Strong convergence theorems for fixed point problems, variational inequality problems and system of generalized mixed equilibrium problems

Mathematical and Computer Modelling 54 (2011) 1510–1522

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Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm

Strong convergence theorems for fixed point problems, variational inequality problems and system of generalized mixed equilibrium problems Yekini Shehu Mathematics Institute, African University of Science and Technology, Abuja, Nigeria

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Article history: Received 12 August 2010 Received in revised form 15 April 2011 Accepted 18 April 2011 Keywords: φ -asymptotically nonexpansive mappings Generalized mixed equilibrium problems Variational inequality problem Hybrid method Banach spaces

The purpose of this paper is to construct a new iterative scheme by hybrid methods to approximate a common element of the common fixed points set of a finite family of φ -asymptotically nonexpansive mappings, the solutions set of a variational inequality problem and the solutions set of a system of generalized mixed equilibrium problems in a 2uniformly convex real Banach space which is also uniformly smooth. Then, we prove strong convergence of the scheme to a common element of the three sets. Our results extend many known recent results in the literature. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Let E be a real Banach space with dual E ∗ and C be nonempty, closed and convex subset of E. A mapping T : C → C is called nonexpansive if

‖Tx − Ty‖ ≤ ‖x − y‖,

∀x, y ∈ C .

(1.1)

A point x ∈ C is called a fixed point of T if Tx = x. The set of fixed points of T is denoted by F (T ) := {x ∈ C : Tx = x}. ∗ We denote by J, the normalized duality mapping from E to 2E defined by J (x) = {f ∈ E ∗ : ⟨x, f ⟩ = ‖x‖2 = ‖f ‖2 }. The following properties of J are well known (the reader can consult [1–3] for more details). (1) (2) (3) (4)

If E is uniformly smooth, then J is norm-to-norm uniformly continuous on each bounded subset of E. J (x) ̸= ∅, x ∈ E. If E is reflexive, then J is a mapping from E onto E ∗ . If E is smooth, then J is single valued.

Throughout this paper, we denote by φ , the functional on E × E defined by

φ(x, y) = ‖x‖2 − 2⟨x, J (y)⟩ + ‖y‖2 ,

∀ x, y ∈ E .

(1.2)

Let C be a nonempty subset of E and let T be a mapping from C onto E. A point p ∈ C is said to be an asymptotic fixed point of T if C contains a sequence {xn }∞ n=0 which converges weakly to p and limn→∞ ‖xn − Txn ‖ = 0. The set of asymptotic fixed  points of T is denoted by F (T ). We say that a mapping T is relatively nonexpansive (see, for example, [4–8]) if the following conditions are satisfied: E-mail address: [email protected]. 0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.04.023

Y. Shehu / Mathematical and Computer Modelling 54 (2011) 1510–1522

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(R1) F (T ) ̸= ∅; (R2) φ(p, Tx) ≤ φ(p, x), ∀x ∈ C , p ∈ F (T ); (R3) F (T ) =  F (T ). If T satisfies (R1) and (R2), then T is said to be relatively quasi-nonexpansive. It is easy to see that the class of relatively quasinonexpansive mappings contains the class of relatively nonexpansive mappings. Many authors have studied the methods of approximating the fixed points of relatively quasi-nonexpansive mappings (see, for example, [9–11] and the references cited therein). Clearly, in Hilbert space H, relatively quasi-nonexpansive mappings and quasi-nonexpansive mappings are the same, for φ(x, y) = ‖x − y‖2 , ∀x, y ∈ H and this implies that

φ(p, Tx) ≤ φ(p, x) ⇔ ‖Tx − p‖ ≤ ‖x − p‖,

∀x ∈ C , p ∈ F (T ).

The examples of relatively quasi-nonexpansive mappings are given in [10]. T is called relatively asymptotically nonexpansive if there exists {kn } ⊂ [1, ∞) and F (T ) =  F (T ) such that φ(p, T n x) ≤ kn φ(p, x), ∀n ≥ 1, x ∈ C , p ∈ F (T ), where kn → 1 as n → ∞. A mapping T from C onto itself is said to be φ -nonexpansive (nonextensive) [12] if φ(Tx, Ty) ≤ φ(x, y), ∀x, y ∈ C and it is called φ -asymptotically nonexpansive if there exists {kn } ⊂ [1, ∞) such that φ(T n x, T n y) ≤ kn φ(x, y), ∀n ≥ 1, x, y ∈ C , where kn → 1 as n → ∞. Clearly, the class of relatively nonexpansive mappings is contained in the class of relatively asymptotically nonexpansive mappings and the class of φ -nonexpansive mappings is contained in the class of φ -asymptotically nonexpansive mappings. Moreover, if E = H, a Hilbert space, the class of φ -nonexpansive mappings and the class of φ -asymptotically nonexpansive mappings reduce to the class of nonexpansive and the class of asymptotically nonexpansive mappings respectively. Furthermore, we note that weakly closedness of nonexpansive and asymptotically nonexpansive mappings in Hilbert spaces implies that every nonexpansive mapping (asymptotically nonexpansive mapping) is relatively nonexpansive (relatively asymptotically nonexpansive) mapping. An operator B : C → E ∗ is called α -inverse-strongly monotone, if there exists a positive real number α such that

⟨x − y, Bx − By⟩ ≥ α‖Bx − By‖2 ,

∀ x, y ∈ C ,

(1.3)

and A is said to be monotone if

⟨x − y, Ax − Ay⟩ ≥ 0,

∀x, y ∈ C .

(1.4)

Let B be a monotone operator from C onto E ∗ , the classical variational inequality [13], denoted by VI (C , B), is to find x∗ ∈ C such that

⟨y − x, Bx∗ ⟩ ≥ 0,

∀y ∈ C .

