) can be described as follows:
Take any positive whole number N. If N is even, divide it by 2. If it is odd, multiply it by 3 and add 1. Repeat this process to the result over and over again. Collatz conjecture is the supposition that for any positive integer N, the sequence will invariably reach the value 1. The main contribution of this paper is to present a new approach to Collatz conjecture. The key idea of this new approach is to clearly differentiate the role of the division by two and the role of what we will name here the jump: a = 3n + 1. With this approach, the proof of the conjecture is given as well as informations on generalizations for jumps of the form qn + r and for jumps being polynomials of degree m >1. The proof of Collatz algorithm necessitates only 5 steps:

1- to differentiate the main function and the jumps;
2- to differentiate branches as well as their rst and last terms a and n;

3- to identify that left and irregular right shifts in branches can be replaced by regular shifts in 2m-type columns;
4- to identify the key equation ai = 3ni−1 + 1 = 2m as well as its solutions;
5- to reduce the problem to compare the number of jumps to the number of divisions in a trajectory.

An extended Tanh-function method with Riccati equation is presented for constructing multiple exact travelling wave solutions of some nonlinear evolution equations which are particular cases of a generalized equation. The results of solitary waves are general compact forms with non-zero constants of integration. Taking the full advantage of the Riccati equation improves the applicability and reliability of the Tanh method with its extended form.

This paper presents the one-dimensional, positive temperature coefficient (PTC) thermistor equation, using the hyperbolic-tangent function as an approximation to the electrical conductivity of the device. The hyperbolic-tangent function describes the qualitative behaviour of the evolving solution of the thermistor in the entire domain. The steady state solution using the new approximation yielded a distribution of device temperature over the spatial dimension and all the phases of the temperature distribution of the device without having to look for a moving boundary. The analysis of the steady state solution and the numerical solution of the unsteady state is presented in the paper.