Skip to main content
Log in

System of generalized nonlinear variational-like inclusions and fixed point problems: graph convergence with an application

  • Published:
Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Abstract

In this paper, we are interested in the investigation of the problem of finding a common element of the set of fixed points of a total uniformly L-Lipschitzian mapping and the set of solutions of a system of generalized nonlinear variational-like inclusions. To approximate such a point lying in the above two sets, by employing some total uniformly L-Lipschitzian mappings and the notion of P-\(\eta \)-proximal-point mapping, a new iterative scheme is constructed. We apply the notions of graph convergence and P-\(\eta \)-proximal-point mapping and establish a new equivalence relationship between graph convergence and proximal-point mappings convergence of a sequence of P-\(\eta \)-accretive mappings. As an application of the obtained equivalence relationship, the strong convergence and stability of the sequence generated by our proposed iterative algorithm to a common point of the above-mentioned two sets are proved. The final section is devoted to the investigation and analysis of the results given in [Nonlinear Anal. 67(2007), 917–929]. Some comments relating to them are pointed out. These results are new, and improve and generalize many known corresponding results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Agarwal, R.P., Huang, N.J., Cho, Y.J.: Generalized nonlinear mixed implicit quasi-variational inclusions with set-valued mappings. J. Inequal. Appl. 7, 807–828 (2002)

    MathSciNet  Google Scholar 

  2. Alakoya, T.O., Mewomo, O.T.: Viscosity \(S\)-iteration method with inertial technique and self-adaptive step size for split variational inclusion, equilibrium and fixed point problems. Comp. Appl. Math. 41, 39 (2022). https://doi.org/10.1007/s40314-021-01749-3

    Article  MathSciNet  Google Scholar 

  3. Alakoya, T.O., Uzor, V.A., Mewomo, O.T.: A new projection and contraction method for solving split monotone variational inclusion, pseudomonotone variational inequality, and common fixed point problems. Comp. Appl. Math. 42, 3 (2023). https://doi.org/10.1007/s40314-022-02138-0

    Article  MathSciNet  Google Scholar 

  4. Alakoya, T.O., Uzor, V.A., Mewomo, O.T., Yao, J.-C.: On a system of monotone variational inclusion problems with fixed-point constraint. J. Inequ. Appl. 2022, 47 (2022). https://doi.org/10.1186/s13660-022-02782-4

    Article  MathSciNet  Google Scholar 

  5. Alber, Ya.I., Chidume, C.E., Zegeya, H.: Approximating fixed points of total asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2006, 10673 (2006). https://doi.org/10.1155/FPTA/2006/10673

    Article  MathSciNet  Google Scholar 

  6. Alimohammady, M., Balooee, J., Cho, Y.J., Roohi, M.: New perturbed finite step iterative algorithms for a system of extended generalized nonlinear mixed quasi-variational inclusions. Comput. Math. Appl. 60, 2953–2970 (2010)

    MathSciNet  Google Scholar 

  7. Ansari, Q.H., Balooee, J., Yao, J.-C.: Extended general nonlinear quasi-variational inequalities and projection dynamical systems. Taiwanese J. Math. 17(4), 1321–1352 (2013)

    MathSciNet  Google Scholar 

  8. Ansari, Q.H., Balooee, J., Yao, J.-C.: Iterative algorithms for systems of extended regularized nonconvex variational inequalities and fixed point problems. Appl. Anal. 93(5), 972–993 (2014)

    MathSciNet  Google Scholar 

  9. Attouch, H.: Variational Convergence for Functions and Operators. Applied Mathematics Series, Pitman, London (1984)

    Google Scholar 

  10. Balooee, J.: Iterative algorithm with mixed errors for solving a new system of generalized nonlinear variational-like inclusions and fixed point problems in Banach spaces. Chin. Ann. Math. 34(4), 593–622 (2013)

    MathSciNet  Google Scholar 

  11. Balooee, J., Cho, Y.J.: Algorithms for solutions of extended general mixed variational inequalities and fixed points. Optim. Lett. 7, 1929–1955 (2013)

