Convergence Results for Jungck-type Iterative Processes in Convex Metric Spaces

Olatinwo, Memudu Olaposi

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 51 (2012) no. 1, pp. 79-87

Abstract (VO)

In this paper, the convergence results of [V. Berinde; A convergence theorem for Mann iteration in the class of Zamfirescu operators, Analele Universitatii de Vest, Timisoara, Seria Matematica-Informatica 45 (1) (2007), 33–41], [V. Berinde; On the convergence of Mann iteration for a class of quasi-contractive operators, Preprint, North University of Baia Mare (2003)] and [V. Berinde; On the Convergence of the Ishikawa Iteration in the Class of Quasi-contractive Operators, Acta Math. Univ. Comenianae 73 (1) (2004), 119–126] are extended from arbitrary Banach space setting to the convex metric space by weakening further the conditions on the parameter sequence $\lbrace \alpha _n\rbrace \subset [0,1]$. We establish the convergence of Jungck–Mann and Jungck–Ishikawa iterative processes for two nonselfmappings in a convex metric space setting by employing a general contractive condition. Similar results are also deduced for the Mann and Ishikawa iterations. Our results generalize, extend and improve a multitude of results in the literature including those of Berinde mentioned above.

Détail

Classification : 47H10, 54H25
Keywords: arbitrary Banach space setting; Jungck–Mann and Jungck–Ishikawa iterative processes; convex metric space

Bibliographie

[1] Agarwal, R. P., O’Regan, D., Sahu, D. R.: Fixed Point Theory for Lipschitzian-type Mappings with Applications. Topological Fixed Point Theory and Its Applications, Vol. 6 6 Springer Science & Bussiness Media, 2009. | MR | Zbl

[2] Banach, S.: Sur les operations dans les ensembles abstraits et leur applications aux equations integrales. Fund. Math. 3 (1922), 133–181.

[3] Beg, I.: Structure of the set of fixed points of nonexpansive mappings on convex metric spaces. Ann. Univ. Marie Curie-Sklodowska Sec. 52, Sec. A (1998), 7–14. | MR | Zbl

[4] Beg, I.: Inequalities in metric spaces with applications. Topological Methods in Nonlinear Analysis 17 (2001), 183–190. | MR | Zbl

[5] Berinde, V.: On the convergence of Mann iteration for a class of quasi-contractive operators. North University of Baia Mare, Preprint, 2003.

[6] 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

[7] Berinde, V.: Iterative Approximation of Fixed Points. Springer-Verlag, Berlin–Heidelberg, 2007. | MR | Zbl

[8] Berinde, V.: A convergence theorem for Mann iteration in the class of Zamfirescu operators. Analele Universitatii de Vest, Timisoara, Seria Matematica-Informatica 45, 1 (2007), 33–41. | MR | Zbl

[9] Berinde, V.: Some remarks on a fixed point theorem for Ciric-type almost contractions. Carpathian J. Math. 25, 2 (2009), 157–162. | MR | Zbl

[10] Chatterjea, S. K.: Fixed-point theorems. C. R. Acad. Bulgare Sci. 10 (1972), 727–730. | MR | Zbl

[11] Ciric, Lj. B.: Generalized contractions and fixed point theorems. Publ. Inst. Math. (Beograd) 12, 26 (1971), 19–26. | MR | Zbl

[12] Ciric, Lj. B.: A generalization of Banach’s contraction principle. Proc. Amer. Math. Soc. 45 (1974), 267–273. | DOI | MR | Zbl

[13] Ciric, L.: On some discontinuous fixed point theorems on convex metric spaces. Czechoslovak Math. J. 43 (1993), 319–326. | MR

[14] Guay, M. D., Singh, K. L., Whitfield, J. H. M.: Fxed point theorems for nonexpansive mappings in convex metric spaces. In: Singh, S. P., Barry, J. H. (eds.) Proceedings of Conference on Nonlinear Analysis 80, Marcel Dekker Inc., New York, 1982, 179–189. | MR

[15] Ishikawa, S.: Fixed point by a new iteration method. Proc. Amer. Math. Soc. 44, 1 (1974), 147–150. | DOI | MR

[16] Jungck, G.: Commuting Mappings and Fixed Points. Amer. Math. Monthly 83, 4 (1976), 261–263. | DOI | MR | Zbl

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

[18] Kannan, R.: Some results on fixed points III. Fund. Math. 70, 2 (1971), 169–177. | MR | Zbl

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

[20] Mann, W. R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 44 (1953), 506–510. | DOI | MR | Zbl

[21] Olatinwo, M. O.: Some stability and strong convergence results for the Jungck-Ishikawa iteration process. Creative Math. Inf. 17 (2008), 33–42. | MR | Zbl

[22] Popescu, O.: Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators. Mathematical Communications 12 (2007), 195–202. | MR | Zbl

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

[24] Rhoades, B. E.: Comments on two fixed point iteration methods. J. Math. Anal. Appl. 56, 2 (1976), 741–750. | DOI | MR | Zbl

[25] Rus, I. A.: Generalized Contractions and Applications. Cluj Univ. Press, Cluj Napoca, 2001. | MR | Zbl

[26] 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

[27] Shimizu, T., Takahashi, W.: Fixed point theorems in certain convex metric spaces. Math. Japon. 37 (1992), 855–859. | MR | Zbl

[28] Singh, S. L., Bhatnagar, C., Mishra, S. N.: Stability of Jungck-type iterative procedures. Internat. J. Math. & Math. Sc. 19 (2005), 3035–3043. | DOI | MR | Zbl

[29] Takahashi, W.: A convexity in metric spaces and nonexpansive mapping I. Kodai Math. Sem. Rep. 22 (1970), 142–149. | DOI | MR

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