Four-Step Iteration Scheme to Approximate Fixed Point for Weak Contractions

Fixed point theory is one of the most important subjects in the setting of metric spaces since fixed point theorems can be used to determine the existence and the uniqueness of solutions of such mathematical problems. It is known that many problems in applied sciences and engineering can be formulated as functional equations. Such equations can be transferred to fixed point theorems in an easy manner. Moreover, we use the fixed point theory to prove the existence and uniqueness of solutions of such integral and differential equations. Let X be a non-empty set. A fixed point for a self-mapping T on X is a point ee ∈ XX that satisfying T e=e. One of the most challenging problems in mathematics is to construct some iterations to faster the calculation or approximation of the fixed point of such problems. Some mathematicians constructed and generated some new iteration schemes to calculate or approximate the fixed point of such problems such as Mann et al. [Mann (1953); Ishikawa (1974); Sintunavarat and Pitea (2016); Berinde (2004b); Agarwal, O’Regan and Sahu (2007)]. The main purpose of the present paper is to introduce and construct a new iteration scheme to calculate or approximate the fixed point within a fewer number of steps as much as we can. We prove that our iteration scheme is faster than the iteration schemes given by Sintunavarat et al. [Sintunavarat and Pitea (2016); Agarwal, O’Regan and Sahu (2007); Mann (1953); Ishikawa (1974)]. We give some numerical examples by using MATLAB to compare the efficiency and effectiveness of our iterations scheme with the efficiency of Mann et al. [Mann (1953); Ishikawa (1974); Sintunavarat and Pitea (2016); Abbas and Nazir (2014); Agarwal, O’Regan and Sahu (2007)] schemes. Moreover, we introduce a problem raised from Newton’s law of cooling as an application of our new iteration scheme. Also, we support our application with a numerical example and figures to illustrate the validity of our iterative scheme.


Introduction
Numerical analysis is one of the most important subjects in science because scientists used numerical analysis to approximate solutions of such important problems in our life. Somayeh et al. [Somayeh and Tofigh (2018)] obtained numerical solution of linear regression based on Z-numbers by improved neural network. Liu et al. [Liu, Liu, Luo et al. (2019)] proposed a new algorithm with good optimization performance to enhance exploitation of artificial bee colony algorithm and can improve both the accuracy and the convergence speed. Very recently, Arif et al. [Arif, Raza, Shatanawi et al. (2019)] employed numerical analysis to present numerical analysis for stochastic SLBR model of computer virus over the internet. Numerical analysis played a major role in constructing some iterations to approximate the existence of fixed points of such functions in fewer number of iterations as much as we can. The fixed point theory, which is basically used to study the existence and uniqueness of fixed point for functions under some conditions, has many applications in various scientific science fields such as physics, economics, some engineering topics, and many mathematical branches. In the rest of this paper, ℛ denotes to the set of real numbers, I denotes to an interval in ℛ, and ( ) denotes to the set of all continuous real-valued functions on . In mathematics, some problems can be formulated to fixed point problems, for example the solution of the integral equation where : × ℛ → ℛ is a continuous mapping and ∈ ( ) is equivalent to the fixed point of the mapping : ( ) → ( ) which defined by So, the attraction of a high number of researchers to study fixed point theory is understandable. It is worth mentioning that the Banach contraction theorem [Banach (1922)] is the first result in the subject of the fixed point theory. The Banach contraction theorem has been extended in many directions by modifying contractive conditions or extending the usual metric space to new forms such as G-metric spaces, partial metric spaces, cone metric spaces, Ω-distance mappings etc. Kannan [Kannan (1968)] extended the Banach contraction to a new type of contraction, called Kannan contraction. Moreover, Kannan utilized his contraction to generate and prove many existing results in the frame of the concept of metric spaces. , Ciric [Ciric (1974] presented a generalization of the Banach contraction theorem by presenting a new type of contractive conditions. Kikkawa et al. [Kikkawa and Suzuki (2008)] studied three fixed point theorems for generalized contractions with constants in complete metric spaces as a generalization of Banach contraction theorem. While, Suzuki [Suzuki (2008)] studied a generalization of the Banach contraction principle that characterizes metric completeness.
Aydi et al. [Aydi, Shatanawi and Vetro (2011)] generalized the Banach contraction theorem by generating a weak contraction in G-metric spaces. Berinde [Berinde (2004a)] generalized the Banach contraction theorem by introducing a weak contraction on the concept of metric spaces. The weak contraction in sense of Berinde is given as follows: Definition 1 Let ( , ) be a metric space and : → be a self-mapping. Then T is called a weak contraction if there exist ∈ [0,1) and ≥ 0 such that ( , ) ≤ ( , ) + ( , ), , .
(3) Moreover, Berinde showed that every weak contraction T on a complete metric space has a fixed point. One of the most attractive subjects in mathematics is to approximate the fixed point by using iterations and construct new schemes to faster the approximation of the fixed point of such problems in fewer number of steps. Recently, many authors introduced some iterations to speed the rate of convergence of the fixed point. In this subject, Berinde [Berinde (2004b)] introduced the following important definitions regarding the rate of convergence. Definition 2 Let ( ) and ( ) be two sequences of real numbers. Also, let , , ∈ ℛ be such that 1. If l=0, then it can be said that ( ) converges to faster than ( ) to .
2. If 0 < < ∞, then it can be said that ( ) and ( ) have the same rate of convergence.

