On a hybrid algorithm for a family of total quasi-ϕ-asymptotically nonexpansive mappings in Banach spaces

Abstract

The purpose of this article is to introduce the concept of total quasi- ϕ-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases and we prove a strong convergence theorem by using a hybrid method for finding a common element of the set of solutions for a generalized mixed equilibrium problems, the set of fixed points of a family of total quasi- ϕ-asymptotically nonexpansive mappings in uniformly smooth and strictly convex Banach space with the Kadec-Klee property. The results presented in the article improve and extend some recent results.

MR (2000) Subject Classification: 47H09; 47H05.

1 Introduction

Let E be a Banach space and C be a closed convex subsets of E. Let F be an equilibrium bifunction from C × C into R, ψ: C → R be a real-valued function and A: C → E* be a nonlinear mapping. The "so-called" generalized mixed equilibrium problem is to find z ∈ C such that

$$F\left(z,y\right)+〈Az,y-z〉+\psi \left(y\right)-\psi \left(z\right)\ge 0,\forall y\in C.$$

(1.1)

The set of solutions of (1.1) is denoted by GMEP, i.e.,

$$GMEP=\left\{z\in C:F\left(z,y\right)+〈Az,y-z〉+\psi \left(y\right)-\psi \left(z\right)\ge 0,\forall y\in C\right\}.$$

Special examples:

• (I) If A = 0, then the problem (1.1) is equivalent to find z ∈ C such that

  $$F\left(z,y\right)+\psi \left(y\right)-\psi \left(z\right)\ge 0,\forall y\in C.$$

  (1.2)

which is called the mixed equilibrium problem, see [1]. The set of solutions of (1.2) is denoted by MEP.

• (II) If F = 0, then the problem (1.1) is equivalent to find z ∈ C such that

  $$〈Az,y-z〉+\psi \left(y\right)-\psi \left(z\right)\ge 0,\forall y\in C.$$

  (1.3)

which is called the mixed variational inequality of Browder type. The set of solutions of (1.3) is denoted by V I(C, A, ψ).

• (III) If ψ = 0, then the problem (1.1) is equivalent to find z ∈ C such that

  $$F\left(z,y\right)+〈Az,y-z〉\ge 0,\forall y\in C.$$

  (1.4)

which is called the generalized equilibrium problem, see [2]. The set of solutions of (1.4) is denoted by EP.

• (IV) If A = 0, ψ = 0, then the problem (1.1) is equivalent to find z ∈ C such that

  $$F\left(z,y\right)\ge 0,\forall y\in C.$$

  (1.5)

which is called the equilibrium problem. The set of solutions of (1.5) is denoted by EP(F).

These show that the problem (1.1) is very general in the sense that numerous problems in physics, optimization, and economics reduce to finding a solution of (1.1). Recently, some methods have been proposed for the generalized mixed equilibrium problem in Banach spaces (see, for example [1–7]).

Let E be a smooth, strictly convex, and reflexive Banach spaces and C be a nonempty closed convex subsets of E. Throughout this article, we denote by ϕ the function defined by

$$\varphi \left(y,x\right)={\left|\right|y\left|\right|}^2-2〈y,Jx〉+{\left|\right|x\left|\right|}^2,\forall x,y\in E,$$

(1.6)

where J: E → 2 ^E* is the normalized duality mapping.

Let T: C → C be a mapping and F (T) be the set of fixed points of T .

Recall that a point p ∈ C is said to be an asymptotic fixed point of T if there exists {x _n } in C which converges weakly to p and lim_n→∞||x _n - Tx _n || = 0. We denote the set of all asymptotic fixed point of T by $\tilde{F}\left(T\right)$. A point p ∈ C is said to be a strong asymptotic fixed point of T if there exists {x _n } in C such that x _n → p and lim_n→∞||x _n - Tx _n || = 0 We denote the set of all strongly asymptotic fixed point of T by $\widehat{F}\left(T\right)$.

A mapping T: C → C is said to be nonexpansive, if

$$\left|\right|Tx-Ty\left|\right|\le \left|\right|x-y\left|\right|,\forall x,y\in C.$$

A mapping T: C → C is said to be relatively nonexpansive if F (T) ≠ Ø, $F\left(T\right)=\tilde{F}\left(T\right)$, and

$$\varphi \left(u,Tx\right)\le \varphi \left(u,x\right),\forall u\in F\left(T\right),x\in C$$

A mapping T: C → C is said to be weak relatively nonexpansive if F(T) ≠ Ø, $F\left(T\right)=\widehat{F}\left(T\right)$, and

$$\varphi \left(u,Tx\right)\le \varphi \left(u,x\right),\forall u\in F\left(T\right),x\in C$$

A mapping T: C → C is said to be quasi-ϕ-nonexpansive, if F(T) ≠ Ø and

$$\varphi \left(u,Tx\right)\le \varphi \left(u,x\right),\forall x\in C,\forall u\in F\left(T\right)$$

A mapping T: C → C is said to be quasi-ϕ-asymptotically nonexpansive, if there exists some real sequence {k _n } with k _n ≥ 1 and k _n → 1 and F (T) ≠ ∅ such that

$$\varphi \left(u,T^{n}x\right)\le k_n\varphi \left(u,x\right),\forall n\ge 1,x\in C,u\in F\left(T\right)$$

(1.7)

A mapping T: C → C is said to be closed, if for any sequence {x _n } ⊂ C with x _n → x and Tx _n → y, then Tx = y.

