The Equivalence between the Mann and Ishikawa Iterations for Generalized Contraction Mappings in a Cone

Generally, the iteration techniques of W.R.Mann [1] and Shiro Ishikwa [2] are used to find the approximation of fixed point of a contraction mapping. These iterations are quite useful even for the cases of where Picard iteration fails. In this paper, we see the equivalence between these Mann and Ishikawa iterations for a generalized contraction mapping in a cone. First, we recall the definition of a cone (refer Huang Long-guang and Zhang Xian [3]) and some of its properties. Definition 1.1: Let be a real Banach space and a subset of is said to be a cone if satisfies the following: E P E


Introduction
Generally, the iteration techniques of W.R.Mann [1] and Shiro Ishikwa [2] are used to find the approximation of fixed point of a contraction mapping.These iterations are quite useful even for the cases of where Picard iteration fails.In this paper, we see the equivalence between these Mann and Ishikawa iterations for a generalized contraction mapping in a cone.First, we recall the definition of a cone (refer Huang Long-guang and Zhang Xian [3]) and some of its properties.
Definition 1.1: Let be a real Banach space and a subset of is said to be a cone if satisfies the following: where  and ( , ) M x y satisfy the following: 1) :[0, ) [0, ) is a real-valued, nondecreasing, right continuous function; 2) ( ) < for each > 0; x y x Tx y Ty x Ty y Tx ( T satisfying above conditions is said to be a Generalized contraction.Below, we see the definition of the two iteration schemes due to Mann [1] and Ishikawa [2].Further, these two iterations are applied to a class of generalized contraction mapping which is mentioned just above.Let 0 0 x u P   .The Mann iteration is defined by 1 = (1 ) The Ishikawa iteration is defined by , ) and from the defi- nition of cone.
Let   n w li be a sequence in P which is a subset of a real Banach space.We say that converges to where .


is the norm associated with .E The main aim of this paper is to show that the convergence of Mann iteration is equivalent to the convergence of Ishikawa iteration in the cone .P Below, we sate two results without proof which are very much useful for our analysis.for proof, one may refer [4] and [5] respectively.

Lemma 1 [4]
Let   n a be a nonnegative sequence which satisfies n the following inequality: n and and from the definition of cone.Here, is a closed and convex subset of E which also follows from the definition of cone.Therefore, the above lemma can be verified for .
 n  P P

Main Result
In this section, we discuss the main result which gives the equivalence of Mann and Ishikawa iterations in the cone.The analysis is similar to the work of Rhoades and Soltuz [6].
We then have the following From the definition of and all above inequalities imply that, is monotone non-increasing in and positive, i.e., bounded below.Hence, there exists . We wish to show that .0 r  = 0 r Suppose not that, .From (2.3), we get the following, > 0 r In general, we have that Therefore, on summing we obtain, The right-hand side is bounded and the left-hand side is unbounded, which leads to a contradiction.Thus = .r o Therefore, we have We now show that both the iteration schemes are equivalent.Suppose the Mann iteration converges,then we have This implies that Ishikawa iteration also converges.Suppose the Ishikawa iteration converges, then we have .
Let be a nonempty closed convex subset of a Banach space , and T a self-map of satisfying (1.1).Let P

.
Denote by x  the unique fixed point of T. Mann iteration (1.7) converges to x  ; 2) the Ishikawa iteration (1.8) converges to x  .Proof: By Lemma 2, both Mann and Ishikawa iterations are bounded.we have to prove the equivalence between (1.7) and (1.8).We need to prove that This implies that Mann iteration converges.Hence the theorem.