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

Sow, T. M. M. ¹

¹ Department of Mathematics, Gaston Berger University, Saint Louis, Senegal

Journal of nonlinear sciences and its applications, Tome 13 (2020) no. 3, p. 119-132.

Résumé

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.

Détail

DOI : 10.22436/jnsa.013.03.01

Classification : 47H09, 65K05, 47J05
Keywords: Fixed points problem, convex minimization problem, equilibrium problem, variational inequality problem

Bibliographie

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

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

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

[4] Chidume, C. Geometric Properties of Banach spaces and Nonlinear Iterations, Springer-Verlag London, London, 2009 | Zbl | DOI

[5] Chidume, C. E.; Djitté, N. Strong convergence theorems for zeros of bounded maximal monotone nonlinear operators, Abstr. Appl. Anal., Volume 2012 (2012), pp. 1-19 | Zbl | DOI

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

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

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

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

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

[11] Lions, J. L.; Stampacchia, G. Variational inequalities, Comm. Pure Appl. Math., Volume 20 (1967), pp. 493-519 | DOI

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

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

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

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

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

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

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

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

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

[21] 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., Volume 4 (2010), pp. 631-643 | Zbl | DOI

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

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

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

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

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

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

[28] Tian, M.; Tong, M. A self-adaptive Armijo-like step size method for solving monotone variational inequality problems in Hilbert spaces, J. Nonlinear Funct. Anal., Volume 2019 (2019), pp. 1-15

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

[30] Xu, H.-K. Inequalities in Banach spaces with applications, Nonlinear Anal., Volume 16 (1991), pp. 1127-1138 | Zbl | DOI

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

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

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

[34] Xu, H.-K.; Alghamdi, M. A.; Shahzad, N. The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., Volume 2015 (2015), pp. 1-12 | Zbl | DOI

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