# A new iterative algorithm for solving some nonlinear problems in Hilbert spaces

Volume 13, Issue 3, pp 119--132
Publication Date: November 13, 2019 Submission Date: September 02, 2019 Revision Date: September 28, 2019 Accteptance Date: October 04, 2019
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### Authors

T. M. M. Sow - Department of Mathematics, Gaston Berger University, Saint Louis, Senegal.

### Abstract

In this paper, a new iterative algorithm for finding a common element of the set of minimizers of a convex function, the set of solutions of variational inequality problem, the set of solutions of equilibrium problems and the set of fixed points of demicontractive mappings is constructed. Convergence theorems are also proved in Hilbert spaces without any compactness assumption. Furthermore, a numerical example is given to demonstrate the implementability of our algorithm. Our theorems are significant improvements in several important recent results.

### Keywords

• Fixed points problem
• convex minimization problem
• equilibrium problem
• variational inequality problem

•  47H09
•  65K05
•  47J05

### References

• [1] L. Ambrosio, N. Gigli, G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Birkhäuser Verlag, Basel (2008)

• [2] P. N. Anh, L. T. H. An, The subgradient extragradient method extended to equilibrium problems, Optimization, 64 (2015), 225--248

• [3] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123--145

• [4] C. Chidume, Geometric Properties of Banach spaces and Nonlinear Iterations, Springer-Verlag London, London (2009)

• [5] C. E. Chidume, N. Djitté, Strong convergence theorems for zeros of bounded maximal monotone nonlinear operators, Abstr. Appl. Anal., 2012 (2012), 19 pages

• [6] P. L. Combettes, J. C. Pesquet, Proximal splitting methods in signal processing, in: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, 2011 (2011), 185--212

• [7] K. Fan, A minimax inequality and applications, in: Inequalities III, 1972 (1972), 103--113

• [8] O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29 (1991), 403--419

• [9] H. Iiduka, W. Takahashi, M. Toyoda, Approximation of solutions of variational inequalities for monotone mappings, PanAmer. Math. J., 14 (2004), 49--61

• [10] S. Kamimura, W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim., 13 (2002), 938--945

• [11] J. L. Lions, G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493--519

• [12] P.-E. Maingé, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499--1515

• [13] P.-E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899--912

• [14] W. A. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506--510

• [15] G. Marino, B. Scardamaglia, R. Zaccone, A general viscosity explicit midpoint rule for quasi--nonexpansive mappings, J. Nonlinear Convex Anal., 18 (2017), 137--148

• [16] G. Marino, H.-K. Xu, A general iterative method for nonexpansive mappings in Hibert spaces, J. Math. Anal. Appl., 318 (2006), 43--52

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

• [18] B. Martinet, Regularisation, d'inéquations variationelles par approximations succesives, Rev. Franaise Informat. Recherche Opérationnelle, 4 (1970), 154--158

• [19] I. Miyadera, Nonlinear semigroups, American Mathematical Society, Providence (1992)

• [20] A. Moudafi, Viscosity approximation methods for fixed point problems, J. Math. Anal. Appl., 241 (2000), 46--55

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

• [22] X. L. Qin, Y. J. Cho, S. M. Kang, H. Zhou, Convergence of a modified Halpern-type iteration algorithm for quasi-$\phi$-nonexpansive mappings, Appl. Math. Lett., 22 (2009), 1051--1055

• [23] R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 1970 (149), 75--88

• [24] T. M. M. Sow, An iterative algorithm for solving equilibrium problems, variational inequality problems and fixed point problems with multivalued quasi-nonexpansive mappings, Appl. Set-Valued Anal. Optim., 1 (2019), 171--185

• [25] S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 331 (2007), 506--515

• [26] W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417--428

• [27] M. Tian, B.-N. Jiang, Weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space, Optimization, 66 (2017), 1689--1698

• [28] M. Tian, M. Tong, A self-adaptive Armijo-like step size method for solving monotone variational inequality problems in Hilbert spaces, J. Nonlinear Funct. Anal., 2019 (2019), 15 pages

• [29] S. Wang, A general iterative method for an infinite family of strictly pseudo-contractive mappings in Hilbert spaces, Appl. Math. Lett., 24 (2011), 901--907

• [30] H.-K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127--1138

• [31] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2), 66 (2002), 240--256

• [32] H.-K. Xu, A variable Krasnoselskii--Mann algorithm and the multiple set split feasiblity problem, Inverse Problem, 26 (2006), 2021--2034

• [33] H.-K. Xu, Iterative methods for the split feasiblity problem in infinite-dimensional Hilbert spaces, Inverse Problem, 26 (2010), 1--17

• [34] H.-K. Xu, M. A. Alghamdi, N. Shahzad, The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2015 (2015), 12 pages

• [35] Y. Yao, H. Zhou, Y. C. Liou, Strong convergence of modified Krasnoselskii-Mann iterative algorithm for nonexpansive mappings, J. Math. Anal. Appl. Comput., 29 (2009), 383--389