(1.5)

The variational inequality (1.5) is connected with the convex minimization problem, the complementarity problem, the problem of finding a point x∗ ∈ E such that Bx∗ = 0 and so on. In this paper, we shall assume that (B1) B is α -inverse strongly monotone; (B2) ‖By‖ ≤ ‖By − Bu‖ for all y ∈ C and u ∈ VI (C , B); (B3) VI (C , B) ̸= ∅. Let ϕ : C → R be a real-valued function, let A : C → E ∗ be a nonlinear mapping, let F : C × C → R be a bifunction. The generalized mixed equilibrium problem is to find x ∈ C (see, for example, [14–16]) such that F (x, y) + ϕ(y) − ϕ(x) + ⟨Ax, y − x⟩ ≥ 0,

(1.6)

for all y ∈ C . We shall denote the solutions set of (1.6) by GMEP(F , ϕ). Thus GMEP(F , ϕ) := {x∗ ∈ C : F (x∗ , y) + ϕ(y) − ϕ(x∗ ) + ⟨Ax∗ , y − x∗ ⟩ ≥ 0, ∀y ∈ C }. If ϕ = 0 and A = 0, then problem (1.6) reduces to an equilibrium problem studied by many authors (see, for example, [17– 24]), which is to find x∗ ∈ C such that F (x∗ , y) ≥ 0

(1.7)

for all y ∈ C . If ϕ = 0 and E = H (a real Hilbert space), then problem (1.6) reduces to a generalized equilibrium problem studied by many authors (see, for example, [25–27]), which is to find x∗ ∈ C such that F (x∗ , y) + ⟨Ax∗ , y − x∗ ⟩ ≥ 0

(1.8)

for all y ∈ C . If A = 0 and E = H, then problem (1.6) reduces to mixed equilibrium problem considered by many authors (see, for example, [28–30]), which is to find x∗ ∈ C such that F (x∗ , y) + ϕ(y) − ϕ(x∗ ) ≥ 0 for all y ∈ C .

(1.9)

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The mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems and equilibrium problems as special cases (see, for example, [31]). Some methods have been proposed to solve the mixed equilibrium problem (see, for example, [29,30,32]). Numerous problems in Physics, optimization and economics reduce to find a solution of problem (1.8). Iiduka et al. [33] studied the following iterative scheme: x0 ∈ C ,

 y = PC (xn − λn Bxn ),   n Cn = {w ∈ C : ‖yn − w‖ ≤ ‖xn − w‖} (1.10)  Qn = {w ∈ C : ⟨xn − w, x0 − xn ⟩ ≥ 0} xn+1 = ΠCn Qn x0 , n ≥ 0, where {λn } is a sequence in [0, 2α]. They proved that the sequence {xn } generated by (1.10) converges strongly PVI (C ,B) x0 , where PVI (C ,B) is a metric projection of C onto VI (C , B). In [34], Iiduka and Takahashi studied the following iterative scheme for finding a zero of a monotone mapping B in a 2-uniformly convex and uniformly smooth Banach space E: x0 ∈ C ,

 −1  yn = J (Jxn − λn JBxn ), Cn = {w ∈ C : φ(w, yn ) ≤ φ(w, xn )}  Qn = {w ∈ C : ⟨xn − w, Jx0 − Jxn ⟩ ≥ 0} xn+1 = ΠCn Qn x0 , n ≥ 0,

(1.11)

where ΠCn  Qn is the generalized projection of E onto Cn Qn , J is the duality mapping from E onto E ∗ and {λn } is a positive real sequence. Then, they proved that the sequence {xn } generated by (1.11) converges strongly to an element x∗ of B−1 (0) provided B is α -inverse strongly monotone and B−1 (0) ̸= ∅. Moreover, under additional assumption (B2), it is proved that {xn } converges strongly to x∗ ∈ VI (C , B). In [35], Kim and Xu introduced and studied the following iteration scheme for asymptotically nonexpansive mapping defined on a nonempty closed convex and bounded subset C of a Hilbert space:



 x0 ∈ C ,    yn = αn xn + (1 − αn )T n xn , Cn = {v ∈ C : ‖yn − v‖2 ≤ ‖xn − v‖2 + θn }   Qn = {v ∈ C : ⟨xn − v, x0 − xn ⟩ ≥ 0},  xn+1 = PCn ∩Qn (x0 ), ∀ n ∈ N,

(1.12)

where θn = (1 − αn )(k2n − 1)(diam(C ))2 → 0 as n → ∞. Kim and Xu [35] proved that if αn ≤ a for all n ∈ N and for some a ∈ (0, 1), then the sequence {xn }n≥0 defined by (1.12) converges strongly to PF (T ) (x0 ). Recently, Zegeye and Shahzad [36] (see also [37]) introduced the following hybrid iterative scheme for approximation of common fixed point of finite family of asymptotically nonexpansive mappings {Ti }ri=1 in a real Hilbert space:

 x ∈ C,   0  yn = αn0 xn + αn1 T1n xn + αn2 T2n xn + · · · + αnr Trn xn , Cn = {v ∈ C : ‖yn − v‖2 ≤ ‖xn − v‖2 + θn }    Qn = {v ∈ C : ⟨xn − v, x0 − xn ⟩ ≥ 0}, xn+1 = PCn ∩Qn (x0 ), ∀n ∈ N,

(1.13)

where θn = [(k2n1 − 1)αn1 + (k2n1 − 1)αn1 + · · · + (k2nr − 1)αnr ](diam(C ))2 → 0 as n → ∞. They proved that if {αni }∞ n=0 ⊂ [ϵ, 1 − ϵ] for some ϵ > 0 and αn0 + αn1 + αn2 + · · · + αnr = 1 for all n ≥ 0, then the sequence {xn }n≥0 defined by (1.13) converges strongly to PF (x0 ), where F := ∩ri=1 F (Ti ). Very recently, Zegeye and Shahzad [38] introduced a hybrid iterative scheme for approximation of fixed point of continuous φ -asymptotically nonexpansive mapping T which is also a solution of an equilibrium problem and a variational inequality problem in a 2-uniformly convex real Banach space which is also uniformly smooth: x0 ∈ C0 = C

 υn = ΠC J −1 (Jxn − λn Bxn )    −1 n   yn = J (αn Jxn + (1 − αn )JT υn ), 1 F (un , y) + ⟨y − un , Jun − Jyn ⟩ ≥ 0, ∀y ∈ C  rn    C = {w ∈ Cn : φ(w, un ) ≤ φ(w, xn ) + θn },  n + 1  xn+1 = ΠCn+1 x0 , n ≥ 1, where θ = (1 − αn )(kn − 1)(2diamC )2 . Then, they proved that {xn }∞ n=0 converges strongly to ΠF x0 , where F := F (T ) ∩ VI (C , B) ∩ EP (F ) ̸= ∅. Motivated by the results Kim and Xu [35], Zegeye and Shahzad [36–38], we prove a strong convergence theorems for approximation of common fixed point of finite family of continuous φ -asymptotically nonexpansive mappings which is also a solution to a variational inequality problem and a common solution to a system of generalized mixed equilibrium problems in a 2-uniformly convex real Banach space which is also uniformly smooth. Our results improve and extend many recent known results in the literature, for example, the results of Kim and Xu [35], Zegeye and Shahzad [36–38].

