A new iterative scheme for equilibrium problems, fixed point problems for nonexpansive mappings and maximal monotone operators

Abstract

In this article, we introduce a new iterative scheme for finding a common element of the set of fixed points of strongly relatively nonexpansive mapping, the set of solutions for equilibrium problems and the set of zero points of maximal monotone operators in a uniformly smooth and uniformly convex Banach space. Consequently, we obtain new strong convergence theorems in the frame work of Banach spaces. Our theorems extend and improve the recent results of Wei et al., Takahashi and Zembayashi, and some recent results.

1 Introduction

Let E be a real Banach space with norm ∥ · ∥ and let C be a nonempty closed convex subset of E. Let E* be the dual space of E and 〈·,·〉 denote the pairing between E and E*. We consider the problem for finding:

$$v\in E\mathsf{\text{such}}\mathsf{\text{that}}0\in Av,$$

(1.1)

where A is an operator from E into E*, such that v ∈ E is called a zero point of A, i.e., A^-10 = {v ∈ E : Av = 0}. Such a problem contains numerous problems in economics, optimization and physics. Many authors studied this problem see, for example [1–5] and references therein.

Let F : C × C → ℝ be a bifunction, where ℝ is the set of real numbers. The equilibrium problem for F : C × C → ℝ is to find x ∈ C such that

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

(1.2)

The set of solutions of is denote by EP(F). The above formulation (1.2) was shown in [6] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, vector equilibrium problems, Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point problem and optimization problem, which can also be written in the form of an EP(F). In other words, the EP(F) is an unifying model for several problems arising in physics, engineering, science, optimization, economics, see, for example [6–9] and references therein.

In 2009, Takahashi and Zembayashi [10], proposed the iteration in a uniformly smooth and uniformly convex Banach space: as sequence {x _n } generated by x₁ ∈ E,

$$\left\{\begin{array}{c}F\left(u_n,y\right)+\frac{1}{r_n}⟨y-u_n,J{u}_n-J{x}_n⟩\ge 0,\forall y\in C,\hfill \\ x_{n+1}=J^{-1}\left({\alpha }_{n}J{u}_n+\left(1-{\alpha }_n\right)JS{u}_n\right),\hfill \end{array}\right.$$

(1.3)

for every n ∈ ℕ, where J is duality mappings form E to E*, S is a relatively nonexpansive self mapping on C and {α _n } is appropriate positive real sequence. They proved that if J is weakly sequentially continuous, then {x _n } converges weakly to some element in EP(F) ∩ F(S), where F(S) is the fixed point set of S i.e., F(S) := {x ∈ C : Sx = x}.

In 2010, Wei et al. [5], constructed the following iterative scheme to approximate the common element of the set of fixed points of a relatively nonexpansive mapping S : E → E and the set of zero points of a maximal monotone operator A : E → 2^E*:

$$\left\{\begin{array}{c}{x}_1\in E,\hfill \\ y_n=J^{-1}\left({\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{Q}_{\lambda }^A{x}_n\right),\hfill \\ x_{n+1}=J^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)JS{y}_n\right),\hfill \end{array}\right.$$

(1.4)

for every n ∈ ℕ, where $Q_{\lambda }^A:E\to E^{*}$ define by $Q_{\lambda }^{A}x={\left(J+\lambda A\right)}^{-1}Jx$ for all x ∈ E and {α _n }, {β _n } are sequence in [0,1). They proved that if J is weakly sequentially continuous, then {x _n } converges weakly to some element in F(S) ∩ A^-10.

Recently, Nilsrakoo [11], proved a strong convergence theorem for finding a common element of the fixed points set of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a uniformly convex and uniformly smooth Banach space. In this article, motivated by the above results and the iterative schemes considered of Wei, et al. [5], Takahashi and Zembayashi [10], we present a new iterative scheme for approximation of a common element in the intersection of the set of solutions for equilibrium problems, the set of zero points of maximal monotone operators and set of fixed points for relatively nonexpansive mapping in a uniformly smooth and uniformly convex Banach space. We prove a strong convergence theorem under some mind conditions. The results presented in this article extend and improve the results of Wei et al. [5], Takahashi and Zembayashi [10], and some authors.

2 Preliminaries

Let E be a real Banach space and let E* be the dual space of E. For q > 1, the generalized duality mapping J _q : E → 2^E*is defined by

$$J_q\left(x\right)=\left\{f\in E^{*}:⟨x,f⟩={∥x∥}^q,∥f∥={∥x∥}^{q-1}\right\}$$

for all x ∈ E. In particular, if q = 2, the mapping J₂ is called the normalized duality mapping and usually write J₂ = J.

Let U = {x ∈ E : ∥x∥ = 1}. A Banach space E is said to be strictly convex if $∥\frac{x+y}{2}∥<1$ for all x, y ∈ U and x ≠ y. A Banach space E is said to be uniformly convex if, for any ϵ ∈ (0, 2], there exists δ > 0 such that, for any x, y ∈ U, ∥x - y∥ ≥ ϵ implies $∥\frac{x+y}{2}∥\le 1-\delta$. It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space E is said to be smooth if the limit $\underset{t\to 0}{\text{lim}}\frac{∥x+ty∥-∥x∥}{t}$ exists or all x, y ∈ U. It is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ U. The modulus of smoothness of E is defined by

$$\rho \left(\tau \right)=\text{sup}\left\{\frac{1}{2}\left(∥x+y∥+∥x-y∥\right)-1:x,y\in E,∥x∥=1,∥y∥=\tau \right\},$$

where ρ : [0, ∞) → [0, ∞) is a function. It is known that E is uniformly smooth if and only if ${\text{lim}}_{\tau \to 0}\frac{\rho \left(\tau \right)}{\tau }=0$. Let q be a fixed real number with 1 < q ≤ 2.

