Strong Convergence and Stability of Jungck-Multistep-SP Iteration for Generalized Contractive-Like Inequality Operators

Hudson Akewe

Abstract


We introduce the Jungck-multistep-SP iteration and prove some convergence as well as stabiilty results for a pair of weakly compatible generalized contractive-like inequality operators defined on a Banach space. As corollaries, the results show that the Jungck-SP and Jungck-Mann iterations can also be used to approximate the common fixed points of such operators. The results are improvements, generalizations and extensions of the work of Chugh and Kumar (2011). Consequently, several results in literature are generalized.

Key words: Jungck-multistep-SP iteration


Keywords


Jungck-multistep-SP iteration

References


Abbas, M., & Jungck, G. (2008). Common Fixed Point Results for Noncommuting Mappings Without Continuity in Cone Metric Spaces. J. Math. Anal. Appl., 341, 416-420.

Berinde, V. (2004). On the Convergence of Ishikawa Iteration in the Class of Quasicontractive Operators. Acta Math. Univ. Comenianae, LXXIII(1), 119-126.

Bosede, A.O. (2010). Strong Convergence Results for the Jungck-Ishikawa and Jungck-Mann Iteration Processes. Bulletin Math. Anal. Appl., 2(3), 65-73.

Bhagwati, P., & Ritu, S. (2011). Weak Stability Results for Jungck-Ishikawa Iteration. Inter. J. Comp. Appl., 16(4).

Chugh, R., & Kumar, V. (2011). Strong Convergence and Stability Results for Jungck-SP Iterative Scheme. Inter. J. Comp. Appl. (0975-8887), 36(12).

Chatterjea, S.K. (1972). Fixed Point Theorems. Comptes rendus de l’Academic bulgare des Sciences, 25(6), 727-730.

Das, K.M., & Naik, K.V. (1979). Common Fixed Point Theorems for Commuting Maps on Metric Spaces. Proc. Amer. Math. Soc., 77, 369-373.

Ishikawa, S. (1974). Fixed Points by a New Iteration Method. Proc. Amer. Math. Soc., 149, 147-150.

Jungck, G. (1976). Commuting Mappings and Fixed Points, Amer. Math. Monthly, 83, 261-263.

Kannan, R. (1969). Some Results on Fixed Points II. Amer. Math. Monthly, 76, 405-408.

Mann, W.R. (1953). Mean Value Methods in Iteration. Proc. Amer. Math. Soc., 4, 506-510.

Noor, M.A. (2000). New Approximation Schemes for General Variational Inequalities. Journal of Mathematical Analysis and Applications, 251(1), 217-229.

Olaleru, J.O. (2006). On the Convergence of the Mann Iteration in Locally Convex Spaces. Carpathian Journal of Mathematics, 22(1-2), 115-120.

Olaleru, J.O. (2007). On the Equivalence of Picard, Mann and Ishikawa Iterations for a Class of Quasi-Contractive Operators. J. Nig. Assoc. Math. Phys., 11, 51-56.

Olaleru, J.O., & Akewe, H. (2010). The Convergence of Jungck-Type Iterative Schemes for Generalized Contractive-Like Operators. Fasciculi Mathematici, (45), 87-98.

Olatinwo, M.O., & Imoru, C.O. (2008). Some Convergence Results for the Jungck-Mann and Jungck-Ishikawa Iteration Process in the Class of Generalized Zamfirescu Operators. Acta Math. Univ. Comenianae, 77(2), 299-304

Olatinwo, M.O. (2008). Some Stability and Strong Convergence Results for the Jungck-Ishikawa Iteration Process. Creative Math. and Info., 17, 33-42.

Olatinwo, M.O. (2008). A Generalization of Some Convergence Results Using the Jungck-Noor Three Step Iteration Process in Arbitrary Banach Space. Fasciculi Mathematici, 40, 37-43.

Osilike, M.O. (1995). Stability Results for Ishikawa Fixed Point Iteration Procedure. Indian J. Pure Appl. Math., 26(10), 937-941.

Rafiq, A. (2006). On the Equivalence of Mann and Ishikawa Iteration Methods with Errors. Math. Comm., 11, 143-152.

Rhoades, B.E. (1976). Comments on Two Fixed Points Iteration Methods. J. Math. Anal.Appl., 56(2), 741-750.

Rhoades, B.E. & Soltuz, S.M. (2004). The Equivalence Between Mann-Ishikawa Iterations and Multi-Step Iteration. Nonlinear Anal., 58, 219-228.

Popescu, O. (2007). Picard Iteration Converges Faster Than Mann Iteration for a Class of Quasi-Contractive Operators. Math. Comm., 12, 195-202.

Singh, S.L. (1977). On Common Fixed Points of Commuting Maps. Math. Sem. Notes Kobe Univ., 5, 131-134.

Singh, S.L., Bhatnagar, C.S. & Mishra, N. (2005). Stability of Jungck-type Iterative Procedures. International J. Math. and Math. Sc., 19, 3035-3043.

Zhiqun, X. (2007). Remarks of Equivalence Among Picard,Mann and Ishikawa Iterations in Normed Space. Fixed Point Theory and Applications, (2007), 5pp.

Zamfirescu, T. (1972). Fixed Point Theorems in Metric Spaces. Arch. Math. (Basel), 23, 292-298.





DOI: http://dx.doi.org/10.3968/j.ans.1715787020120503.1512

DOI (PDF): http://dx.doi.org/10.3968/g3033

DOI (indexed/included/archived): http://dx.doi.org/10.3968/g4610

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