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2. Preliminaries Let E be a real Banach space. The modulus of smoothness of E is the function ρE : [0, ∞) → [0, ∞) defined by

ρE (τ ) := sup



1 2



(‖x + y‖ + ‖x − y‖) − 1 : ‖x‖ ≤ 1, ‖y‖ ≤ τ .

E is uniformly smooth if and only if

ρE (τ ) = 0. τ Let dimE ≥ 2. The modulus of convexity of E is the function δE : (0, 2] → [0, 1] defined by     x + y  : ‖x‖ = ‖y‖ = 1; ϵ = ‖x − y‖ . δE (ϵ) := inf 1 −   2  lim

τ →0

E is uniformly convex if for any ϵ ∈ (0, 2], there exists a δ = δ(ϵ) > 0 such that if x, y ∈ E with ‖x‖ ≤ 1, ‖y‖ ≤ 1 and ‖x − y‖ ≥ ϵ , then ‖ 12 (x + y)‖ ≤ 1 − δ . Equivalently, E is uniformly convex if and only if δE (ϵ) > 0 for all ϵ ∈ (0, 2]. A normed space E is called strictly convex if for all x, y ∈ E , x ̸= y, ‖x‖ = ‖y‖ = 1, we have ‖λx + (1 − λ)y‖ < 1, ∀λ ∈ (0, 1). E is said to be 2-uniformly convex if there exists constant c > 0 such that δE (ϵ) > c ϵ 2 for all ϵ ∈ (0, 2]. The constant 1c is called the 2-uniformly convexity a constant of E. We know that a 2-uniformly convex Banach space is uniformly convex. We know that the following lemma holds in a 2-uniformly convex Banach space. Lemma 2.1 (Beauzamy [39]). Let E be a 2-uniformly convex Banach space, then for all x, y from any bounded set of E and jx ∈ Jx, jy ∈ Jy, we have

⟨x − y, jx − jy⟩ ≥ where

1 c

c2 2

‖x − y‖2 ,

is the 2-uniformly constant of E.

Let E be a smooth, strictly convex and reflexive real Banach space and let C be a nonempty, closed and convex subset of E. Following Alber [40], the generalized projection ΠC from E onto C is defined by

ΠC (x) := arg min φ(y, x),

∀x ∈ E .

y∈C

The existence and uniqueness of ΠC follows from the property of the functional φ(x, y) and strict monotonicity of the mapping J (see, for example, [3,40–43]). If E is a Hilbert space, then ΠC is the metric projection of H onto C . From [43], in uniformly convex and uniformly smooth Banach spaces, we have

(‖x‖ − ‖y‖)2 ≤ φ(x, y) ≤ (‖x‖ + ‖y‖)2 ,

∀ x, y ∈ E .

(2.1)

We know that the following lemmas hold for generalized projections. Lemma 2.2 (Alber [40], Kamimura and Takahashi [43]). Let C be a nonempty, closed and convex subset of a smooth, strictly convex and reflexive Banach space E. Then

φ(x, ΠC y) + φ(ΠC y, y) ≤ φ(x, y),

∀x ∈ C , ∀y ∈ E .

Lemma 2.3 (Alber [40], Kamimura and Takahashi [43]). Let C be a nonempty, closed and convex subset of a smooth, strictly convex and reflexive Banach space E. Let x ∈ E and z ∈ C . Then z = ΠC x ⇔ ⟨y − z , J (x) − J (z )⟩ ≤ 0,

∀y ∈ C .

The fixed points set F (T ) of a continuous φ -asymptotically nonexpansive mapping is closed and convex as given in the following lemma. Lemma 2.4 (Zegeye and Shahzad [38]). Let C be a nonempty, closed and convex subset of a smooth, uniformly convex Banach space E. Let T be a continuous φ -asymptotically nonexpansive mapping of C onto itself. Then F (T ) is closed and convex. Also, this following lemma will be used in what follows. Lemma 2.5 (Kamimura and Takahashi [43]). Let C be a nonempty, closed and convex subset of a smooth, uniformly convex ∞ ∞ ∞ Banach space E. Let {xn }∞ n=1 and {yn }n=1 be sequences in E such that either {xn }n=1 or {yn }n=1 is bounded. If limn→∞ φ(xn , yn ) = 0, then limn→∞ ‖xn − yn ‖ = 0.

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Lemma 2.6 (Zegeye [44]). Let E be a uniformly convex real Banach space. For arbitrary r > 0, let BL (0) := {x ∈ E : ‖x‖ ≤ L}. Then, there exists a continuous strictly increasing convex function g : [0, 2r ] → R,

g (0) = 0

such that for every x, y ∈ Br (0), the following inequality holds:

‖α0 x0 + α1 x1 + · · · + αr xr ‖2 ≤

r −

αi ‖xi ‖2 − αs αt g (‖xs − xt ‖),

i=1

for any s, t ∈ {0, 1, 2, . . . , r } and for any xi ∈ BL (0), i = 0, 1, 2, . . . , r with

∑r

i =1

αi = 1.

Let C be a nonempty, closed and convex subset of a smooth, uniformly convex Banach space E and J be the duality mapping from E onto E ∗ . Then J −1 is single valued, one–one and surjective and it is the duality mapping from E ∗ onto E. We make use of the following function V as studied by Alber [40]: V (x, x∗ ) = ‖x‖2 − 2⟨x, x∗ ⟩ + ‖x∗ ‖2 for all x ∈ E and x ∈ E . Thus, V (x, x ) = φ(x, J Alber [40]. ∗

∗

∗

(2.2) −1

(x )) for all x ∈ E and x ∈ E . We know the following lemma from ∗

∗

∗

Lemma 2.7 (Alber [40]). Let E be a real reflexive, strictly convex and Banach space and V be as in (2.2). Then V (x, x∗ ) + 2⟨J −1 (x∗ ) − x, y∗ ⟩ ≤ V (x, x∗ + y∗ ) for all x ∈ E and x∗ , y∗ ∈ E ∗ . For solving the equilibrium problem for a bifunction F : C × C → R, let us assume that F satisfies the following conditions: (A1) (A2) (A3) (A4)

F (x, x) = 0 for all x ∈ C ; F is monotone, i.e., F (x, y) + F (y, x) ≤ 0 for all x, y, ∈ C ; for each x, y, z ∈ C , lim supt ↓0 F (tz + (1 − t )x, y) ≤ F (x, y); for each x ∈ C , y → F (x, y) is convex and lower semicontinuous.

