zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the rate of convergence of Kirk-type iterative schemes. (English) Zbl 1325.47120
Summary: The purpose of this paper is to introduce Kirk-type new iterative schemes called Kirk-SP and Kirk-CR schemes and to study the convergence of these iterative schemes by employing certain quasi-contractive operators. By taking an example, we will compare Kirk-SP, Kirk-CR, Kirk-Mann, Kirk-Ishikawa, and Kirk-Noor iterative schemes for aforementioned class of operators. Also, using computer programs in C++, we compare the above-mentioned iterative schemes through examples of increasing, decreasing, sublinear, superlinear, and oscillatory functions.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
WorldCat.org
Full Text: DOI
References:
[1] W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506-510, 1953. · Zbl 0050.11603 · doi:10.2307/2032162
[2] W. A. Kirk, “On successive approximations for nonexpansive mappings in Banach spaces,” Glasgow Mathematical Journal, vol. 12, pp. 6-9, 1971. · Zbl 0223.47024 · doi:10.1017/S0017089500001063
[3] S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, pp. 147-150, 1974. · Zbl 0286.47036 · doi:10.2307/2039245
[4] M. A. Noor, “New approximation schemes for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217-229, 2000. · Zbl 0964.49007 · doi:10.1006/jmaa.2000.7042
[5] M. O. Olatinwo, “Some stability results for two hybrid fixed point iterative algorithms in normed linear space,” Matematichki Vesnik, vol. 61, no. 4, pp. 247-256, 2009. · Zbl 1268.47082
[6] R. Chugh and V. Kumar, “Stability of hybrid fixed point iterative algorithms of Kirk-Noor type in normed linear space for self and nonself operators,” International Journal of Contemporary Mathematical sciences, vol. 7, no. 24, pp. 1165-1184, 2012. · Zbl 1253.47043
[7] W. Phuengrattana and S. Suantai, “On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval,” Journal of Computational and Applied Mathematics, vol. 235, no. 9, pp. 3006-3014, 2011. · Zbl 1215.65095 · doi:10.1016/j.cam.2010.12.022
[8] R. Chugh, V. Kumar, and S. Kumar, “Strong Convergence of a new three step iterative scheme in Banach spaces,” (communicated). · Zbl 06297070
[9] T. Zamfirescu, “Fix point theorems in metric spaces,” Archiv der Mathematik, vol. 23, pp. 292-298, 1972. · Zbl 0239.54030 · doi:10.1007/BF01304884
[10] V. Berinde, “On the convergence of the Ishikawa iteration in the class of quasi contractive operators,” Acta Mathematica Universitatis Comenianae, vol. 73, no. 1, pp. 119-126, 2004. · Zbl 1100.47054 · emis:journals/AMUC/_vol-73/_no_1/_berinde/berinde.html · eudml:126710
[11] A. Rafiq, “On the convergence of the three-step iteration process in the class of quasi-contractive operators,” Acta Mathematica, vol. 22, no. 3, pp. 305-309, 2006. · Zbl 1120.47314 · eudml:53429
[12] V. Berinde, Iterative Approximation of Fixed Points, Lecture Notes in Mathematics, Springer, Berlin, Germany, 2007. · Zbl 1165.47047
[13] S. M. \cSolutz, “The equivalence of Picard, Mann and Ishikawa iterations dealing with quasi-contractive operators,” Mathematical Communications, vol. 10, no. 1, pp. 81-88, 2005. · Zbl 1089.47051
[14] S. M. \cSolutz, “The equivalence between Krasnoselskij, Mann, Ishikawa, Noor and multistep iterations,” Mathematical Communications, vol. 12, no. 1, pp. 53-61, 2007. · Zbl 1143.47043
[15] B. E. Rhoades and S. M. \cSolutz, “The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps,” Journal of Mathematical Analysis and Applications, vol. 289, no. 1, pp. 266-278, 2004. · Zbl 1053.47055 · doi:10.1016/j.jmaa.2003.09.057
[16] B. E. Rhoades and S. M. \cSoltuz, “The equivalence between Mann-Ishikawa iterations and multistep iteration,” Nonlinear Analysis, vol. 58, no. 1-2, pp. 219-228, 2004. · Zbl 1064.47070 · doi:10.1016/j.na.2003.11.013
[17] R. Chugh and V. Kumar, “Strong convergence of SP iterative scheme for quasi-contractive operators,” International Journal of Computer Applications, vol. 31, no. 5, pp. 21-27, 2011. · doi:10.5120/3820-5294
[18] B. E. Rhoades, “Comments on two fixed point iteration methods,” Journal of Mathematical Analysis and Applications, vol. 56, no. 3, pp. 741-750, 1976. · Zbl 0353.47029 · doi:10.1016/0022-247X(76)90038-X
[19] S. L. Singh, “A new approach in numerical praxis,” Progress of Mathematics, vol. 32, no. 2, pp. 75-89, 1998. · Zbl 1188.65077
[20] V. Berinde, “Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators,” Fixed Point Theory and Applications, no. 2, pp. 97-105, 2004. · Zbl 1090.47053 · doi:10.1155/S1687182004311058 · eudml:52445
[21] Y. Qing and B. E. Rhoades, “Comments on the rate of convergence between Mann and Ishikawa iterations applied to Zamfirescu operators,” Fixed Point Theory and Applications, vol. 2008, Article ID 387504, 3 pages, 2008. · Zbl 1203.47076 · doi:10.1155/2008/387504 · eudml:45257
[22] L. B. Ciric, B. S. Lee, and A. Rafiq, “Faster Noor iterations,” Indian Journal of Mathematics, vol. 52, no. 3, pp. 429-436, 2010. · Zbl 1219.47107
[23] N. Hussain, A. Rafiq, B. Damjanović, and R. Lazović, “On rate of convergence of various iterative schemes,” Fixed Point Theory and Applications, vol. 45, 6 pages, 2011. · Zbl 1315.47065