Some iterative schemes for generalized vector equilibrium problems and relatively nonexpansive mappings in Banach spaces

Abstract

In this paper, we introduce two iterative schemes for finding a common solution of a generalized vector equilibrium problem and relatively nonexpansive mappings in a real Banach space. We study the strong and weak convergence of the sequences generated by the proposed iterative schemes. The results presented in this paper are the supplement, extension, and generalization of the previously known results in this area.

Introduction

Throughout the paper unless otherwise stated, let E be a real Banach space with its dual space E^∗, let 〈.,.〉 denote the duality pairing between E and E^∗, and let ∥.∥ denote the norm of E as well as of E^∗. Let C be a nonempty, closed, and convex subset of E, and let 2^E denote the set of all nonempty subsets of E. Let Y be a Hausdorff topological space, and let P be a pointed, proper, closed, and convex cone of Y with int P≠∅. We denote the strong convergence and the weak convergence of a sequence {x _n } to x in E by x _n →x and $x_n\rightharpoonup x$, respectively.

The normalized duality mapping $J:E\to 2^{E^{\ast }}$ is defined by

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

for every x∈E. It follows from the Hahn-Banach theorem that J(x) is nonempty. A Banach space E is said to be strictly convex if $\frac{\parallel x+y\parallel }{2}<1$ for x,y∈E with ∥x∥=∥y∥=1 and x≠y. It is also said to be uniformly convex if for each є∈(0,2], there exists δ>0 such that $\frac{\parallel x+y\parallel }{2}\le 1-\delta$ for x,y∈E with ∥x∥=∥y∥=1 and ∥x−y∥≥є. The space E is said to be smooth if the limit $\underset{t\to 0}{lim}\frac{\parallel x+\mathit{\text{ty}}\parallel -\parallel x\parallel }{t}$ exists for all x,y∈M(E)={z∈E:∥z∥=1}. It is also said to be uniformly smooth if the limit exists uniformly in x,y∈M(E). We note that if E is smooth, strictly convex, and reflexive, then the normalized duality mapping J is single-valued, one-to-one, and onto. The normalized duality mapping J is said to be weakly sequentially continuous if $x_n\rightharpoonup x$ implies that $J{x}_n\rightharpoonup \mathit{\text{Jx}}$.In 1994, Blum and Oettli [1] introduced and studied the following equilibrium problem (EP): Find x∈C such that

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

(1.1)

where $F:C\times C\to ℝ$ is a bifunction.

The EP(1.1) includes variational inequality problems, optimization problems, Nash equilibrium problems, saddle point problems, fixed point problems, and complementary problems as special cases. In other words, EP(1.1) is a unified model for several problems arising in science, engineering, optimization, economics, etc.

In the last two decades, EP(1.1) has been generalized and extensively studied in many directions due to its importance (see, for example, [2–6] and references therein for the literature on the existence of solution of the various generalizations of EP(1.1)). Some iterative methods have been studied for solving various classes of equilibrium problems (see, for example, [7–17] and references therein).

In this paper, we introduce and study the following generalized vector equilibrium problem (GVEP). Let F:C×C→Y be a nonlinear bimapping, and let ψ:C→Y be a nonlinear mapping; then, GVEP is to find x^∗∈C such that

$$F(x^{\ast },x)+\psi \left(x\right)-\psi \left(x^{\ast }\right)\in P,\forall x\in \mathrm{C.}$$

(1.2)

The solution set of GVEP(1.2) is denoted by Sol(GVEP(1.2)).

Example 1.1

Let $E=ℝ$, the set of all real numbers, with the inner product defined by $〈x,y〉=\mathit{\text{xy}},\forall x,y\in ℝ$. Let $Y=ℝ$, then P=[0,+∞) and let C=[0,2]. Let F and ψ be defined by F(x,y)=x²−y and ψ(x)=x² ∀x,y∈C, respectively; then, it is observed that Sol(GVEP(1.2))=[1, 2] ≠∅.

If ψ=0, then GVEP(1.2) reduces to the strong vector equilibrium problem (SVEP): Find x^∗∈C such that

$$F(x^{\ast },x)\in P,\forall x\in C,$$

(1.3)

which has been studied by Kazmi and Khan [18]. It is well known that the vector equilibrium problem provides a unified model of several problems, for example, vector optimization, vector variational inequality, vector complementary problem, and vector saddle point problem [5, 6]. In recent years, the vector equilibrium problem has been intensively studied by many authors (see, for example, [2, 4–6, 18, 19] and the references therein).

If $Y=ℝ$, then P= [0,+∞), and hence, GVEP(1.2) reduces to the following generalized equilibrium problem (GEP): Find x∈C such that

$$F(x^{\ast },x)+\psi \left(x\right)-\psi \left(x^{\ast }\right)\ge 0,\forall x\in C,$$

(1.4)

where $\psi :C\to ℝ\cup \{+\infty \}$ be a proper extended real-valued function. GEP(1.4) has been studied by Ceng and Yao [7].

Next, we recall that a mapping T:C→C is said to be nonexpansive if ∥T x−T y∥≤∥x−y∥, ∀x,y∈C.

The fixed point problem (FPP) for a nonexpansive mapping T is to

$$\text{Find}x\in C\text{such that}x\in \text{Fix}\left(T\right),$$

(1.5)

where Fix(T) is the fixed point set of the nonexpansive mapping T. It is well known that Fix(T) is closed and convex.

Let E be a smooth, strictly convex, and reflexive Banach space.

Following Takahashi and Zembayashi [17], a point p∈C is said to be an asymptotic fixed point of T if C contains a sequence {x _n } which converges weakly to p such that limn→∞∥x _n −T x _n ∥=0. The set of asymptotic fixed points of T is denoted by $\widehat{\text{Fix}}\left(T\right)$. A mapping T from C into itself is said to be relatively nonexpansive if Fix(T)≠∅, $\widehat{\text{Fix}}\left(T\right)=\text{Fix}\left(T\right)$, and ϕ(p,T x)≤ϕ(p,x) for all x∈C and p∈Fix(T), where $\varphi :E\times E\to {ℝ}_{+}$ is the Lyapunov functional defined by

$$\varphi (x,y)=\parallel x{\parallel }^2-2〈x,\mathit{\text{Jy}}〉+\parallel y{\parallel }^2,\forall x,y\in \mathrm{E.}$$

(1.6)

In 2007, Tada and Takahashi [15] and Takahashi and Takahashi [16] proved weak and strong convergence theorems for finding a common solution of EP(1.1) and FPP(1.5) of a nonexpansive mapping in a Hilbert space (for further related work, see Ceng and Yao [7] and Shan and Huang [19]).

In 2009, Takahashi and Zembayashi [17] proved weak and strong convergence theorems for finding a common solution of EP(1.1) and FPP(1.5) of a relatively nonexpansive mapping in real Banach space. Further, Petrot et al. [20] extended and generalized some results of Takahashi and Zembayashi [17].

Motivated by the work of Takahashi and Zembayashi [17], Shan and Haung [19], and Petrot et al. [20] and by the ongoing research in this direction, we introduce and study two iterative schemes for finding a common solution of GVEP(1.2) and FPPs for two relatively nonexpansive mappings in real Banach space. We study the strong and weak convergence of the sequences generated by the proposed iterative schemes. The results presented in this paper extend and generalize many previously known results in this research area (see, for instance, [17, 20]).

Preliminaries

We recall some concepts and results which are needed in sequel.

Following Alber [21], the generalized projection Π _C from E onto C is defined by

$${\Pi }_C\left(x\right)=\underset{y\in C}{inf}\varphi \left(x,y\right),\forall x\in E,$$

where ϕ(x,y) is obtained by (1.6).