Definition 1.1 (1) A mapping T: C → C is said to be total quasi-ϕ-asymptotically nonexpansive, if F (T) ≠ ∅ and there exist nonnegative real sequences {υ _n }, {μ _n } with υ _n → 0, μ _n → 0 (as n → ∞) and a strictly increasing continuous function ζ: ℜ⁺ → ℜ ⁺ with ζ(0) = 0 such that for all x ∈ C, p ∈ F(T)

$$\varphi \left(p,T^{n}x\right)\le \varphi \left(p,x\right)+v_n\zeta \left(\varphi \left(p,x\right)\right)\text{ + }{\mu }_n,\forall n\ge \text{1}.$$

(1.8)

• (2) A family of mappings {T _λ }_λ∈Λ: C → C is said to be uniformly total quasi-ϕ-asymptotically nonexpansive, if ∩_λ∈Λ F (T _λ ) ≠ ∅ and there exist nonnegative real sequences {υ _n }, {μ _n } with υ _n → 0, μ _n → 0(as n → ∞) and a strictly increasing continuous function ζ: ℜ⁺ → ℜ ⁺ with ζ(0) = 0 such that for all x∈ C, p∈∩ _λ∈Λ F (T _λ )

  $$\varphi \left(p,T_i^{n}x\right)\le \varphi \left(p,x\right)+v_n\zeta \left(\varphi \left(p,x\right)\right)\text{ + }{\mu }_n,\forall n\ge \text{1}.$$

  (1.9)

Remark 1.1 From the definitions, it is easy to know that

• (1) Each relatively nonexpansive mapping is closed;

• (2) Taking ζ(t) = t, t ≥ 0, υ _n = (k _n - 1) and μ _n = 0,, then (1.7) can be rewritten as

  $$\varphi \left(p,T^{n}x\right)\le \varphi \left(p,x\right)+v_n\zeta \left(\varphi \left(p,x\right)\right)\text{ + }{\mu }_n,\forall n\ge \text{1},x\in C,p\in F\left(T\right).$$

  (1.10)

This implies that each quasi-ϕ-asymptotically nonexpansive mapping must be a total quasi-ϕ-asymptotically nonexpansive mapping, but the converse is not true;

• (3) The class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of quasi-ϕ-nonexpansive mappings as a subclass, but the converse is not true;

• (4) The class of quasi-ϕ-nonexpansive mappings contains properly the class of weak relatively nonexpansive mappings as a subclass, but the converse is not true;

• (5) The class of weak relatively nonexpansive contains properly the class of relatively nonexpansive mappings as a subclass, but the converse is not true.

Iterative approximation of fixed points for relatively nonexpansive mappings, weak relatively nonexpansive mappings, quasi-ϕ nonexpansive mappings, quasi-ϕ-asymptotically non-expansive mappings in the setting of Banach spaces has been studied extensively by many authors (see [5–13]).

Motivated by the above, the purpose of this article is to introduce the concept of total quasi-ϕ-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases and we prove a strong convergence theorem by using a hybrid method for finding a common element of the set of solutions for a generalized mixed equilibrium problems, the set of fixed points of a family of total quasi- ϕ-asymptotically nonexpansive mappings in uniformly smooth and strictly convex Banach space with the Kadec-Klee property. The results presented in the paper improve and extend some recent results.

2 Preliminaries

Throughout this article, we assume that all the Banach spaces are real. We denote by ℕ and ℝ the sets of positive integers and real numbers, respectively. Let E be a Banach space and let E* be the topological dual of E. For all x ∈ E and x* ∈ E* , we denote by 〈x, x*〉 the value of x* at x. The mapping J: E → 2^E*defined by

$$J\left(x\right)=\left\{x^{*}\in E^{*}:〈x,x^{*}〉={\left|\right|x\left|\right|}^2={\left|\right|x^{*}\left|\right|}^2\right\},x\in E,$$

(2.1)

is normalized duality mapping. We denote the weak convergence and the strong convergence of a sequence {x _n } to x by x _n ⇀ x and x _n → x, respectively.

A Banach spaces E is said to be strictly convex if $\frac{\left|\right|x+y\left|\right|}{2}<1$ for x, y∈ S(E) = {z ∈ E: ||z|| = 1} with x ≠ y. It is said to be uniformly convex if for any given ε ∈ (0, 2], there exists δ > 0 such that $\frac{\left|\right|x+y\left|\right|}{2}<1-\delta$ for x, y ∈ S(E) with ||x - y|| ≥ ε. E is said to have the Kadec-Klee property, if for any sequence {x _n } ⊂ E such that x _n ⇀ x∈ E and ||x _n || → ||x||, then x _n → x.

Define f: S(E) × S(E) × ℝ \ {0} → ℝ by

$$f\left(x,y,t\right)=\frac{\parallel x+ty\parallel -\parallel x\parallel }{t}$$

for x, y ∈ S(E) and t ∈ℝ \ {0}. A norm of E is said to be Gâteaux differentiable if lim_{t→ 0}f(x, y, t) has a limit for each x, y ∈ S(E). In this case, E is said to be smooth. We know that if E is smooth, strictly convex, and reflexive, then the duality mapping J is single valued, one to one, and onto. In this case, the inverse mapping J^-1 coincides with the duality mapping J* on E*. See [14] for more details.

Remark 2.1 If E is a reflexive and strictly convex Banach space, then J^-1 is hemi-continuous, i.e., J^-1 is norm-weak-continuous.

Let {C _n } be a sequence of nonempty closed convex subset of a reflexive Banach space E. We define two subsets s - Li _n C _n and w - Ls _n C _n as follows: x ∈ s - Li _n C _n if and only if there exists {x _n } ⊂ E such that {x _n } converges strongly to x and that x _n ∈ C _n for all n ∈ ℕ. Similarly, y ∈ w - Ls _n C _n if and only if there exists a subsequence $\left\{C_{n_i}\right\}$ of {C _n } and a sequence {y _i } ⊂ E such that {y _i } converges weakly to y and $y_i\in C_{n_i}$ for all i ∈ ℕ. We define the Mosco convergence [15] of {C _n } as follows: If C₀ = s - Li _n C _n = w - Ls _n C _n , then {C _n } is said to be convergent to C₀ in the sense of Mosco and we write C₀ = M - lim_n→∞C _n .