Lemma 2.8 (Liu et al. [45], Zhang [46]). Let C be a nonempty, closed and convex subset of a smooth, strictly convex and reflexive Banach space E. Assume that F : C × C → R satisfies (A1)–(A4), A : C → E ∗ be a continuous and monotone mapping and ϕ : C → R be a lower semicontinuous and convex functional. For r > 0 and x ∈ E, there exists z ∈ C such that G(z , y) +

1 r

⟨y − z , Jz − Jx⟩ ≥ 0 for all y ∈ C ,

where G(z , y) = F (z , y) + ⟨Az , y − z ⟩ + ϕ(y) − ϕ(z ), z , y ∈ C . Furthermore, define a mapping TrG : E → C as follows: TrG (x) =



z ∈ C : G(z , y) +

1 r

 ⟨y − z , Jz − Jx⟩ ≥ 0, ∀y ∈ C .

Then, the following hold: (i) TrG is single valued; (ii) For any x, y ∈ E,

⟨TrG x − TrG y, JTrG x − JTrG y⟩ ≤ ⟨TrG x − TrG y, Jx − Jy⟩; (iii) F (TrG ) = GMEP(F , A, ϕ); (iv) GMEP(F , A, ϕ) is closed and convex. Lemma 2.9 (Zhang [46]). Let C be a nonempty, closed and convex subset of a smooth, strictly convex and reflexive Banach space E. Assume that F : C × C → R satisfies (A1)–(A4) and let r > 0. Then for each x ∈ E and q ∈ F (TrG ),

φ(q, TrG x) + φ(TrG x, x) ≤ φ(q, x). For the rest of this paper, the sequence {xn }∞ n=1 converges strongly to p shall be denoted by xn → p as n → ∞. For each k = 1, 2, . . . , m, let Fk be a bifunction from C × C satisfying (A1)–(A4), ϕk : C → R be a lower semicontinuous and convex functional and Ak : C → E ∗ be a continuous and monotone mapping, also for the rest of this paper, we define TrGkk (x) =



z ∈ C : Gk (z , y) +

1 rk

 ⟨y − z , Jz − Jx⟩ ≥ 0, ∀y ∈ C ,

where Gk (z , y) = Fk (z , y) + ϕk (y) − ϕk (z ) + ⟨Ak z , y − z ⟩, z , y ∈ C .

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3. Main results Theorem 3.1. Let E be a 2-uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty closed and convex subset of E. For each k = 1, 2, . . . , m, let Fk be a bifunction from C × C satisfying (A1)–(A4), ϕk : C → R be a lower semicontinuous and convex functional and Ak : C → E ∗ be a continuous and monotone mapping. Suppose B : C → E ∗ is an operator satisfying (B1)–(B3), and {Ti }ri=1 is a finite family of continuous φ -asymptotically nonexpansive r mappings of C onto itself with constants {kni } such that F := VI (C , B) ∩ (∩m k=1 GMEP(Fk , ϕk )) ∩ ∩i=1 F (Ti ) ̸= ∅. Assume that ∞ ∗ ∗ R := sup{‖x ‖ : x ∈ F } < ∞ and suppose kn := max{kni : i = 1, 2, . . . , r } and let {xn }n=0 be iteratively generated by x0 ∈ C , C1 = C , x1 = ΠC1 x0 ,

 −1 (Jxn − λn Bxn )  CJ υ n = Π  −1  yn = J (αn0 J υn + αn1 JT1n υn + αn2 JT2n υn + · · · + αnr JTrn υn ), G un = TrGmm,n Trmm−−11,n · · · TrG22,n TrG11,n yn    C = {w ∈ Cn : φ(w, un ) ≤ φ(w, xn ) + θn },   n+1 xn+1 = ΠCn+1 x0 , n ≥ 1,

(3.1)

where J is the duality mapping on E and θn = (1 − αn0 )(kn − 1) supx∗ ∈F φ(x∗ , υn ) for all υn ∈ C . Suppose {αni }∞ n=1 for each i = 1, 2, . . . , r is a sequence in (0, 1) such that lim infn→∞ αn0 αni > 0, i = 1, 2, . . . , r , αn0 + αn1 + αn2 + · · · + αnr = c α 1, {λn }∞ , where 1c is the 2-uniformly convexity constant of E and n=1 ⊂ [a, b] for some a, b with 0 < a < b < 2 {rk,n }∞ ⊂ ( 0 , ∞), ( k = 1 , 2 , . . . , m ) satisfying lim inf r > 0, (k = 1, 2, . . . , m). Then, {xn }∞ n→∞ k,n n=1 n=0 converges strongly to ΠF x0 . 2

Proof. We first show that Cn , ∀n ≥ 1 is closed and convex. It is obvious that C1 = C is closed and convex. Suppose that Cn is closed and convex for some n > 1. From the definition of Cn+1 , we have that z ∈ Cn+1 implies φ(z , un ) ≤ φ(z , xn ) + θn . This is equivalent to 2(⟨z , Jxn ⟩ − ⟨z , Jun ⟩) ≤ ‖xn ‖2 − ‖un ‖2 + θn . This implies that Cn+1 is closed and convex for the same n > 1. Hence, Cn is closed and convex ∀n ≥ 1. This shows that ΠCn+1 x0 is well defined for all n ≥ 0. G

G

G

G

k−1 By taking θnk = Trkk,n Trk− · · · Tr22,n Tr11,n , k = 1, 2, . . . , m and θn0 = I for all n ≥ 1, we obtain un = θnm yn . 1 ,n

G

We next show that F ⊂ Cn , ∀n ≥ 1. From Lemmas 2.8 and 2.9, one has that Trkk,n , k = 1, 2, . . . , m is relatively nonexpansive mapping. For n = 1, we have F ⊂ C = C1 . Then for each x∗ ∈ F , we obtain

