# Strong convergence theorem for total quasi-ϕ-asymptotically nonexpansive mappings in a Banach space

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## Abstract

In this paper, we prove strong convergence theorems to a point which is a fixed point of multi-valued mappings, a zero of an α-inverse-strongly monotone operator and a solution of the equilibrium problem. Next, we obtain strong convergence theorems to a solution of the variational inequality problem, a fixed point of multi-valued mappings and a solution of the equilibrium problem. The results presented in this paper are improvement and generalization of the previously known results.

## 1 Introduction

Let E be a real Banach space with dual $E ∗$, and let C be a nonempty closed convex subset of E. Let $A:C→ E ∗$ be an operator. A is called monotone if

$〈Ax−Ay,x−y〉≥0,∀x,y∈C;$

α-inverse-strongly monotone if there exists a constant $α>0$ such that

$〈Ax−Ay,x−y〉≥α ∥ A x − A y ∥ 2 ,∀x,y∈C;$

L-Lipschitz continuous if there exists a constant $L>0$ such that

$∥Ax−Ay∥≤L∥x−y∥,∀x,y∈C.$

If A is α-inverse strongly monotone, then it is $1 α$-Lipschitz continuous, i.e.,

$∥Ax−Ay∥≤ 1 α ∥x−y∥,∀x,y∈C.$

A monotone operator A is said to be maximal if its graph $G(A)={(x, x ∗ ): x ∗ ∈Ax}$ is not properly contained in the graph of any other monotone operator.

Let A be a monotone operator. We consider the problem of finding $x∈E$ such that

$0∈Ax,$
(1.1)

a point $x∈E$ is called a zero point of A. Denote by $A − 1 0$ the set of all points $x∈E$ such that $0∈Ax$. This problem is very important in optimization theory and related fields.

Let A be a monotone operator. The classical variational inequality problem for an operator A is to find $z ˆ ∈C$ such that

$〈A z ˆ ,y− z ˆ 〉≥0,∀y∈C.$
(1.2)

The set of solutions of (1.2) is denoted by $VI(A,C)$. This problem is connected with the convex minimization problem, the complementary problem, the problem of finding a point $x∈E$ satisfying $Ax=0$.

The value of $x ∗ ∈ E ∗$ at $x∈E$ will be denoted by $〈x, x ∗ 〉$ or $x ∗ (x)$. For each $p>1$, the generalized duality mapping $J p :E→ 2 E ∗$ is defined by

$J p (x)= { x ∗ ∈ E ∗ : 〈 x , x ∗ 〉 = ∥ x ∥ p , ∥ x ∗ ∥ = ∥ x ∥ p − 1 }$

for all $x∈E$. In particular, $J= J 2$ is called the normalized duality mapping. If E is a Hilbert space, then $J=I$, where I is the identity mapping.

Consider the functional defined by

(1.3)

where J is the normalized duality mapping. It is obvious from the definition of ϕ that

$( ∥ y ∥ − ∥ x ∥ ) 2 ≤ϕ(y,x)≤ ( ∥ y ∥ + ∥ x ∥ ) 2 ,∀x,y∈E.$
(1.4)

Alber  introduced that the generalized projection $Π C :E→C$ is a map that assigns to an arbitrary point $x∈E$ the minimum point of the functional $ϕ(x,y)$, that is, $Π C x= x ¯$, where $x ¯$ is the solution of the minimization problem

$ϕ( x ¯ ,x)= inf y ∈ C ϕ(y,x),$
(1.5)

existence and uniqueness of the operator $Π C$ follows from the properties of the functional $ϕ(x,y)$ and strict monotonicity of the mapping J.