For more details, see [16].

Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Then, for arbitrarily fixed x ∈ E, the function y α ||x - y||²: C → R⁺ has a unique minimizer y _x ∈ C. Using such a point, we define the metric projection P _C by P _C x = y _x = arg min_y∈C||x - y||² for every x ∈ E. In a similar fashion, we can see that the function y α ϕ(x, y): C → R⁺ has a unique minimizer z _x ∈ C. The generalized projection Π _C of E onto C is defined by Π _C = z _x = arg min_y∈Cϕ(x, y) for every x ∈ E; see [17].

The generalized projection Π _C from E onto C is well defined, single valued and satisfies

$${\left(\left|\right|x\left|\right|-\left|\right|y\left|\right|\right)}^2\le \varphi \left(y,x\right)\le {\left(\left|\right|x\left|\right|+\left|\right|y\left|\right|\right)}^2,\forall x,y\in E$$

(2.2)

If E is a Hilbert space, then ϕ(y, x) = ||y - x||² and Π _C is the metric projection P _C of E onto C.

It is well-known that the following conclusions hold:

Lemma 2.1[17, 18]. Let C be a nonempty closed convex subsets of a smooth, strictly convex, and reflexive Banach spaces. Then

$$\varphi \left(x,{\mathrm{\Pi }}_{C}y\right)+\varphi \left({\mathrm{\Pi }}_{C}y,y\right)\le \varphi \left(x,y\right),\forall x\in C,y\in E.$$

(2.3)

Lemma 2.2. Let C be a nonempty closed convex subsets of a smooth, strictly convex, and reflexive Banach spaces E, let x ∈ E and z ∈ C. Then the following conclusions hold:

• (a) z = ∏ _C x⇔〈y - z, Jx - jz〉 ≤ 0, ∀y ∈ C.

• (b) For x, y ∈ E, ϕ (x, y) = 0 if and only if x = y.

The following theorem proved by Tsukada [19] plays an important role in our results.

Theorem 2.1. Let E be a smooth, reflexive, and strictly convex Banach spaces having the Kadec-Klee property. Let {K _n } be a sequence of nonempty closed convex subsets of E. If K₀ = M - lim_n→∞K _n exists and is nonempty, then $\left\{P_{K_n}x\right\}$ converges strongly to $P_{K_0}x$ for each x ∈ C.

Theorem 2.1 is still valid if we replace the metric projections with the generalized pro-jections as follows:

Theorem 2.2 Let E be a smooth, reflexive, and strictly convex Banach spaces having the Kadec-Klee property. Let {K _n } be a sequence of nonempty closed convex subsets of E. If K₀ = M - lim_n→∞K _n exists and is nonempty, then $\left\{{\prod }_{K_n}x\right\}$ converges strongly to ${\prod }_{K_0}x$ for each x ∈ C.

For solving the equilibrium problem for bifunction F: C × C → ℝ, let us assume that F satisfies the following conditions:

(A₁) F (x, x) = 0 for all x ∈ C;

(A₂) F is monotone, i.e., F (x, y) + F (y, x) ≤ 0 for all x, y ∈ C;

(A₃) for each x, y, z ∈ C,

$$\underset{t\downarrow 0}{\text{lim sup}}F\left(tz+\left(1-t\right)x,y\right)\le F\left(x,y\right);$$

(A₄) for each x ∈ C, y α F (x, y) is a convex and lower semicontinuous.

If an equilibrium bifunction F: C × C → R satisfies conditions (A₁)-(A₄), then we have the following results.

Lemma 2.3[20]. Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach spaces E, let F be an equilibrium bifunction from C × C to ℝ satisfying conditions (A₁)-(A₄), let r > 0 and let x ∈ E. Then, there exists z ∈ C such that

$$F\left(z,y\right)+\frac{1}{r}〈y-z,Jz-Jx〉\ge 0,\forall y\in C.$$

Lemma 2.4[21]. Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach spaces E, let F: C × C → ℝ be an equilibrium bifunction satisfying conditions (A₁)-(A₄). For r > 0 and x ∈ E, define a mapping T _r : E → C as follows:

$$T_r\left(x\right)=\left\{z\in C:F\left(z,y\right)+\frac{1}{r}〈y-z,Jz-Jx〉\ge 0,\forall y\in C\right\}.$$

for all x ∈ E. Then, the following hold:

• (1) T _r is single-valued;

• (2) T _r is a firmly nonexpansive-type mapping, i.e., for any x, y ∈ E,

  $$〈T_{r}x-T_{r}y,J{T}_{r}x-J{T}_{r}y〉\le 〈T_{r}x-T_{r}y,Jx-Jy〉;$$

• (3) $F\left(T_r\right)=\widehat{F}\left(T_r\right)=EP\left(F\right)$;

• (4) EP (F) is a closed and convex set.