Lemma 2.1

[[21],[22]]. Let E be a smooth, strictly convex, and reflexive Banach space, and let C be a nonempty closed convex subset of E. Then, the following conclusions hold:

• (i) ϕ(x,Π _C y)+ϕ(Π _C y,y)≤ϕ(x,y), ∀x∈C,y∈E;

• (ii) Let x∈E and z∈C, then

  $$z={\Pi }_C\left(x\right)\iff 〈z-y,\mathit{\text{Jx}}-\mathit{\text{Jz}}〉\ge 0,\forall y\in \mathrm{C.}$$

Remark 2.1

[17]

• (i) From the definition of ϕ, we have

  $${(\parallel x\parallel -\parallel y\parallel )}^2\le \varphi (x,y)\le {(\parallel x\parallel +\parallel y\parallel )}^2,\forall x,y\in \mathrm{E.}$$

• (ii) If E is a real Hilbert space H, then ϕ(x,y)=(∥x∥−∥y∥)², and Π _C is the metric projection P _C of H onto C.

• (iii) If E is a smooth, strictly convex, and reflexive Banach space, then for x,y∈E, ϕ(x,y)=0 if and only if x=y.

Lemma 2.2

[[23]]. Let C be a nonempty, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space E, and let T be a relatively nonexpansive mapping from C into itself. Then, Fix(T) is closed and convex.

Lemma 2.3

[[22]]. Let E be a smooth and uniformly convex Banach space, and let {x _n } and {y _n } be sequences in E such that either {x _n } or {y _n } is bounded. If limn→∞ ϕ(x _n ,y _n )=0, then limn→∞∥x _n −y _n ∥=0.

Lemma 2.4

[[24],[25]]. Let E be a uniformly convex Banach space, and let r>0. Then, there exists a strictly increasing, continuous, and convex function $g:[0,2r]\to ℝ$ such that g(0)=0 and

$$\parallel \mathit{\text{tx}}+(1-t)y{\parallel }^2\le t\parallel x{\parallel }^2+(1-t)\parallel y{\parallel }^2-t(1-t)g(\parallel x-y\parallel )$$

for all x,y∈B _r and t∈[0,1], where B _r ={z∈E:∥z∥≤r}.

Lemma 2.5

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

$$g(\parallel x-y\parallel )\le \varphi (x,y),\forall x,y\in B_r.$$

Definition 2.1

[[26],[27]. Let X and Y be two Hausdorff topological spaces, and let D be a nonempty, convex subset of X and P be a pointed, proper, closed, and convex cone of Y with intP≠∅. Let 0 be the zero point of Y, $\mathbb{U}\left(0\right)$ be the neighborhood set of 0, $\mathbb{U}\left(x_0\right)$ be the neighborhood set of x₀, and f:D→Y be a mapping.

• (i) If, for any $V\in \mathbb{U}\left(0\right)$ in Y, there exists $U\in \mathbb{U}\left(x_0\right)$ such that

  $$f\left(x\right)\in f\left(x_0\right)+V+P,\forall x\in U\cap D,$$

then f is called upper P-continuous on x₀. If f is upper P-continuous for all x∈D, then f is called upper P-continuous on D;

• (ii) If, for any $V\in \mathbb{U}\left(0\right)$ in Y, there exists $U\in \mathbb{U}\left(x_0\right)$ such that

  $$f\left(x\right)\in f\left(x_0\right)+V-P,\forall x\in U\cap D,$$

then f is called lower P-continuous on x₀. If f is lower P-continuous for all x∈D, then f is called lower P-continuous on D;

• (iii) If, for any x,y∈D and t∈[0,1], the mapping f satisfies

  $$f\left(x\right)\in f(\mathit{\text{tx}}+(1-t\left)y\right)+P\mathit{\text{or}}f\left(y\right)\in f(\mathit{\text{tx}}+(1-t\left)y\right)+P,$$

then f is called proper P-quasiconvex;

• (iv) If, for any x ₁,x ₂∈D and t∈[0,1], the mapping f satisfies

  $$\mathit{\text{tf}}\left(x_1\right)+(1-t)f\left(x_2\right)\in f(\mathit{\text{tx}}+(1-t\left)y\right)+P,$$

then f is called P-convex.

Lemma 2.6

[[28]]. Let X and Y be two real Hausdorff topological spaces; D is a nonempty, compact, and convex subset of X, and P is a pointed, proper, closed, and convex cone of Y with intP≠∅. Assume that g:D×D→Y and Φ:D→Y are two nonlinear mappings. Suppose that g and ϕ satisfy

• (i) g(x,x)∈P, ∀x∈D;

• (ii) ϕ is upper P-continuous on D;

• (iii) g(.,y) is lower P-continuous, ∀x∈D;

• (iv) g(x,.)+Φ(.) is proper P-quasiconvex, ∀x∈D.

Then, there exists a point x∈D which satisfies

$$G(x,y)\in P\setminus \left\{0\right\},\forall y\in D,$$

where

$$G(x,y)=g(x,y)+\Phi \left(y\right)-\Phi \left(x\right),\forall x,y\in \mathrm{D.}$$

Let F:C×C→Y and ψ:C→Y be two mappings. For any z∈E, define a mapping G _z :C×C→Y as follows:

$$G_z(x,y)=F(x,y)+\psi \left(y\right)-\psi \left(x\right)+\frac{e}{r}〈y-x,\mathit{\text{Jx}}-\mathit{\text{Jz}}〉,$$

(2.1)

where r is a positive real number and e∈intP.

Assumption 2.1

Let G _z , F, ψ satisfy the following conditions:

• (i) For all x∈C, F(x,x)=0;

• (ii) F is monotone, i.e., F(x,y)+F(y,x)∈−P, ∀x,y∈C;

• (iii) F(.,y) is continuous, ∀y∈C;

• (iv) F(x,.) is weakly continuous and P-convex, i.e.,

  $$\begin{array}{c}\mathit{\text{tF}}(x,y_1)+(1-t)F(x,y_2)\in F(x,t{y}_1+(1-t\left)y_2\right)\\ +P,\forall x,y_1,y_2\in C,\forall t\in [0,1];\end{array}$$

• (v) G _z (.,y) is lower P-continuous, ∀y∈C and z∈E;

• (vi) ψ(.) is P-convex and weakly continuous;

• (vii) G _z (x,.) is proper P-quasiconvex, ∀x∈C and z∈E.

Results

First, we prove the following technical result:

Theorem 3.1

Let E be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty, compact, and convex subset of E. Assume that P is a pointed, proper, closed, and convex cone of a real Hausdorff topological space Y with intP≠∅. Let G _z :C×C→Y be defined by (2.1). Let F:C×C→Y, ψ:C→Y and G _z satisfy Assumption 2.1. Define a mapping T _r (z):E→C as follows:

$$\begin{array}{c}{T}_r\left(z\right)=\{x\in C:F(x,y)+\psi (y)-\psi (x)\\ +\frac{e}{r}〈y-x,\mathit{\text{Jx}}-\mathit{\text{Jz}}〉\in P,\forall y\in C\},\end{array}$$

where e∈intP, and r is a positive real number. Then,

• (i) T _r (z)≠∅, ∀z∈E;

• (ii) T _r is single-valued;

• (iii) T _r is a firmly nonexpansive-type mapping, i.e., for all z ₁,z ₂∈E,

  $$\begin{array}{l}〈T_r{z}_1-T_r{z}_2,J{T}_r{z}_1-J{T}_r{z}_2〉\le 〈T_r{z}_1\hfill \\ -T_r{z}_2,J{z}_1-J{z}_2〉;\end{array}$$

• (iv) Fix (T _r )=Sol(GVEP(1.2));

• (v) Sol(GVEP(1.2)) is closed and convex.

Proof.

