Austine Efut Ofem     (Department of Mathematics, University of Uyo, Uyo, Nigeria)
Unwana Effiong Udofia     (Department of Mathematics and Statistics, Akwa Ibom State University, Ikot Akpaden, Mkpatenin, Nigeria)
Donatus Ikechi Igbokwe     (Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Nigeria)


This paper presents a new iterative algorithm for approximating the fixed points of multivalued generalized \(\alpha\)–nonexpansive mappings. We study the stability result of our new iterative algorithm for a larger concept of stability known as weak \(w^2\)–stability. Weak and strong convergence results of the proposed iterative algorithm are also established. Furthermore, we show numerically that our new iterative algorithm outperforms several known iterative algorithms for multivalued generalized \(\alpha\)–nonexpansive mappings. Again, as an application, we use our proposed iterative algorithm to find the solution of nonlinear Volterra delay integro-differential equations. Finally, we provide an illustrative example to validate the mild conditions used in the result of the application part of this study. Our results improve, generalize and unify several results in the existing literature.


Banach space, Uniformly convex Banach space, Multivalued generalized \(\alpha\)-nonexpansive mapping, Convergence, Nonlinear Volterra delay integro-differential equations.

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  1. Abbas M., Nazir T. A new faster iteration process applied to constrained minimization and feasibility problems. Mat. Vesnik, 2014. Vol. 66, No. 2. P. 223–234. URL: http://hdl.handle.net/2263/43663 
  2. Abkar A., Eslamian M. A fixed point theorem for generalized nonexpansive multivalued mappings. Fixed Point Theory, 2011. Vol. 12, No. 2. P. 241–246. 
  3. Agarwal R.P., O’Regan D., Sahu D.R. Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal., 2007. Vol. 8, No. 1. P. 61–79. 
  4. Ali F., Ali J, Nieto J.J. Some observations on generalized non-expansive mappings with an application. Comp. Appl. Math., 2020. Vol. 39. Art. no. 74. P. 1–20. DOI: 10.1007/s40314-020-1101-4
  5. Aoyama K., Kohsaka F. Fixed point theorem for α-nonexpansive mappings in Banach spaces. Nonlinear Anal., 2011. Vol. 74, No. 13. P. 4387–4391. DOI: 10.1016/j.na.2011.03.057 
  6. Berinde V. Iterative Approximation of Fixed Points. Berlin, Heidelberg: Springer, 2007. 326 p. DOI: 10.1007/978-3-540-72234-2
  7. Browder F.E. Nonexpansive nonlinear operators in a Banach space. Proc. Nat. Acad. Sci. USA., 1965. Vol. 54, No. 4. P. 1041–1044. DOI: 10.1073/pnas.54.4.1041
  8. Cardinali T., Rubbioni P. A generalization of the Caristi fixed point theorem in metric spaces. Fixed Point Theory, 2010. Vol. 11, No. 1. P. 3–10. 
  9. Garodia C., Uddin I. A new fixed point algorithm for finding the solution of a delay differential equation. AIMS Mathematics, 2020. Vol. 5, No. 4. P. 3182–3200. DOI: 10.3934/math.2020205 
  10. Göhde D. Zum Prinzip der kontraktiven Abbildung. Math. Nachr., 1965. Vol. 30, No. 3–4. P. 251–258. (in German) DOI: 10.1002/mana.19650300312 
  11. Gunduz B., Alagoz O., Akbulut S. Convergence theorems of a faster iteration process including multi-valued mappings with analytical and numerical examples. Filomat, 2018. Vol. 32, No. 16. P. 5665–5677. DOI: 10.2298/FIL1816665G
  12. Gürsoy F., Karakaya V. A Picard–S Hybrid Type Iteration Method for Solving a Differential Equation with Retarded Argument. 2014. 16 p. arXiv:1403.2546v2 [math.FA]
  13. Harder A.M. Fixed Point Theory and Stability Results for Fixed Point Iteration Procedures. Ph.D. thesis. Missouri: University of Missouri-Rolla, 1987. 70 p. 
  14. Harder A.M., Hicks T.L. A stable iteration procedure for nonexpansive mappings. Math. Japonica, 1988. Vol. 33, No. 5. P. 687–692. 
  15. Iqbal H., Abbas M., Husnine S.M. Existence and approximation of fixed points of multivalued generalized α-nonexpansive mappings in Banach spaces. Numer. Algor., 2020. Vol. 85. P. 1029–1049. DOI: 10.1007/s11075-019-00854-z 