φ(x∗ , un ) ≤ φ(x∗ , θnm yn ) ≤ φ(x∗ , yn ) = φ(x∗ , J −1 (αn0 J υn + αn1 JT1n υn + αn2 JT2n υn + · · · + αnr JTrn υn )) = ‖x∗ ‖2 − 2αn0 ⟨x∗ , J υn ⟩ − 2αn1 ⟨x∗ , JT1n υn ⟩ − 2αn2 ⟨x∗ , JT2n υn ⟩ − · · · − 2αnr ⟨x∗ , JTrn υn ⟩ + ‖αn0 J υn + αn1 JT1n υn + αn2 JT2n υn + · · · + αnr JTrn υn ‖2 ≤ ‖x∗ ‖2 − 2αn0 ⟨x∗ , J υn ⟩ − 2αn1 ⟨x∗ , JT1n υn ⟩ − 2αn2 ⟨x∗ , JT2n υn ⟩ − · · · − 2αnr ⟨x∗ , JTrn υn ⟩ + αn0 ‖J υn ‖2 + αn1 ‖JT1n υn ‖2 + αn2 ‖JT2n υn ‖2 + · · · + αnr ‖JTrn υn ‖2 = αn0 φ(x∗ , υn ) + αn1 φ(x∗ , T1n υn ) + αn2 φ(x∗ , T2n υn ) + · · · + αnr φ(x∗ , Trn υn ) ≤ αn0 φ(x∗ , υn ) + αn1 kn1 φ(x∗ , υn ) + αn2 kn2 φ(x∗ , υn ) + · · · + αnr knr φ(x∗ , υn ) ≤ αn0 φ(x∗ , υn ) + αn1 kn φ(x∗ , υn ) + αn2 kn φ(x∗ , υn ) + · · · + αnr kn φ(x∗ , υn ) = αn0 φ(x∗ , υn ) + (1 − αn0 )kn φ(x∗ , υn ) = φ(x∗ , υn ) + (1 − αn0 )(kn − 1)φ(x∗ , υn ) ≤ φ(x∗ , υn ) + θn .

(3.2)

Now, by Lemmas 2.2 and 2.7, we obtain

φ(x∗ , υn ) = ≤ = ≤ =

φ(x∗ , ΠC J −1 (Jxn − λn Bxn )) φ(x∗ , J −1 (Jxn − λn Bxn )) V (x∗ , Jxn − λn Bxn ) V (x∗ , (Jxn − λn Bxn ) + λn Bxn ) − 2⟨J −1 (Jxn − λn Bxn ) − x∗ , λn Bxn ⟩ V (x∗ , Jxn ) − 2λn ⟨J −1 (Jxn − λn Bxn ) − x∗ , Bxn ⟩

= φ(x∗ , xn ) − 2λn ⟨xn − x∗ , Bxn ⟩ + 2⟨J −1 (Jxn − λn Bxn ) − xn , −λn Bxn ⟩.

(3.3)

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Y. Shehu / Mathematical and Computer Modelling 54 (2011) 1510–1522

From condition (B1) and x∗ ∈ VI (C , B), we obtain

− 2λn ⟨xn − x∗ , Bxn ⟩ = −2λn ⟨xn − x∗ , Bxn − Bx∗ ⟩ − 2λn ⟨xn − x∗ , Bx∗ ⟩ ≤ −2αλn ‖Bxn − Bx∗ ‖2 .

(3.4)

By Lemma 2.1 and condition (B2), we also obtain 2⟨J −1 (Jxn − λn Bxn ) − xn , −λn Bxn ⟩ = 2⟨J −1 (Jxn − λn Bxn ) − J −1 (Jxn ), −λn Bxn ⟩

≤ 2‖J −1 (Jxn − λn Bxn ) − J −1 (Jxn )‖‖λn Bxn ‖ ≤ = ≤

4 c2 4 c2 4 c2

Combining (3.3)–(3.5) and 0 < a ≤ λn ≤ b <

‖(Jxn − λn Bxn ) − (Jxn )‖‖λn Bxn ‖ λ2n ‖Bxn ‖2 λ2n ‖Bxn − Bx∗ ‖2 . c2α , 2

(3.5)

we obtain 4

φ(x∗ , υn ) ≤ φ(x∗ , xn ) − 2αλn ‖Bxn − Bx∗ ‖2 + 2 λ2n ‖Bxn − Bx∗ ‖2 c   2 ∗ = φ(x , xn ) + 2λn 2 λn − α ‖Bxn − Bx∗ ‖2 c

≤ φ(x , xn ). ∗

(3.6)

Combining (3.2) and (3.6), we have

φ(x∗ , un ) ≤ φ(x∗ , xn ) + θn . So, x∗ ∈ Cn . This implies that F ⊂ Cn , ∀n ≥ 1 and the sequence {xn }∞ n=0 generated by (3.1) is well defined. We now show that limn→∞ φ(xn , x0 ) exists. From (3.1), we have xn = ΠCn x0 which implies that

⟨xn − z , Jx0 − Jxn ⟩ ≥ 0,

∀z ∈ Cn

(3.7)

⟨xn − u, Jx0 − Jxn ⟩ ≥ 0,

∀u ∈ F .

(3.8)

and

By Lemma 2.2, we have

φ(xn , x0 ) = φ(ΠCn x0 , x0 ) ≤ φ(u, x0 ) − φ(u, xn ) ≤ φ(u, x0 ) for each u ∈ F ⊂ Cn , n ≥ 1. Hence, the sequence {φ(xn , x0 )}∞ n=0 is bounded. Since xn = ΠCn x0 and xn+1 = ΠCn+1 x0 ∈ Cn+1 ⊂ Cn , we have

φ(xn , x0 ) ≤ φ(xn+1 , x0 ),

∀n ≥ 0.

{φ(xn , x0 )}∞ n=0

Therefore, is nondecreasing. It follows that the limit of {φ(xn , x0 )}∞ n=0 exists. Now, we show that {xn }∞ n=0 is Cauchy. By the construction of Cn , we have that Cm ⊂ Cn and xm = ΠCm x0 ∈ Cn for any positive integer m ≥ n. It then follows that

φ(xm , xn ) = φ(xm , ΠCn x0 ) ≤ φ(xm , x0 ) − φ(ΠCn x0 , x0 ) = φ(xm , x0 ) − φ(xn , x0 ) → 0 as m, n → ∞. It then follows from Lemma 2.5 that ‖xm − xn ‖ → 0 as m, n → ∞. Hence, {xn }∞ n=0 is Cauchy. Since E is a Banach space and C is closed and convex, then there exists p ∈ C such that xn → p as n → ∞. Now since φ(xm , xn ) → 0 as m, n → ∞ we have in particular that φ(xn+1 , xn ) → 0 as n → ∞ and this further implies that limn→∞ ‖xn+1 − xn ‖ = 0. Since xn+1 = ΠCn+1 x0 ∈ Cn+1 , we have

φ(xn+1 , un ) ≤ φ(xn+1 , xn ) + θn , Since θn → 0 as n → ∞, we obtain lim φ(xn+1 , un ) = 0.

n→∞

∀n ≥ 0.

