Strong Convergence Theorems for Countable Families of Uniformly Quasi-φ-Asymptotically Nonexpansive Mappings and a System of Generalized Mixed Equilibrium Problems

and Applied Analysis 3 If θ ≡ 0, the problem 1.4 reduces into the minimize problem, denoted by arg min φ , which is to find x ∈ C such that φ ( y ) − φ x ≥ 0, ∀y ∈ C. 1.7 The above formulation 1.5 was shown in 11 to covermonotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In other words, the EP θ is an unifyingmodel for several problems arising in physics, engineering, science, optimization, economics, and so forth. Some solution methods have been proposed to solve the EP θ ; see, for example, 11–24 and references therein. A point x ∈ C is a fixed point of a mapping S : C → C if Sx x, by F S denote the set of fixed points of S; that is, F S {x ∈ C : Sx x}. Recall that S is said to be nonexpansive if ∥Sx − Sy∥ ≤ ∥x − y∥, ∀x, y ∈ C. 1.8 S is said to be quasi-nonexpansive if F S / ∅ and ∥x − Sy∥≤ ∥x − y∥, ∀x ∈ F S , y ∈ C. 1.9 S is said to be asymptotically nonexpansive if there exists a sequence {kn} ⊂ 1,∞ with kn → 1 as n → ∞ such that ∥Snx − Sny∥ ≤ kn ∥x − y∥, ∀x, y ∈ C, ∀n ≥ 1. 1.10 S is said to be asymptotically quasi-nonexpansive if F S / ∅ and there exists a sequence {kn} ⊂ 1,∞ with kn → 1 as n → ∞ such that ∥x − Sny∥≤ kn ∥x − y∥, ∀x ∈ F S , y ∈ C, ∀n ≥ 1. 1.11 Recall that a point p in C is said to be an asymptotic fixed point of S 25 if C contains a sequence {xn} which converges weakly to p such that limn→∞‖xn − Sxn‖ 0. The set of asymptotic fixed points of S will be denoted by F̃ S . Let E be a real Banach space with norm ‖ · ‖, let C be a nonempty closed convex subset of E, and let E∗ denote the dual of E. Let 〈·, ·〉 denote the duality pairing of E∗ and E. If E is a Hilbert space, 〈·, ·〉 denotes an inner product on E. Consider the functional defined by φ ( x, y ) ‖x‖ − 2x, Jy ∥y∥2, for x, y ∈ E, 1.12 where J : E → 2E is the normalized duality mapping. A mapping S from C into itself is said to be relatively nonexpansive 26–28 if F̃ S F S / ∅ and φ ( p, Sx ) ≤ φp, x, ∀x ∈ C, p ∈ F S . 1.13 4 Abstract and Applied Analysis S is said to be relatively asymptotic nonexpansive 29 if F̃ S F S / ∅ and there exists a sequence {kn} ⊂ 1,∞ with kn → 1 as n → ∞ such that φ ( p, Sx ) ≤ knφ ( p, x ) , ∀x ∈ C, p ∈ F S , n ≥ 1. 1.14 The asymptotic behavior of a relatively nonexpansive mapping was studied in 30–32 . S is said to be φ-nonexpansive if φ ( Sx, Sy ) ≤ φx, y, ∀x, y ∈ C. 1.15 S is said to be quasi φ-nonexpansive 17, 33, 34 if F S / ∅ and φ ( p, Sx ) ≤ φp, x, ∀x ∈ C, p ∈ F S . 1.16 S is said to be φ-asymptotically nonexpansive if there exists a real sequence {kn} ⊂ 1,∞ with kn → 1 as n → ∞ such that φ ( Sx, Sy ) ≤ knφ ( x, y ) , ∀x, y ∈ C. 1.17 S is said to be quasi φ-asymptotically nonexpansive 34, 35 if F S / ∅ and there exists a real sequence {kn} ⊂ 1,∞ with kn → 1 as n → ∞ such that φ ( p, Sx ) ≤ knφ ( p, x ) , ∀x ∈ C, p ∈ F S , n ≥ 1. 