A Banach space E is said to be q-uniformly smooth if there exists a constant c > 0 such that ρ(τ) ≤ cτ^qfor all τ > 0.

We note that E is a uniformly smooth Banach space if and only if J _q is single-valued and uniformly continuous on any bounded subset of E. Typical examples of both uniformly convex and uniformly smooth Banach spaces are L^p, where p > 1. More precisely, L^pis min{p, 2}-uniformly smooth for every p > 1.

A multi-valued operator A : E → 2^E*;

• (i) The graph of A, G(A) = {(u, v) | u ∈ E and v ∈ A(u)};

• (ii) A is said to be monotone if 〈x ₁ - x ₂, y ₁ - y ₂〉 ≥ 0, x ₁, x ₂ ∈ D(A), y ₁ ∈ Ax ₁, y ₂ ∈ Ax ₂;

• (iii) A is maximal monotone if it is monotone and its graph is maximal with respect to this property, i.e., it is not properly contained in the graph of any other monotone operator.

Example. The mapping A : ℝ → 2^ℝ defined by

$$A\left(x\right)=\left\{\begin{array}{cc}x-a,\hfill & \hfill x<0,\hfill \\ \left[-a,a\right],\hfill & \hfill x=0,\hfill \\ x+a,\hfill & \hfill x>0.\hfill \end{array}\right.$$

(2.1)

Then, A is a monotone mapping.

Note. A is maximal monotone if and only if A(x) = [-a, a], when x = 0.

Let E be smooth Banach space and J the normalized duality mapping from E to E*. Alber [12] considered the following functional φ : E × E → [0, ∞) defined by

$$\phi \left(x,y\right)={∥x∥}^2-2⟨x,Jy⟩+{∥y∥}^2,x,y\in E.$$

(2.2)

It is obvious from the definition of the function φ that

$${\left(∥x∥-∥y∥\right)}^2\le \phi \left(x,y\right)\le {\left(∥x∥+∥y∥\right)}^2$$

and

$$\phi \left(x,J^{-1}\left(tJy+\left(1-t\right)Jz\right)\right)\le t\phi \left(x,y\right)+\left(1-t\right)\phi \left(x,z\right)$$

(2.3)

for all t ∈ [0,1] and x, y, z ∈ E. The following lemma is an analogue of Xu's inequality with respect to φ.

Lemma 2.1. [11]Let E be a uniformly smooth Banach space and r > 0. Then there exists a continuous, strictly increasing, and convex function g : [0, 2r] → [0, ∞) such that g(0) = 0 and

$$\phi \left(x,J^{-1}\left(tJy+\left(1-t\right)Jz\right)\right)\le t\phi \left(x,y\right)+\left(1-t\right)\phi \left(x,z\right)-t\left(1-t\right)g\left(∥Jy-Jz∥\right)$$

(2.4)

for all t ∈ [0,1], x ∈ E and y, z ∈ B _r .

Let C be a closed convex subset of E, and S be a mapping from C into itself. Let F(S) = {x ∈ C : Sx = x} be the set of fixed points of S. A point p ∈ C is said to be an asymptotic fixed point S if C contains a sequence {x _n } which converges weakly to p such that lim_n→∞(x _n - Sx _n ) = 0. The set of asymptotic fixed points of S will denoted by $\widehat{F\left(s\right)}$.

A mapping S from C into itself is said to be relatively nonexpansive if

(C1) $F\left(s\right)\ne \varnothing$;

(C2) φ(p, Sx) ≤ φ(p, x) for all x ∈ C and p ∈ F(S);

(C3) $\widehat{F\left(S\right)}=F\left(S\right)$.

A mapping S from C into itself is said to be strongly relatively nonexpansive if

(D1) S is relatively nonexpansive;

(D2) φ(Sx _n ,x _n ) → 0 whenever {x _n } is bounded sequence in C such that φ(p, x _n ) - φ(p, Sx _n ) → 0 for some p ∈ F(S).

Lemma 2.2. [13]The duality mapping J has the following properties:

(i) If E is a real reflexive and smooth Banach space, then J : E → E* is single-valued;

(ii) For all x ∈ E and λ > 0, J(λx) = λJx;

(iii) If E is strictly convex, then J is one to one and strictly monotone, that is, 〈x - y, x* - y*〉 > 0 hold for all x* ∈ Jx and y* ∈ Jy with x ≠ y;

(iv) If E is a real uniformly convex and uniformly smooth Banach space, then J^-1 : E* → E is also a duality mapping. Moreover, both J and J^-1are uniformly continuous on each bounded subset of E or E*, respectively.

Lemma 2.3. [8, 12]Let E be a real reflexive, strictly convex and smooth Banach space, let C be a nonempty closed and convex subset of E, and x ∈ E. Then there exists a unique element x₀ ∈ C such that φ(x₀,x) = min{φ(z,x) : z ∈ C}.

In this case, the mapping Π _C of E onto C defined by ${\Pi }_{C^x}=x_0$ for all x ∈ E is called the generalized projection operator.

Lemma 2.4. [14]Let E and C be the same as those in Lemma 2.3. Let x ∈ E and$\widehat{x}\in C$. Then,

• (a) $\widehat{x}={\Pi }_{C^x}$ if and only if $⟨y-\widehat{x},Jx-J\widehat{x}⟩\le 0$, for all y ∈ C;

• (b) $\phi \left(y,{\Pi }_{C^x}\right)+\phi \left({\Pi }_{C^x},x\right)\le \phi \left(y,x\right)$, for all y ∈ C.

Lemma 2.5. [14]Let E be a real smooth and uniformly convex Banach space and let {x _n } and {y _n } be two sequences of E. If either {x _n } or {y _n } is bounded and φ(x _n , y _n ) → 0 as n → ∞, then x _n - y _n → 0 as n → ∞.

