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Iterative algorithms of generalized nonexpansive mappings and monotone operators with application to convex minimization problem

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Abstract

In this paper, we present some convergence results for various iterative algorithms built from Hardy-Rogers type generalized nonexpansive mappings and monotone operators in Hilbert spaces. We obtain some comparison results for the rates of convergence of these algorithms to the solution of variational inequality problem including Hardy–Rogers type generalized nonexpansive mappings and monotone operators. A numerical example is given to validate these results. We apply the iterative algorithms handled herein to solve convex minimization problem and illustrate this result by providing a non-trivial numerical example. The presented results considerably improve the corresponding results in Ali et al. (Comput Appl Math 39:74, 2020. https://doi.org/10.1007/s40314-020-1101-4).

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References

  1. Abbas, M., Gürsoy, F., Ibrahim, Y., Khan, A.R.: Split variational inclusions for Bregman multivalued maximal monotone operators. RAIRO-Oper. Res. 55, S2417–S2431 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. Ali, F., Ali, J., Nieto, J.J.: Some observations on generalized non-expansive mappings with an application. Comp. Appl. Math. 39, 74 (2020). https://doi.org/10.1007/s40314-020-1101-4

    Article  MathSciNet  MATH  Google Scholar 

  4. Ali, J., Ali, F.: Approximation of common fixed points and the solution of image recovery problem. Results Math. 74, 130 (2019). https://doi.org/10.1007/s00025-019-1053-4

    Article  MathSciNet  MATH  Google Scholar 

  5. Atalan, Y.: On a new fixed point iterative algorithm for general variational inequalities. J. Nonlinear Convex Anal. 2011, 2371–2386 (2019)

    MathSciNet  MATH  Google Scholar 

  6. Ansari, Q.H., Balooee, J., Dogan, K.: Iterative schemes for solving regularized nonconvex mixed equilibrium problems. J. Nonlinear Convex Anal. 184, 607–622 (2017)

    MathSciNet  MATH  Google Scholar 

  7. Berinde, V.: Iterative Approximation of Fixed Points, 2nd edn. Springer-Verlag, Berlin (2007)

    MATH  Google Scholar 

  8. Berinde, V.: On a notion of rapidity of convergence used in the study of fixed point iterative methods. Creat. Math. Inform. 25, 29–40 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chatterjea, S.K.: Fixed point theorems. C. R. Acad. Bulgare Sci. 25, 727–730 (1972)

    MathSciNet  MATH  Google Scholar 

  10. Ćirić, L.B.: Generalized contractions and fixed-point theorems. Publ. Inst. Math. (Beograd) (N.S.) 12, 19–26 (1971)

    MathSciNet  MATH  Google Scholar 

  11. Dixit, A., Sahu, D.R., Singh, A.K., Som, T.: Application of a new accelerated algorithm to regression problems. Soft Comput. 24, 1539–1552 (2020)

    Article  MATH  Google Scholar 

  12. Ertürk, M., Gürsoy, F., Ansari, Q.H., Karakaya, V.: Picard type iterative method with applications to minimization problems and split feasibility problems. J. Nonlinear Convex Anal. 21, 943–951 (2020)

    MathSciNet  MATH  Google Scholar 

  13. Ertürk, M., Khan, A.R., Karakaya, V., Gürsoy, F.: Convergence and data dependence results for hemicontractive operators. J. Nonlinear Convex Anal. 18, 697–708 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Ertürk, M., Gürsoy, F., Şimşek, N.: \(S-\)iterative algorithm for solving variational inequalities. Int. J. Comput. Math. 98, 435–448 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  15. Garodia, C., Uddin, I.: A new fixed point algorithm for finding the solution of a delay differential equation. AIMS Math. 54, 3182–3200 (2020)

    Article  MathSciNet  Google Scholar 

  16. Garodia, C., Uddin, I., Khan, S.H.: Approximating common fixed points by a new faster iteration process. Filomat 346, 2047–2060 (2020)

    Article  MathSciNet  Google Scholar 

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

  18. Gürsoy, F.: A Picard-S iterative method for approximating fixed point of weak-contraction mappings. Filomat 30, 2829–2845 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gürsoy, F., Khan, A.R., Ertürk, M., Karakaya, V.: Convergence and data dependency of normal-\(S\) iterative method for discontinuous operators on Banach space. Numer. Funct. Anal. Optim. 39, 322–345 (2018)

