Fixed point theory takes a large amount of literature, since it provides useful tools to solve many problems that have applications in different fields like engineering, economics, chemistry and game theory etc. However, once the existence of a fixed point of some mapping is established, then to find the value of that fixed point is not an easy task that is why we use iteration processes for computing them. By time, many iteration processes have been developed and it is impossible to cover them all. The well-known Banach contraction theorem use Picard iteration process for approximation of fixed point. Some of the other well-known iteration processes are Mann (Mann 1953), Ishikawa (Ishikawa 1974), Agarwal (Agarwal et al. 2007), Noor (Noor 2000), Abbas (Abbas and Nazir 2014), SP (Phuengrattana and Suantai 2011), S(^{*}) (Karahan and Ozdemir 2013), CR (Chugh et al. 2012), Normal-S (Sahu and Petrusel 2011), Picard Mann (Khan 2013), Picard-S (Gursoy and Karakaya 2014), Thakur New (Thakur et al. 2016) and Vatan Two-step (Karakaya et al. 2015).
Fastness and stability play important role for an iteration process to be preferred on another iteration process. In 1991 , Rhoades mentioned that the Mann iteration process for decreasing function converge faster than the Ishikawa iteration process and for increasing function the Ishikawa iteration process is better than the Mann iteration process. Also the Mann iteration process appears to be independent of the initial guess (see also Rhoades 1977). In Agarwal et al. (2007), the authors claimed that Agarwal iteration process converges at a rate same as that of the Picard iteration process and faster than the Mann iteration process for contraction mappings. In Abbas and Nazir (2014), the authors claimed that Abbas iteration process converge faster than Agarwal iteration process. In Chugh et al. (2012), the authors claimed that CR iteration process is equivalent to and faster than Picard, Mann, Ishikawa, Agarwal, Noor and SP iteration processes for quasi-contractive operators in Banach spaces. Also in Karakaya et al. (2014) the authors proved that CR iteration process converge faster than the S(^{*}) iterative process for the class of contraction mappings. In Gursoy and Karakaya (2014), authors claimed that Picard-S iteration process converge faster than all Picard, Mann, Ishikawa, Noor, SP, CR, Agarwal, S(^{*},) Abbas and Normal-S iteration processes for contraction mappings. In Thakur et al. (2016), the authors proved with the help of numerical example that Thakur New iteration process converge faster than Picard, Mann, Ishikawa, Agarwal, Noor and Abbas iteration processes for the class of Suzuki generalized nonexpansive mappings. Similarly, in Karakaya et al. (2015), the authors proved that Vatan Two-step iteration process is faster than Picard-S, CR, SP and Picard-Mann iteration processes for weak contraction mappings. For fragmentation models and processes see Goufo (2014), Goufo and Noutchie (2013). Similarly, for local convergence of Chebyshev–Halley methods with six and eight order of convergence to approximate a locally unique solution of a nonlinear equation see Magrenan and Argyros (2016).
Motivated by above, in this paper, we introduce a new iteration process known as AK iteration process and prove analytically that our process is stable. Then we prove that AK iteration process converges faster than Vatan Two-step iteration process which is faster than all Picard, Mann, Ishikawa, Noor, SP, CR, S, S(^{*},) Abbas, Normal-S and Two-step Mann iteration processes for contraction mappings. Numerically we compare the convergence of the AK iteration process with the three most leading iteration processes in the existing literature for contraction mappings. The data dependence result for fixed point of contraction mappings by employing AK iteration process is also proved.
We now recall some definitions, propositions and lemmas to be used in the next two sections.
A point p is called fixed point of a mapping T if (T(p)=p), and F(T) represents the set of all fixed points of a mapping T. Let C be a nonempty subset of a Banach space X. A mapping (T:Crightarrow C) is called contraction if there exists (theta in (0,1)) such that (left| Tx-Tyright| le theta left| x-yright| ,) for all (x,yin C.)
Definition 2
(Berinde 2007) Let ({u_{n}}_{n=0}^{infty }) and ({v_{n}}_{n=0}^{infty }) be two fixed point iteration procedure sequences that converge to the same fixed point p. If (left| u_{n}-pright| le a_{n}) and (left| v_{n}-pright| le b_{n},) for all (nge 0), where ({a_{n}}_{n=0}^{infty }) and ({b_{n}}_{n=0}^{infty }) are two sequences of positive numbers (converging to zero). Then we say that ({u_{n}}_{n=0}^{infty }) converge faster than ({v_{n}}_{n=0}^{infty }) to p if ({a_{n}}_{n=0}^{infty }) converge faster than ({b_{n}}_{n=0}^{infty }).
Definition 3
(Harder 1987) Let ({t_{n}}_{n=0}^{infty }) be an arbitrary sequence in C. Then, an iteration procedure (x_{n+1}=f(T,x_{n}),) converging to fixed point p, is said to be T-stable or stable with respect to T, if for (epsilon _{n}=left| t_{n+1}-f(T,t_{n})right| ,)
(n=0,1,2,3, ldots,) we have