A Strong Convergence Theorem for a Common Fixed Point of Two Sequences of Strictly Pseudocontractive Mappings in Hilbert Spaces and Applications

and Applied Analysis 3 In 2007, Marino and Xu 11 proved the following strong convergence theorem by using the hybrid projection method for a strict pseudocontraction. They defined a sequence as follows: x0 ∈ C chosen arbitrarily, yn αnxn 1 − αn Txn, Cn { z ∈ C : ∥∥yn − z ∥ ∥ 2 ‖xn − z‖ 1 − αn k − αn ‖xn − Txn‖ } , Qn {z ∈ C : 〈xn − z, x0 − xn〉 0}, xn 1 PCn∩Qn x0 , n 0, 1, 2, . . . , 1.5 They proved that if 0 αn < 1, then {xn} defined by 1.5 converges strongly to PF T x0 . Motivated and inspired by the above-mentioned results, it is the purpose of this paper to improve and generalize the algorithm 1.5 to the new general process of two sequences of strictly pseudocontractive mappings in Hilbert spaces. Let C be a closed convex subset of a Hilbert spaceH and Tn, Sn : C → C two sequences of strictly pseudocontractive mappings such that ⋂∞ n 0 F Tn ∩ ⋂∞ n 0 F Sn / ∅. Define {xn} in the following ways: x0 ∈ C chosen arbitrarily, yn αnxn 1 − αn zn, zn βnTnxn ( 1 − βn ) Snxn, Ĉn { z ∈ C : ∥∥yn − z ∥ ∥ 2 ‖xn − z‖ 1 − αn βn ( k T − αn )‖xn − Tnxn‖ 1 − αn ( 1 − βn )( k S − αn )‖xn − Snxn‖ − 1 − αn βn ( 1 − βn )‖Tnxn − Snxn‖ } , Qn {z ∈ C : 〈xn − z, x0 − xn〉 0}, xn 1 PĈn∩Qn x0 , 1.6 where {αn}, {βn} are sequences in 0, 1 . We prove that the algorithm 1.6 converges strongly to a common fixed point of two sequences of strictly pseudocontractive mappings {Tn} and {Sn} provided that {Tn}, {Sn}, {αn} and {βn} satisfy some appropriate conditions, and then we apply the result for finding a common element of the set of fixed points of a strict pseudocontraction and the set of solutions of an equilibrium problem in Hilbert spaces. Our results extend and improve the corresponding ones announced by Marino and Xu 11 and others. Throughout the paper, we will use the following notation: i → for strong convergence and ⇀ for weak convergence, ii ωw xn {x : ∃xnr ⇀ x} denotes the weak ω-limit set of {xn}. 4 Abstract and Applied Analysis 2. Preliminaries This section collects some definitions and lemmas which will be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive. Lemma 2.1. LetH be a real Hilbert space. There holds the following identity: i ‖x − y‖2 ‖x‖2 − ‖y‖2 − 2〈x − y, y〉 for all x, y ∈ H. ii ‖tx 1 − t y‖2 t‖x‖2 1 − t ‖y‖2 − t 1 − t ‖x − y‖2 for all t ∈ 0, 1 , for all x, y ∈ H. Lemma 2.2. Let H be a real Hilbert space. Given a closed convex subset C ⊂ H and x, y, z ∈ H. Given also a real number a ∈ R. The set { v ∈ C : ∥∥y − v∥∥2 ‖x − v‖ 〈z, v〉 a } 2.1 is convex (and closed). Recall that given a closed convex subset C of a real Hilbert space H, the nearest point projection PC fromH onto C assigns to each x ∈ H its nearest point denoted by PCx which is a unique point in C with the property ‖x − PCx‖ ‖x − z‖ ∀z ∈ C. 2.2 Lemma 2.3. Let C be a closed convex subset of real Hilbert space H. Given x ∈ H and z ∈ C. Then, z PCx if and only if there holds the relation 〈 x − z, z − y〉 0 ∀y ∈ C. 2.3 Lemma 2.4 Martinez-Yanes and Xu 8 . Let C be a closed convex subset of real Hilbert spaceH. Let {xn} be a sequence in H and u ∈ H. Let q PCu. If {xn} is such that ωw xn ⊂ C and satisfies the condition ‖xn − u‖ ∥ ∥u − q∥∥ ∀n. 2.4 Then, xn → q. Given a closed convex subset C of a real Hilbert space H and a mapping T : C → C. Recall that T is said to be a quasistrict pseudocontraction if F T is nonempty and there exists a constant 0 k < 1 such that ∥ ∥Tx − p∥∥2 ∥∥x − p∥∥2 k‖x − Tx‖ 2.5 for all x ∈ C and p ∈ F T . Abstract and Applied Analysis 5 Proposition 2.5 Marino and Xu 11, Proposition 2.1 . Assume C is a closed convex subset of a Hilbert spaceH, and let T : C → C be a self-mapping of C.and Applied Analysis 5 Proposition 2.5 Marino and Xu 11, Proposition 2.1 . Assume C is a closed convex subset of a Hilbert spaceH, and let T : C → C be a self-mapping of C. i If T is a k-strict pseudocontraction, then T satisfies Lipschitz condition ∥ ∥Tx − Ty∥∥ 1 k 1 − k ∥ ∥x − y∥∥ ∀x, y ∈ C. 2.6 ii If T is a k-strict pseudocontraction, then the mapping I − T is demiclosed (at 0). That is, if {xn} is a sequence in C such that xn ⇀ x̂ and I − T xn → 0, then I − T x̂ 0. iii If T is a k-quasistrict pseudocontraction, then the fixed-point set F T of T is closed and convex so that the projection PF T is well defined. Lemma 2.6 Plubtieng and Ungchittrakool 12, Lemma 3.1 . Let C be a nonempty subset of a Banach space E and {Tn} a sequence of mappings from C into E. Suppose that for any bounded subset B of C there exists continuous increasing function hB from R into R such that hB 0 0 and lim k,l→∞ ρ l 0, 2.7