Lemma 2.6. [8]Let E be a real smooth and strictly convex Banach space and let C be a closed convex subset of E, and let S be a relatively nonexpansive mapping from C into itself. Then F(S) is convex and closed.

Lemma 2.7. [14]Let E be a real smooth Banach space, let C be a convex subset of E, let x ∈ E and x₀ ∈ C. Then φ(x₀,x) = inf{φ(z, x) : z ∈ C} if and only if 〈z - x₀, Jx₀ - Jx〉 ≥ 0 for all z ∈ C.

Lemma 2.8. [13, 15]Let E be a real smooth and uniformly convex Banach space and let A : E → 2^E*be a maximal monotone operator. Then A^-10 is a closed and convex subset of E and the graph G(A) of A, is demi-closed in the following sense: for all {x _n } ⊂ D(A) with x _n ⇀ x ∈ E and y _n ∈ Ax _n with y _n → y ∈ E*, we have x ∈ D(A) and y ∈ Ax.

Definition 2.9. Let E and A be the same as these in Lemma 2.8. For all λ > 0, define the operator $Q_{\lambda }^A:E\to E$ by $Q_{\lambda }^{A}x={\left(J+\lambda A\right)}^{-1}Jx$ for all x ∈ E.

Lemma 2.10. [16]Let E be a real reflexive, strictly convex and smooth Banach space and let A : E → 2^E*be a maximal monotone operator with A^-10. Then, for all x ∈ E, y ∈ A^-10 and λ > 0, we have$\phi \left(y,Q_{\lambda }^{A}x\right)+\phi \left(Q_{\lambda }^{A}x,x\right)\le \phi \left(y,x\right)$.

Let E be a reflexive, strictly convex and smooth Banach space. The duality mapping J* from E* onto E** = E coincides with the inverse of the duality mapping J from E onto E*, that is, J* = J^-1. We make use of the following mapping V : E × E → ℝ studied in Alber [12]:

$$V\left(x,x^{*}\right)={∥x∥}^2-2⟨x,x^{*}⟩+{∥x^{*}∥}^2$$

(2.5)

for all x ∈ E and x* ∈ E*. Obviously, V(x, x*) = φ(x, J^-1(x*)). We know the following lemma.

Lemma 2.11. [11]Let E be a reflexive, strictly convex and smooth Banach space and let V be as in (2.5). Then

$$V\left(x,x^{*}\right)+2⟨J^{-1}\left(x^{*}\right)-x,y^{*}⟩\le V\left(x,x^{*}+y^{*}\right)$$

for all x ∈ E and x*, y* ∈ E*.

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

(A1) F(x, x) = 0, ∀x ∈ C;

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

(A3) for each x, y, z ∈ C, $\underset{t\downarrow 0}{\text{lim}}F\left(tz+\left(1-t\right)x,y\right)\le F\left(x,y\right)$;

(A4) for each x ∈ C, y ↦ F(x, y) is convex and lower semi-continuous.

In what follows, we shall make use of the following lemmas.

Lemma 2.12. [10]Let C be a closed convex subset of smooth, strictly convex and reflexive Banach space E, let F : C × C → ℝ be a bifunction satisfies (A1)-(A4) and let r > 0 and 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 2.13. [10]Let C be nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space E. Let F : C × C → ℝ be a bifunction satisfies (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 ∈ C. Then, the following conclusions 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, JT _r x - JT _r y〉 ≤ 〈T _r x - T _r y, Jx - Jy〉;

(3) F(T _r ) = EP(F);

(4) EP(F) is closed and convex.

Lemma 2.14. [17]Let C be a nonempty closed convex subset of a Banach space E, let F : C × C → ℝ. be a bifunction satisfying conditions (A1)-(A4) and z ∈ C. Then z ∈ EP(F) if and only if F(y, z) ≤ 0, ∀y ∈ C.

Remark 2.15. [18] Let C be a nonempty subset of a smooth Banach space E. If S : C → E is firmly nonexpansive-type mapping, then

$$\phi \left(z,Sx\right)\le \phi \left(z,Sx\right)+\phi \left(Sx,x\right)\le \phi \left(z,x\right)$$

for all x ∈ C and z ∈ F(S). In particular, S satisfies condition (C2).

Lemma 2.16. [19]Let {a _n } is a sequence of nonnegative real numbers such that

$$a_{n+1}\le \left(1-b_n\right)a_n+b_n{c}_n,n\ge 0$$

where {b _n } is a sequence in (0,1) and {c _n } is a sequence in ℝ such that

(1)${\sum }_{n=1}^{\infty }{b}_n=\infty$, lim_n→∞b _n = 0;

(2) lim sup_n→∞c _n ≤ 0.

Then lim_n→∞a _n = 0.

Lemma 2.17. [20]Let {a _n } be a sequence of real numbers such that there exists a subsequence {n _i } of {n} such that$a_{n_i}<a_{n_i+1}$for all i∈ℕ. Then there exists a nondecreasing sequence {m _k } ⊂ ℕ such that m _k → ∞,

$$a_{m_k}\le a_{m_k+1}and{a}_k\le a_{m_k+1},$$

(2.6)

for all k ∈ ℕ. In fact, m _k = max{j ≤ k : a _j < a_j+1}.

3 Main result

In this section, we prove a strong convergence theorem for finding a common element of the set of solutions for equilibrium problems, the set of zero points of maximal monotone operators and set of fixed points for strongly relatively nonexpansive mapping in a uniformly convex and uniformly smooth Banach space.