• (i) Let g(x,y)=G _z (x,y) and Φ(y)=0 for all x,y∈C and z∈E. It is easy to observe that g(x,y) and Φ(y) satisfy all the conditions of Lemma 2.6. Then, there exists a point x∈C such that

  $$G_z(x,y)+\Phi \left(y\right)-\Phi \left(x\right)\in P,\forall y\in C,$$

and thus T _r (z)≠∅, ∀z∈E.

• (ii) For each z∈E, T _r (z)≠∅, let x ₁,x ₂∈T _r (z). Then,

$$F(x_1,y)+\psi \left(y\right)-\psi \left(x_1\right)+\frac{e}{r}〈y-x_1,J{x}_1-\mathit{\text{Jz}}〉\in P,\forall y\in C$$

(3.1)

and

$$F(x_2,y)+\psi \left(y\right)-\psi \left(x_2\right)+\frac{e}{r}〈y-x_2,J{x}_2-\mathit{\text{Jz}}〉\in P,\forall y\in \mathrm{C.}$$

(3.2)

Letting y=x₂ in (3.1) and y=x₁ in (3.2), and then adding, we have

$$\begin{array}{l}F(x_1,x_2)+F(x_2,x_1)+\frac{e}{r}〈x_2-x_1,J{x}_1-J{x}_2〉\in \mathrm{P.}\end{array}$$

Since F is monotone, e∈intP, r>0 and P is a closed and convex cone, we have

$$〈x_2-x_1,J{x}_1-J{x}_2〉\ge 0.$$

Since E is strictly convex, the preceding inequality implies x₁=x₂. Hence, T _r is single-valued.

• (iii) For any z ₁,z ₂∈E, let x ₁=T _r (z ₁) and x ₂=T _r (z ₂). Then,

  $$\begin{array}{c}F(x_1,y)+\psi \left(y\right)-\psi \left(x_1\right)\\ +\frac{e}{r}〈y-x_1,J{x}_1-J{z}_1〉\in P,\forall y\in C\end{array}$$

  (3.3)

and

$$\begin{array}{l}F(x_2,y)+\psi \left(y\right)-\psi \left(x_2\right)+\frac{e}{r}〈y-x_2,J{x}_2-J{z}_2〉\in P,\hfill \\ \forall y\in \mathrm{C.}\end{array}$$

(3.4)

Letting y=x₂ in (3.3) and y=x₁ in (3.4), and then adding, we have

$$\begin{array}{ll}F(x_1,x_2)& +F(x_2,x_1)+\frac{e}{r}〈x_2-x_1,J{x}_1-J{x}_2\\ -J{z}_1+J{z}_2〉\in \mathrm{P.}\end{array}$$

Again, since F is monotone, e∈intP, r>0 and P is closed and convex cone, we have

$$〈x_2-x_1,J{x}_2-J{x}_1〉\le 〈x_2-x_1,J{z}_2-J{z}_1〉,$$

or

$$\begin{array}{c}〈T_r\left(z_1\right)-T_r\left(z_2\right),J{T}_r\left(z_1\right)-J{T}_r\left(z_2\right)〉\le 〈T_r\left(z_1\right)\\ -T_r\left(z_2\right),J{z}_1-J{z}_2〉.\hfill \end{array}$$

(3.5)

Hence, T _r is firmly nonexpansive-type mapping.

• (iv) Let x∈Fix (T _r ). Then,

  $$\begin{array}{l}F(x,y)+\psi \left(y\right)-\psi \left(x\right)+\frac{e}{r}〈y-x,\mathit{\text{Jx}}-\mathit{\text{Jx}}〉\in P,\forall y\in C\end{array}$$

and so

$$F(x,y)+\psi \left(y\right)-\psi \left(x\right)\in P,\forall y\in \mathrm{C.}$$

Thus, x∈Sol(G V E P(1.2)).

Let x∈Sol(G V E P(1.2)). Then,

$$\begin{array}{c}\hfill F(x,y)+\psi \left(y\right)-\psi \left(x\right)\in P,\forall y\in C\hfill \end{array}$$

and so

$$\begin{array}{l}F(x,y)+\psi \left(y\right)-\psi \left(x\right)+\frac{e}{r}〈y-x,\mathit{\text{Jx}}-\mathit{\text{Jx}}〉\in P,\forall y\in \mathrm{C.}\end{array}$$

Hence, x∈ Fix (T _r ). Thus, Fix(T _r )=Sol(G V E P(1.2)).

(v) As in the proof of Lemma 2.8 in [17], we have

$$\begin{array}{l}\varphi \left(T_r\right(z_1),T_r(z_2\left)\right)+\varphi \left(T_r\right(z_2),T_r(z_1\left)\right)\le \hfill \\ \varphi (T_r{z}_1,z_2)+\varphi (T_r{z}_2,z_1),\end{array}$$

for z₁,z₂∈C. Taking z₂=u∈ Fix (T _r ), we have

$$\varphi (u,T_r{z}_1)\le \varphi (u,z_1).$$

Next, we show that $\widehat{\text{Fix}}\left(T_r\right)=\text{Sol}(\mathit{\text{GVEP}}$(1.2)). Indeed, let $p\in \widehat{\text{Fix}}\left(T_r\right)$. Then, there exists {z _n }⊂E such that $z_n\rightharpoonup p$ and limn→∞(z _n −T _r z _n )=0. Moreover, we get $T_r{z}_n\rightharpoonup p$. Hence, we have p∈C. Since J is uniformly continuous on bounded sets, we have

$$\underset{n\to \infty }{\lim }\frac{\parallel J{z}_n-J{T}_r{z}_n\parallel }{r}=0,r>0.$$

(3.6)

From the definition of T _r , we have

$$\begin{array}{l}F(T_r{z}_n,y)+\psi \left(y\right)-\psi \left(T_r{z}_n\right)\hfill \\ +\frac{e}{r}〈y-T_r{z}_n,J{T}_r{z}_n-J{z}_n〉\in P,\forall y\in C\end{array}$$

$$\begin{array}{l}0\in F(y,T_r{z}_n)-\left(\psi \right(y)-\psi (T_r{z}_n\left)\right)\hfill \\ -\frac{e}{r}〈y-T_r{z}_n,J{T}_r{z}_n-J{z}_n〉+P,\\ \forall y\in \mathrm{C.}\end{array}$$

Let y _t =(1−t)p+t y, ∀t∈(0,1]. Since y∈C and p∈C, we get y _t ∈C and hence

$$\begin{array}{l}\begin{array}{c}0\in F(y_t,T_r{z}_n)-\left(\psi \right(y_t)-\psi (T_r{z}_n\left)\right)\\ -\frac{e}{r}〈y_t-T_r{z}_n,J{T}_r{z}_n-J{z}_n〉+P\\ =F(y_t,T_r{z}_n)-\left(\psi \right(y_t)-\psi (T_r{z}_n\left)\right)\\ -e〈y_t-T_r{z}_n,\frac{J{T}_r{z}_n-J{z}_n}{r}〉+\mathrm{P.}\end{array}\end{array}$$

(3.7)

Since F(x,.) and ψ(.) are weakly continuous for all x∈C, then it follows from (3.6) and (3.7) that

$$0\in F(y_t,p)-\psi \left(y_t\right)+\psi \left(p\right)+\mathrm{P.}$$

(3.8)

Further, it follows from Assumption 2.1 (i), (iv), (vi) that t F(y _t ,y)+(1−t)F(y _t ,p)+t ψ(y)+(1−t)ψ(p)−ψ(y _t )

$$\begin{array}{c}\in F(y_t,y_t)+\psi \left(y_t\right)-\psi \left(y_t\right)+P\\ \in P,\hfill \end{array}$$

or

$$\begin{array}{c}-t\left[F\right(y_t,y)+\psi (y)-\psi (y_t\left)\right]-(1-t)\left[F\right(y_t,p)\\ +\psi \left(p\right)-\psi \left(y_t\right)]\in -\mathrm{P.}\end{array}$$

(3.9)

Using (3.8) in (3.9), we have

$$\begin{array}{l}-t\left[F\right(y_t,y)+\psi (y)-\psi (y_t\left)\right]\in -P\\ F(y_t,y)+\psi \left(y\right)-\psi \left(y_t\right)\in \mathrm{P.}\end{array}$$

Letting t→0, we obtain

$$F(p,y)+\psi \left(y\right)-\psi \left(p\right)\in P,\forall y\in C,$$

i.e., p∈Sol(G V E P(1.2)). So, we get Fix(T _r )=Sol(G V E P(1.2)) $=\widehat{\text{Fix}}\left(T_r\right)$. Therefore, T _r is a relatively nonexpansive mapping. Further, it follows from Lemma 2.2 that Sol(G V E P(1.2)) =Fix(T _r ) is closed and convex. This completes the proof. □

Next, we have the following lemma whose proof is on the similar lines of the proof of Lemma 2.9 [17] and hence omitted.