1.18 A mapping S is said to be closed if for any sequence {xn} ⊂ C with xn → x and Sxn → y, then Sx y. Remark 1.1. It is easy to know that each relatively nonexpansive mapping is closed. The class of quasi-φ-asymptotically nonexpansive mappings contains properly the class of quasiφ-nonexpansive mappings as a subclass, and the class of quasi-φ-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse is not true see for more detail 30–32, 36 . As well known if C is a nonempty closed convex subset of a Hilbert space H and PC : H → C is the metric projection of H onto C, then PC is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently, it is not available in more general Banach spaces. In this connection, Alber 1 recently introduced the generalized projectionΠC : E → C is a map that assigns to an arbitrary point x ∈ E the minimum point of the functional φ x, y ; that is, ΠCx x, where x is the solution to the minimization problem φ x, x inf y∈C φ ( y, x ) . 1.19 Abstract and Applied Analysis 5 The existence and uniqueness of the operatorΠC follows from the properties of the functional φ y, x and the strict monotonicity of the mapping J see, e.g., 1, 37–40 . It is obvious from the definition of function φ thatand Applied Analysis 5 The existence and uniqueness of the operatorΠC follows from the properties of the functional φ y, x and the strict monotonicity of the mapping J see, e.g., 1, 37–40 . It is obvious from the definition of function φ that ∥y ∥ ∥ − ‖x‖2 ≤ φy, x ≤ ∥y∥ − ‖x‖2, ∀x, y ∈ E. 1.20 If E is a Hilbert space, then φ y, x ‖y − x‖ andΠC becomes the metric projection of E onto C. Next we recall the concept of the generalized f-projection operator. Let G : C × E∗ → R ∪ { ∞} be a functional defined as follows: G ξ, ‖ξ‖ − 2〈ξ, 〉 ‖ ‖ 2ρf ξ , 1.21 where ξ ∈ C, ∈ E∗, ρ is positive number, and f : C → R ∪ { ∞} is proper, convex, and lower semicontinuous. From definitions of G and f , it is easy to see the following properties: 1 G ξ, is convex and continuous with respect to when ξ is fixed; 2 G ξ, is convex and lower semicontinuous with respect to ξ when is fixed. Let E be a real Banach space with its dual E∗. Let C be a nonempty closed convex subset of E. We say that π C : E ∗ → 2 is generalized f-projection operator if π f C { u ∈ C : G u, inf ξ∈C G ξ, , ∀ ∈ E∗ } . 1.22 In 2005, Matsushita and Takahashi 36 proposed the following hybrid iteration method it is also called the CQ method with generalized projection for relatively nonexpansive mapping S in a Banach space E: x0 ∈ C, chosen arbitrarily, yn J−1 αnJxn 1 − αn JSxn , Cn { z ∈ C : φz, yn ) ≤ φ z, xn } , Qn {z ∈ C : 〈xn − z, Jx0 − Jxn〉 ≥ 0}, xn 1 ΠCn∩Qnx0. 1.23 They proved that {xn} converges strongly to ΠF S x0, where ΠF S is the generalized projection from C onto F S . Motivated by the results of Takahashi and Zembayashi 41 , Cholamjiak and Suantai 12 proved the following strong convergence theorem by the hybrid iterative scheme for 6 Abstract and Applied Analysis approximation of common fixed point of countable families of relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space: x0 ∈ E, x1 ΠC1x0, C1 C, yn,i J−1 αnJxn 1 − αn JSxn , un,i T θm rm,nT θm−1 rm−1,n · · · T1 r1,nyn,i, Cn 1 { z ∈ Cn : supi>1φ z, Jun,i ≤ φ z, Jxn } , xn 1 ΠCn 1x0, n ≥ 1. 1.24 Then, they proved that under certain appropriate conditions imposed on {αn} and {rn,i}, the sequence {xn} converges strongly to ΠCn 1x0. In 2010, Li et al. 42 introduced the following hybrid iterative scheme for the approximation of fixed point of relatively nonexpansive mapping using the properties of generalized f-projection operator in a uniformly smooth real Banach space which is also uniformly convex: x0 ∈ C, yn J−1 αnJxn 1 − αn JSxn , Cn 1 { w ∈ Cn : G ( w, Jyn ) ≤ G w, Jxn } , xn 1 Π f Cn 1 x0, n ≥ 0, 1.25 whereΠfC : E → 2 is generalized f-projection