Strong convergence theorems for minimization, variational inequality and fixed point problems for quasi-nonexpansive mappings using modified proximal point algorithms in real Hilbert spaces

Document Type : Research Paper


Amadou Mahtar Mbow University, Senegal


In this paper, we investigate the problem of finding a common element of the solution set of convex minimization problem, the solution set of variational inequality problem and the  solution set of fixed point problem with an infinite family of quasi-nonexpansive mappings in  real Hilbert spaces. Based on the well-known proximal point algorithm and viscosity approximation method, we propose and analyze a  new iterative algorithm for computing a common element. Under very mild assumptions, we obtain a strong convergence theorem for the sequence generated by the proposed method. Application to convex minimization and variational inequality problems coupled with inclusion problem  is provided to support our main results.\,Our proposed method is quite general and includes the iterative methods considered in the earlier and recent literature as special cases.


[1] P.N. Anh and L.T. H. An, The subgradient extragradient method extended to equilibrium problems, Optim. 64 (2015) 225–248.
[2] L. Ambrosio, N. Gigli and G. Savar´e, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second edition, Lectures in Mathematics ETH Z¨urich, Birkh¨auser Verlag, Basel, 2008.
[3] K. Aoyama, I. H. Koji and W. Takahashi, Weak convergence of an iterative sequence for accretive operators in Banach spaces, Fixed Point Theory Appl. (2006), Art. ID 35390, 13 pp.
[4] K. O. Aremu, C. Izuchukwu, G. C. Ugwunnadi and O. T. Mewomo, On the proximal point algorithm and demimetric mappings in CAT (0) spaces, Demonstr. Math. 51(1) (2018) 277–294.
[5] F. E. Browder, Convergenge theorem for sequence of nonlinear operator in Banach spaces, Math. Z. 100 (1967) 201–225.
[6] L. C Ceng, N. Hadjisavvas and N. C. Wong, Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems, J. Glob. Optim. 46 (2010) 635–646.
[7] S.S. Chang, H.W.J. Lee and C.K. Chan, A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization, Nonlinear Anal. 70 (2009) 3307–3319.
[8] W. Cholamjiak, Shrinking projection methods for a split equilibrium problem and a nonspreading-type multivalued mapping, J. Nonlinear Sci. Appl., (in press).
[9] C. E. Chidume, Geometric Properties of Banach spaces and Nonlinear Iterations, Springer Verlag Series: Lecture Notes in Mathematics, Vol. 1965, 2009.
[10] C. E. Chidume and N. Djitte, Strong convergence theorems for zeros of bounded maximal monotone nonlinear operators, J. Abst. Appl. Anal. Volume 2012, Article ID 681348, 19 pages.
[11] H. Dehghan, C. Izuchukwu, O. T. Mewomo, D. A. Taba and G. C. Ugwunnadi, Iterative algorithm for a family of monotone inclusion problems in CAT(0) spaces, Quaest. Math., (2019),
[12] H. Iiduka, W. Takahashi and M. Toyoda, Approximation of solutions of variational inequalities for monotone mappings, PanAmer. Math. J. 14 (2004) 49–61.
[13] O. G¨uler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim.29 (1991) 403–419.
[14] C. Izuchukwu, H. A. Abass and O. T. Mewomo, Viscosity approximation method for solving minimization problem and fixed point problem for nonexpansive multivalued mapping in CAT(0) spaces, Ann. Acad. Rom. Sci. Ser. Math. Appl. 11(1)(2019) 131–158.
[15] C. Izuchukwu, A. A. Mebawondu, K. O. Aremu, H. A. Abass and O. T. Mewomo, Viscosity iterative techniques for approximating a common zero of monotone operators in an Hadamard space, Rend. Circ. Mat. Palermo II, (2019). in p-uniformly convex metric space, Numer. Algorithms., (2018). DOI: 10.1007/s11075-018-0633-9.
[16] C. Izuchukwu, K.O. Aremu, A.A. Mebawondu and O.T. Mewomo, A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space, Appl. Gen. Topol. 20(1) (2019) 193–210.
[17] L.O. Jolaoso, T.O. Alakoya, A. Taiwo and O.T. Mewomo, A parallel combination extragradient method with Armijo line searching for finding common solution of finite families of equilibrium and fixed point problems, Rend. Circ. Mat. Palermo II, (2019), DOI:10.1007/s12215-019-00431-2
[18] J. Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature, Comment. Math. Helv. 70 (1995) 659–673.
[19] S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim. 13(3) (5003) 938–945.
[20] H. He, S. Liu and H. Zhou, An explicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of an infinite family of nonexpansive mappings, Nonlinear Anal. 72 (2010) 3124 3135.
[21] P. E. Mainge, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim. 47 (2008) 1499–1515.
[22] P.E. Maing´e, Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints, Eur. J. Oper. Res. 205 (2010) 501–506.
[23] B. Martinet, R´egularisation d’in´equations variationnelles par approximations successives, (French) Rev. Franaise Informat. Recherche Op´erationnelle, 4 (1970) 154–158.
[24] A. Moudafi, Viscosity approximation methods for fixed point problems, J. Math. Anal. Appl. 241 (2000) 46–55.
[25] I. Miyadera, Nonlinear Semigroups, Translations of Mathematical Monographs, American Mathematical Society, Providence, 1992.
[26] W. Nilsrakoo and S. Saejung, Weak and strong convergence theorems for countable Lipschitzian mappings and its applications,, Nonlinear Anal. 69 (2008) 2695–2708.
[27] J. W. Peng and J.C. Yao, Strong convergence theorems of an iterative scheme based on extragradient method for mixed equilibruim problem and fixed point problems, Math. Com. Model. 49 (2009) 1816–1828.
[28] R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970) 7588.
[29] Y. Shehu and O.T. Mewomo, Further investigation into split common fixed point problem for demicontractive operators, Acta Math, Sin. (Engl. Ser.), 32(11) (2016) 1357–1376.
[30] K. Shimoji and W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math. 5 (2001).
[31] A. Taiwo, L. O. Jolaoso and O. T. Mewomo, General alternative regularization method for solvng split equality common fixed point problem for quasi-pseudocontractive mappings in Hilbert spaces, Ricerche Mat. (2019), DOI: 10.1007/s11587-019-00460-0.
[32] A. Taiwo, L. O. Jolaoso and O. T. Mewomo, Parallel hybrid algorithm for solving pseudomonotone equilibrium and Split Common Fixed point problems, Bull. Malays. Math. Sci. Soc., (2019), DOI: 10.1007/s40840-019-00781-1.
[33] H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66(2) (2002) 240–256.
[34] L.-C. Zeng and J.-C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese J. Math. 10(5) (2006) 1293-1303.
Volume 12, Issue 2
November 2021
Pages 511-526
  • Receive Date: 30 June 2019
  • Revise Date: 29 October 2019
  • Accept Date: 31 October 2019