Iiduka and Takahashi  introduced the following iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly monotone operator A in a 2-uniformly convex and uniformly smooth Banach space E: $x 1 =x∈C$ and

$x n + 1 = Π C J − 1 (J x n − λ n A x n ),∀n≥1,$
(1.6)

where $Π C$ is the generalized projection from E onto C, J is the duality mapping from E into $E ∗$ and ${ λ n }$ is a sequence of positive real numbers. They proved that the sequence ${ x n }$ generated by (1.6) converges weakly to some element of $VI(A,C)$. In connection, Iiduka and Takahashi  studied the following iterative scheme for finding a zero point of a monotone operator A in a 2-uniformly convex and uniformly smooth Banach space E:

${ x 1 = x ∈ E chosen arbitrarily , y n = J − 1 ( J x n − λ n A x n ) , X n = { z ∈ E : ϕ ( z , y n ) ≤ ϕ ( z , x n ) } , Y n + 1 = { z ∈ E : 〈 x n − z , J x − J x n 〉 ≥ 0 } , x n + 1 = Π X n ∩ Y n ( x ) ,$
(1.7)

where $Π X n ∩ Y n$ is the generalized projection from E onto $X n ∩ Y n$, J is the duality mapping from E into $E ∗$ and ${ λ n }$ is a sequence of positive real numbers. They proved that the sequence ${ x n }$ converges strongly to an element of $A − 1 0$. Moreover, under the additional suitable assumption they proved that the sequence ${ x n }$ converges strongly to some element of $VI(A,C)$. Some solution methods have been proposed to solve the variational inequality problem; see, for instance, .

A mapping $T:C→C$ is said to be ϕ-nonexpansive [7, 8] if

$ϕ(Tx,Ty)≤ϕ(x,y),∀x,y∈C.$

T is said to be quasi-ϕ-nonexpansive [7, 8] if $F(T)≠∅$ and

$ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T).$

T is said to be total quasi-ϕ-asymptotically nonexpansive, if $F(T)≠∅$ and there exist nonnegative real sequences $ν n$, $μ n$ with $ν n →0$, $μ n →0$ as $n→∞$ and a strictly increasing continuous function $φ: R + → R +$ with $φ(0)=0$ such that

$ϕ ( p , T n x ) ≤ϕ(p,x)+ ν n φ ( ϕ ( p , x ) ) + μ n ,∀n≥1,∀x∈C,p∈F(T).$

Let $2 C$ be the family of all nonempty subsets of C, and let $S:C→ 2 C$ be a multi-valued mapping. For a point $q∈C$, $n≥1$ define an iterative sequence as follows:

$S q : = { q 1 : q 1 ∈ S q } , S 2 q = S S q : = ⋃ q 1 ∈ S q S q 1 , S 3 q = S S 2 q : = ⋃ q 2 ∈ T 2 q S q 2 , ⋮ S n q = S S n − 1 q : = ⋃ q n − 1 ∈ S n − 1 q S q n − 1 .$

A point $p∈C$ is said to be an asymptotic fixed point of S if there exists a sequence ${ x n }$ in C such that ${ x n }$ converges weakly to p and

$lim n → ∞ d( x n ,S x n ):= lim n → ∞ inf x ∈ S x n ∥ x n −x∥=0.$

The asymptotic fixed point set of S is denoted by $F ˆ (S)$.

A multi-valued mapping S is said to be total quasi-ϕ-asymptotically nonexpansive if $F(S)≠∅$ and there exist nonnegative real sequences $ν n$, $μ n$ with $ν n →0$, $μ n →0$ as $n→∞$ and a strictly increasing continuous function $φ: R + → R +$ with $φ(0)=0$ such that for all $x∈C$, $p∈F(S)$,

$ϕ(p, w n )≤ϕ(p,x)+ ν n φ ( ϕ ( p , x ) ) + μ n ,∀n≥1, w n ∈ S n x.$

S is said to be closed if for any sequence ${ x n }$ and ${ w n }$ in C with $w n ∈S x n$ if $x n →x$ and $w n →w$, then $w∈Sx$.

A multi-valued mapping S is said to be uniformly asymptotically regular on C if

$lim n → ∞ ( sup x ∈ C ∥ s n + 1 − s n ∥ ) =0, s n ∈ S n x.$

Every quasi-ϕ-asymptotically nonexpansive multi-valued mapping implies a quasi-ϕ-asymptotically nonexpansive mapping but the converse is not true.

