 Applied Mathematics, 2011, 2, 1369-1371 doi:10.4236/am.2011.211192 Published Online November 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM The Equivalence between the Mann and Ishikawa Iterations for Generalized Contraction Mappings in a Cone L. Jones Tarcius Doss, T. Esakkiappan Department of Mat hematics, Anna University, Chennai, India E-mail: tesakki@gmail.com Received July 22, 201 1; revised Septe mber 13, 2011; accepted September 20, 2011 Abstract In this paper, equivalence between the Mann and Ishikawa iterations for a generalized contraction mapping in cone subset of a real Banach space is discussed. Keywords: Mann Iteration, Ishikawa Iterations, Generalized Contraction, Cone 1. Introduction Generally, the iteration techniques of W.R.Mann  and Shiro Ishikwa  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 contra- ction mapping in a cone. First, we recall the definition of a cone (refer Huang Long-guang and Zhang Xian ) 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: EP E1) , P is closed and ; Pax by {0}P,2) for every PxyP and ; ,0ab3) . ()=PP{0}The partial ordering with respect to the cone P is defines by xy if and only if yxP. We shall write 0;ttt (1.3) 3) isnondecreasingon(0,); (1.4) 4) ():=(())isnonincreasingon(0,);gtt tt (1.5) 5) (,):=max, ,,,MxyxyxTxyTyxTyyTx (1.6) T satisfying above conditions is said to be a Generalized contraction. Below, we see the definition of the two iteration schemes due to Mann  and Ishikawa . Further, these two iterations are applied to a class of generalized contraction mapping which is mentioned just above. Let 00xuP. The Mann iteration is defined by 1=(1)nnnnuunTu. (1.7) The Ishikawa iteration is defined by 1=(1),=(1),nnnnnnnnnnxxTyyxTx (1.8) where (0,1), 0,1nn. Clearly, the sequences nx,nu and ny are in because 00P=xuP and (0,1)n and [0n,1) and from the defi- nition of cone. Let nwli be a sequence in P which is a subset of a real Banach space. We say that converges to nwwand write if m =nnww lim= 0nnww where . is the norm associated with . EThe main aim of this paper is to show that the con- vergence of Mann iteration is equivalent to the con- vergence of Ishikawa iteration in the cone . PBelow, we sate two results without proof which are very much useful for our analysis. for proof, one may refer  and  respectively. Lemma 1  Let na be a nonnegative sequence which satisfies L. J. T. DOSS ET AL. 1370 nthe following inequality: 1(1 ),nnnaa (1.9) where (0,1)n=( )n for all , and 01,=nnnnno. Then . lim nna = 0Lemma 2  Let be a nonempty closed convex subset of a Banach space , and T a self-map of satisfying (1.1). Let P{nE}P satisfy the conditions >0n for all and . Then the sequences 0n,,nn1n,n=nn ,nxyuTxTy and  are bounded. nTuClearly, the sequences and ,nnuxny are in because 00P=xuP and (0,1)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 . [0,1)nPP 2. 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 . THEOREM 2.1 Let be a cone subset of a Banach space , and a self-map of satisfying (1.1)-(1.6). Let {}P ET Pn satisfy the conditions 0n for all and 0n1=nn. Denote by x the unique fixed point of T. Then for , the following are equivalent: 00uxP1) the Mann iteration (1.7) converges to x; 2) the Ishikawa iteration (1.8) converges to x. Proof: By Lemma 2, both Mann and Ishikawa itera- tions are bounded. we have to prove the equivalence between (1.7) and (1. 8). We need to prove that lim= 0nnnxu . (2.1) Set = maxsup:sup:sup: sup:nnj njnj njrxTyjnuTuxTu jnuTy jn  jn (2.2) We then have the following 11 1111 11111 111(,1();njnnjnn jnnnn jnnn n111 111 11111 111(,1();njnnjnnnnnn jnnn nu TuuTuTuTurMuurr      )j 11 1111 11111 111(1();njnnjnnnnnn jnnn n,)jxTux TuTy TurMyurr       111 111 11111 111(,1().njnnjnn jnnnn jnnn nu TyuTyTuTyrMuyrr      ) From the definition of and all above inequalities imply that, nr111 111111(0()nnnnnnnnn nrrrrrrr ). (2.3) Therefore, nr is monotone non-increasing in and positive, i.e., bounded below. Hence, there exists nlim nnr, denoted by . We wish to show that . 0r=0rSuppose not that, . From (2.3), we get the following, >0r11 11()nn nnnrrrgr 1111() ()nnnnnngr grrr rrrr 1n. In general, we have that 1() .kkkgr rrr Therefore, on summing we obtain, 10=0 =0() ()=.nnkkkkkgr grrr rrrr  1n The right-hand side is bounded and the left-hand side is unbounded, which leads to a contradiction. Thus =.roTherefore, we have =0 =0lim limnn nnnnxTu uTy  (2.4) =0 =0lim limnn nnnnxTy uTu  (2.5) )xTyx TyTy TyrMyyrr       We now show that both the iteration schemes are equivalent. Suppose the Mann iteration converges,then we have 1111.nn nnnnnnnnn n nnnnxux uTyTuxuTyxxTu     Using (2.4), (2.5) , Lemma 1 and above eq uations with Copyright © 2011 SciRes. AM L. J. T. DOSS ET AL. Copyright © 2011 SciRes. AM 1371the following :::, ,()  nnnnnnnnnnnx uTyxxTuo we have lim= 0nn , that is (2.1) holds. Then, the relation 0.nnnnxxxuxu This implies that Ishikawa iteration also converges. Suppose the Ishikawa iteration converges, then we have 11(1 )(1 ).nn nnnnnnnn nnn nnnxuxuTyx uTy uu Tu  Tu Using (2.4), (2.5) , Lemma 1 and above eq uations with the following :=, :=,=( ),nnnnnnnnnnnxuTyu uTuo we have lim= 0nn , that is (2.1) holds. Then, the relation 0.nnnnuxxu xx This implies that Mann iteration converges. Hence the theorem. 3. Acknowledgements The second author Mr. T. Esakkiappan would like to thank the referee for his valid suggestions. Further, the same author would like to thank his research supervisor Prof.P.Vijayaraju for his valuable guidance and support. 4. References  W. R. Mann, “Mean Value Methods in Iteration,” Pro-ceedings of the American Mathematical Society, Vol. 4, No. 3, 1953, pp. 506-510. doi:10.1090/S0002-9939-1953-0054846-3  S. Ishikawa, “Fixed Points by a New Iteration Method,” Proceedings of the American Mathematical Society, Vol. 44, No. 1, 1974, pp. 147-150. doi:10.1090/S0002-9939-1974-0336469-5  L.-G. Huang and X. Zhang, “Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 332, No. 2, 2007, pp.1468-1476. doi:10.1016/j.jmaa.2005.03.087  X. Weng, “Fixed Point Iteration for Local Strictly Pseudo- Contractive Mapping,” Proceedings of the American Ma- thematical Society, Vol. 113, No. 3, 1991, pp. 727-731. doi:10.1090/S0002-9939-1991-1086345-8  B. E. Rhoades, “Convergence of an Isikawa-Type Itera-tion Scheme for a Generalized Contraction,” Journal of Mathematical Analysis and Applications, Vol. 185, No. 2, 1994, pp. 350-355. doi:10.1006/jmaa.1994.1253  B. E. Rhoades and S. M. Soltuz, “The Eqivalence be-tween Mann and Ishikawa Iterations Dealing with Gener- alized Contractions,” International Journal of Mathemat- ics and Mathematical Sciences, Vol. 2006, 2006, pp. 1-5.