operator. They proved the strong convergence theorem for finding an element in the fixed point set of S. We remark here that the results of Li et al. 42 extended and improved on the results of Matsushita and Takahashi 36 . Recently, Shehu 43 introduced a new iterative scheme by hybrid methods and proved strong convergence theorem for the approximation of a common fixed point of two countable families of weak relatively nonexpansive mappings which is also a solution to a system of generalized mixed equilibrium problems in a uniformly convex and uniformly smooth Banach space by using the properties of the generalized f-projection operator. Chang et al. 44 used the modified block iterative method to propose an iterative algorithm for solving the convex feasibility problems for an infinite family of quasi-φ-asymptotically nonexpansive mappings. Very recently, Kim 45 and Saewan and Kumam 46 considered the shrinking projection methods for asymptotically quasi-φ-nonexpansive mappings in a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property. In this paper, we introduce a new hybrid block iterative scheme of the generalized fprojection operator for finding a common element of the fixed point set of uniformly quasi-φasymptotically nonexpansive mappings and the set of solutions of the system of generalized mixed equilibrium problems in a uniformly smooth and strictly convex Banach spacewith the Kadec-Klee property. Then, we prove that our new iterative scheme converges strongly to a common element of the aforementioned sets. The results presented in this paper improve and extend the results of Shehu 43 , Chang et al. 44 , Li et al. 42 , Takahashi and Zembayashi 41 , Cholamjiak and Suantai 12 , and many authors. Abstract and Applied Analysis 7 2. Preliminaries A Banach space E is said to be strictly convex if ‖ x y /2‖ < 1 for all x, y ∈ E with ‖x‖ ‖y‖ 1 and x / y. Let U {x ∈ E : ‖x‖ 1} be the unit sphere of E. Then a Banach space E is said to be smooth if the limit limt→ 0 ‖x ty‖ − ‖x‖ /t exists for each x, y ∈ U. It is also said to be uniformly smooth if the limit exists uniformly in x, y ∈ U. Let E be a Banach space. The modulus of smoothness of E is the function ρE : 0,∞ → 0,∞ defined by ρE t sup{ ‖x y‖ ‖x − y‖ /2 − 1 : ‖x‖ 1, ‖y‖ ≤ t}. The modulus of convexity of E is the function δE : 0, 2 → 0, 1 defined by δE ε inf{1 − ‖ x y /2‖ : x, y ∈ E, ‖x‖ ‖y‖ 1, ‖x − y‖ ≥ ε}. The normalized duality mapping J : E → 2E is defined by J x {x∗ ∈ E∗ : 〈x, x∗〉 ‖x‖, ‖x∗‖ ‖x‖}. If E is a Hilbert space, then J I, where I is the identity mapping. Remark 2.1. If E is a reflexive, strictly convex and smooth Banach space, then for x, y ∈ E, φ x, y 0 if and only if x y. It is sufficient to show that if φ x, y 0, then x y. From 1.12 , we have ‖x‖ ‖y‖. This implies that 〈x, Jy〉 ‖x‖ ‖Jy‖. From the definition of J , one has Jx Jy. Therefore, we have x y; see 38, 40 for more details. Recall that a Banach space E has the Kadec-Klee property 38, 40, 47 , if for any sequence {xn} ⊂ E and x ∈ E with xn ⇀ x and ‖xn‖ → ‖x‖, then ‖xn − x‖ → 0 as n → ∞. It is well known that if E is a uniformly convex Banach space, then E has the Kadec-Klee property. Remark 2.2. Let E be a Banach space. Then we know thatand Applied Analysis 7 2. Preliminaries A Banach space E is said to be strictly convex if ‖ x y /2‖ < 1 for all x, y ∈ E with ‖x‖ ‖y‖ 1 and x / y. Let U {x ∈ E : ‖x‖ 