In 2012, Chang et al.  introduced the concept of total quasi-ϕ-asymptotically nonexpansive multi-valued mapping and then proved some strong convergence theorem by using the hybrid shrinking projection method.

Let $f:C×C→R$ be a bifunction, the equilibrium problem is to find $x∈C$ such that

$f(x,y)≥0,∀y∈C.$
(1.8)

The set of solutions of (1.8) is denoted by $EP(f)$. The equilibrium problem is very general in the sense that it includes, as special cases, optimization problems, variational inequality problems, min-max problems, saddle point problem, fixed point problem, Nash EP. In 2008, Takahashi and Zembayashi [10, 11] introduced iterative sequences for finding a common solution of an equilibrium problem and a fixed point problem. Some solution methods have been proposed to solve the equilibrium problem; see, for instance, .

For a mapping $A:C→ E ∗$, let $f(x,y)=〈Ax,y−x〉$ for all $x,y∈C$. Then $x∈EP(f)$ if and only if $〈Tx,y−x〉≥0$ for all $y∈C$; i.e., x is a solution of the variational inequality.

Motivated and inspired by the work mentioned above, in this paper, we introduce and prove strong convergence of a new hybrid projection algorithm for a fixed point of total quasi-ϕ-asymptotically nonexpansive multi-valued mappings, the solution of the equilibrium problem, a zero point of monotone operators. Moreover, we prove strong convergence to the solution of the variation inequality in a uniformly smooth and 2-uniformly convex Banach space.

## 2 Preliminaries

A Banach space E with the norm $∥⋅∥$ is called strictly convex if $∥ x + y 2 ∥<1$ for all $x,y∈E$ with $∥x∥=∥y∥=1$ and $x≠y$. Let $U={x∈E:∥x∥=1}$ be the unit sphere of E. A Banach space E is called smooth if the limit $lim t → 0 ∥ x + t y ∥ − ∥ x ∥ t$ exists for each $x,y∈U$. It is also called uniformly smooth if the limit exists uniformly for all $x,y∈U$. The modulus of convexity of E is the function $δ:[0,2]→[0,1]$ defined by

$δ(ε)=inf { 1 − ∥ x + y 2 ∥ : x , y ∈ E , ∥ x ∥ = ∥ y ∥ = 1 , ∥ x − y ∥ ≥ ε } .$

A Banach space E is uniformly convex if and only if $δ(ε)>0$ for all $ε∈(0,2]$. Let p be a fixed real number with $p≥2$. A Banach space E is said to be p-uniformly convex if there exists a constant $c>0$ such that $δ(ε)≥c ε p$ for all $ε∈[0,2]$. Observe that every p-uniform convex is uniformly convex. Every uniformly convex Banach space E has the Kadec-Klee property, that is, for any sequence ${ x n }⊂E$, if $x n ⇀x∈E$ and $∥ x n ∥→∥x∥$, then $x n →x$.

Let E be a real Banach space with dual $E ∗$, E is uniformly smooth if and only if $E ∗$ is a uniformly convex Banach space. If E is a uniformly smooth Banach space, then E is a smooth and reflexive Banach space.

Remark 2.1

• If E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.

• If E is reflexive smooth and strictly convex, then the normalized duality mapping J is single-valued, one-to-one and onto.

• If E is a reflexive strictly convex and smooth Banach space and J is the duality mapping from E into $E ∗$, then $J − 1$ is also single-valued, bijective and is also the duality mapping from $E ∗$ into E and thus $J J − 1 = I E ∗$ and $J − 1 J= I E$.

See  for more details.

Remark 2.2 If E is a reflexive, strictly convex and smooth Banach space, then $ϕ(x,y)=0$ if and only if $x=y$. It is sufficient to show that if $ϕ(x,y)=0$, then $x=y$. From (1.3) we have $∥x∥=∥y∥$. This implies that $〈x,Jy〉= ∥ x ∥ 2 = ∥ J y ∥ 2$. From the definition of J, one has $Jx=Jy$. Therefore, we have $x=y$ (see [22, 23] for more details).