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1517

Since E is uniformly convex and smooth, we have from Lemma 2.5 that lim ‖xn+1 − xn ‖ = 0 = lim ‖xn+1 − un ‖.

n→∞

n→∞

So,

‖xn − un ‖ ≤ ‖xn+1 − xn ‖ + ‖xn+1 − un ‖. Hence, lim ‖xn − un ‖ = 0.

n→∞

Since J is uniformly norm-to-norm continuous on bounded sets and limn→∞ ‖xn − un ‖ = 0, we obtain lim ‖Jxn − Jun ‖ = 0.

n→∞

Let L = supn≥1 {‖υn ‖, ‖T1n υn ‖, ‖T2n υn ‖, . . . , ‖Trn υn ‖}. Since E is uniformly smooth, we know that E ∗ is uniformly convex. Then from Lemma 2.6, we have

φ(x∗ , un ) = φ(x∗ , θnm yn ) ≤ φ(x∗ , yn ) = φ(x∗ , J −1 (αn0 J υn + αn1 JT1n υn + αn2 JT2n υn + · · · + αnr JTrn υn )) = ‖x∗ ‖2 − 2αn0 ⟨x∗ , J υn ⟩ − 2αn1 ⟨x∗ , JT1n υn ⟩ − 2αn2 ⟨x∗ , JT2n υn ⟩ − · · · − 2αnr ⟨x∗ , JTrn υn ⟩ + ‖αn0 J υn + αn1 JT1n υn + αn2 JT2n υn + · · · + αnr JTrn υn ‖2 ≤ ‖x∗ ‖2 − 2αn0 ⟨x∗ , J υn ⟩ − 2αn1 ⟨x∗ , JT1n υn ⟩ − 2αn2 ⟨x∗ , JT2n υn ⟩ − · · · − 2αnr ⟨x∗ , JTrn υn ⟩ + αn0 ‖J υn ‖2 + αn1 ‖JT1n υn ‖2 + αn2 ‖JT2n υn ‖2 + · · · + αnr ‖JTrn υn ‖2 − αn0 αn1 g (‖J υn − JT1n υn ‖) = αn0 φ(x∗ , υn ) + αn1 φ(x∗ , T1n υn ) + αn2 φ(x∗ , T2n υn ) + · · · + αnr φ(x∗ , Trn υn ) − αn0 αn1 g (‖J υn − JT1n υn ‖) ≤ φ(x∗ , υn ) + (1 − αn0 )(kn − 1)φ(x∗ , υn ) − αn0 αn1 g (‖J υn − JT1n υn ‖). It then follows that

αn0 αn1 g (‖J υn − JT1n υn ‖) ≤ φ(x∗ , xn ) − φ(x∗ , un ) + (1 − αn0 )(kn − 1)φ(x∗ , υn ). On the other hand,

φ(x∗ , xn ) − φ(x∗ , un ) = ≤ ≤ ≤

‖xn ‖2 − ‖un ‖2 − 2⟨x∗ , Jxn − Jun ⟩ |‖xn ‖2 − ‖un ‖2 | + 2|⟨x∗ , Jxn − Jun ⟩| |‖xn ‖ − ‖un ‖|(‖xn ‖ + ‖un ‖) + 2‖x∗ ‖‖Jxn − Jun ‖ ‖xn − un ‖(‖xn ‖ + ‖un ‖) + 2‖x∗ ‖‖Jxn − Jun ‖.

From limn→∞ ‖xn − un ‖ = 0 and limn→∞ ‖Jxn − Jun ‖ = 0, we obtain

φ(x∗ , xn ) − φ(x∗ , un ) → 0,

n → ∞.

Using the condition lim infn→∞ αn0 αn1 > 0 and since kn → 1 as n → ∞, we have lim g (‖J υn − JT1n υn ‖) = 0.

n→∞

By property of g, we have limn→∞ ‖J υn − JT1n υn ‖ = 0. Since J −1 is also uniformly norm-to-norm continuous on bounded sets, we have lim ‖υn − T1n υn ‖ = 0.

(3.9)

n→∞

Similarly, we can obtain lim ‖υn − Tin υn ‖ = 0,

n→∞

i = 2, 3, . . . , r .

We know that 4

φ(x∗ , υn ) ≤ φ(x∗ , xn ) − 2αλn ‖Bxn − Bx∗ ‖2 + 2 λ2n ‖Bxn − Bx∗ ‖2 c   2 = φ(x∗ , xn ) + 2λn 2 λn − α ‖Bxn − Bx∗ ‖2 . c

(3.10)

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By (3.2) and (3.10), we obtain

−2λn



 λ − α ‖Bxn − Bx∗ ‖2 ≤ φ(x∗ , xn ) − φ(x∗ , un ) → 0, n 2

2 c

n → ∞.

Hence, lim ‖Bxn − Bx∗ ‖ = 0.

n→∞

By Lemmas 2.2 and 2.7 and (3.5), we have

φ(xn , υn ) = φ(xn , ΠC J −1 (Jxn − λn Bxn )) ≤ φ(xn , J −1 (Jxn − λn Bxn )) = V (xn , Jxn − λn Bxn ) ≤ V (xn , (Jxn − λn Bxn ) + λn Bxn ) − 2⟨J −1 (Jxn − λn Bxn ) − xn , λn Bxn ⟩ = φ(xn , xn ) + 2⟨J −1 (Jxn − λn Bxn ) − xn , −λn Bxn ⟩ = 2⟨J −1 (Jxn − λn Bxn ) − xn , −λn Bxn ⟩ ≤

4 c2

b2 ‖Bxn − Bx∗ ‖2 → 0,

n → ∞.

(3.11)

It then follows from Lemma 2.5 that limn→∞ ‖xn − υn ‖ = 0. Since xn → p, we obtain that υn → p as n → ∞. Observe that for all i = 1, 2, . . . , r, we have

‖Tin p − p‖ ≤ ‖Tin p − Tin υn ‖ + ‖υn − Tin υn ‖ + ‖υn − p‖.

(3.12)

By φ -asymptotic nonexpansiveness of Ti , the fact that υn → p as n → ∞, we have that

φ(Tin p, Tin υn ) ≤ kni φ(p, υn ) → 0 as n → ∞.

(3.13)

Therefore, from (3.9), (3.12), (3.13) and Lemma 2.5, we obtain that ‖Tin p − p‖ → 0 as n → ∞. Now, by continuity of Ti , we get that p = limn→∞ (Tin p) = limn→∞ Ti (Tin−1 p) = Ti limn→∞ (Tin−1 p) = Ti p, that is p ∈ ∩ri=1 F (Ti ). By following the same line of arguments as that in the proof of Theorem 3.1 of [47], we have p ∈ VI (C , B). Finally, we show that p ∈ ∩m k=1 GMEP(Fk , ϕk ). Now, by Lemma 2.9 we obtain

φ(un , yn ) = φ(θnm yn , yn ) ≤ φ(x∗ , yn ) − φ(x∗ , θnm yn ) ≤ φ(x∗ , xn ) − φ(x∗ , un ) → 0,

n → ∞.

