Document Type: Research Paper

Authors

1 Department of Mathematics, Sarab Branch, Islamic Azad University, Sarab, Iran.

2 Department of Mathematics, Faculty of Mathematical Science, University of Tabriz, Tabriz, Iran.

3 Department of Mathematics, University of Bonab, Bonab, Iran.

Abstract

In this paper, we study the iterations of quasi $\phi$-nonexpansive mappings and its applications in Banach spaces. At the first, we prove strong convergence of the sequence generated by the hybrid proximal point method to a common fixed point of a family of quasi $\phi$-nonexpansive mappings.  Then, we give  applications of our main results in equilibrium problems.

Keywords

[1] Y.I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications. Theory and applications of nonlinear operators of accretive and monotone type, Lecture Notes in Pure and Appl. Math., 178, Dekker, New York, 1996, pp. 15-50.

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

[3] A.N. Iusem and M. Nasri, Inexact proximal point methods for equilibrium problems in Banach spaces, Numer. Funct. Anal. Optim., 28 (2007), pp. 1279-1308.

[4] S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim., 13 (2002), no. 3, pp. 938-945.

[5] H. Khatibzadeh and V. Mohebbi, On the iterations of a sequence of strongly quasi-nonexpansive mappings with applications, Numer. Funct. Anal. Optim., (2019) doi: 10.1080/01630563.2019.1626419.

[6] H. Khatibzadeh and V. Mohebbi, On the proximal point method for an infinite family of equilibrium problems in Banach spaces, Bull. Korean Math. Soc., 56 (2019), pp. 757--777.

[7] F. Kohsaka and W. Takahashi, Strong convergence of an iterative sequence for maximal monotone operators in a Banach space, Abstr. Appl. Anal., (2004), pp. 239-249.

[8] Z. Ma, L. Wang and S. Chang, Strong convergence theorem for quasi-$phi$-asymptotically nonexpansive mappings in the intermediate sense in Banach spaces, J. Inequal. Appl., (2013) 2013:306, 13 pp.

[9] B. Martinet, Regularisation d'Inequations Variationnelles par Approximations Successives, Revue Francaise d'Informatique et de Recherche Operationnelle, 3 (1970), pp. 154-158.

[10] S. Reich, A weak convergence theorem for the alternating method with Bregman distances, Theory and applications of nonlinear operators of accretive and monotone type, Lecture Notes in Pure and Appl. Math., 178, Dekker, New York, (1996), pp. 313-318.

[11] R.T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), pp. 497-510.

[12] R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877-898.

[13] R.T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), pp. 209-216.