1} be the unit sphere of E. Then a Banach space E is said to be smooth if the limit limt→ 0 ‖x ty‖ − ‖x‖ /t exists for each x, y ∈ U. It is also said to be uniformly smooth if the limit exists uniformly in x, y ∈ U. Let E be a Banach space. The modulus of smoothness of E is the function ρE : 0,∞ → 0,∞ defined by ρE t sup{ ‖x y‖ ‖x − y‖ /2 − 1 : ‖x‖ 1, ‖y‖ ≤ t}. The modulus of convexity of E is the function δE : 0, 2 → 0, 1 defined by δE ε inf{1 − ‖ x y /2‖ : x, y ∈ E, ‖x‖ ‖y‖ 1, ‖x − y‖ ≥ ε}. The normalized duality mapping J : E → 2E is defined by J x {x∗ ∈ E∗ : 〈x, x∗〉 ‖x‖, ‖x∗‖ ‖x‖}. If E is a Hilbert space, then J I, where I is the identity mapping. Remark 2.1. If E is a reflexive, strictly convex and smooth Banach space, then for x, y ∈ E, φ x, y 0 if and only if x y. It is sufficient to show that if φ x, y 0, then x y. From 1.12 , we have ‖x‖ ‖y‖. This implies that 〈x, Jy〉 ‖x‖ ‖Jy‖. From the definition of J , one has Jx Jy. Therefore, we have x y; see 38, 40 for more details. Recall that a Banach space E has the Kadec-Klee property 38, 40, 47 , if for any sequence {xn} ⊂ E and x ∈ E with xn ⇀ x and ‖xn‖ → ‖x‖, then ‖xn − x‖ → 0 as n → ∞. It is well known that if E is a uniformly convex Banach space, then E has the Kadec-Klee property. Remark 2.2. Let E be a Banach space. Then we know that 1 if E is an arbitrary Banach space, then J is monotone and bounded; 2 if E is strictly convex, then J is strictly monotone; 3 if E is smooth, then J is single valued and semicontinuous; 4 if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E; 5 E is reflexive, smooth, and strictly convex, then the normalized duality mapping J J2 is single valued, one-to-one, and onto; 6 if E is uniformly smooth, then E is smooth and reflexive; 7 E is uniformly smooth if and only if E∗ is uniformly convex; see 38 for more details. We also need the following lemmas for the proof of our main results. For solving the equilibrium problem for a bifunction θ : C×C → R, let us assume that θ satisfies the following conditions. A1 θ x, x 0, for all x ∈ C. A2 θ is monotone; that is, θ x, y θ y, x ≤ 0, for all x, y ∈ C. 8 Abstract and Applied Analysis A3 for each x, y, z ∈ C, lim t↓0 θ ( tz 1 − t x, y ≤ θx, y. 2.1 A4 for each x ∈ C, y → θ x, y is convex and lower semicontinuous. For example, let A be a continuous and monotone operator of C into E∗ and define θ ( x, y ) 〈 Ax, y − x, ∀x, y ∈ C. 2.2 Then, θ satisfies A1 – A4 . The following result is in Blum and Oettli 11 . Motivated by Combettes and Hirstoaga 13 in a Hilbert space and Takahashi and Zembayashi 48 in a Banach space, Zhang 49 obtained the following lemma. Lemma 2.3 Liu et al. 50 , Zhang 49, Lemma 1.5 . Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Let θ be a bifunction from C × C to R satisfying (A1)– (A4), let A : C → E∗ be a continuous and monotone mapping, let φ : C → R be a semicontinuous and convex functional, for r > 0, and let x ∈ E. Then, there exists z ∈ C such that Q ( z, y ) 1 r 〈 y − z, Jz − Jx ≥ 0, ∀y ∈ C, 2.3 where Q z, y θ z, y 〈Az, y − z〉 φ y − φ z . Furthermore, define a mapping Tr : E → C as follows: Trx { z ∈ C : Qz, y 1 r 〈 y − z, Jz − Jx ≥ 0, ∀y ∈ C } . 2.4 Then the following holds. 1 Tr is single-valued. 2 Tr is firmly nonexpansive; that is, for all x, y ∈ E, 〈Trx − Try, JTrx − JTry〉 ≤ 〈Trx − Try, Jx − Jy〉. 3 F Tr F̃ Tr GMEP θ,A, φ . 4 GMEP θ,A, φ is closed and convex. 