Geodesic
Some common fixed point theorems in normed linear spaces
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 49 (2010) no. 1, pp. 17-24

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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].
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].
Classification : 47H10, 54H25
Keywords: Common fixed point; contractive condition; Mann and Ishikawa iterations 
Bosede, Alfred Olufemi. Some common fixed point theorems in normed linear spaces. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 49 (2010) no. 1, pp. 17-24. http://geodesic.mathdoc.fr/item/AUPO_2010_49_1_a1/
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[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 | Zbl

[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 | Zbl

[4] Chatterjea, S. K.: Fixed point theorems. C. R. Acad. Bulgare Sci. 25 (1972), 727–730. | MR | Zbl

[5] Das, G., Debata, J. P.: Fixed points of Quasi-nonexpansive mappings. Indian J. Pure and Appl. Math. 17 (1986), 1263–1269. | MR

[6] Ishikawa, S.: Fixed points by a new iteration method. Proc. Amer. Math. Soc. 44 (1974), 147–150. | DOI | MR | Zbl

[7] Kannan, R.: Some results on fixed points. Bull. Calcutta Math. Soc. 10 (1968), 71–76. | MR | Zbl

[8] Kannan, R.: Some results on fixed points III. Fund. Math. 70 (1971), 169–177. | MR | Zbl

[9] Kannan, R.: Construction of fixed points of class of nonlinear mapppings. J. Math. Anal. Appl. 41 (1973), 430–438. | DOI | MR

[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 | MR | Zbl

[11] Mann, W. R.: Mean value methods in iterations. Proc. Amer. Math. Soc. 4 (1953), 506–510. | DOI | MR

[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 | Zbl

[13] Osilike, M. O.: Stability results for fixed point iteration procedures. J. Nigerian Math. Soc. 14/15 (1995/96), 17–29. | MR

[14] Rafiq, A.: Common fixed points of quasi-contractive-operators. General Mathematics 16, 2 (2008), 49–58. | MR | Zbl

[15] Rhoades, B. E.: Fixed point iteration using infinite matrices. Trans. Amer. Math. Soc. 196 (1974), 161–176. | DOI | MR | Zbl

[16] Rhoades, B. E.: Comments on two fixed point iteration method. J. Math. Anal. Appl. 50, 2 (1976), 741–750. | DOI | MR

[17] Rhoades, B. E.: A comparison of various definitions of contractive mappings. Trans. Amer. Math. Soc. 226 (1977), 257–290. | DOI | MR | Zbl

[18] Rus, I. A.: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca, 2001. | MR | Zbl

[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 | Zbl

[20] Takahashi, W.: Iterative methods for approximation of fixed points and their applications. J. Oper. Res. Soc. Jpn. 43, 1 (2000), 87–108. | DOI | MR | Zbl

[21] Takahashi, W., Tamura, T.: Convergence theorems for a pair of nonexpansive mappings. J. Convex Analysis 5, 1 (1995), 45–58. | MR

[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

[23] Zamfirescu, T.: Fix point theorems in metric spaces. Arch. Math. (Basel) 23 (1972), 292–298. | DOI | MR | Zbl

[24] Zeidler, E.: Nonlinear Functional Analysis and its Applications, I. Fixed Point Theorems, Springer, New York, 1986. | MR | Zbl