General Viscosity Iterative Process for Solving Variational Inclusion and Fixed Point Problems Involving Multi-Valued Quasi-Non-Expansive and Demicontractive Operators With Application

Document Type : Original Article

Author
Thierno Mohamadane Mansour Sow
Department of Mathematics, Gaston Berger University, Saint Louis, Senegal
10.29252/maco.1.1.9
Abstract
In this paper, we  introduce and study a new iterative method which is based on viscosity general algorithm and forward-backward splitting method  for finding a common element of the set of common fixed points of multivalued demicontractive and quasi-nonexpansive mappings and the set of solutions of  variational inclusion with set-valued maximal monotone mapping and inverse strongly monotone mappings in  real Hilbert spaces. We prove that the sequence $x_n$ which is generated by the proposed iterative algorithm converges strongly to a common element of two sets above. Finally, our theorems are applied to approximate a common solution of fixed point problems with set-valued operators and the composite convex minimization problem. Our theorems are significant improvements on several important recent results.

Keywords

[1] S. Adly, Perturbed algorithms and sensitivity analysis for a general class of variational inclusions, Journal of Mathematical Analysis and Applications, 201(2):609–630, 1996.
[2] J. B. Baillon and G. Haddad, Quelques proprits des oprateurs angle-borns et n-cycliquement monotones, Israel J. Math., 26, 137–150 ,1977.
[3] K. Bredies, A forward-backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space, Inverse Probl. 25(1): 015005, 20 pp, 2009.
[4] F. E. Browder, Convergenge theorem for sequence of nonlinear operator in Banach spaces, Mathematische Zeitschrift, 100, 201–225, 1967.
[5] S.S. Chang, D.P. Wu, L. Wang, G. Wang, Proximal point algorithms involving fixed point of nonspreadingtype multivalued mappings in Hilbert spaces, J. Nonlinear Sci. Appl., 9(10), 5561–5569, 2016.
[6] G.H-G.Chen, R.T. Rockafellar, Convergence rates in forward-backward splitting, SIAM J. Optim., 7(2), 421–444, 1997.
[7] C. E. Chidume, Geometric Properties of Banach spaces and Nonlinear Iterations, Springer Verlag Series: Lecture Notes in Mathématiciennes, ISBN: 978-1-84882-189-7, 2009.
[8] C. E. Chidume, C. O. Chidume, N. Djitte and M. S. Minjibir, Convergence theorems for fixed points of multi-valued strictly pseudo-contractive mappings in Hilbert Spaces, Abstract and Applied Analysis, 629–468, 2013.
[9] J. Geanakoplos, Nash and Walras equilibrium via Brouwer, Economic Theory, 21, 585–603, 2003.
[10] A. Gibali, D.V. Thong, Tseng type methods for solving inclusion problems and its applications, Calcolo, 55(4), 2018.
[11] S. Kakutani, A generalization of Brouwers fixed point theorem, Duke Mathematical Journal 8(3): 457–459, 1941.
[12] B. Lemaire, Which fixed point does the iteration method select? Recent advances in optimization (Trier,1996), Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, 54-167, 1997.
[13] P.L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16, 964–979, 1979.
[14] Marino G., and Xu H. K., A general iterative method for nonexpansive mappings in Hibert spaces, J. Math. Anal. Appl., 318, 43-52, 2006.
[15] A. Moudafi, Viscosity approximation methods for fixed point problems, J. Math. Anal. Appl., 241, 46–55, 2000.
[16] G.J. Minty, Monotone (nonlinear) operator in Hilbert space, Duke Math., 29, 341–346, 1962.
[17] Jr. Nadla, Multivalued contraction mappings, Pacific J. Math., 30, 475–488, 1969.
[18] J.F. Nash, Non-cooperative games, Annals of Mathematics, Second series 54, 286–295, 1951.
[19] J.F. Nash, Equilibrium points in n-person games, Proceedings of the National Academy of Sciences of the United States of America, 36(1): 48–49, 1950.
[20] B. Panyanak, Endpoints of multivalued nonexpansive mappings in geodesic spaces, Fixed Point Theory Appl., 11 pages, 2015
[21] G.B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert spaces, J. Math. Anal. Appl., 72, 383–390, 1979.
[22] F.O. Isiogugu and M. O. Osilike, Convergence theorems for new classes of multivalued hemicontractivetype mappings, Fixed Point Theory and Applications, 2014, 93:2014.
[23] T.L. Hicks, J.D. Kubicek, On the Mann iteration process in a Hilbert space, J. Math. Anal. Appl., 59, 498–504, 1977.
[24] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization, 14(5): 877–898, 1976.
[25] Y. Song, Y.J. Cho, Some notes on Ishikawa iteration for multi-valued mappings, Bull. Korean Math. Soc., 48(3): 575–584, 2011.
[26] T. M. M. Sow, M. Sène, N. Djitté, Strong convergence theorems for a common fixed point of a finite family of multi-valued Mappings in certain Banach Spaces, Int. J. Math. Anal., 9: 437–452, 2015.
[27] P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38(2): 431–446, 2000.
[28] H.K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116: 659–678, 2003.
[29] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16(12): 1127–1138, 1991.
[30] Wang, S., A general iterative method for an infinite family of strictly pseudo-contractive mappings in Hilbert spaces, Applied Mathematics Letters, 24: 901–907, 2011.
[31] Y. Wang, H.K. Xu, Strong convergence for the proximal-gradient method, J.Nonlinear Convex Anal. 15(3): 581–593, 2014.
Mathematical Analysis and Convex Optimization
Volume 1, Issue 1
June 2020
Pages 75-91