# An explicit iterative algorithm for $k$-strictly pseudo-contractive mappings in Banach spaces

Volume 9, Issue 7, pp 5021--5028 Publication Date: July 23, 2016
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### Authors

Qinwei Fan - School of Science, Xi'an Polytechnic University, Xi'an, Shaanxi 710048, China. Xiaoyin Wang - Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China.

### Abstract

Let $E$ be a real uniformly smooth Banach space. Let $K$ be a nonempty bounded closed and convex subset of $E$. Let $T : K \rightarrow K$ be a strictly pseudo-contractive map and $f$ be a contraction on $K$. Assume $F(T) := \{x \in K : Tx = x\} \neq\emptyset$. Consider the following iterative algorithm in $K$ given by $x_{n+1} = \alpha_nf(x_n) + \beta_nx_n +\gamma_nS_nx_n,$ where $S_n : K \rightarrow K$ is a mapping defined by $S_nx := (1 -\delta_n)x + \delta_nTx$. It is proved that the sequence $\{x_n\}$ generated by the above iterative algorithm converges strongly to a fixed point of $T$. Our results mainly extend and improve the results of [C. O. Chidume, G. De Souza, Nonlinear Anal., 69 (2008), 2286-2292] and [J. Balooee, Y. J. Cho, M. Roohi, Numer. Funct. Anal. Optim., 37 (2016), 284-303].

### Keywords

• Strictly pseudo-contractive mappings
• iterative algorithm
• strong convergence
• fixed point
• Banach spaces.

•  47J25
•  47H09

### References

• [1] J. Balooee, Y. J. Cho, M. Roohi, Convergence theorems for pointwise asymptotically strict pseudo-contractions in Hilbert spaces, Numer. Funct. Anal. Optim., 37 (2016), 284--303

• [2] F. E. Browder, W. E. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197--228

• [3] C. E. Chidume, Iterative approximation of fixed points of Lipschitz pseudocontractive maps, Proc. Amer. Math. Soc., 129 (2001), 2245--2251

• [4] C. O. Chidume, G. De Souza, Convergence of a Halpern-type iteration algorithm for a class of pseudo-contractive mappings, Nonlinear Anal., 69 (2008), 2286--2292

• [5] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957--961

• [6] V. E. Ingbianfam, F. A. Tsav, I. S. Iornumbe , Weak and strong convergence of an iterative algorithm for Lipschitz pseudo-contractive maps in Hilbert spaces, Adv. Fixed Point Theory, 6 (2016), 194--206

• [7] J. S. Jung, A general composite iterative method for strictly pseudocontractive mappings in Hilbert spaces, Fixed Point Theory Appl., 2014 (2014), 21 pages

• [8] T. H. Kim, H. K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal., 61 (2005), 51-60

• [9] P. L. Lions, Approximation de points fixes de contractions, (French) C. R. Acad. Sci. Paris Sér. A-B, 284 (1977), 1357--1359

• [10] W. R. Mann, Mean value methods in iterations, Proc. Amer. Math. Soc., 4 (1953), 506--510

• [11] G. Marino, H. K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl., 329 (2007), 336--349

• [12] K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279 (2003), 372--379

• [13] J. A. Park, Mann-iteration process for the fixed point of strictly pseudocontractive mapping in some Banach spaces, J. Korean Math. Soc., 31 (1994), 333--337

• [14] X. Qin, S. Y. Cho, J. K. Kim, On the weak convergence of iterative sequences for generalized equilibrium problems and strictly pseudocontractive mappings, Optimization, 61 (2012), 805--821

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

• [16] S. Reich, Approximating fixed points of nonexpansive mappings, Panamer. Math. J., 4 (1994), 23--28

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

• [18] R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. (Basel), 58 (1992), 486--491

• [19] H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240--256

• [20] H. K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116 (2003), 659--678

• [21] H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279--291

• [22] Y. Yao, R. P. Agarwal, M. Postolache, Y. C. Liou, Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem, Fixed Point Theory Appl., 2014 (2014), 14 pages

• [23] Y. Yao, Y. C. Liou, J. C. Yao , Split common fixed point problem for two quasi-pseudo-contractive operators and its algorithm construction , Fixed Point Theory Appl., 2015 (2015), 19 pages

• [24] Y. Yao, G. Marino, H. K. Xu, Y. C. Liou, Construction of minimum-norm fixed points of pseudocontractions in Hilbert spaces, J. Inequal. Appl., 2014 (2014), 14 pages

• [25] Y. Yao, M. Postolache, S. M. Kang, Strong convergence of approximated iterations for asymptotically pseudocontractive mappings , Fixed Point Theory Appl., 2014 (2014), 13 pages

• [26] Y. Yao, M. Postolache, Y. C. Liou, Coupling Ishikawa algorithms with hybrid techniques for pseudocontractive mappings, Fixed Point Theory Appl., 2013 (2013), 8 pages

• [27] Y. Yao, M. Postolache, Y. C. Liou, Strong convergence of a self-adaptive method for the split feasibility problem, Fixed Point Theory Appl., 2013 (2013), 12 pages

• [28] H. Zegeye, N. Shahzad, An algorithm for a common fixed point of a family of pseudocontractive mappings, Fixed Point Theory Appl., 2013 (2013), 14 pages

• [29] H. Zhou, Convergence theorems of fixed points for $\kappa$-strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., 69 (2008), 456--462