Definition 3
Let ( , ∥. ∥) be a normed linear space and let ( ) ( ) be two sequences in X. Suppose that ( ) ( ) converge to a point ∈ and the error estimates ∥ − ∥≤ and ∥ − ∥≤ are available, where ( ) and ( ) are nonnegative real sequences that converging to zero. If ( ) converges to zero faster than ( ), then it can be said that ( ) converges to z faster than ( ). From now on, Y represents to a normed linear space, represents to a convex subset of Y and ( ), ( ) and ( ) represent real sequences in the interval [0, 1]. In 1953, Mann [Mann (1953)] presented an iteration process ( ) by the sequence ( ) to approximate a fixed point, which is defined as follows: While, Ishikawa [Ishikawa (1974)] presented an iteration process ( ) by the sequence ( ) which is defined as follows: It is worth mentioning that the iteration process of Ishikawa is faster than the iteration process of Mann.
In the last decade, Agarwal et al. [Agarwal, O'Regan and Sahu (2007)] presented an iteration process ( ) by the sequence ( ) which is defined as follows: It is worth noting that the iteration scheme of Agarwal et al. is faster than Mann [Mann (1953)] and Ishikawa [Ishikawa (1974)] iteration schemes.
Recently, Abbas et al. [Abbas and Nazir (2014)] introduced the following iteration scheme to speed the rate of convergence as much as they can. Also, they proved that their iteration scheme is faster than the iteration scheme given by Agarwal et al. [Agarwal, O'Regan and Sahu (2007)]: In 2016, Sintunavarat et al. [Sintunavarat and Pitea (2016)] presented an iteration scheme ( ) by the sequence ( ) as follows: The aim of the present paper is to introduce a new scheme to speed the approximation of a fixed point of such problems as much as we can. Also, we present some numerical examples to show the efficiency and effectiveness of our new scheme.

New iterative scheme with analytic proof
In this section, we introduce a four-step iterative scheme to approximate a fixed point for contraction mappings of weak type. Let be a nonempty closed convex subset of a Banach space Y and T be a self-mapping on C. We define the iteration scheme ( ) by the sequence ( ) as follows: Theorem 1 Let ( , ||. ||) be a Banach space. Suppose that is a closed convex subset of Y and : → be a self-mapping. Assume that T satisfies condition (3)  , t hen the iteration process ( ) converges to faster than ( ). Proof: For each natural number n, ( ) implies that By (10) and (11), we have Therefore, On the other hand, the iterative process gives that  6.149975887561380 6.017364383359820 6.000003535729840 6.000000533210640 6.000000000000110 21 6.115237489175310 6.011811992843360 6.000001586743620 6.000000215759680 6.000000000000020 22 6.088549543658810 6.008032810428890 6.000000711785820 6.000000087243640 6.000000000000000 23 6.068044636341280 6.005461356839480 6.000000319168790 6.000000035254280 6.000000000000000 24 6.052289287574550 6.003712186247620 6.000000143064610 6.000000014237210 6.000000000000000 25 6.040182784523080 6.002522678812220 6.000000064105560 6.000000005746330 6.000000000000000 For more details, see the below table which gives the results to approximate the fixed point of the function T with start point 0 = 2. Note that by using MATLAB we can see that L has the fixed point ~0.299786903282166.

Applications
In this section, we shall prove that the initial value problem: has a unique solution. Also, we will give some examples to show the validity of our iterative scheme.
In order to prove that, we need the following lemma Lemma 1 [Plaat (1971)] ( ) is a solution for the initial value problem if and only if Let ∥. ∥ ∞ be the superior norm on ( ) which is defined by ∥ ∥ ∞ = sup ∈ ( ). Now, we have the following theorem: Theorem 2 Let : × ℛ → ℛ be a continuous function on × ℛ and let 0 be an interior point of I. Suppose that there is 0 > 0 such that H satisfies the following condition: For all 1 , 2 ∈ ℛ and for all ∈ . Then the initial value problem (16) has a unique solution ∈ ( ). Proof: Let ≥ 0 be a real number such that 0 < 1. Define the mapping : ( ) → ( ) by Therefore, condition (3) is satisfied and hence the result is satisfied. Newton's law of cooling is a differential equation that foresees the cooling of somebody that placed in a colder environment which may be written as follows: represents to the temperature of the object at the time t, represents to the temperature of the environment, and l represents to the proportionality constant. If ( 0 ) = 0 , then we have the following initial value problem Let ( , ) = − ( ( ) − ). Then, one can show that H satisfies condition (19). By Theorem 4.2, the initial value problem (20) has a unique solution.

Example 3
An orange pie was taken out of the oven at a temperature of 170 o . Then it was left to cool down with an air temperature of 20 o . If the temperature of the pie decreases initially at a rate of 3/minute, find the relation between the time and the temperature of the pie. Suppose that the pie temperature obeys Newton's law of cooling. Then, we get the following initial value problem:  (19), and so by Theorem 2, has a unique fixed point. The following figures illustrate the result of approximating the fixed point of with the initial point 0 ( ) = + 2 . We use MATLAB to get the solution obtained from the above iteration schemes at the 10 th iteration:

Conclusion
We introduced a new scheme to speed the approximation of a fixed point of such problems. We applied our scheme to some numerical examples. Also, we compared our scheme with some known schemes such as Mann's scheme [Mann (1953)], Ishikawa's scheme [Ishikawa (1974)], Agarwal et al.'s scheme [Agarwal, O'Regan and Sahu (2007)] and Sintunavarat et al. scheme [Sintunavarat and Pitea (2016)] to show the efficiency and effectiveness of our new scheme. Also, we gave an application that raised from Newton's law of cooling to show the applicability of our new scheme.