5 φ p, Trz φ Trz, z ≤ φ p, z , for all p ∈ F Tr and z ∈ E. For the generalized f-projection operator, Wu and Huang 4 proved the following basic properties. Lemma 2.4 Wu and Huang 4 . Let E be a reflexive Banach space with its dual E∗ and let C be a nonempty closed convex subset of E. The following statements hold. 1 π C is nonempty closed convex subset of C for all ∈ E∗. Abstract and Applied Analysis 9 2 If E is smooth, then for all ∈ E∗, x ∈ π C if and only ifand Applied Analysis 9 2 If E is smooth, then for all ∈ E∗, x ∈ π C if and only if 〈 x − y, − Jx ρfy − ρf x ≥ 0, ∀y ∈ C. 2.5 3 If E is strictly convex and f : C → R∪{ ∞} is positive homogeneous (i.e., f tx tf x for all t > 0 such that tx ∈ C where x ∈ C), then π C is single-valued mapping. Recently, Fan et al. 5 have shown that the condition f which is positive homogeneous and appeared in 5, Lemma 2.1 iii can be removed. Lemma 2.5 Fan et al. 5 . Let E be a reflexive Banach space with its dual E∗, and let C be a nonempty closed convex subset of E. If E is strictly convex, then π C is single valued. Recall that J is single value mapping when E is a smooth Banach space. There exists a unique element ∈ E∗ such that Jx where x ∈ E. This substitution for 1.21 gives G ξ, Jx ‖ξ‖ − 2〈ξ, Jx〉 ‖x‖ 2ρf ξ . 2.6 Now we consider the second generalized f-projection operator in Banach spaces see 42 . Definition 2.6. Let E be a real smooth Banach space, and let C be a nonempty closed convex subset of E. We say that ΠfC : E → 2 is generalized f-projection operator if ΠfCx { u ∈ C : G u, Jx inf ξ∈C G ξ, Jx , ∀x ∈ E } . 2.7 Lemma 2.7 Deimling 51 . Let E be a Banach space, and let f : E → R ∪ { ∞} be a lower semicontinuous convex functional. Then there exist x∗ ∈ E∗ and α ∈ R such that f x ≥ 〈x, x∗〉 α, ∀x ∈ E. 2.8 Lemma 2.8 Li et al. 42 . Let E be a reflexive smooth Banach space, and C let be a nonempty closed convex subset of E. The following statements hold. 1 ΠfCx is nonempty closed convex subset of C for all x ∈ E. 2 For all x ∈ E, x̂ ∈ ΠfCx if and only if 〈 x̂ − y, Jx − Jx̂ ρfy − ρf x̂ ≥ 0, ∀y ∈ C. 2.9 3 If E is strictly convex, thenΠfC is single-valued mapping. 10 Abstract and Applied Analysis Lemma 2.9 Li et al. 42 . Let E be a reflexive smooth Banach space and let C be a nonempty closed convex subset of E, and let x ∈ E, x̂ ∈ ΠfCx. Then φ ( y, x̂ ) G x̂, Jx ≤ Gy, Jx, ∀y ∈ C. 2.10 Remark 2.10. Let E be a uniformly convex and uniformly smooth Banach space, and let f x 0 for all x ∈ E. Then Lemma 2.9 reduces to the property of the generalized projection operator considered by Alber 1 . Lemma 2.11 Li et al. 42 . Let E be a Banach space, and let f : E → R ∪ { ∞} be a proper, convex, and lower semicontinuous mapping with convex domainD f . If {xn} is a sequence inD f such that xn ⇀ x̂ ∈ D f and limn→∞G xn, Jy G x̂, Jy , then limn→∞‖xn‖ ‖x̂‖. Lemma 2.12 Chang et al. 44 . Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and let C be a nonempty closed convex subset of E. Let S : C → C be a closed and quasi-φ-asymptotically nonexpansive mapping with a sequence {kn} ⊂ 1,∞ , kn → 1. Then F S is a closed convex subset of C. Lemma 2.13 Chang et al. 44 . Let E be a uniformly convex Banach space, let r > 0 be a positive number, and let Br 0 be a closed ball of E. Then, for any given sequence {xi}i 1 ⊂ Br 0 and for any given sequence {λi}i 1 of positive number with ∑∞ n 1 λn 1, there exists a continuous, strictly increasing, and convex function g : 0, 2r → 0,∞ with g 0 0 such that, for any positive integer i, j with i < j, ∥∥∥∥ ∞ ∑ n 1 λnxn ∥∥∥∥ 2 ≤ ∞ ∑ n 1 λn‖xn‖ − λiλjg ∥xi − xj ∥. 