Using Lemma 2.5, we have limn→∞ ‖un − yn ‖ = 0. Furthermore,

‖xn − yn ‖ ≤ ‖xn − un ‖ + ‖un − yn ‖ → 0,

n → ∞.

Since xn → p as n → ∞ and ‖xn − yn ‖ → 0 as n → ∞, then yn → p as n → ∞. By the fact that θnk , k = 1, 2, . . . , m is relatively nonexpansive and using Lemma 2.9 again, we have that

φ(θnk yn , yn ) ≤ φ(x∗ , yn ) − φ(x∗ , θnk yn ) ≤ φ(x∗ , xn ) − φ(x∗ , θnk yn ).

(3.14)

Observe that

φ(x∗ , un ) = φ(x∗ , θnm yn ) G

G

= φ(x∗ , TrGmm,n Trmm−−11,n · · · TrGkk,n Trk−k−11,n · · · TrG22,n TrG11,n yn ) G

= φ(x∗ , TrGmm,n Trmm−−11,n · · · θnk yn ) ≤ φ(x∗ , θnk yn ).

(3.15)

Using (3.15) in (3.14), we obtain

φ(θnk yn , yn ) ≤ φ(x∗ , xn ) − φ(x∗ , un ) → 0, Then Lemma 2.5 implies that limn→∞ ‖yn − θ

k n yn

n → ∞.

‖ = 0, k = 1, 2, . . . , m. Now

‖p − θnk yn ‖ ≤ ‖yn − θnk yn ‖ + ‖yn − p‖ → 0,

n → ∞, k = 1, 2, . . . , m.

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Similarly, limn→∞ ‖p − θnk−1 yn ‖ = 0, k = 1, 2, . . . , m. This further implies that lim ‖θnk yn − θnk−1 yn ‖ = 0.

(3.16)

n→∞

Also, since J is uniformly norm-to-norm continuous on bounded sets and using (3.16), we obtain lim ‖J θnk yn − J θnk−1 yn ‖ = 0.

n→∞

Since lim infn→∞ rk,n > 0, (k = 1, 2, . . . , m), lim

‖J θnk yn − J θnk−1 yn ‖ rk,n

n→∞

= 0.

(3.17)

By Lemma 2.8, we have that for each k = 1, 2, . . . , m Gk (θnk yn , y) +

1 rk,n

⟨y − θnk yn , J θnk yn − J θnk−1 yn ⟩ ≥ 0,

∀y ∈ C .

Furthermore, using (A2) we obtain 1 rk,n

⟨y − θnk yn , J θnk yn − J θnk−1 yn ⟩ ≥ Gk (y, θnk yn ).

(3.18)

By (A4), (3.17) and θnk yn → p, we have for each k = 1, 2, . . . , m Gk (y, p) ≤ 0,

∀y ∈ C .

For fixed y ∈ C , let zt ,y := ty + (1 − t )p for all t ∈ (0, 1]. This implies that zt ∈ C . This yields that Gk (zt , p) ≤ 0. It follows from (A1) and (A4) that 0 = Gk (zt , zt ) ≤ tGk (zt , y) + (1 − t )Gk (zt , p)

≤ tGk (zt , y) and hence 0 ≤ Gk (zt , y). From condition (A3), we obtain Gk (p, y) ≥ 0,

∀y ∈ C .

This implies that p ∈ GMEP(Fk , ϕk ), k = 1, 2, . . . , m. Thus, p ∈ k=1 GMEP(Fk , ϕk ). Hence, we have p ∈ F := m  r VI (C , B) ∩ k=1 GMEP(Fk , ϕk ) ∩ ∩i=1 F (Ti ). Finally, we show that p = ΠF x0 . Now by taking the limit in (3.7), we have

m

⟨p − z , Jx0 − Jp⟩ ≥ 0,

∀z ∈ F .

By Lemma 2.3, we have p = ΠF x0 .



Remark 3.2. If {Ti }ri=1 is a finite family of φ -nonexpansive mappings, we get the following result. Theorem 3.3. Let E be a 2-uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty closed and convex subset of E. For each k = 1, 2, . . . , m, let Fk be a bifunction from C × C satisfying (A1)–(A4), ϕk : C → R be a lower semicontinuous and convex functional and Ak : C → E ∗ be a continuous and monotone mapping. Suppose B : C → E ∗ is an operator satisfying (B1)–(B3), and {Ti }ri=1 is a finite family of continuous φ -nonexpansive mappings of C onto itself such that r ∞ F := VI (C , B) ∩ (∩m k=1 GMEP(Fk , ϕk )) ∩ ∩i=1 F (Ti ) ̸= ∅. Let {xn }n=0 be iteratively generated by x0 ∈ C , C1 = C , x1 = ΠC1 x0 ,

 υn = ΠC J −1 (Jxn − λn Bxn )     yn = J −1 (αn0 J υn + αn1 JT1 υn + αn2 JT2 υn + · · · + αnr JTr υn ), G un = TrGmm,n Trmm−−11,n · · · TrG22,n TrG11,n yn    C = {w ∈ Cn : φ(w, un ) ≤ φ(w, xn )},   n+1 xn+1 = ΠCn+1 x0 , n ≥ 1, where J is the duality mapping on E. Suppose {αni }∞ = 1, 2, . . . , r is a sequence in (0, 1) such that n=1 for each i lim infn→∞ αn0 αni > 0, i = 1, 2, . . . , r , αn0 + αn1 + αn2 + · · · + αnr = 1, {λn }∞ n=1 ⊂ [a, b] for some a, b with 0 < a < b < c 2α , where 1c is the 2-uniformly convexity constant of E and {rk,n }∞ n=1 ⊂ (0, ∞), (k = 1, 2, . . . , m) satisfying lim infn→∞ rk,n > 0, (k = 1, 2, . . . , m). Then, {xn }∞ n=0 converges strongly to ΠF x0 . 2

