A novel approach for convergence and stability of Jungck-Kirk-Type algorithms for common fixed point problems in Hilbert spaces

Document Type : Research Paper


Department of Mathematics, Micheal Okpara University of Agriculture, Umudike, Umuahia Abia State, Nigeria


In this paper, two novel iteration algorithms called Jungck-DI-Noor-multistep and Jungck-DI-SP-multistep iterative schemes are introduced and studied. Using their strong convergence, a common fixed point of nonself mappings was achieved without any imposition of 'sum conditions' on the control sequences. Further, we studied and proved the stability results of our new iterative schemes in the setting of a real Hilbert space. Our results improve, generalize and unify several known results currently in the literature.


[1] I.K. Agwu and D.I. Igbokwe, Convergence and stability of new approximation algorithms for certain contractivetype mappings, Eur. J. Math. Anal. 2 (2022), 1–1.
[2] I.K. Agwu and D.I. Igbokwe, Fixed points and stability of new approximation algorithms for contractive-type operators in Hilbert spaces, Int. J. Nonlinear Anal. Appl. In Press, doi: 10.22075/ijnaa.2021.22923.2432
[3] I.K. Agwu and D.I. Igbokwe, New iteration algorithm for equilibrium problems and fixed point problems of two finite families of asymptotically demicontractive multivalued mappings, In Press.
[4] H. Akewe and A. Mogbademu, Common fixed point of Jungck-Kirk-type iteration for nonself operators in normed linear spaces, Fasciculi Math. 2016 (2016), 29–41.
[5] H. Akewe and H. Olaoluwa, On the convergence of modified iteration process for generalise contractive-like operators, Bull. Math. Anal. Appl. 4 (2012), no. 3, 78–86.
[6] H. Akewe, G.A. Okeeke and A. Olayiwola, Strong convergence and stability of Kirk-multistep-type iterative schemes for contractive-type operators, Fixed Point Theory Appl. 2014 (2014), 45.
[7] H. Akewe, Approximation of fixed and common fixed points of generalised contractive-like operators, PhD Thesis, University of Lagos, Nigeria, 2010.
[8] V. Berinde, On the stability of some fixed point problems, Bull. Stint. Univ. Bala Mare, Ser. B Fasc. Mat-inform. XVIII(1) 14 (2002), 7–14.
[9] V. Berinde, Iterative approximation of fixed points for pseudo-contractive operators, Seminar on Fixed Point Theory, 2002.
[10] R. Chugh and V. Kummar, Stability of hybrid fixed point iterative algorithm of Kirk-Noor-type in nonlinear spaces for self and nonself operators, Int. J. Contemp. Math. Sci. 7 (2012), no. 24, 1165–1184.
[11] R. Chugh and V. Kummar, Strong convergence of SP iterative scheme for quasi-contractive operators, Int. J. Comput. Appl. 31 (2011), no. 5, 21–27.
[12] A.M. Harder and T.L. Hicks, Stability results for fixed point iterative procedures, Math. Jpn. 33 (1988), no. 5, 693–706.
[13] N. Hussain, R. Chugh, V. Kummar and A. Rafig, On the convergence of Kirk-type iterative schemes, J. Appl. Math. 2012 (2012), Article ID 526503, 22 pages.
[14] S. Ishikawa, Fixed points by a new iteration methods, Proc. Amer. Math. Soc. 44 (1974), 147–150.
[15] C.O. Imoru and M.O. Olatinwo, On the stability of Picard’s and Mann’s iteration, Carpath. J. Math. 19 (2003), 155–160.
[16] F.O. Isiogugu., C. Izuchukwu and C.C. Okeke, New iteration scheme for approximating a common fixed point of a finite family of mappings, J. Math. 2020 (2020), Article ID 3287968.
[17] G. Jungck, Commuting mappings and fixed points, Amer. Math. Month. 83 (1976), no. 4, 261–263.
[18] W.A. Kick, On successive approximations for nonexpansive mappings in Banach spaces, Glasg. Math. J. 12 (1971), 6–9.
[19] W.R. Mann, Mean value method in iteration, Proc. Amer. Math. Soc. 44 (2000), 506–510.
[20] M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), 217–229.
[21] M.O. Olatinwo, A generalization of some convergence results using a Jungck-Noor three-step iteration process in arbitrary Banach space, Fasciculi Math. 40 (2008), 37–43.
[22] J.O. Olaleru and H. Akewe, On the convergence of Jungck-type iterative schemes for generalized contractive-like operators, Fasciculi Math. 45 (2010), 87–98.
[23] M.O. Olatinwo, Stability results for Jungck-kirk-Mann and Jungck-kirk hybrid iterative algorithms, Anal. Theory Appl. 29 (2013), 12–20.
[24] M.O. Olatinwo, Convergence results for Jungck-type iterative process in convex metric spaces, Acta Univ. Palacki Olomue, Fac. Rer. Nat. Math. 51 (2012), 79–87.
[25] M.O. Olatinwo, Some stability and strong convergence results for the Jungck-Ishikawa iteration process, Creative Math. Inf. 17 (2008), 33–42.
[26] J.O. Olaeru and H. Akewe, An extension of Gregus fixed point theorem, Fixed Point Theory Appl. 2007 (2007), Article ID 78628.
[27] M.O. Olutinwo, Some stability results for two hybrid fixed point iterative algorithms in normed linear space, Mat. vesnik 61 (2009), no. 4, 247–256.
[28] M.O. Osilike and A. Udoemene, A short proof of stability resultsfor fixed point iteration procedures for a class of contractive-type mappings, Indian J. Pure Appl. Math. 30 (1999), 1229–1234.
[29] M.O. Osilike, Stability results for lshikawa fixed point iteration procedure, Indian J. Pure Appl. Math. 26 (1996), no. 10, 937–941.
[30] A.M. Ostrowski, The round off stability of iterations, Z. Angew Math. Mech. 47 (1967), 77–81.
[31] B.E. Rhoade, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure Appl. Math. 24 (1993), no. 11, 691-703.
[32] B.E. Rhoade, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure Appl. Math. 21 (1990), 1–9.
[33] A. Ratiq, A convergence theprem for Mann’s iteration procedure, Appl. Math. E-Note 6 (2006), 289–293.
[34] B.E. Rhoade, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 266 (1977), 257–290.
[35] B.E. Rhoade, Comments on two fixed point iteration methods, Trans. Am. Math. Soc. 56 (1976), 741–750.
[36] B.E. Rhoade, Fixed point iteration using infinite matrices, Trans. Am. Math. Soc. 196 (1974), 161–176.
[37] A. Ratiq, On the convergence of the three step iteration process in the class of quasi-contractive operators, Acta. Math. Acad. Paedagag Nayhazi 22 (2006), 300–309.
[38] S.L. Singh, C. Bhatnaga and S.N Mishra, Stability of Jungck-type iteration procedures, Int. J. Math. Math. Sci. 19 (2005), 3035–3043.
[39] T. Zamfirescu, Fixed point theorems in metric spaces, Arch. Math. 23 (1972), 292–298.
Volume 14, Issue 4
April 2023
Pages 297-312
  • Receive Date: 11 March 2021
  • Revise Date: 16 May 2021
  • Accept Date: 29 May 2021