Theorem 3.1. Let C be nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, F: C × C → ℝ bea bifunction satisfying conditions (A1)-(A4), S : C → C be a strongly relatively nonexpansive mapping, and let A : E → 2^E*be a maximal monotone operator with$\Omega =:F\left(S\right)\cap EP\left(F\right)\cap A^{-1}\left(0\right)\ne \varnothing$. For a positive number λ, let the sequence {x _n } be generated by x₀ ∈ C, x₁ ∈ E and

$$\left\{\begin{array}{c}F\left(u_n,y\right)+\frac{1}{r_n}⟨y-u_n,J{u}_n-J{x}_n⟩\ge 0,\forall y\in C,\hfill \\ y_n={\Pi }_C{J}^{-1}\left({\beta }_{n}J{x}_0+\left(1-{\beta }_n\right)J{Q}_{\lambda }^A{u}_n\right),\hfill \\ x_{n+1}=J^{-1}\left({\alpha }_{n}J{u}_n+\left(1-{\alpha }_n\right)JS{y}_n\right),\hfill \end{array}\right.$$

(3.1)

for every n ≥ 1, where {α _n }, {β _n } are sequences in [0,1), and {r _n } ⊂ (0, ∞). If the control sequences satisfy the following restrictions:

(i)${\sum }_{n=1}^{\infty }{\beta }_n=\infty$and lim_n→∞β _n = 0;

(ii) 0 < lim inf_n→∞α _n ≤ lim sup_n→∞α _n < 1;

(iii) lim inf_n→∞r _n > 0.

Then {x _n } and {u _n } converge strongly to p = Π_Ω(x₀), where Π_Ωis the generalized projection from E onto Ω.

Proof. First, taking p = Π_Ω(x₀), since ${\Pi }_C,T_{r_n},Q_{\lambda }^A$ and S satisfy the condition (C2) and (2.4), $u_n=T_{r_n}{x}_n$, it follows that

$$\begin{array}{ll}\hfill \phi \left(p,y_n\right)& =\phi \left(p,{\Pi }_C{J}^{-1}\left({\beta }_{n}J{x}_0+\left(1-{\beta }_n\right)J{Q}_{\lambda }^A{u}_n\right)\right)\\ \le \phi \left(p,J^{-1}\left({\beta }_{n}J{x}_0+\left(1-{\beta }_n\right)J{Q}_{\lambda }^A{u}_n\right)\right)\\ \le {\beta }_n\phi \left(p,x_0\right)+\left(1-{\beta }_n\right)\phi \left(p,Q_{\lambda }^A{u}_n\right)\\ \le {\beta }_n\phi \left(p,x_0\right)+\left(1-{\beta }_n\right)\phi \left(p,u_n\right)\\ ={\beta }_n\phi \left(p,x_0\right)+\left(1-{\beta }_n\right)\phi \left(p,T_{r_n}{x}_n\right)\\ \le {\beta }_n\phi \left(p,x_0\right)+\left(1-{\beta }_n\right)\phi \left(p,x_n\right)\end{array}$$

and

$$\begin{array}{ll}\hfill \phi \left(p,x_{n+1}\right)& =\phi \left(p,J^{-1}\left({\alpha }_{n}J{u}_n+\left(1-{\alpha }_n\right)JS{y}_n\right)\right)\\ \le {\alpha }_n\phi \left(p,u_n\right)+\left(1-{\alpha }_n\right)\phi \left(p,S{y}_n\right)\\ ={\alpha }_n\phi \left(p,T_{r_n}{x}_n\right)+\left(1-{\alpha }_n\right)\phi \left(p,S{y}_n\right)\\ \le {\alpha }_n\phi \left(p,x_n\right)+\left(1-{\alpha }_n\right)\phi \left(p,y_n\right)\\ \le {\alpha }_n\phi \left(p,x_n\right)+\left(1-{\alpha }_n\right){\beta }_n\phi \left(p,x_0\right)+\left(1-{\alpha }_n\right)\left(1-{\beta }_n\right)\phi \left(p,x_n\right)\\ =\left(1-{\alpha }_n\right){\beta }_n\phi \left(p,x_0\right)+\left(1-{\beta }_n\left(1-{\alpha }_n\right)\right)\phi \left(p,x_n\right)\\ \le \underset{n\ge 1}{\text{max}}\left\{\phi \left(p,x_0\right),\phi \left(p,x_n\right)\right\}\\ \le \cdots \\ \le \underset{n\ge 1}{\text{max}}\left\{\phi \left(p,x_0\right),\phi \left(p,x_1\right)\right\}.\end{array}$$

This show that {x _n } is bounded. Hence {u _n }, {y _n } and {Sy _n } are also bounded.

Put $v_n=J^{-1}\left({\beta }_{n}J{x}_0+\left(1-{\beta }_n\right)J{Q}_{\lambda }^A{u}_n\right)$.

Using Lemma 2.11 gives

$$\begin{array}{ll}\hfill \phi \left(p,y_n\right)& =\phi \left(p,{\Pi }_C{v}_n\right)\\ \le \phi \left(p,v_n\right)=V\left(p,J{v}_n\right)\\ \le V\left(p,J{v}_n-{\beta }_n\left(J{x}_0-Jp\right)\right)-2⟨v_n-p,-{\beta }_n\left(J{x}_0-Jp\right)⟩\\ =\phi \left(p,J^{-1}\left({\beta }_{n}Jp+\left(1-{\beta }_n\right)J{Q}_{\lambda }^A{u}_n\right)\right)+2{\beta }_n⟨v_n-p,J{x}_0-Jp⟩\\ \le {\beta }_n\phi \left(p,p\right)+\left(1-{\beta }_n\right)\phi \left(p,Q_{\lambda }^A{u}_n\right)+2{\beta }_n⟨v_n-p,J{x}_0-Jp⟩\\ \le {\beta }_n\phi \left(p,p\right)+\left(1-{\beta }_n\right)\phi \left(p,u_n\right)+2{\beta }_n⟨v_n-p,J{x}_0-Jp⟩\\ ={\beta }_n\phi \left(p,p\right)+\left(1-{\beta }_n\right)\phi \left(p,T_{r_n}{x}_n\right)+2{\beta }_n⟨v_n-p,J{x}_0-Jp⟩\\ \le \left(1-{\beta }_n\right)\phi \left(p,x_n\right)+2{\beta }_n⟨v_n-p,J{x}_0-Jp⟩.\end{array}$$

(3.2)

Let g : [0, 2r] → [0, ∞) be a function satisfying the properties of Lemma 2.1, where r = sup_n≥1{∥u _n ∥, ∥Sy _n ∥}.