2.11 Definition 2.14. Chang et al. 44 . 1 Let {Si}i 1 : C → C be a sequence of mappings. {Si}i 1 is said to be a family of uniformly quasi-φ-asymptotically nonexpansive mappings, if F : ∩i 1F Si / ∅, and there exists a sequence {kn} ⊂ 1,∞ with kn → 1 such that for each i ≥ 1 φ ( p, Sni x ) ≤ knφ ( p, x ) , ∀p ∈ F, x ∈ C, ∀n ≥ 1. 2.12 2 Amapping S : C → C is said to be uniformly L-Lipschitz continuous, if there exists a constant L > 0 such that ∥Snx − Sny∥ ≤ L∥x − y∥, ∀x, y ∈ C. 2.13 If f x ≥ 0, it is clearly by the definition of mappings {Si}i 1 is a family of uniformly quasi-φ-asymptotically nonexpansive is equivalent to if ∩i 1F Si / ∅ and there exists a sequence {kn} ⊂ 1,∞ with kn → 1 such that for each i ≥ 1, G ( p, JSni x ) ≤ knG ( p, Jx ) , ∀p ∈ F, x ∈ C, ∀n ≥ 1. 2.14 Abstract and Applied Analysis 11 3. Strong Convergence Theorem Now we state and prove our main result. Theorem 3.1. Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let {Si}i 1 : C → C be an infinite family of closed uniformly Li-Lipschitz continuous and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence {kn} ⊂ 1,∞ , kn → 1, and let f : E → R be a convex lower semicontinuous mapping with C ⊂ int D f . For each j 1, 2, . . . , m, let θj be a bifunction from C × C to R which satisfies conditions (A1)–(A4), let Aj : C → E∗ be a continuous and monotone mapping, and let φj : C → R be a lower semicontinuous and convex function. Assume that F : ∩i 1F Si ∩ ∩j 1 GMEP θj ,Aj , φj / ∅. For an initial point x0 ∈ E with x1 Π f C1 x0 and C1 C, we define the sequence {xn} as follows: zn J−1 ( αn,0Jxn ∞ ∑and Applied Analysis 11 3. Strong Convergence Theorem Now we state and prove our main result. Theorem 3.1. Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let {Si}i 1 : C → C be an infinite family of closed uniformly Li-Lipschitz continuous and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence {kn} ⊂ 1,∞ , kn → 1, and let f : E → R be a convex lower semicontinuous mapping with C ⊂ int D f . For each j 1, 2, . . . , m, let θj be a bifunction from C × C to R which satisfies conditions (A1)–(A4), let Aj : C → E∗ be a continuous and monotone mapping, and let φj : C → R be a lower semicontinuous and convex function. Assume that F : ∩i 1F Si ∩ ∩j 1 GMEP θj ,Aj , φj / ∅. For an initial point x0 ∈ E with x1 Π f C1 x0 and C1 C, we define the sequence {xn} as follows: zn J−1 ( αn,0Jxn ∞ ∑ i 1 αn,iJS n i xn ) , yn J−1 ( βnJxn ( 1 − βn ) Jzn ) , un T Qm rm,nT Qm−1 rm−1,n · · · T2 r2,nTQ r1,nyn, Cn 1 {z ∈ Cn : G z, Jun ≤ G z, Jxn kn − 1 Mn}, xn 1 Π f Cn 1 x0, ∀n ≥ 1, 3.1 whereMn supq∈F{G q, Jxn }, {αn,i}, {βn} are sequences in 0, 1 , and ∑∞ i 0 αn,i 1, for all n ≥ 0, satisfy the following conditions. i {rj,n} ⊂ d,∞ for some d > 0. ii lim infn→∞αn,0αn,i > 0 for all i ≥ 1, and lim infn→∞ 1 − βn > 0. iii f x ≥ 0 for all x ∈ C. and f 0 0. Then {xn} converges strongly to p ∈ F, where p ΠfFx0. Proof. We split the proof into six steps. Step 1. We first show that Cn 1 is closed and convex for each n ≥ 1. Clearly C1 C is closed and convex. Suppose that Cn is closed and convex for each n ∈ N. Since for any z ∈ Cn, we know that G z, Jun ≤ G z, Jxn kn − 1 Mn is equivalent to 2〈z, Jxn − Jun 〉 ≤ ‖xn‖ − ‖un‖ kn − 1 Mn, 3.2