Lemma 3.1

Let E, C, F, ψ, G _z be same as in Theorem 3.1, and let r>0. Then, for x∈E and q∈Fix(T _r ), we have

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

Now, we prove a strong convergence theorem for finding a common solution of GVEP(1.2) and the fixed point problems of two relatively nonexpansive mappings in a Banach space.

Theorem 3.2

Let E be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty, compact, and convex subset of E. Assume that P is a pointed, proper, closed, and convex cone of a real Hausdorff topological space Y with intP≠∅. Let the mappings F:C×C→Y and ψ:C→Y satisfy Assumption 2.1, and let S, T be relatively nonexpansive mappings from C into itself such that Γ:=Fix(T)∩Fix(S)∩Sol(G V E P(1.2)) ≠∅. Let {x _n } be a sequence generated by the scheme:

$$\begin{array}{ll}{x}_0& =x\in C,\\ y_n& =J^{-1}({\delta }_{n}J{x}_n+(1-{\delta }_n\left)\mathit{\text{JT}}{z}_n\right),\\ z_n& =J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n\left)\mathit{\text{JS}}{x}_n\right),\end{array}$$

$$\begin{array}{l}{u}_n\in C\mathrm{such}\mathit{\text{that}}F(u_n,y)+\psi \left(y\right)-\psi \left(u_n\right)\hfill \\ +\frac{e}{r}〈y-u_n,J{u}_n-J{y}_n〉\in P,\forall y\in C,\end{array}$$

(3.10)

$$\begin{array}{ll}{H}_n& =\{z\in C:\varphi (z,u_n)\le \varphi (z,x_n\left)\right\},\\ W_n& =\{z\in C:〈x_n-z,\mathit{\text{Jx}}-J{x}_n〉\ge 0\},\\ x_{n+1}& ={\prod }_{H_n\cap W_n}x,\mathrm{for}\mathit{\text{every}}n\in N\cup \left\{0\right\},\end{array}$$

where e∈intP, J is the normalized duality mapping on E, and r∈[a,∞) for some a>0. Assume that {α _n } and {δ _n } are sequences in [0,1] satisfying the conditions:

• (i) limsup_n→∞ δ _n <1;

• (ii) 0< liminf_n→∞ α _n ≤ limsup_n→∞ α _n <1.

Then, {x _n } converges strongly to $\prod _{\Gamma }x$, where $\prod _{\Gamma }x$ is the generalized projection of E onto Γ.

Proof

Since S and T are relatively nonexpansive mappings from C into itself, it follows from Lemma 2.2 and Theorem 3.1(v) that Γ is closed and convex. Now, we show that H _n ∩W _n is closed and convex. From the definition of W _n , it is obvious that W _n is closed and convex. Further, from the definition of ϕ, we observe that H _n is closed and

$$\begin{array}{ll}\varphi (z,u_n)& \le \varphi (z,x_n)\iff \parallel u_n{\parallel }^2-\parallel x_n{\parallel }^2\\ -2〈z,J{u}_n-J{x}_n〉\ge 0,\end{array}$$

and hence H _n is convex. So, H _n ∩W _n is a closed convex subset of E for all n∈N∪{0}.

Let u∈Γ. It follows from Theorem 3.1 that (3.10) is equivalent to u _n =T _r y _n for all n∈N∪{0}, and T _r is relatively nonexpansive. Since S and T are relatively nonexpansive, we have

$$\begin{array}{l}\varphi (u,u_n)=\varphi (u,T_r{y}_n)\hfill \\ \le \varphi (u,y_n)\\ \le \varphi (u,J^{-1}({\delta }_{n}J{x}_n+(1-{\delta }_n)\mathit{\text{JT}}{z}_n\left)\right)\\ =\parallel u{\parallel }^2-2〈u,{\delta }_{n}J{x}_n+(1-{\delta }_n)\mathit{\text{JT}}{z}_n〉\\ +\parallel {\delta }_{n}J{x}_n+(1-{\delta }_n)\mathit{\text{JT}}{z}_n{\parallel }^2\\ \le \parallel u{\parallel }^2-2{\delta }_n〈u,J{x}_n〉-2(1-{\delta }_n)〈u,\mathit{\text{JT}}{z}_n〉\\ +{\delta }_n\parallel x_n{\parallel }^2+(1-{\delta }_n)\parallel T{z}_n{\parallel }^2\\ \le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\varphi (u,T{z}_n)\\ \le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\varphi (u,T{z}_n)\\ \le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\varphi (u,z_n),\end{array}$$

(3.11)

and

$$\begin{array}{l}\varphi (u,z_n)=\varphi (u,J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)\mathit{\text{JS}}{x}_n\left)\right)\hfill \\ =\parallel u{\parallel }^2-2〈u,{\alpha }_{n}J{x}_n+(1-{\alpha }_n)\mathit{\text{JS}}{x}_n〉\\ +\parallel {\alpha }_{n}J{x}_n+(1-{\alpha }_n)\mathit{\text{JS}}{x}_n{\parallel }^2\\ \le \parallel u{\parallel }^2-2{\alpha }_n〈u,J{x}_n〉-2(1-{\alpha }_n)〈u,\mathit{\text{JS}}{x}_n〉\\ +{\alpha }_n\parallel x_n{\parallel }^2+(1-{\alpha }_n)\parallel S{x}_n{\parallel }^2\\ \le {\alpha }_n\varphi (u,x_n)+(1-{\alpha }_n)\varphi (u,S{x}_n)\\ \le {\alpha }_n\varphi (u,x_n)+(1-{\alpha }_n)\varphi (u,x_n)\\ \le \varphi (u,x_n).\end{array}$$

(3.12)

Using (3.12) in (3.11), we have

$$\begin{array}{ll}\varphi (u,u_n)& \le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\varphi (u,x_n)\\ \le \varphi (u,x_n).\end{array}$$

Hence, we have u∈H _n . This implies that Γ⊂H _n , ∀n∈N∪{0}.

Next, we show by induction that Γ⊂H _n ∩W _n , ∀n∈N∪{0}. From W₀=C, we have Γ⊂H₀∩W₀. Suppose that Γ⊂H _k ∩W _k , for some k∈N∪{0}. Then, there exists x_k+1∈H _k ∩W _k such that $x_{k+1}=\prod _{H_k\cap W_k}x$. From the definition of x_k+1, we have, for all z∈H _k ∩W _k ,

$$〈x_{k+1}-z,\mathit{\text{Jx}}-J{x}_{k+1}〉\ge 0.$$

Since Γ⊂H _k ∩W _k , we have

$$〈x_{k+1}-z,\mathit{\text{Jx}}-J{x}_{k+1}〉\ge 0,\forall z\in \Gamma ,$$

(3.13)

and hence z∈W_k+1. So, we have Γ⊂W_k+1. Therefore, we have Γ⊂H_k+1∩W_k+1.