    Article  MathSciNet  Google Scholar 

  20. Gürsoy, F., Sahu, D.R., Ansari, Q.H.: S iteration process for variational inclusions and its rate of convergence. J. Nonlinear Convex Anal. 17, 1753–1767 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Gürsoy, F., Ertürk, M., Abbas, M.: A Picard-type iterative algorithm for general variational inequalities and nonexpansive mappings. Numer. Algor. 83, 867–883 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hardy, G.E., Rogers, T.D.: A generalization of a fixed point theorem of Reich. Can. Math. Bull. 16, 201–206 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hishinuma, K., Iiduka, H.: Fixed point quasiconvex subgradient method. Eur. J. Oper. Res. 282, 428–437 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ishikawa, S.: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147–150 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  25. Iqbal, H., Abbas, M., Husnine, S.M.: Existence and approximation of fixed points of multivalued generalized \(\alpha \)-nonexpansive mappings in Banach spaces. Numer. Algorithms 85, 1029–1049 (2020)

    Article  MathSciNet  Google Scholar 

  26. Kannan, R.: Some results on fixed point. Bull. Calcutta Math. Soc. 60, 71–76 (1968)

    MathSciNet  MATH  Google Scholar 

  27. Karakaya, V., Gürsoy, F., Ertürk, M.: Some convergence and data dependence results for various fixed point iterative methods. Kuwait J. Sci. 43, 112–128 (2016)

    MathSciNet  MATH  Google Scholar 

  28. Karakaya, V., Dogan, K., Atalan, Y., Bouzara, N.E.H.: The local and semilocal convergence analysis of new Newton-like iteration methods. Turk. J. Math. 423, 735–751 (2018)

    MathSciNet  MATH  Google Scholar 

  29. Khan, A.R., Gürsoy, F., Karakaya, V.: Jungck Khan iterative scheme and higher convergence rate. Int. J. Comput. Math. 93, 2092–2105 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Khan, A.R., Gürsoy, F., Kumar, V.: Stability and data dependence results for Jungck Khan iterative scheme. Turk. J. Math. 40, 631–640 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Khatoon, S., Uddin, I., Baleanu, D.: Approximation of fixed point and its application to fractional differential equation. J. Appl. Math. Comput. 1–19 (2020)

  32. Kumam, W., Khammahawong, K., Kumam, P.: Error estimate of data dependence for discontinuous operators by new iteration process with convergence analysis. Numer. Funct. Anal. Optim. 40, 1644–1677 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. La Cruz, W.: A residual algorithm for finding a fixed point of a nonexpansive mapping. J. Fixed Point Theory Appl. 20, 116 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Noor, M.A., Huang, Z.: Three-step methods for nonexpansive mappings and variational inequalities. Appl. Math. Comput. 187, 680–685 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Reich, S.: Kannan’s fixed point theorem. Boll. Un. Mat. Ital. 4, 1–11 (1971)

    MathSciNet  MATH  Google Scholar 

  38. Rus, I.A.: Some fixed point theorems in metric spaces. Rend. Ist. Mat. Univ. Trieste 3, 169–172 (1972)

    MathSciNet  MATH  Google Scholar 

  39. Suparatulatorn, R., Cholamjiak, W., Suantai, S.: A modified S-iteration process for G-nonexpansive mappings in Banach spaces with graphs. Numer. Algorithms 77, 479–490 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  40. Stampacchia, G.: Formes bilineaires coercivities sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413–4416 (1964)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. Uddin, I., Khatoon, S., Mlaiki, N., Abdeljawad, T.: A modified iteration for total asymptotically nonexpansive mappings in Hadamard spaces. AIMS Math. 65, 4758–4770 (2021)

    Article  MathSciNet  Google Scholar 

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  1. Department of Mathematics, Aksaray University, 68100, Aksaray, Turkey

    Samet Maldar

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Maldar, S. Iterative algorithms of generalized nonexpansive mappings and monotone operators with application to convex minimization problem. J. Appl. Math. Comput. 68, 1841–1868 (2022). https://doi.org/10.1007/s12190-021-01593-y

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  • DOI: https://doi.org/10.1007/s12190-021-01593-y

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