Lemma 2.3 (Beauzamy  and Xu )

If E is a 2-uniformly convex Banach space, then, for all $x,y∈E$, we have

$∥x−y∥≤ 2 c 2 ∥Jx−Jy∥,$

where J is the normalized duality mapping of E and $0.

The best constant $1 c$ in the lemma is called the p-uniformly convex constant of E.

Lemma 2.4 (Beauzamy  and Zalinescu )

If E is a p-uniformly convex Banach space, and let p be a given real number with $p≥2$, then, for all $x,y∈E$, $J x ∈ J p (x)$ and $J y ∈ J p (y)$,

$〈x−y, J x − J y 〉≥ c p 2 p − 2 p ∥ x − y ∥ p ,$

where $J p$ is the generalized duality mapping of E and $1 c$ is the p-uniformly convex constant of E.

Lemma 2.5 (Kamimura and Takahashi )

Let E be a uniformly convex and smooth Banach space, and let ${ x n }$, ${ y n }$ be two sequences of E. If $ϕ( x n , y n )→0$ and either ${ x n }$ or ${ y n }$ is bounded, then $∥ x n − y n ∥→0$.

Lemma 2.6 (Alber )

Let C be a nonempty closed convex subset of a smooth Banach space E, and let $x∈E$. Then $x 0 = Π C x$ if and only if

$〈 x 0 −y,Jx−J x 0 〉≥0,∀y∈C.$

Lemma 2.7 (Alber )

Let E be a reflexive strictly convex and smooth Banach space, C be a nonempty closed convex subset of E, and let $x∈E$. Then

$ϕ(y, Π C x)+ϕ( Π C x,x)≤ϕ(y,x),∀y∈C.$

Lemma 2.8 (Chang et al. )

Let C be a nonempty, closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let $S:C→ 2 C$ be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequence $ν n$ and $μ n$ with $ν n →0$, $μ n →0$ as $n→∞$ and a strictly increasing continuous function $φ: R + → R +$ with $φ(0)=0$. If $μ 1 =0$, then the fixed point set $F(S)$ is a closed convex subset of C.

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

1. (A1)

$f(x,x)=0$ for all $x∈C$;

2. (A2)

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

3. (A3)

for each $x,y,z∈C$,

$lim t ↓ 0 f ( t z + ( 1 − t ) x , y ) ≤f(x,y);$
4. (A4)

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

Lemma 2.9 (Blum and Oettli )

Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E, let f be a bifunction from $C×C$ to satisfying (A1)-(A4), and let $r>0$ and $x∈E$. Then there exists $z∈C$ such that

$f(z,y)+ 1 r 〈y−z,Jz−Jx〉≥0,∀y∈C.$

Lemma 2.10 (Takahashi and Zembayashi )

Let C be a closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space E, and let f be a bifunction from $C×C$ to satisfying conditions (A1)-(A4). For all $r>0$ and $x∈E$, define a mapping $T r :E→C$ as follows:

$T r x= { z ∈ C : f ( z , y ) + 1 r 〈 y − z , J z − J x 〉 ≥ 0 , ∀ y ∈ C } .$

Then the following hold:

1. (1)

$T r$ is single-valued;

2. (2)

$T r$ is a firmly nonexpansive-type mapping , that is, for all $x,y∈E$,

$〈 T r x− T r y,J T r x−J T r y〉≤〈 T r x− T r y,Jx−Jy〉;$
3. (3)

$F( T r )=EP(f)$;

4. (4)

$EP(f)$ is closed and convex.

