Title:
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Stability of Noor Iteration for a General Class of Functions in Banach Spaces (English) |
Author:
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Bosede, Alfred Olufemi |
Language:
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English |
Journal:
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Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
ISSN:
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0231-9721 |
Volume:
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51 |
Issue:
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2 |
Year:
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2012 |
Pages:
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19-25 |
Summary lang:
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English |
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Category:
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math |
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Summary:
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In this paper, we prove the stability of Noor iteration considered in Banach spaces by employing the notion of a general class of functions introduced by Bosede and Rhoades [6]. We also establish similar result on Ishikawa iteration as a special case. Our results improve and unify some of the known stability results in literature. (English) |
Keyword:
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stability |
Keyword:
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Noor and Ishikawa iterations |
MSC:
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47H10 |
MSC:
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47J25 |
MSC:
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54H25 |
idZBL:
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Zbl 06204927 |
idMR:
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MR3058870 |
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Date available:
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2012-11-26T10:14:26Z |
Last updated:
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2014-03-12 |
Stable URL:
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http://hdl.handle.net/10338.dmlcz/143064 |
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Reference:
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[1] Agarwal, R. P., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge, 2001. Zbl 0960.54027, MR 1825411 |
Reference:
|
[2] Berinde, V.: Iterative Approximation of Fixed Points. Editura Efemeride, Baia Mare, 2002. Zbl 1036.47037, MR 1995230 |
Reference:
|
[3] Bosede, A. O.: Noor iterations associated with Zamfirescu mappings in uniformly convex Banach spaces. Fasciculi Mathematici 42 (2009), 29–38. Zbl 1178.47042, MR 2573523 |
Reference:
|
[4] Bosede, A. O.: Some common fixed point theorems in normed linear spaces. Acta Univ. Palacki. Olomuc., Fac. rer. nat., Math. 49, 1 (2010), 19–26. MR 2797519 |
Reference:
|
[5] Bosede, A. O.: Strong convergence results for the Jungck–Ishikawa and Jungck-Mann iteration processes. Bulletin of Mathematical Analysis and Applications 2, 3 (2010), 65–73. MR 2718198 |
Reference:
|
[6] Bosede, A. O., Rhoades, B. E.: Stability of Picard and Mann iterations for a general class of functions. Journal of Advanced Mathematical Studies 3, 2 (2010), 23–25. MR 2722440 |
Reference:
|
[7] Bosede, A. O., Akinbo, G.: Some stability theorems associated with $A$-distance and $E$-distance in uniform spaces. Acta Universitatis Apulensis 26 (2011), 121–128. Zbl 1274.54117, MR 2850603 |
Reference:
|
[8] Ciric, L. B.: Fixed point theorems in Banach spaces. Publ. Inst. Math. (Beograd) 47, 61 (1990), 85–87. Zbl 0722.47046, MR 1103533 |
Reference:
|
[9] Ciric, L. B.: Fixed Point Theory. Contraction Mapping Principle. FME Press, Beograd, 2003. |
Reference:
|
[10] Harder, A. M., Hicks, T. L.: Stability results for fixed point iteration procedures. Math. Japonica 33 (1988), 693–706. Zbl 0655.47045, MR 0972379 |
Reference:
|
[11] Imoru, C. O., Akinbo, G., Bosede, A. O.: On the fixed points for weak compatible type and parametrically $\varphi (\epsilon , \delta ; a)$-contraction mappings. Math. Sci. Res. Journal 10, 10 (2006), 259–267. MR 2277967 |
Reference:
|
[12] Imoru, C. O., Olatinwo, M. O.: On the stability of Picard and Mann iteration processes. Carpathian J. Math. 19, 2 (2003), 155–160. Zbl 1086.47512, MR 2069844 |
Reference:
|
[13] Imoru, C. O., Olatinwo, M. O., Akinbo, G., Bosede, A. O.: On a version of the Banach’s fixed point theorem. General Mathematics 16, 1 (2008), 25–32. Zbl 1235.54038, MR 2438519 |
Reference:
|
[14] Ishikawa, S.: Fixed points by a new iteration method. Proc. Amer. Math. Soc. 44 (1974), 147–150. Zbl 0286.47036, MR 0336469, 10.1090/S0002-9939-1974-0336469-5 |
Reference:
|
[15] Mann, W. R.: Mean value methods in iterations. Proc. Amer. Math. Soc. 4 (1953), 506–510. MR 0054846, 10.1090/S0002-9939-1953-0054846-3 |
Reference:
|
[16] Noor, M. A.: General variational inequalities. Appl. Math. Letters 1 (1988), 119–121. Zbl 0655.49005, MR 0953368, 10.1016/0893-9659(88)90054-7 |
Reference:
|
[17] Noor, M. A.: New approximations schemes for general variational inequalities. J. Math. Anal. Appl. 251 (2000), 217–299. MR 1790406, 10.1006/jmaa.2000.7042 |
Reference:
|
[18] Noor, M. A.: Some new developments in general variational inequalities. Appl. Math. Computation 152 (2004), 199–277. MR 2050063 |
Reference:
|
[19] Noor, M. A., Noor, K. I., Rassias, T. M.: Some aspects of variational inequalities. J. Comput. Appl. Math. 47 (1993), 493–512. Zbl 0788.65074, MR 1251844 |
Reference:
|
[20] Osilike, M. O.: Stability results for fixed point iteration procedures. J. Nigerian Math. Soc. 14/15 (1995/1996), 17–29. MR 1775011 |
Reference:
|
[21] Rhoades, B. E.: Fixed point iteration using infinite matrices. Trans. Amer. Math. Soc. 196 (1974), 161–176. Zbl 0267.47032, MR 0348565, 10.1090/S0002-9947-1974-0348565-1 |
Reference:
|
[22] Rhoades, B. E.: Comments on two fixed point iteration methods. J. Math. Anal. Appl. 56, 2 (1976), 741–750. Zbl 0353.47029, MR 0430880, 10.1016/0022-247X(76)90038-X |
Reference:
|
[23] Rhoades, B. E.: Fixed point theorems and stability results for fixed point iteration procedures. Indian J. Pure Appl. Math. 21, 1 (1990), 1–9. Zbl 0692.54027, MR 1048010 |
Reference:
|
[24] Rus, I. A., Petrusel, A., Petrusel, G.: Fixed Point Theory, 1950–2000, Romanian Contributions. House of the Book of Science, Cluj-Napoca, 2002. Zbl 1005.54037, MR 1947195 |
Reference:
|
[25] Zamfirescu, T.: Fix point theorems in metric spaces. Arch. Math. 23 (1972), 292–298. Zbl 0239.54030, MR 0310859, 10.1007/BF01304884 |
Reference:
|
[26] Zeidler, E.: Nonlinear Functional Analysis and its Applications: Fixed Point Theorems. Springer, New York, 1986. MR 0816732 |
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