# Proximal forward-backward splitting method for zeros of sum accretive operators for a fixed point set and inverse problems

Volume 17, Issue 4, pp 506-526 Publication Date: October 27, 2017       Article History
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### Authors

K. Sitthithakerngkiet - Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut K. Promluang - KMUTTFixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut P. Thounthong - Renewable Energy Research Centre $\&$ Department of Teacher Training in Electrical Engineering, Faculty of Technical Education, King Mongkut P. Kumam - KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut

### Abstract

In this paper, we investigate regularization method via a proximal point algorithm for solving treating sum of two accretive operators and fixed point problems. Strong convergence theorems are established in the framework of Banach spaces. Also we apply our result to variational inequalities and equilibrium problems. Furthermore, an illustrative numerical example is presented.

### Keywords

• Regularization method
• proximal point algorithm
• zero points
• accretive operators
• inverse problems

•  47H09
•  47H17
•  47J25
•  49J40

### References

• [1] K. Aoyama, H. Iiduka, W. Takahashi, Weak convergence of an iterative sequence for accretive operators in Banach spaces, Fixed Point Theory Appl., 2006 (2006 ), 13 pages.

• [2] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Translated from the Romanian, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden (1976)

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

• [4] F. E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1272–1276.

• [5] S. Y. Cho, X.-L. Qin, L. Wang, Strong convergence of a splitting algorithm for treating monotone operators , Fixed Point Theory Appl., 2014 (2014 ), 15 pages.

• [6] P. L. Combettes, S. A. Hirstoaga, Equilibrium programming in Hilbert spaces , J. Nonlinear Convex Anal., 6 (2005), 117–136.

• [7] J.-P. Gossez, E. Lami Dazo, Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math., 40 (1972), 565–573.

• [8] S. Kitahara, W. Takahashi, Image recovery by convex combinations of sunny nonexpansive retractions, Topol. Methods Nonlinear Anal., 2 (1993), 333–342.

• [9] N. Lehdili, A. Moudafi, Combining the proximal algorithm and Tikhonov regularization, Optimization, 37 (1996), 239– 252.

• [10] L.-S. Liu, Ishikawa-type and Mann-type iterative processes with errors for constructing solutions of nonlinear equations involving m-accretive operators in Banach spaces , Nonlinear Anal. , 34 (1998), 307–317.

• [11] G. López, V. Martín-Márquez, F.-H. Wang, H.-K. Xu, Forward-backward splitting methods for accretive operators in Banach spaces, Abstr. Appl. Anal., 2012 (2012 ), 25 pages.

• [12] H. Manaka, W. Takahashi, Weak convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space, Cubo, 13 (2011), 11–24.

• [13] B. Martinet, Régularisation d’inéquations variationnelles par approximations successives, (French) Rev. Française Informat. Recherche Opérationnelle, 4 (1970), 154–159.

• [14] Y. Qing, S. Y. Cho, A regularization algorithm for zero points of accretive operators , Fixed Point Theory Appl., 2013 (2013 ), 9 pages.

• [15] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces , J. Math. Anal. Appl., 67 (1979), 274–276.

• [16] R. T. Rockafellar , Monotone operators and the proximal point algorithm, SIAM J. Control Optimization, 14 (1976), 877–898.

• [17] D. R. Sahu, J. C. Yao, The prox-Tikhonov regularization method for the proximal point algorithm in Banach spaces, J. Global Optim., 51 (2011), 641–655.

• [18] Y.-L. Song, L.-C. Ceng, Weak and strong convergence theorems for zeroes of accretive operators in Banach spaces, J. Appl. Math., 2014 (2014 ), 11 pages.

• [19] T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227–239.

• [20] W. Takahashi, Viscosity approximation methods for resolvents of accretive operators in Banach spaces , J. Fixed Point Theory Appl., 1 (2007), 135–147.

• [21] W. Takahashi, Introduction to nonlinear and convex analysis, Yokohama Publishers, Yokohama (2009)

• [22] S. Takahashi, W. Takahashi, M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces , J. Optim. Theory Appl., 147 (2010), 27–41.

• [23] A. N. Tikhonov, Solution of incorrectly formulated problems and the regularization method, Soviet Meth. Dokl., 4 (1963), 1035–1038.

• [24] A. N. Tikhonov, Improper problems of optimal planning and stable methods of their solution, Soviet Meth. Dokl., 6 (1965), 1264–1267.

• [25] A. N. Tikhonov, V. Y. Arsenin , Solutions of ill-posed problems, Translated from the Russian, Preface by translation editor Fritz John, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York-Toronto, Ont.-London (1977)