An Algorithm to Solve Equilibrium Problems and Fixed Points Problems Involving a Finite Family of Multivalued Strictly Pseudo-Contractive Mappings

1. Berinde, V., Pǎcurar, M.: The role of the Pompeiu–Hausdorff metric in fixed point theory. Creat. Math. Inform. 22, 143–150 (2013)

   MathSciNet  MATH  Google Scholar 

2. Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 20, 197–228 (1967)

   Article  MathSciNet  Google Scholar 

3. Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Nonlinear Programming Techniques for Equilibria. Springer Nature, Switzerland (2019)

   Book  Google Scholar 

4. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)

   MathSciNet  MATH  Google Scholar 

5. Combettes, P.L.: A block-iterative surrogate constraint splitting method for quadratic signal recovery. IEEE Trans. Signal Process. 51, 1771–1782 (2003)

   Article  MathSciNet  Google Scholar 

6. Chang, S., Tang, Y., Wang, L., Xu, Y., Zhao, Y., Wang, G.: Convergence theorems for some multi-valued generalized nonexpansive mappings. Fixed Point Theory Appl. 2014, 33 (2014)

   Article  MathSciNet  Google Scholar 

7. Chidume, C.E.: Geometric Properties of Banach Spaces and Nonlinear Iterations. Lecture Notes in Mathematics, vol. 1965. Springer, London (2009)

   Google Scholar 

8. Chidume, C.E., Chidume, C.O., Djitte, N., Minjibir, M.S.: Krasnoselskii-type algorithm for fixed points of multi-valued strictly pseudo-contractive mappings. Fixed Point Theory Appl. 2013, 58 (2013)

   Article  MathSciNet  Google Scholar 

9. Fan, K.: A minimax inequality and applications. In: Shisha, O (ed.) Inequalities III, pp 103–113. Academic Press, San Diego (1972)

10. Fan, Q.-W., Wu, W., Zurada, J.M.: Convergence of batch gradient learning with smoothing regularization and adaptive momentum for neural networks. SpringerPLus 5, 295 (2016)

    Article  Google Scholar 

11. Iiduka, H.: Convergence analysis of iterative methods for nonsmooth convex optimization over fixed point sets of quasi-nonexpansive mappings. Math. Program. 159, 509–538 (2016)

    Article  MathSciNet  Google Scholar 

12. Hung, P.G., Muu, L.D.: On inexact Tikhonov and proximal point regularization methods for pseudomonotone equilibrium problems. Vietnam J. Math. 40, 255–274 (2012)

    MathSciNet  MATH  Google Scholar 

13. García-Falset, J., Llorens-Fuster, E., Suzuki, T.: Fixed point theory for a class of generalized nonexpansive mappings. J. Math. Anal. Appl. 375, 185–195 (2011)

    Article  MathSciNet  Google Scholar 

14. Kakutani, S.: A generalization of Brouwer’s fixed point theorem. Duke Math. J. 8, 457–459 (1941)

    Article  MathSciNet  Google Scholar 

15. Khan, S.H., Yildirim, I., Rhoades, B.E.: RETRACTED: A One-step iterative process for two multivalued nonexpansive mappings in Banach spaces. Comput. Math. Appl. 61, 3172–3178 (2011)

    Article  MathSciNet  Google Scholar 

16. Moudafi, A.: Viscosity approximation methods for fixed-point problems. J. Math. Anal. Appl. 241, 46–55 (2000)

    Article  MathSciNet  Google Scholar 

17. Marino, G., Xu, H.-K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Math. Appl. 329, 336–346 (2007)

    MathSciNet  MATH  Google Scholar 

18. Muu, L.D., Quoc, T.D.: Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash–Cournot equilibrium model. J. Optim. Theory Appl. 142, 185–204 (2009)

    Article  MathSciNet  Google Scholar 

19. Nash, J.F.: Equilibrium points in n-person games. Proc. Natl. Acad. Sci. U. S. A. 36, 48–49 (1950)

    Article  MathSciNet  Google Scholar 

20. Nash, J.F.: Non-cooperative games. Ann. Math. 2nd Ser. 54, 286–295 (1951)

    Article  MathSciNet  Google Scholar 

21. Panyanak, B.: Mann and Ishikawa iteration processes for multivalued mappings in Banach spaces. Comput. Math. Appl. 54, 872–877 (2007)

    Article  MathSciNet  Google Scholar 

22. Petrot, N., Wattanawitoon, K., Kumam, P.: A hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces. Nonlinear Anal. Hybrid Syst. 4, 631–643 (2010)

    Article  MathSciNet  Google Scholar 

23. Qin, X., Cho, Y.J., Kang, S.M., Zhou, H.: Convergence of a modified Halpern-type iteration algorithm for quasi-ϕ-nonexpansive mappings. Appl. Math. Lett. 22, 1051–1055 (2009)

    Article  MathSciNet  Google Scholar 

24. Sene, M., Faye, P., Djitté, N.: A Krasnoselskii-type algorithm for approximating a common fixed point of a finite family of multivalued strictly pseudo contractive mappings in Hilbert spaces. J. Math. Sci. Adv. Appl. 27, 59–80 (2014)

    Google Scholar 

25. Song, Y., Cho, Y.-J.: Some notes on Ishikawa iteration for multi-valued mappings. Bull. Korean Math. Soc. 48, 575–584 (2011)

    Article  MathSciNet  Google Scholar 

26. Sabach, S.: Iterative Methods for Solving Optimization Problems. Research Thesis, Technion-Israel Institute of Technology, Haifa (2012)

27. Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506–515 (2007)

    Article  MathSciNet  Google Scholar 

28. Xu, H.-K.: A variable Krasnosel’skii–Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 26, 2021–2034 (2006)

    Article  Google Scholar 

29. Xu, H.-K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)

    Article  MathSciNet  Google Scholar 

30. Xu, H.-K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26, 105018 (2010)

    Article  MathSciNet  Google Scholar 

31. Yao, Y., Zhou, H., Liou, Y.-C.: Strong convergence of modified Krasnoselskii–Mann iterative algorithm for non-expansive mappings. J. Appl. math. Comput. 29, 383–389 (2009)

    Article  MathSciNet  Google Scholar 

Download references