An explicit iterative algorithm for \(k\)-strictly pseudo-contractive mappings in Banach spaces


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.


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].