Lemma 2.5[21]. Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach spaces E, let F: C × C → ℝ be an equilibrium bifunction satisfying conditions (A₁)-(A₄). For r > 0 and x ∈ E and q ∈ F (T _r ),

$$\varphi \left(q,T_{r}x\right)+\varphi \left(T_{r}x,x\right)\le \varphi \left(q,x\right)$$

Lemma 2.6[22]. Let E be a uniformly convex Banach space and let r > 0. Then there exists a strictly increasing, continuous, and convex function g: [0, 2r] → R such that g(0) = 0 and

$$\parallel tx+\left(1-t\right)y{\parallel }^2\le t\parallel x{\parallel }^2+\left(1-t\right)\parallel y{\parallel }^2-t\left(1-t\right)g\left(\parallel x-y\parallel \right).$$

(2.4)

3 The main results

Theorem 3.1. Let E be a uniformly smooth and strictly convex Banach space with Kadec-Klee property and C be a nonempty closed convex subset of E. A: C → E* be a continuous and monotone mapping, ψ: C → ℝ be a lower semi-continuous and convex function and F be a bifunction from C ×C to ℝ which satisfies the conditions (A₁)-(A₄). Let {T _λ }(λ ∈ Λ): C → C be a family of uniformly L _λ -Lipschitzian continuous and uniformly total quasi-ϕ-asymptotically nonexpansive mappings such that $픉={\bigcap }_{\lambda \in \mathrm{\Lambda }}F\left(T_{\lambda }\right)\bigcap GMEP\ne \varnothing$. Assume that K = sup{||u||: u ∈ F} < ∞. Let {x _n } be the sequence generated by x₁ = x ∈ C, C₁ = C and

$$\left\{\begin{array}{c}{y}_{\lambda ,n}=J^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{T}_{\lambda }^n{x}_n\right),\lambda \in \mathrm{\Lambda },\hfill \\ u_{\lambda ,n}\in Csuchthat\hfill \\ F\left(u_{\lambda ,n},y\right)+〈A{u}_{\lambda ,n},y-u_{\lambda ,n}〉+\psi \left(y\right)-\psi \left(u_{\lambda ,n}\right)\hfill \\ +\frac{1}{r_{\lambda ,n}}〈y-u_{\lambda ,n},J{u}_{\lambda ,n}-J{y}_{\lambda ,n}〉\ge 0,\forall y\in C,\lambda \in \mathrm{\Lambda }\hfill \\ C_{n+1}=\left\{z\in C_n:\underset{\lambda \in \mathrm{\Lambda }}{\text{sup}}\varphi \left(z,u_{\lambda ,n}\right)\le \varphi \left(z,x_n\right)+{\xi }_n\right\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}x,\forall n\ge 0.\hfill \end{array}\right.$$

(3.1)

where ξ _n = (1 - α _n )(ν _n sup_u∈Fζ(ϕ(u, x _n )) + μ _n ),{α _n } is a sequence in [0, 1] such that lim inf_n→∞α _n (1 - α _n ) > 0, lim inf _n→∞ α _n < 1 and {r _λ , _n } [a, ∞) for some a > 0, then {x _n } converge strongly to some point x* in F.

Proof. We define a bifunction G: C × C → R by

$$G\left(z,y\right)=F\left(z,y\right)+〈Az,y-z〉+\psi \left(y\right)-\psi \left(z\right),\forall z,y\in C.$$

It is easy to prove that the bifunction G satisfies conditions(A₁)-(A₄).

Therefore, the generalized mixed equilibrium problem (1.1) is equivalent to the following equilibrium problem: find z ∈ C such that

$$G\left(z,y\right)\ge 0,\forall y\in C,$$

and GMEP = EP (G), F = GMEP ∩⋂_λ∈∈Λ=EP(G) ∩⋂_λ∈ΛF(T _λ ). So, (3.1) can be written as:

$$\left\{\begin{array}{c}{y}_{\lambda ,n}=J^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{T}_{\lambda }^n{x}_n\right)\text{for}\text{all}\lambda \in \mathrm{\Lambda },\hfill \\ u_{\lambda ,n}\in C\text{such}\text{that}G\left(u_{\lambda ,n},y\right)+\frac{1}{r_{\lambda ,n}}〈y-u_{\lambda ,n},J{u}_{\lambda ,n}-J{y}_{\lambda ,n}〉\ge 0,\forall y\in C,\lambda \in \mathrm{\Lambda }\hfill \\ C_{n+1}=\left\{z\in C_n:\underset{\lambda \in \mathrm{\Lambda }}{\text{sup}}\varphi \left(z,u_{\lambda ,n}\right)\le \varphi \left(z,x_n\right)+{\xi }_n\right\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}x,\forall n\ge 0.\hfill \end{array}\right.$$

(3.2)

Since the bifunction G satisfies conditions (A₁)-(A₄), from Lemma 2.4, for given r > 0 and x ∈ E, the mapping W _r : E → C defined by

$$W_r\left(x\right)=\left\{z\in C:G\left(z,y\right)+\frac{1}{r}〈y-z,Jz-Jx〉\ge 0,\forall y\in C\right\}.$$

has the same properties as in Lemma 2.4.

Putting $u_{\lambda ,n}=W_{r_{\lambda ,n}}{y}_{\lambda ,n}$ for all n ∈ ℕ, we have from Lemmas 2.4 and 2.5 that $W_{r_{\lambda ,n}}$ is relatively nonexpansive.

We divide the proof of Theorem 3.1 into five steps:

Step 1. We first show that C _n is closed and convex for every n ∈ ℕ.

From the definition of ϕ, we may show that

$$\begin{array}{cc}\hfill C_{n+1}& =\left\{z\in C_n:{\text{sup}}_{\lambda \in \mathrm{\Lambda }}\varphi \left(z,u_{\lambda ,n}\right)\le \varphi \left(z,x_n\right)+{\xi }_n\right\}\hfill \\ ={\cap }_{\lambda \in \mathrm{\Lambda }}\left\{z\in C_n:\varphi \left(z,u_{\lambda ,n}\right)\le \varphi \left(z,x_n\right)+{\xi }_n\right\}\hfill \\ ={\cap }_{\lambda \in \mathrm{\Lambda }}\left\{z\in C:2〈z,J{x}_n-J{u}_{\lambda ,n}〉+\parallel u_{\lambda ,n}{\parallel }^2-\parallel x_n{\parallel }^2-{\xi }_n\le 0\right\}\cap C_n,\hfill \end{array}$$

and thus C _n is closed and convex for every n ∈ ℕ.