Thus, we have that Γ⊂H _n ∩W _n for all n∈N∪{0}. This means that {x _n } is well defined. From the definition of W _n , we have $x_n=\prod _{W_n}x$.

Using $x_n=\prod _{W_n}x$, from Lemma 2.1, we have $\varphi (x_n,x)=\varphi (\prod _{W_n}x,x)\le \varphi (u,x)-\varphi (u,\prod _{W_n}x)\le \varphi (u,x),\forall u\in \Gamma \subset W_n$.

Then, {ϕ(x _n ,x)} is bounded. Therefore, {x _n } and {S x _n } are bounded.

Since $x_{n+1}=\prod _{H_n\cap W_n}x\in H_n\cap W_n\subset W_n$ and $x_n=\prod _{W_n}x$, from the definition of $\prod _{W_n}$, we have

$$\varphi (x_n,x)\le \varphi (x_{n+1},x),\forall n\in N\cup \left\{0\right\}.$$

Thus, {ϕ(x _n ,x)} is nondecreasing. So, the limit of {ϕ(x _n ,x)} exists. By the construction of W _n , we have W _m ⊂W _n and $x_m=\prod _{W_m}x\in W_n$ for any positive integer m≥n. It follows that

$$\begin{array}{l}\varphi (x_m,x_n)=\varphi \left(x_m,\prod _{W_n}x\right)\hfill \\ \le \varphi (x_m,x)-\varphi \left(\prod _{W_n}x,x\right)\\ =\varphi (x_m,x)-\varphi (x_n,x).\end{array}$$

(3.14)

Letting m,n→∞ in (3.14), we have ϕ(x _m ,x _n )→0. It follows from Lemma 2.3 that ∥x _m −x _n ∥→0 as m,n→∞. Hence, {x _n } is a Cauchy sequence. Since E is a Banach space and C is closed and convex, one can assume that $x_n\to \widehat{x}\in C$ as n→∞. From (3.14), we have

$$\varphi (x_{n+1},x_n)\le \varphi (x_{n+1},x)-\varphi (x_n,x),\forall n\in N\cup \left\{0\right\}$$

which implies

$$\underset{n\to \infty }{\lim }\varphi (x_{n+1},x_n)=0.$$

Further, from $x_{n+1}=\prod _{H_n\cap W_n}x\in H_n$, we have

$$\varphi (x_{n+1},u_n)\le \varphi (x_{n+1},x_n),\forall n\in N\cup \left\{0\right\}$$

and hence

$$\underset{n\to \infty }{\lim }\varphi (x_{n+1},u_n)=0.$$

Since

$$\underset{n\to \infty }{\lim }\varphi (x_{n+1},x_n)=\underset{n\to \infty }{lim}\varphi (x_{n+1},u_n)=0,$$

and E is uniformly convex and smooth, then from Lemma 2.3, we have

$$\underset{n\to \infty }{\lim }\parallel x_{n+1}-x_n\parallel =\parallel x_{n+1}-u_n\parallel =0,$$

and hence, we have

$$\underset{n\to \infty }{\lim }\parallel x_n-u_n\parallel =0.$$

Since J is uniformly norm-to-norm continuous on bounded sets, we have

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

because E is a uniformly smooth Banach space and E^∗ is a uniformly convex Banach space.

Since {x _n } and {S x _n } are bounded and z _n =J⁻¹(α _n J x _n +(1−α _n )J S x _n ), then we can easily see that {z _n } is a bounded sequence, and hence, {T z _n } is bounded.

Let r= supn∈N∪{0}{∥x _n ∥,∥T z _n ∥,∥S x _n ∥}. From Lemma 2.4, we have

$$\begin{array}{ll}\varphi (u,z_n)& =\varphi (u,J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)\mathit{\text{JS}}{x}_n\left)\right)\\ =\parallel u{\parallel }^2-2〈u,{\alpha }_{n}J{x}_n+(1-{\alpha }_n)\mathit{\text{JS}}{x}_n〉\\ +\parallel {\alpha }_{n}J{x}_n+(1-{\alpha }_n)\mathit{\text{JS}}{x}_n{\parallel }^2\\ \le \parallel u{\parallel }^2-2{\alpha }_n〈u,J{x}_n〉-2(1-{\alpha }_n)〈u,\mathit{\text{JS}}{x}_n〉\\ +{\alpha }_n\parallel x_n{\parallel }^2+(1-{\alpha }_n)\parallel S{x}_n{\parallel }^2\\ -{\alpha }_n(1-{\alpha }_n)g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel )\\ \le {\alpha }_n\varphi (u,x_n)+(1-{\alpha }_n)\varphi (u,S{x}_n)\\ -{\alpha }_n(1-{\alpha }_n)g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel )\\ \le {\alpha }_n\varphi (u,x_n)+(1-{\alpha }_n)\varphi (u,x_n)\\ -{\alpha }_n(1-{\alpha }_n)g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel )\\ \le \varphi (u,x_n)-{\alpha }_n(1-{\alpha }_n)g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel ).\end{array}$$

It follows from (3.11) that

$$\begin{array}{ll}\varphi (u,u_n)& \le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\varphi (u,z_n)\\ \le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\left[\varphi \right(u,x_n)\\ -{\alpha }_n(1-{\alpha }_n)g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel )]\\ \le \varphi (u,x_n)-{\alpha }_n(1-{\alpha }_n)(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel ),\end{array}$$

or

$${\alpha }_n(1-{\alpha }_n)(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel )\le \varphi (u,x_n)-\varphi (u,u_n).$$

(3.15)

Further, we have

$$\begin{array}{ll}\varphi (u,x_n)-\varphi (u,u_n)& =\parallel x_n{\parallel }^2-\parallel u_n{\parallel }^2-2〈u,J{x}_n-J{u}_n〉\\ \le |\parallel x_n{\parallel }^2-\parallel u_n{\parallel }^2|+2\parallel 〈u,J{x}_n-J{u}_n〉\parallel \\ \le |\parallel x_n\parallel -\parallel u_n\parallel |(\parallel x_n\parallel +\parallel u_n\parallel )\\ +2\parallel u\parallel \parallel J{x}_n-J{u}_n\parallel \\ \le \parallel x_n-u_n\parallel (\parallel x_n\parallel +\parallel u_n\parallel )\\ +2\parallel u\parallel \parallel J{x}_n-J{u}_n\parallel .\end{array}$$

and hence, it follows from limn→∞∥x _n −u _n ∥=0 and limn→∞∥J x _n −J u _n ∥=0 that

$$\underset{n\to \infty }{\lim }\left(\varphi \right(u,x_n)-\varphi (u,u_n\left)\right)=0.$$

(3.16)

Using conditions (i) and (ii) and (3.16) in (3.15), we have

$$\underset{n\to \infty }{\lim }g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel )=0.$$

Further, it follows from the property of g that

$$\underset{n\to \infty }{\lim }\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel =0.$$

Since J⁻¹ is uniformly norm-to-norm continuous on bounded sets, we have

$$\underset{n\to \infty }{\lim }\parallel x_n-S{x}_n\parallel =0.$$

(3.17)