    MathSciNet  Google Scholar 

  12. Balooee, J., Cho, Y.J.: Convergence and stability of iterative algorithms for mixed equilibrium problems and fixed point problems in Banach spaces. J. Nonlinear Convex Anal. 14(3), 601–626 (2013)

    MathSciNet  Google Scholar 

  13. Cai, G., Shehu, Y., Iyiola, O.S.: Viscosity iterative algorithms for fixed point problems of asymptotically nonexpansive mappings in the intermediate sense and variational inequality problems in Banach spaces. Numer. Algor. 76(2), 521–553 (2017)

    MathSciNet  Google Scholar 

  14. Chang, S.S., Joseph Lee, H.W., Chan, C.K., Wang, L., Qin, L.J.: Split feasibility problem for quasi-nonexpansive multi-valued mappings and total asymptotically strict pseduo-contracive mapping. Appl. Math. Comput. 219(20), 10416–10424 (2013)

    MathSciNet  Google Scholar 

  15. Cegielski, A., Gibali, A., Reich, S., Zalas, R.: An algorithm for solving the variational inequality problem over the fixed point set of a quasi-nonexpansive operator in Euclidean spaces. Numer. Funct. Anal. Optim. 34, 1067–1096 (2013)

    MathSciNet  Google Scholar 

  16. Ceng, L.C., Hadjisavas, N., Wong, N.C.: Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J. Glob. Optim. 46, 635–646 (2010)

    MathSciNet  Google Scholar 

  17. Chidume, C.-E., Kazmi, K.-R., Zegeye, H.: Iterative approximation of a solution of a general variational-like inclusion in Banach spaces. Internat. J. Math. Math. Sci. 22, 1159–1168 (2004)

    MathSciNet  Google Scholar 

  18. Goebel, K., Kirk, W.A.: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Amer. Math. Soc. 35, 171–174 (1972)

    MathSciNet  Google Scholar 

  19. Godwin, E.C., Alakoya, T.O., Mewomo, O.T., Yao, J.-C.: Relaxed inertial Tseng extragradient method for variational inequality and fixed point problems. Appl. Anal. 102(15), 4253–4278 (2023). https://doi.org/10.1080/00036811.2022.2107913

    Article  MathSciNet  Google Scholar 

  20. Guo, D.J.: Nonlinear Functional Analysis, 2nd edn. Shandong Science and Technology Publishing Press, Shandong (2001)

    Google Scholar 

  21. Fang, Y.-P., Huang, N.-J.: \(H\)-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces. Appl. Math. Lett. 17, 647–653 (2004)

    MathSciNet  Google Scholar 

  22. Fang, Y.-P., Huang, N.-J.: \(H\)-monotone operator and resolvent operator technique for variational inclusions. Appl. Math. Comput. 145, 795–803 (2003)

    MathSciNet  Google Scholar 

  23. Fang, Y.-P., Huang, N.-J.: Iterative algorithm for a system of variational inclusions involving \(H\)-accretive operators in Banach spaces. Acta Math. Hungar. 108(3), 183–195 (2005)

    MathSciNet  Google Scholar 

  24. Fang, Y.P., Huang, N.J., Thompson, H.B.: A new system of variational inclusions with \((H,\eta )\)-monotone operators in Hilbert spaces. Comput. Math. Appl. 49, 365–374 (2005)

    MathSciNet  Google Scholar 

  25. Harder, A.M., Hick, T.L.: Stability results for fixed-point iteration procedures. Math. Japonica 33(5), 693–706 (1998)

    MathSciNet  Google Scholar 

  26. Huang, N.J., Fang, Y.P.: Generalized \(m\)-accretive mappings in Banach spaces. J. Sichuan Univ. 38(4), 591–592 (2001)

    Google Scholar 

  27. Huang, N.J., Fang, Y.P.: A new class of general variational inclusions involving maximal \(\eta \)-monotone mappings. Publ. Math. Debrecen 62(1–2), 83–98 (2003)

    MathSciNet  Google Scholar 

  28. Huang, N.J., Fang, Y.P., Cho, Y.J.: Perturbed three-step approximation processes with errors for a class of general implicit variational inclusions. J. Nonlinear Convex Anal. 4, 301–308 (2003)