Step 2. Sequence {x _n } is bounded.

In fact, since $x_n={\mathrm{\Pi }}_{C_n}x$, for any p ∈ F, from Lemma 2.1, we have

$$\varphi \left(x_n,x\right)=\varphi \left({\mathrm{\Pi }}_{C_n}x,x\right)\le \varphi \left(p,x\right)-\varphi \left(p,x_n\right)\le \varphi \left(p,x\right).$$

This implies that the sequence {ϕ (x _n , x)} is bounded, and so {x _n } is bounded.

Step 3. Next we show that $픉\subset C_n$ for each n ∈ ℕ.

For any $u\in 픉$, since $W_{r_{\lambda ,n}}$ is relatively nonexpansive, {T _λ }, λ ∈ Λ is uniformly total quasi-ϕ-asymptotically nonexpansive and E* is uniformly convex, it follows from Lemma 2.6 that

$$\begin{array}{cc}\hfill \varphi \left(u,u_{\lambda ,n}\right)& =\varphi \left(u,W_{r_{\lambda ,n}}{y}_{\lambda ,n}\right)\le \varphi \left(u,y_{\lambda ,n}\right)\hfill \\ =\varphi (u,J^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{T}_{\lambda }^n{x}_n\right)\hfill \\ =\left|\right|u|{|}^2-2〈u,{\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{T}_{\lambda }^n{x}_n〉+||{\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{T}_{\lambda }^n{x}_n|{|}^2\hfill \\ \le \left|\right|u|{|}^2-2{\alpha }_n〈u,J{x}_n〉-2\left(1-{\alpha }_n\right)〈u,J{T}_{\lambda }^n{x}_n〉+{\alpha }_n|\left|J{x}_n\right|{|}^2+\left(1-{\alpha }_n\right)\left|\right|J{T}_{\lambda }^n{x}_n|{|}^2\hfill \\ -{\alpha }_n\left(1-{\alpha }_n\right)g\left(\left|\right|J{x}_n-J{T}_{\lambda }^n{x}_n\left|\right|\right)\hfill \\ ={\alpha }_n\varphi \left(u,x_n\right)+\left(1-{\alpha }_n\right)\varphi \left(u,T_{\lambda }^n{x}_n\right)-{\alpha }_n\left(1-{\alpha }_n\right)g\left(\left|\right|J{x}_n-J{T}_{\lambda }^n{x}_n\left|\right|\right)\hfill \\ \le {\alpha }_n\varphi \left(u,x_n\right)+\left(1-{\alpha }_n\right)\left[\varphi \left(u,x_n\right)+v_n\zeta \left(\varphi \left(u,x_n\right)\right)\text{ + }{\mu }_n\right]-{\alpha }_n\left(\text{1}-{\alpha }_n\right)g\left(\left|\right|J{x}_{n}J{T}_{\lambda }^n{x}_n\left|\right|\right)\hfill \\ =\varphi \left(u,x_n\right)+\left(1-{\alpha }_n\right)\left[v_n\zeta \left(\varphi \left(u,x_n\right)\right)\text{ + }{\mu }_n\right]{-\alpha }_n\left(\text{1}{-\alpha }_n\right)g\left(\left|\right|J{x}_{n}J{T}_{\lambda }^n{x}_n\left|\right|\right)\hfill \\ \le \varphi \left(u,x_n\right)+{\xi }_n-{\alpha }_n\left(1-{\alpha }_n\right)g\left(\left|\right|J{x}_n-J{T}_{\lambda }^n{x}_n\left|\right|\right)\hfill \\ \le \varphi \left(u,x_n\right)+{\xi }_n.\hfill \end{array}$$

(3.3)

This shows that {y _{λ, n} } is bounded and sup_λ∈Λϕ (u, u _{λ, n} ) ≤ ϕ(u, x _n ) + ξ _n , i.e., u ∈ C _n , this implies that

$$픉\subset C_n,\forall n\in \mathbb{N}.$$

Step 4. Now we prove that the limit lim _n→∞ x _n exists.

Since $픉$ is nonempty, C _n is a nonempty closed convex subset of E and thus ${\mathrm{\Pi }}_{C_n}$ exists for every n ∈ ℕ. Hence {x _n } is well defined. Also, since {C _n } is a decreasing sequence of closed convex subsets of C such that $C_0={\bigcap }_{n=1}^{\infty }{C}_n$ is nonempty. Therefore, we have

$$M-\underset{n\to \infty }{\text{lim}}{C}_n=C_0=\bigcap _{n=1}^{\infty }{C}_n\ne \varnothing .$$

By Theorem 2.2, {x _n } = $\left\{{\mathrm{\Pi }}_{C_n}x\right\}$ converges strongly to x* = ${\mathrm{\Pi }}_{C_0}x$. Therefore, we have

$$\parallel x_{n+1}-x_n\parallel \to 0.$$

(3.4)

Step 5. Next we prove $x^{*}\in 픉$.

• (a) First, we prove x*∈⋂_{λ ∈ Λ} F(T _λ ).