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Proof. Since {Ti }ri=1 is a finite family of continuous φ -nonexpansive mappings of C onto itself, it is a finite family of continuous φ -asymptotically nonexpansive mappings of C onto itself with sequence {kni } = {1} and so, θn = (1 − αn0 )(kn − 1) supx∗ ∈F φ(x∗ , υn ) = 0. Therefore, all the conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.3 can now be obtained from Theorem 3.1 immediately.  Corollary 3.4 (Zegeye and Shahzad [38]). Let E be a 2-uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty closed convex and bounded subset of E. Let F be a bifunction from C × C satisfying (A1)–(A4) and suppose B : C → E ∗ is an operator satisfying (B1)–(B3) and T is continuous φ -asymptotically nonexpansive mappings of C onto itself with constant {kn } such that F := VI (C , B) ∩ EP (F ) ∩ F (T ) ̸= ∅. Let {xn }∞ n=0 be iteratively generated by x0 ∈ C0 = C

 υn = ΠC J −1 (Jxn − λn Bxn )    −1 n   yn = J (αn Jxn + (1 − αn )JT υn ), 1 F (un , y) + ⟨y − un , Jun − Jyn ⟩ ≥ 0, ∀y ∈ C  rn    C = {w ∈ Cn : φ(w, un ) ≤ φ(w, xn ) + θn },  n + 1  xn+1 = ΠCn+1 x0 , n ≥ 0, where J is the duality mapping on E and θ = (1 − αn )(kn − 1)(2diamC )2 . Suppose {αn }∞ n=1 is a sequence in (0, 1) such that c α lim infn→∞ αn (1 − αn ) > 0, {λn }∞ , where 1c is the 2-uniformly convexity n=1 ⊂ [a, b] for some a, b with 0 < a < b < 2 ∞ ∞ constant of E and {rn }n=1 ⊂ (0, ∞) satisfying lim infn→∞ rn > 0. Then, {xn }n=0 converges strongly to ΠF x0 . 2

Remark 3.5. Our Theorem 3.1 improves and extends on the results of Zegeye and Shahzad [38] in the following aspects: (1) We improve on the results of Zegeye and Shahzad [38] by weakening the boundedness condition imposed on C in the results of Zegeye and Shahzad [38] (see Corollary 3.4); (2) We extend the results of Zegeye and Shahzad [38] from approximation of fixed point of a single φ -asymptotically nonexpansive mapping which is also a solution to equilibrium problem and variational inequality problem to finite family of continuous φ -asymptotically nonexpansive mappings which at the same time approximate a common solution to system of generalized mixed equilibrium problems and a solution to variational inequality problem. Strong convergence theorem for approximation of common element of the set of common fixed points of finite family of continuous φ -asymptotically nonexpansive mappings and the set of common solution of a system of generalized mixed equilibrium problems may not require E to be 2-uniformly convex. In fact, we have the following theorem. Theorem 3.6. Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty closed and convex subset of E. For each k = 1, 2, . . . , m, let Fk be a bifunction from C × C satisfying (A1)–(A4), ϕk : C → R be a lower semicontinuous and convex functional and Ak : C → E ∗ be a continuous and monotone mapping. Suppose {Ti }ri=1 is a finite family of continuous φ -asymptotically nonexpansive mappings of C onto itself with constants {kni } such that r ∗ ∗ S T F := (∩m k=1 GMEP(Fk , ϕk )) ∩ ∩i=1 F (Ti ) ̸= ∅. Assume that R := sup{‖x ‖ : x ∈ F } < ∞ and suppose kn := max{kn , kn } and ∞ let {xn }n=0 be iteratively generated by x0 ∈ C , C1 = C , x1 = ΠC1 x0 ,

 y = J −1 (αn0 Jxn + αn1 JT1n xn + αn2 JT2n xn + · · · + αnr JTrn xn ),    n G un = TrGmm,n Trmm−−11,n · · · TrG22,n TrG11,n yn  Cn+1 = {w ∈ Cn : φ(w, un ) ≤ φ(w, xn ) + θn },  xn+1 = ΠCn+1 x0 , n ≥ 1, where J is the duality mapping on E and θn = (1 − αn0 )(kn − 1) supx∗ ∈F φ(x∗ , xn ) for all xn ∈ C . Suppose {αni }∞ n=1 for each i = 1, 2, . . . , r is a sequence in (0, 1) such that lim infn→∞ αn0 αni > 0, i = 1, 2, . . . , r , αn0 + αn1 + αn2 + · · · + αnr = 1 and ∞ {rk,n }∞ n=1 ⊂ (0, ∞), (k = 1, 2, . . . , m) satisfying lim infn→∞ rk,n > 0, (k = 1, 2, . . . , m). Then, {xn }n=0 converges strongly to ΠF x0 . Proof. Put B ≡ 0 Theorem 3.1, then, we get that υn = xn . Thus, the method of proof of Theorem 3.1 gives the required method of proof of Theorem 3.6 without the requirement that E be 2-uniformly convex.  Corollary 3.7 (Zegeye and Shahzad [36,37]). Let H be a real Hilbert space and let C be a nonempty closed convex and bounded subset of H. Suppose {Ti }ri=1 is a finite family of asymptotically nonexpansive mappings of C onto itself with constants {kni } such that F := ∩ ∩ri=1 F (Ti ) ̸= ∅. Let {xn }∞ n=0 be iteratively generated by

 x0 ∈ C ,   yn = αn0 xn + αn1 T n xn + αn2 T n xn + · · · + αnr T n xn ,  1 2 r Cn = {v ∈ C : ‖yn − v‖2 ≤ ‖xn − v‖2 + θn }    Qn = {v ∈ C : ⟨xn − v, x0 − xn ⟩ ≥ 0}, xn+1 = PCn ∩Qn (x0 ), ∀n ≥ 0,

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where θn = [(k2n1 − 1)αn1 + (k2n1 − 1)αn1 + · · · + (k2nr − 1)αnr ](diam(C ))2 → 0 as n → ∞. If {αni }∞ n=0 ⊂ [ϵ, 1 − ϵ] for some ϵ > 0 and αn0 + αn1 + αn2 + · · · + αnr = 1 for all n ≥ 0. Then, {xn }∞ n=0 converges strongly to PF x0 . Remark 3.8. Our Theorem 3.1 improves and generalizes the results of Zegeye and Shahzad [36,37] from approximation of common fixed point of finite family of asymptotically nonexpansive mappings in real Hilbert space to approximation of common fixed point of finite family of φ -asymptotically nonexpansive mappings which is also a solution to variational inequality problem and a common solution to system of generalized mixed equilibrium problem in a 2-uniformly convex and uniformly smooth Banach space. Furthermore, our results extend and improve on the results of Kim and Xu [35]. Acknowledgments The author thanks Professor Ervin Y. 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