By Lemma 2.1, Remark 2.15 and (3.2), we get

$$\begin{array}{ll}\hfill \phi \left(p,x_{n+1}\right)& =\phi \left(p,J^{-1}\left({\alpha }_{n}J{u}_n+\left(1-{\alpha }_n\right)JS{y}_n\right)\right)\\ \le {\alpha }_n\phi \left(p,u_n\right)+\left(1-{\alpha }_n\right)\phi \left(p,S{y}_n\right)-{\alpha }_n\left(1-{\alpha }_n\right)g\left(∥J{u}_n-JS{y}_n∥\right)\\ \le {\alpha }_n\left(\phi \left(p,x_n\right)-\phi \left(u_n,x_n\right)\right)+\left(1-{\alpha }_n\right)\phi \left(p,y_n\right)-{\alpha }_n\left(1-{\alpha }_n\right)g\left(∥J{u}_n-JS{y}_n∥\right)\\ \le {\alpha }_n\left(\phi \left(p,x_n\right)-\phi \left(u_n,x_n\right)\right)+\left(1-{\alpha }_n\right)\left(\left(1-{\beta }_n\right)\phi \left(p,x_n\right)+2{\beta }_n⟨v_n-p,J{x}_0-Jp⟩\right)\\ -{\alpha }_n\left(1-{\alpha }_n\right)g\left(∥J{u}_n-JS{y}_n∥\right)\\ =\left(1-\left(1-{\alpha }_n\right){\beta }_n\right)\phi \left(p,x_n\right)+2\left(1-{\alpha }_n\right){\beta }_n⟨v_n-p,J{x}_0-Jp⟩\\ -{\alpha }_n\phi \left(u_n,x_n\right)-{\alpha }_n\left(1-{\alpha }_n\right)g\left(∥J{u}_n-JS{y}_n∥\right)\end{array}$$

(3.3)

$$\le \left(1-\left(1-{\alpha }_n\right){\beta }_n\right)\phi \left(p,x_n\right)+2\left(1-{\alpha }_n\right){\beta }_n⟨v_n-p,J{x}_0-Jp⟩.$$

(3.4)

We divide the proof into two parts:

Case 1. Suppose that there exists n₀ ∈ ℕ such that ${\left\{\phi \left(p,x_n\right)\right\}}_{n=n_0}^{\infty }$ is nonincreasing. In this situation, ${\left\{\phi \left(p,x_n\right)\right\}}_{n=n_0}^{\infty }$ is convergent. Then

$$\phi \left(p,x_n\right)-\phi \left(p,x_{n+1}\right)\to 0,\mathsf{\text{as}}n\to \infty .$$

(3.5)

From (3.3) and β _n → 0, we have

$${\alpha }_n\phi \left(u_n,x_n\right)+{\alpha }_n\left(1-{\alpha }_n\right)g\left(∥J{u}_n-JS{y}_n∥\right)\to 0,\mathsf{\text{as}}n\to \infty .$$

Since {α _n } ⊂ [a, b] ⊂ (0,1), we get

$$\phi \left(u_n,x_n\right)\to 0\mathsf{\text{and}}g\left(∥J{u}_n-JS{y}_n∥\right)\to 0,\mathsf{\text{as}}n\to \infty .$$

By Lemma 2.5, we have

$$u_n-x_n\to 0,J{u}_n-JS{y}_n\to 0\mathsf{\text{and}}{u}_n-S{y}_n\to 0,\mathsf{\text{as}}n\to \infty .$$

By (2.3) and β _n → 0, we have

$$\begin{array}{ll}\hfill \phi \left(u_n,y_n\right)& \le \phi \left(u_n,v_n\right)\\ =\phi \left(u_n,J^{-1}\left({\beta }_{n}J{x}_0+\left(1-{\beta }_n\right)J{Q}_{\lambda }^A{u}_n\right)\right)\\ \le {\beta }_n\phi \left(u_n,x_0\right)+\left(1-{\beta }_n\right)\phi \left(u_n,Q_{\lambda }^A{u}_n\right)\\ \le {\beta }_n\phi \left(u_n,x_0\right)+\left(1-{\beta }_n\right)\phi \left(u_n,u_n\right)={\beta }_n\phi \left(u_n,x_0\right)\to 0\mathsf{\text{as}}n\to \infty .\end{array}$$

By Lemma 2.5, implies that u _n - y _n → 0 and u _n - v _n → 0 as n → ∞.

Hence,

$$v_n-y_n\to 0,y_n-S{y}_n\to 0\mathsf{\text{and}}{y}_n-x_n\to 0,\mathsf{\text{as}}n\to \infty .$$

(3.6)

Now let us show that ω(x _n ) ⊂ EP(F) ∩ F(S) ∩ A^-10, where

$$\omega \left(x_n\right):=\left\{\bar{x}\in C:x_{n_i}\rightharpoonup \bar{x},\exists \left\{n_i\right\}\subset \left\{n\right\}\mathsf{\text{with}}\left\{n_i\right\}\to \infty \right\}.$$

Indeed, since {x _n } is bounded and E is reflexive, we know that $\omega \left(x_n\right)\ne \varnothing$. Take $\bar{x}\in \omega \left(x_n\right)$ arbitrary, there exists a subsequence $\left\{x_{n_i}\right\}$ of {x _n } such that $x_{n_i}\rightharpoonup \bar{x}$. Let us show that $\bar{x}\in A^{-1}0$.