In fact, since x _n → x*, we have

$$\varphi \left(x_{n+1},x_n\right)\to 0.$$

(3.5)

In view of x_n+1∈ C_n+1, from the definition of C_n+1, we have

$$\underset{\lambda \in \mathrm{\Lambda }}{\text{sup}}\varphi \left(x_{n+1},u_{\lambda ,n}\right)\le \varphi \left(x_{n+1},x_n\right)+{\xi }_n.$$

From (3.5) and ξ _n → 0, we have

$$\underset{\lambda \in \mathrm{\Lambda }}{\text{sup}}\varphi \left(x_{n+1},u_{\lambda ,n}\right)\to 0.$$

From (2.2) it yields sup_λ∈Λ(||x_n+1|| - ||u _{λ, n} ||)²→ 0. Since ||x_n+1|| → ||x*||, we have

$$\parallel u_{\lambda ,n}\parallel \to \parallel x^{*}\parallel \left(n\to \infty \right),\forall \lambda \in \mathrm{\Lambda }$$

(3.6)

Hence we have

$$\left|\right|J{u}_{\lambda },n\left|\right|\to \left|\right|J{x}^{*}\left|\right|\left(n\to \infty \right),\forall \lambda \in \mathrm{\Lambda }$$

(3.7)

This implies that {Ju _{λ, n} } is uniformly bounded in E*. Since E is reflexive, and so is E*. We can assume that Ju _{λ, n} ⇀f₀ ∈ E*. In view of the reflexive of E, we see that J(E) = E*. Hence there exists p ∈ E such that Jp = f₀. Since

$$\begin{array}{cc}\hfill \varphi \left(x_{n+1},u_{\lambda ,n}\right)& =\left|\right|x_{n+1}|{|}^2-2〈x_{n+1},J{u}_{\lambda ,n}〉+|\left|u_{\lambda ,n}\right|{|}^2\hfill \\ =\left|\right|x_{n+1}|{|}^2-2〈x_{n+1},J{u}_{\lambda ,n}〉+|\left|u_{\lambda ,n}\right|{|}^2\hfill \end{array}$$

Taking lim inf _n→∞ on the both sides of equality above and in view of the weak lower semicontinuity of norm || · ||, it yields that

$$\begin{array}{cc}\hfill 0& \ge \left|\right|x^{*}|{|}^2-2〈x^{*},f_0〉+|\left|f_0\right|{|}^2=\left|\right|x^{*}|{|}^2-2〈x^{*},Jp〉+|\left|Jp\right|{|}^2\hfill \\ =\left|\right|x^{*}|{|}^2-2〈x^{*},Jp〉+|\left|p\right|{|}^2=\varphi \left(x^{*},p\right)\hfill \end{array}$$

i.e., x* = p. This implies that f₀ = Jx*, and so Ju _{λ, n} ⇀ Jx*, ∀ λ ∈ Λ. It follows from (3.7) and the Kadec-Klee property of E* that Ju _{λ, n} → Jx*(n → ∞). Note that J^-1: E* → E is hemi-continuous, it yields that u _{λ, n} ⇀ x*. In view of (3.6) and the Kadec-Klee property of E, we have

$$\underset{n\to \infty }{\text{lim}}{u}_{\lambda ,n}=x^{*}.\forall \lambda \in \mathrm{\Lambda }$$

(3.8)

From (3.8), we have

$$\underset{n\to \infty }{\text{lim}}\left|\right|x_n-u_{\lambda ,n}\left|\right|=0,\forall \lambda \in \mathrm{\Lambda }$$

(3.9)

Since J is uniformly continuous, we have that

$$\left|\right|J{x}_n-J{u}_{\lambda ,n}\left|\right|\to 0,\forall \lambda \in \mathrm{\Lambda }$$

(3.10)

From (3.9) and (3.10), we have

$$\begin{array}{cc}\hfill \varphi \left(u,x_n\right)-\varphi \left(u,u_{\lambda ,n}\right)& ={\left|\right|x_n\left|\right|}^2-{\left|\right|u_{\lambda ,n}\left|\right|}^2-2〈u,J{x}_n-J{u}_{\lambda ,n}〉\hfill \\ \le \left|\right|\left|x_n\right|{|}^2-\left|\right|u_{\lambda ,n}\left|{|}^2\right|+2\left|〈u,J{x}_n-J{u}_{\lambda ,n}〉\right|\hfill \\ \le \left|\right|\left|x_n\right||-|\left|u_{\lambda ,n}\right|\left|\right|\left(\left|\right|x_n\left|\right|+\left|\right|u_{\lambda ,n}\left|\right|\right)+2\left|\right|u\left|\right|\cdot \left|\right|J{x}_n-J{u}_{\lambda ,n}\left|\right|\hfill \\ \le \left|\right|x_n-u_{\lambda ,n}\left|\right|\left(\left|\right|x_n\left|\right|+\left|\right|u_{\lambda ,n}\left|\right|\right)+2\left|\right|u\left|\right|\cdot \left|\right|J{x}_n-J{u}_{\lambda ,n}\left|\right|.\hfill \\ \to 0\hfill \end{array}$$

(3.11)

It follows from (3.3), (3.11) and ξ _n → 0 that

$${\alpha }_n\left(1-{\alpha }_n\right)g\left(\left|\right|J{x}_n-J{T}_{\lambda }^n{x}_n\left|\right|\right)\le \varphi \left(u,x_n\right)-\varphi (u,u_{\lambda ,n}+{\xi }_n\to 0.$$

(3.12)

In view of condition lim inf _n→∞ α _n (1 - α _n ) > 0, we see that

$$g\left(\left|\right|J{x}_n-J{T}_{\lambda }^n{x}_n\left|\right|\right)\to 0\left(asn\to \infty \right).$$

It follows from the property of g that

$$\left|\right|J{x}_n-J{T}_{\lambda }^n{x}_n\left|\right|\to 0\left(asn\to \infty \right).$$