Next, we have

$$\begin{array}{ll}\varphi (u,u_n)& \le \varphi (u,y_n)\\ \le \varphi (u,J^{-1}({\delta }_{n}J{x}_n+(1-{\delta }_n)\mathit{\text{JT}}{z}_n\left)\right)\\ =\parallel u{\parallel }^2-2〈u,{\delta }_{n}J{x}_n+(1-{\delta }_n)\mathit{\text{JT}}{z}_n〉+\parallel {\delta }_{n}J{x}_n\\ +(1-{\delta }_n)\mathit{\text{JT}}{z}_n{\parallel }^2\\ \le \parallel u{\parallel }^2-2{\delta }_n〈u,J{x}_n〉-2(1-{\delta }_n)〈u,\mathit{\text{JT}}{z}_n〉\\ +{\delta }_n\parallel x_n{\parallel }^2+(1-{\delta }_n)\parallel T{z}_n{\parallel }^2\\ -{\delta }_n(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JT}}{z}_n\parallel )\\ \le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\varphi (u,T{z}_n)\\ -{\delta }_n(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JT}}{z}_n\parallel )\\ \le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\varphi (u,z_n)\\ -{\delta }_n(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JT}}{z}_n\parallel )\\ \le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\varphi (u,x_n)\\ -{\delta }_n(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JT}}{z}_n\parallel )\\ \le \varphi (u,x_n)-{\delta }_n(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JT}}{z}_n\parallel ),\end{array}$$

or

$$\begin{array}{c}{\delta }_n(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JT}}{z}_n\parallel )\le \varphi (u,x_n)\\ -\varphi (u,u_n)\to 0\text{as}n\to \mathrm{\infty .}\end{array}$$

Thus,

$$\underset{n\to \infty }{\lim }g(\parallel J{x}_n-\mathit{\text{JT}}{z}_n\parallel )=0.$$

It follows from the property of g that

$$\underset{n\to \infty }{\lim }\parallel J{x}_n-\mathit{\text{JT}}{z}_n\parallel =0,$$

and hence

$$\underset{n\to \infty }{\lim }\parallel x_n-T{z}_n\parallel =0.$$

(3.18)

Now,

$$\begin{array}{ll}\parallel J{x}_n-J{z}_n\parallel & =\parallel J{x}_n-({\alpha }_{n}J{x}_n+(1-{\alpha }_n\left)\mathit{\text{JS}}{x}_n\right)\parallel \\ =\parallel (1-{\alpha }_n)(J{x}_n-\mathit{\text{JS}}{x}_n)\parallel \\ =(1-{\alpha }_n)\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel .\end{array}$$

Since limn→∞∥J x _n −J S x _n ∥=0, the preceding equality implies that

$$\underset{n\to \infty }{\lim }\parallel J{x}_n-J{z}_n\parallel =0,$$

and hence

$$\underset{n\to \infty }{\lim }\parallel x_n-z_n\parallel =0.$$

(3.19)

It follows from (3.18), (3.19), and the inequality

$$\parallel z_n-T{z}_n\parallel \le \parallel z_n-x_n\parallel +\parallel x_n-T{z}_n\parallel$$

that

$$\underset{n\to \infty }{\lim }\parallel z_n-T{z}_n\parallel =0.$$

Since $x_n\to \widehat{x}$, it follows from (3.17), (3.18), and (3.20) that $\widehat{x}$ is a fixed point of S and T, i.e., $\widehat{x}\in \text{Fix}\left(T\right)\cap \text{Fix}\left(S\right)$.

On the same lines of the proof of Theorem 3.1(v) with (3.10), we can easily prove that $\widehat{x}\in \text{Sol}(\mathit{\text{GVEP}}$(1.2)). Then, $\widehat{x}\in \Gamma$.

Finally, we prove that $\widehat{x}=\prod _{\Gamma }x$. By taking the limit in (3.13), we have

$$〈\widehat{x}-z,\mathit{\text{Jx}}-J\widehat{x}〉\ge 0,\forall z\in \mathrm{\Gamma .}$$

(3.20)

Further, in view of Lemma 2.1, we see that $\widehat{x}=\prod _{\Gamma }x$. This completes the proof. □

Now, we prove the weak convergence theorem for finding the common solution for GVEP (1.2) and the fixed point problems of two relatively nonexpansive mappings. First, we prove the following proposition:

Proposotion 3.1

Let E be a uniformly smooth and uniformly convex Banach space, and let C be a non-empty, compact, and convex subset of E. Assume that P is a pointed, proper, closed, and convex cone of a real Hausdorff topological space Y with intP≠∅. Let F:C×C→Y and ψ:C→Y satisfy Assumption 2.1, and let S, T be relatively nonexpansive mappings from C into itself such that Γ≠∅. Let {x _n } be a sequence generated by the following scheme:

$$\begin{array}{l}{z}_1\in E,\\ x_n\in C\mathrm{such}\mathit{\text{that}}\\ F(x_n,y)& +\psi \left(y\right)-\psi \left(x_n\right)+\frac{e}{r}〈y-x_n,J{x}_n\\ -J{z}_n〉\in P,\forall y\in C,\\ y_n& =J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n\left)\mathit{\text{JS}}{x}_n\right),\\ z_{n+1}& =J^{-1}({\delta }_{n}J{x}_n+(1-{\delta }_n\left)\mathit{\text{JT}}{y}_n\right),\end{array}$$

for every n∈N, where e∈intP, J is the normalized duality mapping on E, and r∈ [ a,∞) for some a>0. Assume that {α _n } and {δ _n } are sequences in [0,1] satisfying the conditions (i) and (ii) of Theorem 3.2. Then, $\left\{\prod _{\Gamma }{x}_n\right\}$ converges strongly to z∈Γ.

Proof

Let u∈Γ. Since x _n =T _r z _n and T _r , S, T are relatively nonexpansive, we have

$$\begin{array}{l}\varphi (u,x_{n+1})=\varphi (u,T_r{z}_{n+1})\hfill \\ \le \varphi (u,z_{n+1})\\ \le \varphi (u,J^{-1}({\delta }_{n}J{x}_n+(1-{\delta }_n)\mathit{\text{JT}}{y}_n\left)\right)\\ =\parallel u{\parallel }^2-2〈u,{\delta }_{n}J{x}_n+(1-{\delta }_n)\mathit{\text{JT}}{y}_n〉\\ +\parallel {\delta }_{n}J{x}_n+(1-{\delta }_n)\mathit{\text{JT}}{y}_n{\parallel }^2\\ \le \parallel u{\parallel }^2-2{\delta }_n〈u,J{x}_n〉-2(1-{\delta }_n)〈u,\mathit{\text{JT}}{y}_n〉\\ +{\delta }_n\parallel x_n{\parallel }^2+(1-{\delta }_n)\parallel T{y}_n{\parallel }^2\\ \le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\varphi (u,T{y}_n)\\ \le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\varphi (u,y_n),\end{array}$$

(3.21)

and

$$\begin{array}{l}\varphi (u,y_n)=\varphi (u,J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)\mathit{\text{JS}}{x}_n\left)\right)\\ =\parallel u{\parallel }^2-2〈u,{\alpha }_{n}J{x}_n+(1-{\alpha }_n)\mathit{\text{JS}}{x}_n〉\\ +\parallel {\alpha }_{n}J{x}_n+(1-{\alpha }_n)\mathit{\text{JS}}{x}_n{\parallel }^2\\ \le \parallel u{\parallel }^2-2{\alpha }_n〈u,J{x}_n〉-2(1-{\alpha }_n)〈u,\mathit{\text{JS}}{x}_n〉\\ +{\alpha }_n\parallel x_n{\parallel }^2+(1-{\alpha }_n)\parallel S{x}_n{\parallel }^2\\ \le {\alpha }_n\varphi (u,x_n)+(1-{\alpha }_n)\varphi (u,S{x}_n)\\ \le {\alpha }_n\varphi (u,x_n)+(1-{\alpha }_n)\varphi (u,x_n)\\ \le \varphi (u,x_n).\end{array}$$

(3.22)

Using (3.22) in (3.21), we have

$$\begin{array}{c}\varphi (u,x_{n+1})\le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\varphi (u,x_n)\\ \varphi (u,x_{n+1})\le \varphi (u,x_n).\hfill \end{array}$$

(3.23)

Therefore, limn→∞ ϕ(u,x _n ) exists, and hence, ϕ(u,x _n ) is bounded. This implies that {x _n } and {S x _n } are bounded. Further, it follows from (3.22) that ϕ(u,y _n ) is also bounded, and hence, {y _n } and {T y _n } are bounded.

