Approximation of fixed points and convergence of generalized Ishikawa iteration (Q1611405)

This article deals with Ishikawa approximations \[ x_{n+1}= (1-\alpha_n) x_n +\alpha_nu_n,\;y_n=(1-\beta_n)x_n+\beta_nv_n\;(u_n \in T_1y_n,\;v_n\in T_2x_n), \] where \(T_1,T_2:K\to 2^K\) are two multi-valued strongly pseudo-contractive mappings of a nonempty bounded closed convex subset \(K\) in a real uniformly smooth Banach space \(X\) with nonempty closed values in \(2^K\). The main result is the following: under the conditions (i) \(0\leq \alpha_n\), \(\beta_n \leq 1\), (ii) \(\sum^\infty_{n=0}\alpha_n=\infty\), \(\beta_n\to 0\), and \(\text{Fix} \,T_1\cap\text{Fix}\,T_1\cap\text{Fix}\,T_2\neq\emptyset\), the Ishikawa iterations \((x_0\in K)\) converge strongly to the unique common fixed point of \(T_1\) and \(T_2\) in \(K\). The authors write: ``we generalize and extend the results of Chang aud Chang, Cho, Lee, Jung and Kang''.