    MathSciNet  Google Scholar 

  29. Jin, M.M.: Convergence and Stability of iterative algorithm for a new system of \((A,\eta )\)-accretive mapping inclusions in Banach spaces. Comput. Math. Appl. 56, 2305–2311 (2008)

    MathSciNet  Google Scholar 

  30. Jin, M.M.: Generalized nonlinear mixed quasi-variational inequalities involving maximal \(\eta \)-monotone mappings. J. Inequal. Pure and Appl. Math. 7, 114 (2006)

    MathSciNet  Google Scholar 

  31. Kazmi, K.R., Bhat, M.I.: Convergence and stability of iterative algorithms of generalized set-valued variational-like inclusions in Banach spaces. Appl. Math. Comput. 166, 164–180 (2005)

    MathSciNet  Google Scholar 

  32. Kazmi, K.-R., Khan, F.-A.: Existence and iterative approximation of solutions of generalized mixed equilibrium problems. Comput. Math. Appl. 56, 1314–1321 (2008)

    MathSciNet  Google Scholar 

  33. Kazmi, K.-R., Khan, F.-A.: Iterative approximation of a unique solution of a system of variational-like inclusions in real \(q\)-uniformly smooth Banach spaces. Nonlinear Anal. (TMA) 67, 917–929 (2007)

    MathSciNet  Google Scholar 

  34. Kazmi, K.-R., Khan, H.H., Ahmad, N.: Existence and iterative approximation of solutions of a system of general variational inclusions. Appl. Math. Comput. 215, 110–117 (2009)

    MathSciNet  Google Scholar 

  35. Kiziltunc, H., Purtas, Y.: On weak and strong convergence of an explicit iteration process for a total asymptotically quasi-nonexpansive mapping in Banach space. Filomat 28(8), 1699–1710 (2014)

    MathSciNet  Google Scholar 

  36. Liu, L.S.: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 194, 114–125 (1995)

    MathSciNet  Google Scholar 

  37. Liu, Z., Liu, M., Kang, S.M., Lee, S.: Perturbed Mann iterative method with errors for a new systems of generalized nonlinear variational-like inclusions. Math. Comput. Model. 51, 63–71 (2010)

    MathSciNet  Google Scholar 

  38. Liu, Z., Ume, J.-S., Kang, S.-M.: General strongly nonlinear quasivariational inequalities with relaxed Lipschitz and relaxed monotone mappings. J. Optim. Theory Appl. 114(3), 639–656 (2002)

    MathSciNet  Google Scholar 

  39. Lou, J., He, X.-F., He, Z.: Iterative methods for solving a system of variational inclusions involving \(H\)-\(\eta \)-monotone operators in Banach spaces. Comput. Math. Appl. 55, 1832–1841 (2008)

    MathSciNet  Google Scholar 

  40. Ogwo, G.N., Izuchukwu, C., Mewomo, O.T.: Relaxed inertial methods for solving split variational inequality problems without product space formulation. Acta Math. Sci. 42, 1701–1733 (2022). https://doi.org/10.1007/s10473-022-0501-5

    Article  MathSciNet  Google Scholar 

  41. Ogwo, G.N., Izuchukwu, C., Shehu, Y., Mewomo, O.T.: Convergence of relaxed inertial subgradient extragradient methods for quasimonotone variational inequality problems. J. Sci. Comput. 90, 10 (2022). https://doi.org/10.1007/s10915-021-01670-1

    Article  MathSciNet  Google Scholar 

  42. Osilike, M.A.: Stability of the Mann and Ishikawa iteration procedures for \(\phi \)-strongly pseudocontractions and nonlinear equations of the \(\phi \)-strongly accretive type. J. Math. Anal. Appl. 277, 319–334 (1998)

    MathSciNet  Google Scholar 

  43. Peng, J.-W., Zhu, D.-L.: A system of variational inclusions with \(P\)-\(\eta \)-accretive operators. J. Comput. Appl. Math. 216, 198–209 (2008)