Define $w_n=\prod _{\Gamma }{x}_n$, for every n∈N. Then, from w _n ∈Γ and (3.23), we have

$$\varphi (w_n,x_{n+1})\le \varphi (w_n,x_n).$$

(3.24)

Since $\prod _{\Gamma }$ is the generalized projection, from Lemma 2.1, we have

$$\begin{array}{l}\varphi (w_{n+1},x_{n+1})=\varphi \left({\prod }_{\Gamma }{x}_{n+1},x_{n+1}\right)\\ \le \varphi (w_n,x_{n+1})-\varphi \left(w_n,{\prod }_{\Gamma }{x}_{n+1}\right)\\ =\varphi (w_n,x_{n+1})-\varphi (w_n,w_{n+1})\\ \le \varphi (w_n,x_{n+1}).\end{array}$$

(3.25)

Hence, from (3.24), we have

$$\varphi (w_{n+1},x_{n+1})\le \varphi (w_n,x_n).$$

Therefore, {ϕ(w _n ,x _n )} is a convergent sequence. We also have from (3.24) that, for all m∈N,

$$\varphi (w_n,x_{n+m})\le \varphi (w_n,x_n).$$

From $w_{n+m}=\prod _{\Gamma }{x}_{n+m}$ and Lemma 2.1, we have

$$\begin{array}{ll}\varphi (w_n,w_{n+m})+& \varphi (w_{n+m},x_{n+m})\\ \le \varphi (w_n,x_{n+m})\le \varphi (w_n,x_n)\end{array}$$

and hence

$$\varphi (w_n,w_{n+m})\le \varphi (w_n,x_n)-\varphi (w_{n+m},x_{n+m}).$$

Let r= supn∈N∥w _n ∥. From Lemma 2.3, there exists a continuous, strictly increasing, and convex function g with g(0)=0 such that g(∥x−y∥)≤ϕ(x,y) for x,y∈B _r . So, we have

$$\begin{array}{ll}g(\parallel w_n& -w_{n+m}\parallel )\le \varphi (w_n,w_{n+m})\le \varphi (w_n,x_n)\\ -\varphi (w_{n+m},x_{n+m}).\end{array}$$

Since {ϕ(w _n ,x _n )} is a convergent sequence, from the property of g, we have that {w _n } is a Cauchy sequence. Since Γ is closed, {w _n } converges strongly to z∈Γ. This completes the proof. □

Now, we are able to prove the following weak convergence theorem.

Theorem 3.3

Let E be a uniformly smooth and uniformly convex Banach space, and let C be a non-empty, compact, and convex subset of E. Assume that P is a pointed, proper, closed, and convex cone of a real Hausdorff topological space Y with intP≠∅. Let F:C×C→Y and ψ:C→Y satisfy Assumption 2.1, and let S, T be relatively nonexpansive mappings from C into itself such that Γ≠∅. Let {x _n } be a sequence generated by the scheme:

$$\begin{array}{l}{z}_1\in E,\\ x_n\in C\text{such that}\\ F(x_n,y)& +\psi \left(y\right)-\psi \left(x_n\right)\\ +\frac{e}{r}〈y-x_n,J{x}_n-J{z}_n〉\in P,\forall y\in C,\\ y_n=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n\left)\mathit{\text{JS}}{x}_n\right),\\ z_{n+1}=J^{-1}({\delta }_{n}J{x}_n+(1-{\delta }_n\left)\mathit{\text{JT}}{y}_n\right),\end{array}$$

for every n∈N, where e∈intP, J is the normalized duality mapping on E, and r∈ [ a,∞) for some a>0. Assume that {α _n } and {δ _n } are sequences in [0,1] satisfying the conditions (i) and (ii) of Theorem 3.2. If J is weakly sequentially continuous, then x _n converges weakly to z∈Γ, where $z=\underset{n\to \infty }{lim}\prod _{\Gamma }{x}_n.$

Proof

As in the proof of Proposition 3.1, we have that {x _n }, {y _n }, {S x _n }, and {T y _n } are bounded sequences. Let r= supn∈N{∥x _n ∥,∥y _n ∥,∥S x _n ∥,∥T y _n ∥}. Let u∈Γ. Since x _n =T _r z _n and T _r , S, T are relatively nonexpansive, using Lemma 2.3, we have

$$\begin{array}{l}\varphi (u,y_n)=\varphi (u,J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)\mathit{\text{JS}}{x}_n\left)\right)\\ =\parallel u{\parallel }^2-2〈u,{\alpha }_{n}J{x}_n+(1-{\alpha }_n)\mathit{\text{JS}}{x}_n〉\\ +\parallel {\alpha }_{n}J{x}_n+(1-{\alpha }_n)\mathit{\text{JS}}{x}_n{\parallel }^2\\ \le \parallel u{\parallel }^2-2{\alpha }_n〈u,J{x}_n〉-2(1-{\alpha }_n)〈u,\mathit{\text{JS}}{x}_n〉\\ +{\alpha }_n\parallel x_n{\parallel }^2+(1-{\alpha }_n)\parallel S{x}_n{\parallel }^2\\ -{\alpha }_n(1-{\alpha }_n)g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel )\\ \le {\alpha }_n\varphi (u,x_n)+(1-{\alpha }_n)\varphi (u,S{x}_n)\\ -{\alpha }_n(1-{\alpha }_n)g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel )\\ \le {\alpha }_n\varphi (u,x_n)+(1-{\alpha }_n)\varphi (u,x_n)\\ -{\alpha }_n(1-{\alpha }_n)g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel )\\ \le \varphi (u,x_n)-{\alpha }_n(1-{\alpha }_n)g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel ).\end{array}$$

(3.26)

Using (3.26) in (3.21), we have

$$\begin{array}{ll}\varphi (u,x_{n+1})& \le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\left[\varphi \right(u,x_n)\\ -{\alpha }_n(1-{\alpha }_n)g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel )]\\ \le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\varphi (u,x_n)\\ -{\alpha }_n(1-{\alpha }_n)(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel )\\ \le \varphi (u,x_n)-{\alpha }_n(1-{\alpha }_n)(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel ),\end{array}$$

or

$$\begin{array}{c}{\alpha }_n(1-{\alpha }_n)(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel )\\ \le \varphi (u,x_n)-\varphi (u,x_{n+1}).\end{array}$$

(3.27)

Since {ϕ(u,x _n )} is convergent and using conditions (i) and (ii) in (3.27), we have

$$\underset{n\to \infty }{\lim }g(\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel )=0.$$

From the property of g, we have

$$\underset{n\to \infty }{\lim }\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel =0.$$

Since J⁻¹ is uniformly norm-to-norm continuous on bounded sets, we have

$$\underset{n\to \infty }{\lim }\parallel x_n-S{x}_n\parallel =0.$$

(3.28)

