Investigation of some fixed point theorems in hyperbolic spaces for a three step iteration process
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Abstract
In the present paper, we investigate the convergence, equivalence of convergence, rate of convergence and data dependence results using a three step iteration process for mappings satisfying certain contractive condition in hyperbolic spaces. Also we give non-trivial examples for the rate of convergence and data dependence results to show effciency of three step iteration process. The results obtained in this paper may be interpreted as a refinement and improvement of the previously known results.
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[1] R. Agarwal, D. O'Regan and D.Sahu, Iterative Construction of Fixed Points of Nearly Asymptotically Nonexpansive Mappings, J. Nonlinear Convex Anal. 8 (2007), 61-79. Google Scholar
[2] M.R. Alfuraidan and M.A. Khamsi, Fixed points of monotone nonexpansive mappings on a hyperbolic metric space with a graph, Fixed Point Theory Appl. 44 (2015), 1-10. Google Scholar
[3] V. Berinde, On The Stability of Some Fixed Point Procedures, Bul. Stiintc.Univ. Baia Mare, Ser. B, Mat.-Inform. 18 (2002), 7-14. Google Scholar
[4] V. Berinde, On The Convergence of The Ishikawa Iteration in The Class of Quasi Contractive Operators, Acta Math. Univ. Comen. 73 (2004), 119-126. Google Scholar
[5] A.O.Bosede and B.E. Rhoades, Stability of Picard and Mann iteration for a General Class of Functions, J. Adv. Math. Stud. 3 (2010), 23-26. Google Scholar
[6] L. Chen, Y. Cui, H. Hudzik and R. Kaczmarek, Ellipsoidal geometry of Banach spaces and applications, J. Nonlinear Convex Anal. 18 (2017), 279-308. Google Scholar
[7] R. Chugh and V. Kumar, Data Dependence of Noor and SP iterative Schemes When Dealing with Quasi-Contractive Operators, Int. J. Comput. Appl. 40 (2011), 41-46. Google Scholar
[8] Y. Cui, H. Hudzik, R. Kaczmarek, H. Ma, Y. Wang and M. Zhang, On Some Applications of Geometry of Banach Spaces and Some New Results Related to the Fixed Point Theory in Orlicz Sequence Spaces, J. Math. Study, 49 (2016) 325-378. Google Scholar
[9] S. Dilworth, D. Kutzarova, G. Lancien and N. Randrianarivony, Asymptotic geometry of Banach spaces and uniform quotient maps, Proc. Amer. Math. Soc., 142 (2014), 2747-2762. Google Scholar
[10] H. Fukhar-ud-din, One Step Iterative Scheme for a Pair of Nonexpansive Mappings in a Convex Metric Space, Hacet. J. Math. Stat. 44 (2015), 1023-1031. Google Scholar
[11] H. Fukhar-ud-din and V. Berinde, Iterative Methods for the Class of Quasi-Contractive Type Operators and Comparsion of Their Rate of Convergence in Convex Metric Spaces, Filomat, 30 (2016), 223-230. Google Scholar
[12] H. Fukhar-ud-din and M. Khan, Convergence Analysis of a General Iteration Schema of Non-Linear Mappings in Hyperbolic Spaces, Fixed Point Theory Appl. 2013 (2013), 1-18. Google Scholar
[13] G. Godefroy, G. Lancien and V. Zizler, The non-linear geometry of Banach spaces after Nigel Kalton, Rocky Mountain J. Math. 44 (2014), 1529-1583. Google Scholar
[14] K. Goebel and W.A.Kirk, Iteration Processes for Nonexpansive Mappings, Topol. Methods Nonlinear Anal. 21 (1983) 115-123. Google Scholar
[15] F. Gursoy, V. Karakaya and B.E. Rhoades, Data Dependence Results of New Multi-step and S-Iterative Schemes for Contractive-like Operators, Fixed Point Theory Appl. 2013 (2013), 1-12. Google Scholar
[16] F. Gursoy, A.R. Khan and H. Fukhar-ud-din, Convergence and data dependence results for quasi-contractive type operators in hyperbolic spaces, Hacet. J. Math. Stat. 46 (2017), 377-388. Google Scholar
[17] F. Gursoy, A Picard-S Iterative Method for Approximating Fixed Point of Weak- Contraction Mappings, Filomat, 30 (2016), 2829-2845. Google Scholar
[18] E. Hacioglu and V. Karakaya, Existence and convergence for a new multivalued hybrid mapping in CAT(k) spaces, Carpathian J. Math. 33 (2017), 319-326. Google Scholar
[19] E. Hacioglu and V. Karakaya, Some Fixed Point Results for A Multivalued Generalization of Generalized Hybrid Mappings in CAT(k)-Spaces, Konuralp J. Math. 6 (2018), 26-34. Google Scholar