From $T_{r_n}{x}_n=u_n$, Lemma 2.10 and (2.3), we have

$$\begin{array}{ll}\hfill \phi \left(p,x_{n+1}\right)& \le {\alpha }_n\phi \left(p,u_n\right)+\left(1-{\alpha }_n\right)\phi \left(p,S{y}_n\right)\\ \le {\alpha }_n\phi \left(p,u_n\right)+\left(1-{\alpha }_n\right)\left[{\beta }_n\phi \left(p,x_0\right)+\left(1-{\beta }_n\right)\phi \left(p,Q_{\lambda }^A{u}_n\right)\right]\\ \le {\alpha }_n\phi \left(p,u_n\right)+\left(1-{\alpha }_n\right){\beta }_n\phi \left(p,x_0\right)+\left(1-{\alpha }_n\right)\left(1-{\beta }_n\right)\left[\phi \left(p,u_n\right)-\phi \left(Q_{\lambda }^A{u}_n,u_n\right)\right]\\ =\left(1-\left(1-{\alpha }_n\right){\beta }_n\right)\phi \left(p,u_n\right)+\left(1-{\alpha }_n\right){\beta }_n\phi \left(p,x_0\right)-\left(1-{\alpha }_n\right)\left(1-{\beta }_n\right)\phi \left(Q_{\lambda }^A{u}_n,u_n\right)\\ =\left(1-\left(1-{\alpha }_n\right){\beta }_n\right)\phi \left(p,T_{r_n}{x}_n\right)+\left(1-{\alpha }_n\right){\beta }_n\phi \left(p,x_0\right)-\left(1-{\alpha }_n\right)\left(1-{\beta }_n\right)\phi \left(Q_{\lambda }^A{u}_n,u_n\right)\\ \le \left(1-\left(1-{\alpha }_n\right){\beta }_n\right)\phi \left(p,x_n\right)+\left(1-{\alpha }_n\right){\beta }_n\phi \left(p,x_0\right)-\left(1-{\alpha }_n\right)\left(1-{\beta }_n\right)\phi \left(Q_{\lambda }^A{u}_n,u_n\right)\\ \le \phi \left(p,x_n\right)+\left(1-{\alpha }_n\right){\beta }_n\phi \left(p,x_0\right)-\left(1-{\alpha }_n\right)\left(1-{\beta }_n\right)\phi \left(Q_{\lambda }^A{u}_n,u_n\right).\end{array}$$

It follows that

$$\left(1-{\alpha }_n\right)\left(1-{\beta }_n\right)\phi \left(Q_{\lambda }^A{u}_n,u_n\right)\le \phi \left(p,x_n\right)-\phi \left(p,x_{n+1}\right)+\left(1-{\alpha }_n\right){\beta }_n\phi \left(p,x_0\right).$$

(3.7)

Since β _n → 0, lim sup_n→∞α _n < 1 and (3.5), we obtain $\phi \left(Q_{\lambda }^A{u}_n,u_n\right)\to 0$ as n → ∞.

From Lemma 2.5, implies that $Q_{\lambda }^A{u}_n-u_n\to 0$ as n → ∞. Since y _n - x _n → 0, then $y_{n_i}\rightharpoonup \bar{x}$ and from Sy _n - y _n → 0 as as n → ∞. Hence $\bar{x}\in \widehat{F\left(S\right)}=F\left(S\right)$.

Since J is uniformly continuous on bound subset of E, we have $J{Q}_{\lambda }^A{u}_n-J{u}_n\to 0$ as n → ∞.

Let $z_n=Q_{\lambda }^A{u}_n$. Then there exists w _n ∈ Az _n such that

$$w_n=\frac{J{u}_n-J{z}_n}{\lambda }=\frac{J{u}_n-J{Q}_{\lambda }^A{u}_n}{\lambda }\to 0,\mathsf{\text{as}}n\to \infty .$$

(3.8)

Since $Q_{\lambda }^A{u}_n-u_n\to 0$, u _n - x _n → 0, and $x_{n_i}\rightharpoonup \bar{x}$, then $Q_{\lambda }^A{u}_{n_i}=z_{n_i}\rightharpoonup \bar{x}$. By Lemma 2.8, $\bar{x}\in A^{-1}0$. Thus ω(x _n ) ⊂ F(S) ∩ A^-10.

Finally, let us show that $\bar{x}\in EP\left(F\right)$. Since x _n - u _n → 0 and $x_{n_i}\rightharpoonup \bar{x}$, we obtain that $u_{n_i}\rightharpoonup \bar{x}$, and

$$\underset{n\to \infty }{\text{lim}}∥J{u}_n-J{x}_n∥=0.$$

From lim inf_n→∞r _n > 0, it follows that

$$\underset{n\to \infty }{\text{lim}}\frac{∥J{u}_n-J{x}_n∥}{r_n}=0.$$

(3.9)

By the definition of $u_n=T_{r_n}{x}_n$, we have

$$F\left(u_n,y\right)+\frac{1}{r_n}⟨y-u_{n}J{u}_n-J{x}_n⟩\ge 0,\forall y\in C.$$

(3.10)

Replacing n by n _i , we have from (A2) that

$$\frac{1}{r_{n_i}}⟨y-u_{n_i},J{u}_{n_i}-J{x}_{n_i}⟩\ge -F\left(u_{n_i},y\right)\ge F\left(y,u_{n_i}\right),\forall y\in C.$$

(3.11)

Letting i → ∞, from (3.9), (A4) and $u_{n_i}\rightharpoonup \bar{x}$ that

$$F\left(y,\bar{x}\right)\le 0,\forall y\in C.$$

(3.12)

From Lemma 2.14, we have ω(x _n ) ⊂ EP(F) ∩ F(S) ∩ A^-10.