Next, we have

$$\begin{array}{ll}\varphi (u,x_{n+1})& \le \varphi (u,z_{n+1})\\ \le \varphi (u,J^{-1}({\delta }_{n}J{x}_n+(1-{\delta }_n)\mathit{\text{JT}}{y}_n\left)\right)\\ =\parallel u{\parallel }^2-2〈u,{\delta }_{n}J{x}_n+(1-{\delta }_n)\mathit{\text{JT}}{y}_n〉\\ +\parallel {\delta }_{n}J{x}_n+(1-{\delta }_n)\mathit{\text{JT}}{y}_n{\parallel }^2\\ \le \parallel u{\parallel }^2-2{\delta }_n〈u,J{x}_n〉-2(1-{\delta }_n)〈u,\mathit{\text{JT}}{y}_n〉\\ +{\delta }_n\parallel x_n{\parallel }^2+(1-{\delta }_n)\parallel T{y}_n{\parallel }^2\\ -{\delta }_n(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JT}}{y}_n\parallel )\\ \le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\varphi (u,T{y}_n)\\ -{\delta }_n(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JT}}{y}_n\parallel )\\ \le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\varphi (u,y_n)\\ -{\delta }_n(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JT}}{y}_n\parallel )\\ \le {\delta }_n\varphi (u,x_n)+(1-{\delta }_n)\varphi (u,x_n)\\ -{\delta }_n(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JT}}{y}_n\parallel )\\ \le \varphi (u,x_n)-{\delta }_n(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JT}}{y}_n\parallel ),\end{array}$$

or

$${\delta }_n(1-{\delta }_n)g(\parallel J{x}_n-\mathit{\text{JT}}{y}_n\parallel )\le \varphi (u,x_n)-\varphi (u,x_{n+1}).$$

(3.29)

Since {ϕ(u,x _n )} is convergent and using condition (i) in (3.29), we have

$$\underset{n\to \infty }{\lim }g(\parallel J{x}_n-\mathit{\text{JT}}{y}_n\parallel )=0.$$

From the property of g, we have

$$\underset{n\to \infty }{\lim }\parallel J{x}_n-\mathit{\text{JT}}{y}_n\parallel =0,$$

and hence

$$\underset{n\to \infty }{\lim }\parallel x_n-T{y}_n\parallel =0.$$

(3.30)

Now,

$$\begin{array}{ll}\parallel J{x}_n-J{y}_n\parallel & =\parallel J{x}_n-({\alpha }_{n}J{x}_n+(1-{\alpha }_n\left)\mathit{\text{JS}}{x}_n\right)\parallel \\ =\parallel (1-{\alpha }_n)(J{x}_n-\mathit{\text{JS}}{x}_n)\parallel \\ =(1-{\alpha }_n)\parallel J{x}_n-\mathit{\text{JS}}{x}_n\parallel ,\end{array}$$

which implies

$$\underset{n\to \infty }{\lim }\parallel J{x}_n-J{y}_n\parallel =0,$$

and hence

$$\underset{n\to \infty }{\lim }\parallel x_n-y_n\parallel =0.$$

(3.31)

It follows from (3.30), (3.31), and the inequality ∥y _n −T y _n ∥≤∥y _n −x _n ∥+∥x _n −T y _n ∥ that

$$\underset{n\to \infty }{\lim }\parallel y_n-T{y}_n\parallel =0.$$

(3.32)

Since {x _n } and {y _n } are bounded and limn→∞∥x _n −y _n ∥=0, there exist subsequences $\left\{x_{n_k}\right\}$ and $\left\{y_{n_k}\right\}$ of {x _n } and {y _n }, respectively, such that $x_{n_k}\rightharpoonup \widehat{x}\in C$ and $y_{n_k}\rightharpoonup \widehat{x}\in C$. It follows from (3.28) and (3.32) that $\widehat{x}\in \widehat{\text{Fix}}\left(S\right)\cap \widehat{\text{Fix}}\left(T\right)=\text{Fix}\left(S\right)\cap \text{Fix}\left(T\right)$, i.e., $\widehat{x}\in \text{Fix}\left(S\right)\cap \text{Fix}\left(T\right).$

Next, we show that $\widehat{x}\in \text{Sol}(\mathit{\text{GVEP}}$(1.2)). Let r= supn∈N{∥x _n ∥,∥z _n ∥}. From Lemma 2.4, there exists a continuous, strictly increasing, and convex function g₁ with g₁(0)=0 such that

$$g_1(\parallel x-y\parallel )\le \varphi (x,y),\forall x,y\in B_r.$$

Since x _n =T _r z _n , we have from Lemma 3.1 that, for u∈Γ,

$$\begin{array}{ll}{g}_1(\parallel x_n-z_n\parallel )& \le \varphi (x_n,z_n)\\ \le \varphi (u,z_n)-\varphi (u,x_n)\\ \le \varphi (u,x_{n-1})-\varphi (u,x_n).\end{array}$$

Since {ϕ(u,x _n )} converges, we have

$$\underset{n\to \infty }{\lim }{g}_1(\parallel x_n-z_n\parallel )=0.$$

From the property of g₁, we have

$$\underset{n\to \infty }{\lim }\parallel x_n-z_n\parallel =0.$$

Since J is uniformly norm-to-norm continuous on bounded sets, we have

$$\underset{n\to \infty }{\lim }\parallel J{x}_n-J{z}_n\parallel =0.$$

From r≥a, we have

$$\underset{n\to \infty }{\lim }\frac{\parallel J{x}_n-J{z}_n\parallel }{r}=0.$$

By x _n =T _r z _n , we have

$$\begin{array}{l}F\left(T_r\right(z_n),y)+\psi \left(y\right)-\psi \left(T_r{z}_n\right)\\ +\frac{e}{r}〈y-T_r{z}_n,J{T}_r{z}_n-J{z}_n〉\in P,\\ \forall y\in C\\ 0\in F(y,T_r(z_n\left)\right)-\psi \left(y\right)-\psi \left(T_r{z}_n\right)\hfill \\ +\frac{e}{r}〈y-T_r{z}_n,J{T}_r{z}_n-J{z}_n〉+P,\\ \forall y\in \mathrm{C.}\end{array}$$

Replacing n by n _i , we have

$$\begin{array}{l}\begin{array}{c}0\in F(y,T_r(z_{n_i}\left)\right)-\psi \left(y\right)-\psi \left(T_r{z}_{n_i}\right)\\ +\frac{e}{r}〈y-T_r{z}_{n_i},J{T}_r{z}_{n_i}-J{z}_{n_i}〉+P,\forall y\in \mathrm{C.}\end{array}\end{array}$$

As in the proof of Theorem 3.1, we have $\widehat{x}\in \text{Sol}(\mathit{\text{GVEP}}$(1.2)). Hence, $\widehat{x}\in \Gamma$.

Let $w_n=\prod _{\Gamma }{x}_n$. From Lemma 2.1 and $\widehat{x}\in \Gamma$, we have

$$〈w_{n_k}-\widehat{x},J{x}_{n_k}-J{w}_{n_k}〉\ge 0.$$

It follows from Proposition 3.1 that {w _n } converges strongly to z∈Γ. Since J is weakly sequentially continuous, we have

$$〈z-\widehat{x},J\widehat{x}-\mathit{\text{Jz}}〉\ge 0\text{as}k\to \mathrm{\infty .}$$

On the other hand, since J is monotone, we have

$$〈z-\widehat{x},J\widehat{x}-\mathit{\text{Jz}}〉\le 0.$$

Hence, we have

$$〈z-\widehat{x},J\widehat{x}-\mathit{\text{Jz}}〉=0.$$

From the strict convexity of E, we have $z=\widehat{x}.$ Therefore, {x _n } converges weakly to $\widehat{x}\in \Gamma$, where $\widehat{x}=\underset{n\to \infty }{lim}\prod _{\Gamma }{x}_n$. This completes the proof. □

Remark 3.1

• (i) If we take ψ=0, then Theorems 3.2 and 3.3 are reduced to the theorems of finding a common solution of SVEP(1.3) and fixed point problems for two relatively nonexpansive mappings.

• (ii) If we take $Y=ℝ,P=[0,+\infty ),T=I$, identity mapping, and δ _n =0, ∀n, then the results presented in this paper are reduced to the corresponding results of Takahashi and Zembayashi [17].

• (iii) The method presented in this paper can be used to extend the results of Shan and Huang [19] and Petrot et al. [20].