(3.13)

Since x _n → x* and so Jx _n → Jx*. From (3.13) we have

$$J{T}_{\lambda }^n{x}_n\to J{x}^{*}\left(asn\to \infty \right).$$

Since J^-1: E* → E is hemi-continuous, it follows that

$$T_{\lambda }^n{x}_n\rightharpoonup x^{*},\forall \lambda \in \mathrm{\Lambda }.$$

(3.14)

On the other hand, for each λ ∈ Λ we have

$$\left|\right|\left|T_{\lambda }^n{x}_n\right||-|\left|x^{*}\right|\left|\right|=\left|\right|\left|J\left(T_{\lambda }^n{x}_n\right)\right||-|\left|J{x}^{*}\right|\left|\right|\le \left|\right|J\left(T_{\lambda }^n{x}_n\right)-J{x}^{*}\left|\right|\to 0\left(asn\to \infty \right).$$

This together with (3.14) shows that

$$T_{\lambda }^n{x}_n\to x^{*},\forall \lambda \in \mathrm{\Lambda }.$$

(3.15)

Furthermore, by the assumption that for each λ ∈ Λ, T _λ is uniformly L _λ -Lipschitz continuous, hence from (3.4) and (3.15), we have

$$\begin{array}{cc}\hfill \left|\right|T_{\lambda }^{n+1}{x}_n-T_{\lambda }^n{x}_n\left|\right|& \le \left|\right|T_{\lambda }^{n+1}{x}_n-T_{\lambda }^{n+1}{x}_{n+1}\left|\right|+\left|\right|T_{\lambda }^{n+1}{x}_{n+1}-x_{n+1}\left|\right|\hfill \\ +\left|\right|x_{n+1}-x_n\left|\right|+\left|\right|x_n-T_{\lambda }^n{x}_n\left|\right|\hfill \\ \le \left(L_{\lambda }+1\right)\left|\right|x_{n+1}-x_n\left|\right|+\left|\right|T_{\lambda }^{n+1}{x}_{n+1}-x_{n+1}\left|\right|+\left|\right|x_n-T_{\lambda }^n{x}_n\left|\right|\hfill \\ \to 0\hfill \end{array}$$

This implies that $T_{\lambda }^{n+1}{x}_n\to x^{*}$, i.e., $T_{\lambda }{T}_{\lambda }^n{x}_n\to x^{*}$. In view of (3.15) and T _λ is uniformly Lipschitzian continuous, it yields that T _λ x* = x*, ∀ λ ∈ Λ. This implies that x*∈⋂_{λ ∈ Λ}F(T _λ ).

• (b) Next, we prove x* ∈ EP(G).

Since

$$\begin{array}{cc}\hfill \varphi \left(u_{\lambda ,n},y_{\lambda ,n}\right)& =\varphi \left(W_{r_{\lambda ,n}}{y}_{\lambda ,n},y_{\lambda ,n}\right)\hfill \\ \le \varphi \left(u,y_{\lambda ,n}\right)-\varphi \left(u,W_{r_{\lambda ,n}}{y}_{\lambda ,n}\right)\hfill \\ \le \varphi \left(u,x_n\right)+{\xi }_n-\varphi \left(u,W_{r_{\lambda ,n}}{y}_{\lambda ,n}\right)\hfill \\ =\varphi \left(u,x_n\right)+{\xi }_n-\varphi \left(u,u_{\lambda ,n}\right).\hfill \end{array}$$

(3.16)

Hence it follows from (3.11) and (3.16) that

$$\underset{n\to \infty }{\text{lim}}\varphi \left(u_{\lambda ,n},y_{\lambda ,n}\right)=0.$$

(3.17)

From (2.2) and (3.17) it yields (||u _{λ, n} || - ||y _{λ, n} ||) ² →0. Since ||u _{λ, n} ||→ ||x*||, we have

$$\left|\right|y_{\lambda ,n}\left|\right|\to \left|\right|x^{*}\left|\right|\left(n\to \infty \right).$$

(3.18)

Hence we have

$$\left|\right|J{y}_{\lambda ,n}\left|\right|\to \left|\right|J{x}^{*}\left|\right|\left(n\to \infty \right).$$

(3.19)

This implies that {Jy _{λ, n} } is bounded in E*. Since E is reflexive, and so is E*. we can assume that Jy _{λ, n} ⇀ g₀ ∈ E*. In view of the reflexive of E, we see that J(E) = E*. Hence there exists y ∈ E such that Jy = g₀. Since

$$\begin{array}{cc}\hfill \varphi \left(u_{\lambda ,n},y_{\lambda ,n}\right)& =\left|\right|u_{\lambda ,n}|{|}^2-2〈u_{\lambda ,n},J{y}_{\lambda ,n}〉+|\left|y_{\lambda ,n}\right|{|}^2\hfill \\ =\left|\right|u_{\lambda ,n}|{|}^2-2〈u_{\lambda ,n},J{y}_{\lambda ,n}〉+|\left|J{y}_{\lambda ,n}\right|{|}^2\hfill \end{array}$$

Taking lim inf _n→∞ on the both sides of equality above and in view of the weak lower semicontinuity of norm || · ||, it yields that

$$\begin{array}{cc}\hfill 0& \ge \left|\right|x^{*}|{|}^2-2〈x^{*},g_0〉+|\left|g_0\right|{|}^2=\left|\right|x^{*}|{|}^2-2〈x^{*},Jy〉+|\left|Jy\right|{|}^2\hfill \\ =\left|\right|x^{*}|{|}^2-2〈x^{*},Jy〉+|\left|y\right|{|}^2=\varphi \left(x^{*},y\right)\hfill \end{array}$$

i.e., x* = y. This implies that g₀ = Jx*, and so Jy _{λ, n} ⇀ Jx*. It follows from (3.19) and the Kadec-Klee property of E* that Jy _n (λ) → Jx*(n → ∞). Note that J^-1: E* → E is hemicontinuous, it yields that y _{λ, n} ⇀ x*. It follows from (3.18) and the Kadec-Klee property of E that

$$\underset{n\to \infty }{\text{lim}}{y}_{\lambda ,n}=x^{*}.$$

(3.20)