Since {y _n } is bounded and E is reflexive, we choose a subsequence $\left\{y_{n_i}\right\}$ of {y _n } such that

$$\underset{n\to \infty }{\text{lim}\text{sup}}⟨y_n-p,J{x}_0-Jp⟩=\underset{i\to \infty }{\text{lim}}⟨y_{n_i}-p,J{x}_0-Jp⟩.$$

Since $y_{n_i}\rightharpoonup \bar{x}\in \Omega$. By Lemma 2.4(a), we obtain that

$$\underset{n\to \infty }{\text{lim}\text{sup}}⟨y_n-p,J{x}_0-Jp⟩=\underset{i\to \infty }{\text{lim}}⟨y_{n_i}-p,J{x}_0-Jp⟩=⟨\bar{x}-p,J{x}_0-Jp⟩\le 0.$$

Since v _n - y _n → 0, we have

$$\underset{n\to \infty }{\text{lim}\text{sup}}⟨v_n-p,J{x}_0-Jp⟩=\underset{n\to \infty }{\text{lim}\text{sup}}⟨y_n-p,J{x}_0-Jp⟩\le 0.$$

(3.13)

By (3.4), it follows that

$$\phi \left(p,x_{n+1}\right)\le \left(1-\left(1-{\alpha }_n\right){\beta }_n\right)\phi \left(p,x_n\right)+2\left(1-{\alpha }_n\right){\beta }_n⟨v_n-p,J{x}_0-Jp⟩.$$

(3.14)

Set b _n = (1 - α _n )β _n and c _n = 2〈v _n - p, Jx₀ - Jp〉. Then we have

$$\phi \left(p,x_{n+1}\right)\le \left(1-b_n\right)\phi \left(p,x_n\right)+b_n{c}_n.$$

(3.15)

From the condition (i) and (3.13), we see that ${\sum }_{n=0}^{\infty }{b}_n=\infty$ and lim sup_n→∞c _n ≤ 0. Therefore, applying Lemma 2.16 to (3.15), we get that φ(p,x _n ) → 0. Then x _n → p and since u _n - x _n → 0, we have u _n → p.

Case 2. Suppose that there exists a subsequence {n _i } of {n} such that

$$\phi \left(p,x_{n_i}\right)<\phi \left(p,x_{n_i+1}\right),\forall i\in \mathbb{N}.$$

(3.16)

By Lemma 2.17, there exists a nondecreasing sequence {m _k } ⊂ ℕ such that m _k → ∞, $\phi \left(p,x_{m_k}\right)<\phi \left(p,x_{m_k+1}\right)$ and $\phi \left(p,x_k\right)<\phi \left(p,x_{m_k+1}\right)$, for all k ∈ ℕ.

From (3.3), condition (i) and (ii), we have

$$\begin{array}{l}{\alpha }_{m_k}\phi \left(u_{m_k},x_{m_k}\right)+{\alpha }_{m_k}\left(1-{\alpha }_{m_k}\right)g\left(∥J{u}_{m_k}-JS{y}_{m_k}∥\right)\\ \le \phi \left(p,x_{m_k}\right)-\phi \left(p,x_{m_k+1}\right)-\left(1-{\alpha }_{m_k}\right){\beta }_{m_k}\phi \left(p,x_{m_k}\right)+2\left(1-{\alpha }_{m_k}\right){\beta }_{m_k}⟨v_{m_k}-p,J{x}_0-Jp⟩\\ \le -\left(1-{\alpha }_{m_k}\right){\beta }_{m_k}\phi \left(p,x_{m_k}\right)+2\left(1-{\alpha }_{m_k}\right){\beta }_{m_k}⟨v_{m_k}-p,J{x}_0-Jp⟩.\end{array}$$

Similary proof of Case 1, we obtain that

$$\underset{k\to \infty }{\text{lim}\text{sup}}⟨v_{m_k}-p,J{x}_0-Jp⟩\le 0.$$

(3.17)

From (3.3), we have

$$\phi \left(p,x_{m_k+1}\right)\le \phi \left(p,x_{m_k}\right)-\left(1-{\alpha }_{m_k}\right){\beta }_{m_k}\phi \left(p,x_{m_k}\right)+2\left(1-{\alpha }_{m_k}\right){\beta }_{m_k}⟨v_{m_k}-p,J{x}_0-Jp⟩.$$

(3.18)

Since $\phi \left(p,x_{m_k}\right)\le \phi \left(p,x_{m_k+1}\right)$, we have

$$\begin{array}{ll}\hfill \left(1-{\alpha }_{m_k}\right){\beta }_{m_k}\phi \left(p,x_{m_k}\right)& \le \phi \left(p,x_{m_k}\right)-\phi \left(p,x_{m_k+1}\right)+2\left(1-{\alpha }_{m_k}\right){\beta }_{m_k}⟨v_{m_k}-p,J{x}_0-Jp⟩\\ \le 2\left(1-{\alpha }_{m_k}\right){\beta }_{m_k}⟨v_{m_k}-p,J{x}_0-Jp⟩.\end{array}$$

Since, $\left(1-{\alpha }_{m_k}\right){\beta }_{m_k}>0$ and (3.17) we have

$$\phi \left(p,x_{m_k}\right)\le 2⟨v_{m_k}-p,J{x}_0-Jp⟩\le 0.$$

(3.19)

It follows that $\phi \left(p,x_{m_k}\right)\to 0$. From (3.18), we gives $\phi \left(p,x_{m_k+1}\right)\to 0$, as k → ∞.

