Iterative algorithm for computing fixed points of demicontractive and zeros points of multivalued accretive operators in certain Banach spaces with application

Year 2020, Volume: 4 Issue: 2, 100 - 111, 30.06.2020

Thierno Sow

https://doi.org/10.31197/atnaa.655466

Abstract

In this paper, an iterative algorithm for finding a common point of the set of common zero of an infinite family of multivalued accretive operators and the set of fixed points of a demicontractive operator is constructed and studied in certain Banach spaces having a weakly continuous duality map. Under suitable control conditions, strong convergence of the sequence generated by proposed algorithm to a common point of the two sets is established. Moreover, application to convex minimization problems involving an infinite family of lower semi-continuous and convex functions are included.The main theorems develop and complement the recent results announced by researchers in this area.

Keywords

Proximal point algorithm, Demicontractive operators, Accretive operators, Weakly continuous duality maps

References

• F. E. Browder, Convergenge theorem for sequence of nonlinear operator in Banach spaces, 100 (1967) 201-225.
• R. E. Bruck Jr, A strongly convergent iterative solution of 0 ∈ U (x) for a maximal monotone operator U in Hilbert spaces, J. Math. Anal. Appl., 48,114-126. (1974).
• R. D. Chen, Z. C. Zhu, Viscosity approximation fixed point for nonexpansive and m-accretive operators, Fixed Point Theory Appl., 2006 (2006), 10 pages.
• I. Cioranescu, Geometry of Banach space, duality mapping and nonlinear problems, Kluwer, Dordrecht, (1990).
• S. Chang, J. K. Kim, X. R. Wang, Modified block iterative algorithm for solving convex feasibility problems in Banach spaces, Journal of Inequalities and Applications, vol. 2010, Article ID 869684, 14 pages.
• C. E. Chidume, Geometric Properties of Banach spaces and Nonlinear Iterations, Springer Verlag Series: Lecture Notes in Mathematics, Vol. 1965,(2009), ISBN 978-1-84882-189. C.E. Chidume, N. Djitte, Strong convergence theorems for zeros of bounded maximal monotone nonlinear operators, J. Abstract and Applied Analysis, Volume 2012, Article ID 681348, 19 pages, doi:10.1155/2012/681348.
• C.E. Chidume, The solution by iteration of nonlinear equations in certain Banach spaces, J. Nigerian Math. Soc., 3 (1984), 57-62.
• K. Goebel and W.A. Kirk, Topics in metric fixed poit theory, Cambridge Studies, in Advanced Mathemathics, Vol. 28, University Cambridge Press, Cambridge 1990
• K. Goebel, S. Reich, (1984). Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York.
• O. Guller, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29 (1991), 403-419.
• B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc., 3 (1967), 957-961.
• Hicks, T.L. Kubicek, J.D., On the Mann iteration process in a Hilbert spaces, J. Math. Anal. Appl. 59 (1977) 498-504.
• S. Ishikawa, Fixed points and iteration of nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc., 73 (1976), 61-71.
• S. Kamimura, W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. of Optimization 13(3) (5003), 938-945.
• N. Lehdili, A. Moudafi, Combining the proximal algorithm and Tikhonov regularization, Optimization 37(1996), 239-252.
• T.C. Lim, H.K. Xu, Fixed point theorems for assymptoticaly nonexpansive mapping, Nonliear Anal 22(1994), no. 11, 1345-1355.
• W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953) 506-510.
• 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.
• B. Martinet, RÃľgularisation dâĂŹinÃľquations variationnelles par approximations successives, (French) Rev. Franaise Informat. Recherche OpÃľrationnelle, 4 (1970), 154-158.
• A. Moudafi, Viscosity approximation methods for fixed-point problems, J. Math Anal. Appl. 241 (2000), 46-55.
• G.J. Minty, Monotone (nonlinear) operator in Hilbert space. Duke Math. 29, 341-V346 (1962).
• Miyadera, I. (1992) Nonlinear semigroups, Translations of Mathematical Monographs, 109 American Mathematical Society, Providence, RI.
• Z. Opial, Weak convergence of sequence of succecive approximation of nonexpansive mapping, Bull Amer. Math.soc. 73 (1967), 591-597.
• R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optimization, 14 (1976), 877-898.
• S. Reich: Weak convergence theorems for nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 4276, 1979.
• T.M.M. Sow, N. DjittÃľ, and C.E. Chidume, A path convergence theorem and construction of fixed points for nonexpansive mappings in certain Banach spaces, Carpathian J.Math.,32(2016),No.2,217-226,2016.
• J. J. Soo, Strong convergence of an iterative algorithm for accretive operators and nonexpansive mappings, Journal of Nonlinear Sciences and Applications, (2016), Vol 9, ISSN 2008-1901
• H. K. Xu , A regularization method for the proximal point algorithm, J. Global. Optim., 36, 115-125. (2006).
• H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002), no. 2, 240 - 256.
• 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.