Strong Convergence by a Hybrid Algorithm for Finding a Common Fixed Point of Lipschitz Pseudocontraction and Strict Pseudocontraction in Hilbert Spaces

and Applied Analysis 3 Remark 1.2. i Since 0 αn βn 1, n 1 and ∑∞ n 1 αnβn ∞, the iterative sequence 1.6 could not be reduced to a Mann iterative sequence 1.4 . Therefore, the iterative sequence 1.6 has some particular cases. ii The iterative sequence 1.6 is usually called the Ishikawa iterative sequence. iii Chidume andMutangadura 17 gave an example to show that the Mann iterative sequence failed to be convergent to a fixed point of Lipschitzian pseudocontractive mapping. In an infinite-dimensional Hilbert spaces, Mann and Ishikawa’s iteration algorithms have only weak convergence, in general, even for nonexpansive mapping. In order to obtain a strong convergence theorem for theMann iteration method 1.4 to nonexpansive mapping, Nakajo and Takahashi 18 modified 1.4 by employing two closed convex sets that are created in order to form the sequence via metric projection so that strong convergence is guaranteed. Later, it is often referred as the hybrid algorithm or the CQ algorithm. After that the hybrid algorithm have been studied extensively by many authors see e.g., 19–23 . Particularly, Martinez-Yanes and Xu 24 and Plubtieng and Ungchittrakool 20 extended the same results of Nakajo and Takahashi 18 to the Ishikawa iteration process. In 2007, Marino and Xu 15 further generalized the hybrid algorithm from nonexpansive mappings to strict pseudocontractive mappings. In 2008, Zhou 25 established the hybrid algorithm for pseudocontractive mapping in the case of the Ishikawa iteration process. Recently, Yao et al. 26 introduced the hybrid iterative algorithm which just involved one closed convex set for pseudocontractive mapping in Hilbert spaces as follows. Let C be a nonempty closed convex subset of a real Hilbert spaceH. Let T : C → C be a pseudocontraction. Let {αn} be a sequence in 0, 1 . Let x0 ∈ H. ForC1 C and x1 PC1 x0 , define a sequence {xn} of C as follows. yn 1 − αn xn αnTzn, Cn 1 { v ∈ Cn : ∥ αn I − T yn ∥ ∥ 2 2αn 〈 xn − v, I − T yn 〉} , xn 1 PCn 1 x0 . 1.7 Theorem 1.3 see 26 . Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a L-Lipschitz pseudocontraction such that F T / ∅. Assume the sequence {αn} ⊂ a, b for some a, b ∈ 0, 1/ L 1 . Then the sequence {xn} generated by 1.7 converges strongly to PF T x0 . Very recently, Tang et al. 27 generalized the hybrid algorithm 1.7 in the case of the Ishikawa iterative precess as follows: yn 1 − αn xn αnTzn, zn ( 1 − βn ) xn βnTxn, Cn 1 { v ∈ Cn : ∥ αn I − T yn ∥ ∥ 2 2αn 〈 xn − v, I − T yn 〉 2αnβnL‖xn − Txn‖ ∥ yn − xn αn I − T yn ∥ ∥ } ,