From $\phi \left(p,x_k\right)\le \phi \left(p,x_{m_k+1}\right)$ and Lemma 2.5, we obtain that x _k → p and u _k → p, as k → ∞.

From Cases 1 and 2, we conclude that {x _n } and {u _n } converge strongly to p. This completes the proof.

Corollary 3.2. Let C be nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, S : C → C be a strongly relatively nonexpansive mapping, and let A : E → 2^E*be a maximal monotone operator with$F\left(S\right)\cap A^{-1}\left(0\right)\ne \mathrm{0̸}$. For a positive number λ, let the sequence {x _n } be generated by x₀ ∈ C, x₁ ∈ E and

$$\left\{\begin{array}{c}{u}_n={\Pi }_C{x}_n,\hfill \\ y_n={\Pi }_C{J}^{-1}\left({\beta }_{n}J{x}_0+\left(1-{\beta }_n\right)J{Q}_{\lambda }^A{u}_n\right),\hfill \\ x_{n+1}=J^{-1}\left({\alpha }_{n}J{u}_n+\left(1-{\alpha }_n\right)JS{y}_n\right),\hfill \end{array}\right.$$

(3.20)

for every n ≥ 1, where {α _n } and {β _n } are sequences in [0,1). If the control sequences satisfy the following restrictions:

(i)${\sum }_{n=0}^{\infty }{\beta }_n=\infty$and lim_n→∞β _n = 0;

(ii) 0 < lim inf_n→∞α _n ≤ lim sup_n→∞α _n < 1.

Then {x _n } converges strongly to p = Π_F(S)∩A^-1(0)(x₀), where Π_F(S)∩A^-1(0)is the generalized projection from E onto F(S) ∩ A^-1(0).

Proof. Put F ≡ 0 and r _n ≡ 1 in Theorem 3.1. Then the conclusion of Corollary 3.2 can be obtained the desired result easily.

Corollary 3.3. Let C be nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, F : C × C → ℝ bea bifunction satisfying conditions (A1)-(A4) and S : C → C be a strongly relatively nonexpansive mapping, with$F\left(S\right)\cap EP\left(F\right)\ne \mathrm{0̸}$. Let the sequence {x _n } be generated by x₀ ∈ C, x₁ ∈ E and

$$\left\{\begin{array}{c}F\left(u_n,y\right)+\frac{1}{r_n}⟨y-u_n,J{u}_n-J{x}_n⟩\ge 0,\forall y\in C,\hfill \\ y_n={\Pi }_C{J}^{-1}\left({\beta }_{n}J{x}_0+\left(1-{\beta }_n\right)J{u}_n\right),\hfill \\ x_{n+1}=J^{-1}\left({\alpha }_{n}J{u}_n+\left(1-{\alpha }_n\right)JS{y}_n\right),\hfill \end{array}\right.$$

(3.21)

for every n ≥ 1, where {α _n }, {β _n } are sequences in [0,1), and {r _n } ⊂ (0, ∞). If the control sequences satisfy the following restrictions:

(i)${\sum }_{n=0}^{\infty }{\beta }_n=\infty$and lim_n→∞β _n = 0;

(ii) 0 < lim inf_n→∞α _n ≤ lim sup_n→∞α _n < 1;

(iii) lim inf_n→∞r _n > 0.

Then {x _n } and {u _n } converge strongly to p = Π_F(S)∩EP(F)(x₀), where Π_F(S)∩EP(F)is the generalized projection from E onto F(S) ∩ EP(F).

Proof. Put A ≡ 0, then $Q_{\lambda }^A$ is an identity mapping, in Theorem 3.1. Then the conclusion of Corollary 3.3 can be obtained the desired result easily.

Corollary 3.4. Let C be nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, F: C × C → ℝ bea bifunction satisfying conditions (A1)-(A4) and let A : E → 2^E*be a maximal monotone operator with$EP\left(F\right)\cap A^{-1}\left(0\right)\ne \mathrm{0̸}$. For a positive number λ, let the sequence {x _n } be generated by x₀ ∈ C, x₁ ∈ E and

$$\left\{\begin{array}{c}F\left(u_n,y\right)+\frac{1}{r_n}⟨y-u_n,J{u}_n-J{x}_n⟩\ge 0,\forall y\in C,\hfill \\ y_n={\Pi }_C{J}^{-1}\left({\beta }_{n}J{x}_0+\left(1-{\beta }_n\right)J{Q}_{\lambda }^A{u}_n\right),\hfill \\ x_{n+1}=J^{-1}\left({\alpha }_{n}J{u}_n+\left(1-{\alpha }_n\right)J{y}_n\right),\hfill \end{array}\right.$$

(3.22)

for every n ≥ 1, where {α _n }, {β _n } are sequences in [0,1), and {r _n } ⊂ (0, ∞). If the control sequences satisfy the following restrictions:

(i)${\sum }_{n=0}^{\infty }{\beta }_n=\infty$and lim_n→∞β _n = 0;

(ii) 0 < lim inf_n→∞α _n ≤ lim sup_n→∞α _n < 1;

(iii) lim inf_n→∞r _n > 0.

Then {x _n } and {u _n } converge strongly to p = Π_EP(F)∩A^-1(0)(x₀), where Π_EP(F)∩A^-1(0)is the generalized projection from E onto EP(F) ∩ A^-1(0).

Proof. Put S ≡ I, is an identity mapping, in Theorem 3.1. Then the conclusion of Corollary 3.4 can be obtained the desired result easily.

Remark 3.5. Our main result extends and improves the recent results of Wei et al. [5], Takahashi and Zembayashi [10] and generalizes the result of Nilsakoo [11].