On Modified Picard−S−AK Hybrid Iterative Algorithm for Approximating Fixed Point of Banach Contraction Map

  • Joshua Olilima Augustine University
  • Hudson Akewe Covenant University
  • Adefemi Adeniran Augustine University
Keywords: Contractive-Type Operators, AK and Picard-S Iterative Algorithm, Fixed Point, Convergence Result, T-Stable, Speed of Convergence, Data Dependence

Abstract

The purpose of this work is to introduce a new iteration called the modified Picard-S-AK hybrid iterative scheme for approximating fixed point for Banach contractive maps. We show that our scheme converges to a unique fixed point p at a rate faster than the recent AK iterative scheme for Banach contractive maps. Furthermore, using Java programming language, we give some numerical examples to justify our claim. Stability and data dependence of the proposed scheme are also explored.

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Author Biographies

Joshua Olilima, Augustine University

Department of Mathematical Sciences, Augustine University, Ilara-Epe, Lagos, Nigeria.

Hudson Akewe, Covenant University

Department of Mathematics, Covenant University, Cananland, KM 10 Idiroko Road, Ota, Ogun State, Nigeria.

Adefemi Adeniran, Augustine University

Department of Mathematical Sciences, Augustine University, Ilara-Epe, Lagos, Nigeria

References

W. A. Kirk, Some recent results in metric fixed point theory, J. Fixed Point Theory Appl, 2 (2007) 195-207. https://doi.org/10.1007/s11784-007-0031-8

M. Abbas, T. Nazir. A new faster iteration process applied to constrained minimization and feasibility problems, Mat Vesn 66 (2014) 223-234.

R. P. Agarwal, D. O. Regan, D. R. Sahu. Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal 8(2007) 61-79.

H. Akewe, G. A. Okeke. Convergence and stability theorems for the Picard-Mann hybrid iterative scheme for a general class of contractive-like operators, Fixed Point Theory and Applications, (2015) 2015:66.

H. Akewe, G. A. Okeke. and A. Olayiwola. Strong convergence and stability of Kirk-multistep-type iterative schemes for contractive-type operators. Fixed Point Theory Appications, (2014), 45 (2014), 24 pages.

S. Banach. Sur les op´erations dans les ensembles abstraits et leur application aux ´equations int´egrales, Fund. Math. 3 (1922), 133-181 (French).

V. Berinde. On the stability of some fixed point procedures, Buletinul Stiintic al Universitatii din Baia Mare. Seria B. Fascicola Mathematica-Informatica, vol. XVIII (1) (2002), 7-14.

V. Berinde. Iterative Approximation of Fixed Points, Springer, Berlin (2007).

F. Gursoy, V. Karakaya. A Picard-S hybrid type iteration method for solving a differential equation with retarded argument. (2014) arXiv:1403.2546v2.

A. M. Harder, T. L. Hicks. Stability results for fixed point iteration procedures, Math. Japonica, 33 (5) (1988), 693-706.

C. O. Imoru, M.O. Olatinwo. On the stability of Picard and Mann iteration, Carpath. J. Math. 19(2003), 155-160.

S. Ishikawa. Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147-150.

I. Karahan, M. Ozdemir. A general iterative method for approximation of fixed points and their applications. Adv Fixed Point Theory 3(2013) 510-526.

V. Karakaya, N. E. H. Bouzara, K. Dogan, Y. Atalan. On different results for a new two-step iteration method under weak-contraction mapping in Banach spaces. arXiv:1507.00200v1

S.H. Khan. A Picard-Mann hybrid iterative process. Fixed Point Theory Appl, (2013) doi:10.1186/1687-1812- 2013-69

W. R. Mann. Mean Value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.

M. A. Noor. New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251(2000), 217-229.

M. O. Olatinwo. Stability results for Jungck-kirk-Mann and Jungck-kirk hybrid iterative algorithms, Anal. Theory Appl., 29 (2013), 12-20.

M. O. Olatinwo, Some Stability and Strong Convergence Results for the Jungck-Ishikawa Iteration Process. Creative Math. and Info., 17 (2008) 33-42.

M. O. Olatinwo, Some stability results for two hybrid fixed point iterative algorithms of Kirk-Ishikawa and Kirk-Mann type, Journal of Advanced Mathematical Studies, vol. 1 (2008), No. 1-2, 87-96.

M. O. Osilike. Some Stability results for fixed point iteration procedures, J. Nigeria Math. Soc., 14/15(1995) 17-29.

M. O. Osilike, Stability of the Mann and Ishikawa Iteration Procedures for φ-Strong Pseudocontractions and Nonlinear Equations of the φ-Strongly Accretive Type, J. of Mathematical Anal. & App., 227 (1998), 319-334, Article No. AY986075

M. O. Osilike, A. Udomene. Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings, Indian J. Pure Appl. Math., 30 (12) (1999), 1229-1234.

A. M. Ostrowski. The round-o stability of iterations, Z. Angew. Math. Mech., 47 (1967), 77-81.

B. E. Rhoades. Fixed Point Theorems and Stability Results for Fixed Point Iteration Procedures, Indian J. Pure Appl. Math. 21 (1990), No. 1, 1-9

B. E. Rhoades. Fixed Point Theorems and Stability Results for Fixed Point Iteration Procedures II, Indian J. Pure Appl. Math. 24 (1993), No. 11, 691-703

S. M. Soltuz, T. Grosan. Data dependence for Ishikawa iteration when dealing with contractive like operators. Fixed Point Theory and Applications 2008: 242916(1-7).

B.S. Thakur, D. Thakur, M. Postolache. A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings. Appl Math Comput 275 (2016) 147-155.

K. Ullah, M. Arshad. On different results for new three step iteration process in Banach spaces, Springerplus (2016) 5:1616 DOI 10.1186/s40064-016-3056-x

X. Weng. Fixed point iteration for local strictly pseudocontractive mapping. Proc. Amer. Math. Soc. 113 (1991) 727-731.

Published
2019-12-18
How to Cite
Olilima, J., Akewe, H., & Adeniran, A. (2019). On Modified Picard−S−AK Hybrid Iterative Algorithm for Approximating Fixed Point of Banach Contraction Map. MathLAB Journal, 4, 111-125. Retrieved from https://purkh.com/index.php/mathlab/article/view/599
Section
Research Articles