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

Fan, Qinwei¹ ; Wang, Xiaoyin²

¹ School of Science, Xi'an Polytechnic University, Xi'an, Shaanxi 710048, China
² Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 7, p. 5021-5028.

Résumé

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].

Détail

DOI : 10.22436/jnsa.009.07.06

Classification : 47J25, 47H09
Keywords: Strictly pseudo-contractive mappings, iterative algorithm, strong convergence, fixed point, Banach spaces.

Bibliographie

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

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

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

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

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

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

[7] Jung, J. S. A general composite iterative method for strictly pseudocontractive mappings in Hilbert spaces, Fixed Point Theory Appl., Volume 2014 (2014), pp. 1-21

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[22] Yao, Y.; Agarwal, R. P.; Postolache, M.; Liou, Y. C. Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem, Fixed Point Theory Appl., Volume 2014 (2014), pp. 1-14

[23] Yao, Y.; Liou, Y. C.; J. C. Yao Split common fixed point problem for two quasi-pseudo-contractive operators and its algorithm construction , Fixed Point Theory Appl., Volume 2015 (2015), pp. 1-19

[24] Yao, Y.; Marino, G.; Xu, H. K.; Liou, Y. C. Construction of minimum-norm fixed points of pseudocontractions in Hilbert spaces, J. Inequal. Appl., Volume 2014 (2014), pp. 1-14

[25] Yao, Y.; Postolache, M.; Kang, S. M. Strong convergence of approximated iterations for asymptotically pseudocontractive mappings , Fixed Point Theory Appl., Volume 2014 (2014), pp. 1-13

[26] Yao, Y.; Postolache, M.; Liou, Y. C. Coupling Ishikawa algorithms with hybrid techniques for pseudocontractive mappings, Fixed Point Theory Appl., Volume 2013 (2013), pp. 1-8

[27] Yao, Y.; Postolache, M.; Liou, Y. C. Strong convergence of a self-adaptive method for the split feasibility problem, Fixed Point Theory Appl., Volume 2013 (2013), pp. 1-12

[28] Zegeye, H.; Shahzad, N. An algorithm for a common fixed point of a family of pseudocontractive mappings, Fixed Point Theory Appl., Volume 2013 (2013), pp. 1-14

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