# Article

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Keywords:
Common fixed point; contractive condition; Mann and Ishikawa iterations
Summary:
In this paper, we establish some generalizations to approximate common fixed points for selfmappings in a normed linear space using the modified Ishikawa iteration process with errors in the sense of Liu [10] and Rafiq [14]. We use a more general contractive condition than those of Rafiq [14] to establish our results. Our results, therefore, not only improve a multitude of common fixed point results in literature but also generalize some of the results of Berinde [3], Rhoades [15] and recent results of Rafiq [14].
References:
[1] Berinde, V.: A priori and a posteriori error estimates for a class of $\varphi$-contractions. Bulletins for Applied and Computing Math. (1999), 183–192.
[2] Berinde, V.: Iterative Approximation of Fixed Points. Editura Efemeride, Baia Mare, 2002. MR 1995230 | Zbl 1036.47037
[3] Berinde, V.: On the convergence of the Ishikawa iteration in the class of quasi-contractive operators. Acta Math. Univ. Comenianae 73, 1 (2004), 119–126. MR 2076050 | Zbl 1100.47054
[4] Chatterjea, S. K.: Fixed point theorems. C. R. Acad. Bulgare Sci. 25 (1972), 727–730. MR 0324493 | Zbl 0274.54033
[5] Das, G., Debata, J. P.: Fixed points of Quasi-nonexpansive mappings. Indian J. Pure and Appl. Math. 17 (1986), 1263–1269. MR 0868962
[6] Ishikawa, S.: Fixed points by a new iteration method. Proc. Amer. Math. Soc. 44 (1974), 147–150. DOI 10.1090/S0002-9939-1974-0336469-5 | MR 0336469 | Zbl 0286.47036
[7] Kannan, R.: Some results on fixed points. Bull. Calcutta Math. Soc. 10 (1968), 71–76. MR 0257837 | Zbl 0209.27104
[8] Kannan, R.: Some results on fixed points III. Fund. Math. 70 (1971), 169–177. MR 0283649 | Zbl 0246.47065
[9] Kannan, R.: Construction of fixed points of class of nonlinear mapppings. J. Math. Anal. Appl. 41 (1973), 430–438. DOI 10.1016/0022-247X(73)90218-7 | MR 0320837
[10] Liu, L. S.: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 194, 1 (1995), 114–125. DOI 10.1006/jmaa.1995.1289 | MR 1353071 | Zbl 0872.47031
[11] Mann, W. R.: Mean value methods in iterations. Proc. Amer. Math. Soc. 4 (1953), 506–510. DOI 10.1090/S0002-9939-1953-0054846-3 | MR 0054846
[12] Osilike, M. O.: Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings. Indian J. Pure Appl. Math. 20, 12 (1999), 1229–1234. MR 1729212 | Zbl 0955.47038
[13] Osilike, M. O.: Stability results for fixed point iteration procedures. J. Nigerian Math. Soc. 14/15 (1995/96), 17–29. MR 1775011
[14] Rafiq, A.: Common fixed points of quasi-contractive-operators. General Mathematics 16, 2 (2008), 49–58. MR 2439234 | Zbl 1235.47073
[15] Rhoades, B. E.: Fixed point iteration using infinite matrices. Trans. Amer. Math. Soc. 196 (1974), 161–176. DOI 10.1090/S0002-9947-1974-0348565-1 | MR 0348565 | Zbl 0267.47032
[16] Rhoades, B. E.: Comments on two fixed point iteration method. J. Math. Anal. Appl. 50, 2 (1976), 741–750. DOI 10.1016/0022-247X(76)90038-X | MR 0430880
[17] Rhoades, B. E.: A comparison of various definitions of contractive mappings. Trans. Amer. Math. Soc. 226 (1977), 257–290. DOI 10.1090/S0002-9947-1977-0433430-4 | MR 0433430 | Zbl 0365.54023
[18] Rus, I. A.: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca, 2001. MR 1947742 | Zbl 0968.54029
[19] Rus, I. A., Petrusel, A., Petrusel, G.: Fixed Point Theory: 1950-2000, Romanian Contributions. House of the Book of Science, Cluj-Napoca, 2002. MR 1947195 | Zbl 1005.54037
[20] Takahashi, W.: Iterative methods for approximation of fixed points and their applications. J. Oper. Res. Soc. Jpn. 43, 1 (2000), 87–108. DOI 10.1016/S0453-4514(00)88753-0 | MR 1768388 | Zbl 1004.65069
[21] Takahashi, W., Tamura, T.: Convergence theorems for a pair of nonexpansive mappings. J. Convex Analysis 5, 1 (1995), 45–58. MR 1649417
[22] Xu, Y.: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations. J. Math. Anal. Appl. 224 (1998), 91–101. DOI 10.1006/jmaa.1998.5987
[23] Zamfirescu, T.: Fix point theorems in metric spaces. Arch. Math. (Basel) 23 (1972), 292–298. DOI 10.1007/BF01304884 | MR 0310859 | Zbl 0239.54030
[24] Zeidler, E.: Nonlinear Functional Analysis and its Applications, I. Fixed Point Theorems, Springer, New York, 1986. MR 0816732 | Zbl 0583.47050

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