[20] A.M. Harder and T.L. Hicks, Stability Results for Fixed Point Iteration Procedures, Math. Japonica, 33 (1988), 693-706. Google Scholar
[21] H. Hudzik, V. Karakaya, M. Mursaleen and N. Simsek, Banach-Saks Type and Gurarii Modulus of Convexity of Some Banach Sequence Spaces, Abstr. Appl. Anal. 2014 (2014) 1-9. Google Scholar
[22] C.O.Imoru and M.O.Olantiwo, On The Stability of Picard and Mann Iteration Processes, Carpath. J. Math. 19 (2003), 155-160. Google Scholar
[23] Z.Z. Jamil and B.A. Ahmed, Convergence and Data Dependence Result for Picard S-Iterative Scheme Using Contractive-Like operators, Amer. Rev. Math. Stat. 3 (2015), 83-86. Google Scholar
[24] V.Karakaya, Y. Atalan, K. Dogan and NEH. Bouzara, Some Fixed Point Results for a New Three Steps Iteration Process in Banach Spaces, Fixed Point Theory, 18 (2017), 625-640. Google Scholar
[25] V. Karakaya, F. Gursoy, K. Dogan and M.Erturk, Data Dependence Results for Multistep and CR Iterative Schemes in the Class of Contractive-Like Operators, Abstr. Appl. Anal. 2013 (2013) 1-7. Google Scholar
[26] S.H. Khan, Fixed Points of Contractive-Like Operators by a Faster Iterative Process, Int. J. Math. Comput. Sci. Eng. 7 (2013), 57-59. Google Scholar
[27] A.R. Khan, H. Fukhar-ud-din and M.A.A. Khan, An Implicit Algorithm for Two Finite Families of Nonexpansive Maps in Hyperbolic spaces, Fixed Point Theory Appl. 54 (2012), 1-12. Google Scholar
[28] JK. Kim, KS. Kim and YM. Nam, Convergence of Stability of Iterative Processes for a Pair of Simultaneously Asymptotically Quasi-Nonexpansive Type Mappings in Convex Metric Spaces, J. Comput. Anal. Appl. 9 (2007), 159-172. Google Scholar
[29] JK. Kim, SA. Chun and YM. Nam, Convergence Theorems of Iterative Sequences for Generalized pp-Quasicontractive Mappings in pp-Convex Metric Spaces, J. Comput. Anal. Appl. 10 (2008), 147-162. Google Scholar
[30] U. Kohlenbach, Some Logical Metatheorems with Applications in Functional Analysis, Trans. Amer. Math. Soc. 357 (2004), 89-128. Google Scholar
[31] J.O. Olaleru and H. Akewe, On Multistep Iterative Scheme for Approximating the Common Fixed Points of Contractive-like Operators, Int. J. Math. Math. Sci. 2010 (2010) 1-11. Google Scholar
[32] M.O. Olatinwo, O.O. Owojori and C.O.Imoru, On Some Stability Results for Fixed Point Iteration Procedure, J. Math. Stat. Sci. 2 (2006), 339-342. Google Scholar
[33] M.O. Olatinwo, Some Stability Results for Nonexpansive and Quasi-Nonexpansive Operators in Uniformly Convex Banach Space Using Two New Iterative Processes of Kirk-Type, Fasc. Math. 43 (2010) 101-114. Google Scholar
[34] M.O. Osilike and 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 (1999), 1229-1234. Google Scholar
[35] W. Pheungrattana and S. Suantai, On the Rate of Convergence of Mann,Ishikawa, Noor and SP Iterations for Continuous on an Arbitrary Interval, J. Comput. Appl. Math. 235 (2011), 3006-3914. Google Scholar
[36] W. Phuengrattana and S. Suantai, Comparison of The Rate of Convergence of Various Iterative Methods for the Class of Weak Contractions in Banach Spaces, Thai J. Math. 11 (2012), 217-226. Google Scholar
[37] S. Reich and I. Shafrir, Nonexpansive Iterations in Hyperbolic Spaces, Nonlinear Anal. Theory Methods Appl. 15 (1990), 37-558. Google Scholar
[38] B. Rhoades and S.M. Soltuz, The Equivalence Between Mann-Ishikawa Iterations and Multistep Iteration, Nonlinear Anal. 58 (2004), 219-228. Google Scholar
[39] S.M. Soltuz and T. Grosan, Data Dependence for Ishikawa Iteration When Dealing with Contractive-Like Operators, Fixed Point Theory Appl. 2008 (2008) 1-7. Google Scholar
[40] W.A.Takahashi, Convexity in Metric Space and Nonexpansive Mappings, Kodai Math. Sem Rep. 22 (1970), 142-149. Google Scholar
[41] X. Weng, Fixed Point Iteration for Local Strictly Pseudo-Contractive Mapping, Proc. Amer. Math. Soc. 113 (1991), 727-731. Google Scholar
[42] Q. I. N. Xiaolong and S. Y. Cho, Convergence analysis of a monotone projection algorithm in reflexive Banach spaces, Acta Math. Sci. 37 (2017), 488-502. Google Scholar
[43] T. Zamrescu, Fix Point Theorems in Metric Spaces, Arch. Math. 23 (1972), 292-298. Google Scholar