  16. Ishikawa S. Fixed points by a new iteration method. Proc. Amer. Math. Soc., 1994. Vol. 44. P. 147–150. DOI: 10.2307/2039245
  17. Kirk W.A. A fixed point theorem for mappings which do not increase distance. Amer. Math. Monthly, 1965. Vol. 72, No. 9. P. 1004–1006. DOI: 10.2307/2313345
  18. Kucche K.D., Shikhare P.U. Ulam Stabilities for nonlinear Volterra delay integro-differential equations. J. Contemp. Math. Anal., 2019. Vol. 54, No. 5. P. 276–287. DOI: 10.3103/S1068362319050042
  19. Mann W.R. Mean value methods in iteration. Proc. Amer. Math. Soc., 1953. Vol. 4. P. 506–510. DOI: 10.1090/S0002-9939-1953-0054846-3
  20. Markin J. A fixed point theorem for set valued mappings. Bull. Amer. Math. Soc., 1968. Vol. 74, No. 1. P. 639–640. 
  21. Nadler S.B. Multi-valued contraction mappings. Pacific J. Math., 1969. Vol. 30, No. 2. P. 475–488. DOI: 10.2140/pjm.1969.30.475 
  22. Noor M.A. New approximation schemes for general variational inequalities. J. Math. Anal. Appl., 2000. Vol. 251, No. 1. P. 217–229. DOI: 10.1006/jmaa.2000.7042 
  23. Ofem A.E., Igbokwe D.I. An efficient iterative method and its applications to a nonlinear integral equation and a delay differential equation in Banach spaces. Turkish J. Ineq., 2020. Vol. 4, No. 2. P. 79–107. 
  24. Ofem A.E., Udofia U.E., Igbokwe D.I. New iterative algorithm for solving constrained convex minimization problem and split feasibility problem. Eur. J. Math. Anal., 2021. Vol. 1, No. 2. P. 106–132. DOI: 10.28924/ada/ma.1.106
  25. Ofem A.E., Udofia U.E. Iterative solutions for common fixed points of nonexpansive mappings and strongly pseudocontractive mappings with applications. Canad. J. Appl. Math., 2021. Vol. 3, No. 1. P. 18–36. 
  26. Okeke G.A. Convergence analysis of the Picard–Ishikawa hybrid iterative process with applications. Afr. Mat., 2019. Vol. 30, No. 5–6. P. 817–835. DOI: 10.1007/s13370-019-00686-z
  27. Okeke G.A., Abbas M.A. A solution of delay differential equations via Picard–Krasnoselskii hybrid iterative process. Arab. J. Math., 2017. Vol. 6. P. 21–29. DOI: 10.1007/s40065-017-0162-8
  28. Okeke G.A., Abbas M.A., de la Sen M. Approximation of the fixed point of multivalued quasinonexpansive mappings via a faster iterative. Process with applications. Discrete Dyn. Nat. Soc., 2020. Vol. 2020. Art. no. 8634050. P. 1–11. DOI: 10.1155/2020/8634050 
  29. Pant D., Shukla R. Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces. Numer. Funct. Anal. Optim., 2017. Vol. 38, No. 2. P. 248–266. DOI: 10.1080/01630563.2016.1276075
  30. Schu J. Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc., 1991. Vol. 43, No. 1. P. 153–159. DOI: 10.1017/S0004972700028884
  31. Senter H.F., Dotson W.G. Approximating fixed points of nonexpansive mappings. Proc. Amer. Math. Soc., 1974. Vol. 44, No. 2. P. 375–380. DOI: 10.2307/2040440
  32. Song Y., Cho Y.J. Some notes on Ishikawa iteration for multivalued mappings. Bull. Korean Math. Soc., 2011. Vol. 48, No. 3. P. 575–584. DOI: 10.4134/BKMS.2011.48.3.575
  33. Suzuki T. Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl., 2008. Vol. 340, No. 2. P. 1088–10995. DOI: 10.1016/j.jmaa.2007.09.023
  34. Thakur B.S., Thakur D., Postolache M. A new iteration scheme for approximating fixed points of nonexpansive mappings. Filomat, 2016. Vol. 30, No. 10. P. 2711–2720. DOI: 10.2298/FIL1610711T
  35. Timis I. On the weak stability of Picard iteration for some contractive type mappings and coincidence theorems. Int. J. Comput. Appl., 2012. Vol. 37, No. 4. P. 9–13. 
  36. Ullah K., Arshad M. Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process. Filomat, 2018. Vol. 32. P. 187–196. DOI: 10.2298/FIL1801187U
  37. Weng X. Fixed point iteration for local strictly pseudo-contractive mapping. Proc. Amer. Math. Soc., 1991. Vol. 113. P. 727–731. DOI: 10.1090/S0002-9939-1991-1086345-8

DOI: http://dx.doi.org/10.15826/umj.2021.2.005

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