Since u _{λ, n} → x*, from (3.20), we have

$$\underset{n\to \infty }{\text{lim}}\left|\right|u_{\lambda ,n}-y_{\lambda ,n}\left|\right|=0.$$

(3.21)

Since J is uniformly norm-to-norm continuous on bounded sets, from (3.21), we have

$$\underset{n\to \infty }{\text{lim}}\left|\right|J{u}_{\lambda ,n}-J{y}_{\lambda ,n}\left|\right|=0.$$

(3.22)

From r _{λ, n} ≥ a, we have

$$\underset{n\to \infty }{\text{lim}}\frac{\left|\right|J{u}_{\lambda ,n}-J{y}_{\lambda ,n}\left|\right|}{r_{\lambda ,n}}=0.$$

(3.23)

By $u_{\lambda ,n}=W_{r_{\lambda ,n}}{y}_{\lambda ,n}$, we have

$$G\left(u_{\lambda ,n},y\right)+\frac{1}{r_{\lambda ,n}}〈y-u_{\lambda ,n},J{u}_{\lambda ,n}-J{y}_{\lambda ,n}〉\ge 0,\forall y\in C.$$

(3.24)

From condition (A₂), we have

$$\frac{1}{r_{\lambda ,n}}〈y-u_{\lambda ,n},J{u}_{\lambda ,n}-J{y}_{\lambda ,n}〉\ge -G\left(u_{\lambda ,n},y\right)\ge G\left(y,u_{\lambda ,n}\right),\forall y\in C.$$

(3.25)

Since G(x,·) is convex and lower semicontinuous, it is also weakly lower semicontinuous, letting n → ∞, we have from (3.25) and (A₄) that

$$G\left(y,x^{*}\right)\le 0,\forall y\in C.$$

(3.26)

For any t with 0 < t ≤ 1 and y ∈ C, let y _t = ty + (1 - t)x*. Since y ∈ C and hence G(y _t , x*) ≤ 0, from conditions (A₁) and (A₄), we have

$$0=G\left(y_t,y_t\right)\le tG\left(y_t,y\right)+\left(1-t\right)G\left(y_t,x^{*}\right)\le tG\left(y_t,y\right)$$

This implies that G(y _t , y) ≥ 0. Hence from condition (A₃), we have G(x*, y) ≥ 0 for all y ∈ C, and hence x* ∈ EP (G).

This completes the proof of Theorem 3.1.

The proof of Theorem 3.1 shows that the properties of generalized projections used in the iterative scheme do not interact with the properties of mappings {T _λ }.

Theorem 3.2. Let E be a uniformly smooth and strictly convex Banach space with Kadec-Klee property and C be a nonempty closed convex subset of E. A: C → E* be a continuous and monotone mapping, ψ: C → ℝ be a lower semi-continuous and convex function and F be a bifunction from C ×C to ℝ which satisfies the conditions (A₁)-(A₄). Let {T _λ }(λ ∈ Λ): C → C be a family of uniformly L _λ -Lipschitzian continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings such that $픉={\bigcap }_{\lambda \in \mathrm{\Lambda }}F\left(T_{\lambda }\right)\bigcap GMEP\ne \varnothing$. Assume that $K=\text{sup}\left\{\left|\right|u\left|\right|:u\in 픉\right\}<\infty$. Let {x _n } be the sequence generated by x₁ = x ∈ C, C₁ = C and

$$\left\{\begin{array}{c}{y}_{\lambda ,n}=J^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{T}_{\lambda }^n{x}_n\right),\lambda \in \mathrm{\Lambda },\hfill \\ u_{\lambda ,n}\in C\text{such}\text{that}\hfill \\ F\left(u_{\lambda ,n},y\right)+〈A{u}_{\lambda ,n},y-u_{\lambda ,n}〉+\psi \left(y\right)-\psi \left(u_{\lambda ,n}\right)\hfill \\ +\frac{1}{r_{\lambda ,n}}〈y-u_{\lambda ,n},J{u}_{\lambda ,n}-J{y}_{\lambda ,n}〉\ge 0,\forall y\in C,\lambda \in \mathrm{\Lambda }\hfill \\ C_{n+1}=\left\{z\in C_n:\underset{\lambda \in \mathrm{\Lambda }}{\text{sup}}\varphi \left(z,u_{\lambda ,n}\right)\le \varphi \left(z,x_n\right)+{\xi }_n\right\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}x,\forall n\ge 0.\hfill \end{array}\right.$$

(3.27)

where ${\xi }_n=\left(1-{\alpha }_n\right)\left(v_n{\text{sup}}_{u\in 픉}\zeta \left(\varphi \left(u,x_n\right)\right)+{\mu }_n\right)$, {α _n } is a sequence in 0[1] such that lim inf _n→∞ α _n (1 - α _n ) > 0, lim inf _n→∞ α _n < 1 and {r _{λ, n} } ⊂ [a, ∞) for some a > 0, then {x _n } converge strongly to some point x* in F.

Proof. In Theorem 3.1 take ζ(t) = t, ν _n = k _n - 1, μ _n = 0. Therefore the conclusion of Theorem 3.2 can be obtained form Theorem 3.1.