On the Convergence and Stability Results for a New General Iterative Process

We put forward a new general iterative process. We prove a convergence result as well as a stability result regarding this new iterative process for weak contraction operators.


Introduction and Preliminaries
Throughout this paper, by N, we denote the set of all positive integers. In this paper, we obtain results on the stability and strong convergence for a new iteration process (3) in an arbitrary Banach space by using weak contraction operator in the sense of Berinde [1]. Also, we obtain that the iteration procedure (3) can be useful method for solution of delay differential equations. To obtain solution of delay differential equation by using fixed point theory, some authors have done different studies. One can find these works in [2,3]. Many results of stability have been established by some authors using different contractive mappings. The first study on the stability of the Picard iteration under Banach contraction condition was done by Ostrowski [4]. Some other remarkable results on the concept of stability can be found in works of the following authors involving Harder and Hicks [5,6], Rhoades [7,8], Osilike [9], Osilike and Udomene [10], and Singh and Prasad [11]. In 1988, Harder and Hicks [5] established applications of stability results to first order differential equations. Osilike and Udomene [10] developed a short proof of stability results for various fixed point iteration processes. Afterward, in following studies, same technique given in [10] has been used, by Berinde [12], Olatinwo [13], Imoru and Olatinwo [14], Karakaya et al. [15], and some authors.
Let ( , ) be complete metric space and : → a self-map on ; and the set of fixed points of in is defined by = { ∈ : = }. Let { } ∈N ⊂ be the sequence generated by an iteration involving which is defined by where 0 ∈ is the initial point and is a proper function. Suppose that sequence { } ∈N converges to a fixed point of . Let { } ∈N ⊂ and set = ( +1 , ( , )) = 0, 1, . . . .
Then, the iteration procedure (1) is said to be stable or stable with respect to if and only if lim → ∞ = 0 implies lim → ∞ = .
Now, let be a convex subset of a normed space and : → a self-map on . We introduce a new twostep iteration process which is a generalization of Ishikawa iteration process as follows: In the following remark, we show that the new iteration process is more general than the Ishikawa and Mann iteration processes.

Then has a unique fixed point and the Picard iteration
converges to for any arbitrary but fixed 0 ∈ .
In 2004, Berinde introduced the definition which is a generalization of the above operators.
Definition 5 (see [1]). A mapping is said to be a weak contraction operator, if there exist ≥ 0 and ∈ (0, 1) such that for all , ∈ .
Definition 7 (see [22] Then, it is said that { } ∈N converges faster than {V } ∈N to fixed point of .
The rate of convergence of the Picard and Mann iteration processes in terms of Zamfirescu operators in arbitrary Banach setting was compared by Berinde [22]. Using this class of operator, the Mann iteration method converges faster than the Ishikawa iteration method that was shown by Babu and Vara Prasad [23]. After a short time, Qing and Rhoades [24] showed that the claim of Babu and Vara Prasad [23] is false. There are many studies which have been made on the rate of convergence as given in [15,25,26] which are just a few of them.

Theorem 8. Let be a nonempty closed convex subset of an arbitrary Banach space and let :
→ be a mapping satisfying (11). Let { } ∈N be defined through the new iteration In addition, Substituting (14) in (13), we have the following estimates: Since ‖ − ‖ = 0, we have for all ∈ N.

Theorem 9. Let ( , ‖ ⋅ ‖) be Banach space and
: → a self-mapping with fixed point with respect to weak contraction condition in the sense of Berinde (11). Let { } ∈N be iteration process (3) converging to fixed point of , where 1] such that 0 < ℘ ≤ ℘ for all . Then two-step iteration process is stable.
Example 10 (see [24]). Let Secondly, we consider the Ishikawa iterative process, and we have Now, taking the above two equalities, we obtain It is clear that Therefore, the proof is completed. Now, we can give Table 1 and Figures 1 and 2 to support and reinforce our claim in the Example 10.
Finally, we check that this iteration procedure can be applied to find the solution of delay differential equations.

Ishikawa
New iteration process functions. It is well known that [ , ] is a real Banach space with respect to ‖ ⋅ ‖ ∞ norm; more details can be found in [2,27]. Now, we will consider a delay differential equation such that and an assumed solution Assume that the following conditions are satisfied: The Scientific World Journal Using weak-contraction mapping, we obtain the following.