    MathSciNet  Google Scholar 

  44. Sahu, D.R.: Fixed Points of demicontinuous nearly Lipschitzian mappings in Banach spaces. Comment. Math. Univ. Carolin 46, 653–666 (2005)

    MathSciNet  Google Scholar 

  45. Shehu, Y., Ogbuisi, F.U.: An iterative method for solving split monotone variational inclusion and fixed point problems. RACSAM 11(2), 503–518 (2016)

    MathSciNet  Google Scholar 

  46. Sunthrayuth, P., Cholamjiak, P.: A modified extragradient method for variational inclusion and fixed point problems in Banach spaces. Appl. Anal. 100(1), 2049–2068 (2021)

    MathSciNet  Google Scholar 

  47. Taiwo, A., Alakoya, T.O., Mewomo, O.T.: Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces. Numer. Algorithms 86, 1359–1389 (2021)

    MathSciNet  Google Scholar 

  48. Uzor, V.A., Alakoya, O.T., Mewomo, O.T.: On split monotone variational inclusion problem with multiple output sets with fixed point constraints. Comput. Methods Appl. Math. 23(3), 729–749 (2023). https://doi.org/10.1515/cmam-2022-0199

    Article  MathSciNet  Google Scholar 

  49. Verma, R.U.: A generalization to variational convergence for operators. Adv. Nonlinear Var. Inequal. 11, 97–101 (2008)

    MathSciNet  Google Scholar 

  50. Verma, R.U.: General class of implicit variational inclusions and graph convergence on \(A\)-maximal relaxed monotonicity. J. Optim. Theory Appl. 155, 196–214 (2012)

    MathSciNet  Google Scholar 

  51. Verma, R.U.: General system of \(A\)-monotone nonlinear variational inclusion problems with applications. J. Optim. Theory Appl. 131(1), 151–157 (2006)

    MathSciNet  Google Scholar 

  52. Verma, R.U.: General system of \((A,\eta )\)-monotone variational inclusion problems based on generalized hybrid iterative algorithm. Nonlinear Anal. Hybrid Syst 1, 326–335 (2007)

    MathSciNet  Google Scholar 

  53. Wen, M., Hu, C., Cui, A., Peng, J.: Algorithms for finding a common element of the set of common fixed points for nonexpansive semigroups, variational inclusions and generalized equilibrium problems. RACSAM 114, 175 (2020)

    MathSciNet  Google Scholar 

  54. Xia, F.-Q., Huang, N.-J.: Variational inclusions with a general \(H\)-monotone operator in Banach spaces. Comput. Math. Appl. 54, 24–30 (2007)

    MathSciNet  Google Scholar 

  55. Xu, H.-K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16(12), 1127–1138 (1991)

    MathSciNet  Google Scholar 

  56. Yang, J., Liu, H.: The subgradient extragradient method extended to pseudomonotone equilibrium problems and fixed point problems in Hilbert space. Optim. Lett. 14, 1803–1816 (2020)

    MathSciNet  Google Scholar 

  57. Yao, Y., Cho, Y.J., Liou, Y.-C.: Iterative algorithms for variational inclusions, mixed equilibrium and fixed point problems with application to optimization problems. Cent. Eur. J. Math. 9(3), 640–656 (2011)

    MathSciNet  Google Scholar 

  58. Zou, Y.-Z., Huang, N.-J.: A new system of variational inclusions involving \(H(.,.)\)-accretive operator in Banach spaces. Appl. Math. Comput. 212(1), 135–144 (2009)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are thankful to the referees for their valuable suggestions and comments to improve the paper.

Funding

This research received no funding.

Author information

Authors and Affiliations

Authors

Contributions

All authors contributed equally to this research

Corresponding author

Correspondence to Mihai Postolache.

Ethics declarations

Conflict of interests

The authors declare no conflict of interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Balooee, J., Postolache, M. & Yao, Y. System of generalized nonlinear variational-like inclusions and fixed point problems: graph convergence with an application. Rend. Circ. Mat. Palermo, II. Ser 73, 1343–1384 (2024). https://doi.org/10.1007/s12215-023-00988-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-023-00988-z

Keywords

Mathematics Subject Classification

Navigation