Abstract

In this paper we introduce the Jungck-Agarwal et al. iteration procedure and obtain strong convergence as well as stability results for a pair of non-self mappings. The results obtained are generalization of some existing results in the literature. In addition, we show that the rate of convergence of this newly defined iteration procedure is better than Jungck- Mann, Jungck-Ishikawa and Jungck- Noor iteration procedures.

References

• Agarwal, R. P. , O'Regan, D. and Sahu, D. R. : Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, Journal of Nonlinear and Convex Analysis 8(1) (2007), 61-79.
• Banach, S. , Sur les Operations dans les Ensembles Abstraits et leur Applications aux Equations Integrals, Fund. Math. 3 (1922), 133-181
• Berinde, V. : On the convergence of the Ishikawa iteration in the class of quasi-contractive operators, Acta Mathematica Universitatis Comenianae, vol. 73, no. 1, pp. 119–126(2004).
• Berinde, V. : Iterative approximation of fixed points. Editura Efemeride, Baia Mare, (2002).
• Berinde, V. : On the stability of some fixed point procedures. Bul. S¸tiint¸. Univ. Baia Mare Ser. B Fasc. Mat. -Inform. , 18(1):7–14, 2002. Dedicated to Costic?a Must?at¸a on his 60th anniversary.
• Harder, A. M. and Hicks, T. L. : Stability Results for Fixed Point Iteration Procedures, Math. Japonica 33 (5) (1988), 693-706.
• Ishikawa, S. , Fixed Point by a New Iteration Method, Proc. Amer. Math. Soc. 44 (1) (1974), 147-150
• Jachymski, J. R. : An extension of A. Ostrowski's theorem on the round-off stability of iterations. Aequationes Math. , 53(3):242–253, 1997.
• Jungck, G. : Commuting mappings and fixed points, The American Mathematical Monthly, vol. 83, no. 4, pp. 261–263(1976).
• Mann, W. R. , Mean Value Methods in Iteration, Proc. Amer. Math. Soc. 44 (1953), 506-510
• Noor, M. A. : New approximation schemes for general variational inequalities, Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217–229(2000).
• Olatinwo , M. O. and Imoru, C. O. : Some convergence results for the Jungck-Mann and the Jungck-Ishikawa iteration processes in the class of generalized Zamfirescu operators, Acta Mathematica Universitatis Comenianae, vol. 77, no. 2, pp. 299–304( 2008).
• Olatinwo, M. O. : Some stability and strong convergence results for the Jungck-Ishikawa iteration process, Creative Mathematics and Informatics, vol. 17, pp. 33–42(2008).
• Olatinwo, M. O. : A generalization of some convergence results using the Jungck-Noor three-step iteration process in an arbitrary Banach space, Fasciculi Mathematici, no. 40, pp. 37–43(2008).
• Osilike, M. O. and Udomene, A. : Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings. Indian J. Pure Appl. Math. , 30(12):1229–1234, (1999).
• Osilike, M. O. : Stability results for fixed point iteration procedures. J. Nigerian Math. Soc. , 14/15:17–29, (1995/96).
• Osilike, M. O. : Stability results for the Ishikawa fixed point iteration procedure, Indian Journal of Pure and Applied Mathematics, vol. 26, no. 10, pp. 937–945 (1995).
• Ostrowski, M. The round-off stability of iterations. Z. Angew. Math. Mech. , 47:77–81, (1967).
• P. , Bhagwati and S. , Ritu: Weak stability results for Jungck-Ishikawa iteration, International Journal of Computer Applications, Volume16, No. 4, February (2011).
• Rhoades, B. E. : Fixed point theorems and stability results for fixed point iteration procedures. Indian J. Pure Appl. Math. , 21(1):1–9, (1990).
• Rhoades, B. E. : Fixed point theorems and stability results for fixed point iteration procedures. II. Indian J. Pure Appl. Math. , 24(11):691–703, (1993).
• Rhoades, B. E. , Fixed Point Iteration using Infinite Matrices, Trans. Amer. Math. Soc. 196 (1974), 161- 176.
• Rhoades, B. E. : Comments on two fixed point iteration methods," Journal of Mathematical Analysis and Applications, vol. 56, no. 3, pp. 741–750(1976).
• Singh, S. L. , Bhatnagar, Charu and Mishra, S. N. : Stability of Jungck-type iterative procedures, International Journal of Mathematics and Mathematical Sciences, no. 19, pp. 3035–3043 (2005).
• Zamfirescu, T. , Fixed Point Theorems in Metric Spaces, Arch. Math. 23(1972), 292-29.