Lemma 2.11 (Takahashi and Zembayashi )

Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E, let f be a bifunction from $C×C$ to satisfying (A1)-(A4), and let $r>0$. Then, for $x∈E$ and $q∈F( T r )$,

$ϕ(q, T r x)+ϕ( T r x,x)≤ϕ(q,x).$

Let A be an inverse-strongly monotone mapping of C into $E ∗$ which is said to be hemicontinuous if for all $x,y∈C$, the mapping h of $[0,1]$ into $E ∗$, defined by $h(t)=A(tx+(1−t)y)$, is continuous with respect to the weak topology of $E ∗$. We define by $N C (v)$ the normal cone for C at a point $v∈C$, that is,

$N C (v)= { x ∗ ∈ E ∗ : 〈 v − y , x ∗ 〉 ≥ 0 , ∀ y ∈ C } .$
(2.1)

Theorem 2.12 (Rockafellar )

Let C be a nonempty, closed convex subset of a Banach space E, and let A be a monotone, hemicontinuous operator of C into $E ∗$. Let $B⊂E× E ∗$ be an operator defined as follows:

$Bv={ A v + N C ( v ) , v ∈ C ; ∅ , otherwise .$
(2.2)

Then B is maximal monotone and $B − 1 0=VI(A,C)$.

Theorem 2.13 (Takahashi )

Let C be a nonempty subset of a Banach space E, and let A be a monotone, hemicontinuous operator of C into $E ∗$ with $C=D(A)$. Then

$VI(A,C)= { u ∈ C : 〈 v − u , A v 〉 ≥ 0 , ∀ v ∈ C } .$
(2.3)

It is obvious that the set $VI(A,C)$ is a closed and convex subset of C and the set $A − 1 0=VI(A,E)$ is a closed and convex subset of E.

Theorem 2.14 (Takahashi )

Let C be a nonempty compact convex subset of a Banach space E, and let A be a monotone, hemicontinuous operator of C into $E ∗$ with $C=D(A)$. Then $VI(A,C)$ is nonempty.

We make use of the following mapping V studied in Alber :

$V ( x , x ∗ ) = ∥ x ∥ 2 −2 〈 x , x ∗ 〉 + ∥ x ∗ ∥ 2 ,∀x∈E, x ∗ ∈ E ∗ ,$
(2.4)

that is, $V(x, x ∗ )=ϕ(x, J − 1 ( x ∗ ))$.

Lemma 2.15 (Alber )

Let E be a reflexive strictly convex smooth Banach space, and let V be as in (2.4). Then we have

$V ( x , x ∗ ) +2 〈 J − 1 ( x ∗ ) − x , y ∗ 〉 ≤V ( x , x ∗ + y ∗ ) ,∀x∈E, x ∗ , y ∗ ∈ E ∗ .$

Lemma 2.16 (Beauzamy  and Xu )

If E is a 2-uniformly convex Banach space, then, for all $x,y∈E$, we have

$∥x−y∥≤ 2 c 2 ∥Jx−Jy∥,$

where J is the normalized duality mapping of E and $0.

Lemma 2.17 (Cho et al. )

Let E be a uniformly convex Banach space, and let $B r (0)={x∈E:∥x∥≤r}$ be a closed ball of E. Then there exists a continuous strictly increasing convex function $g:[0,∞)→[0,∞)$ with $g(0)=0$ such that

$∥ λ x + μ y + γ z ∥ 2 ≤ ∥ λ x ∥ 2 + ∥ μ y ∥ 2 + ∥ γ z ∥ 2 −λμg ( ∥ x − y ∥ )$

for all $x,y,z∈ B r (0)$ and $λ,μ,γ∈[0,1]$ with $λ+μ+γ=1$.

Lemma 2.18 (Pascali and Sburlan )

Let E be a real smooth Banach space, and let $A:E→ 2 E ∗$ be a maximal monotone mapping. Then $A − 1 0$ is a closed and convex subset of E and the graph $G(A)$ of A is demiclosed in the following sense: if ${ x n }⊂D(A)$ with $x n ⇀x∈E$ and $y n ∈A x n$ with $y n →y∈ E ∗$, then $x∈D(A)$ and $y∈Ax$.

## 3 Main results

Theorem 3.1 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from $C×C$ to satisfying conditions (A1)-(A4), and let A be an α-inverse-strongly monotone mapping of E into $E ∗$. Let $S:C→ 2 C$ be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences $ν n$, $μ n$ with $ν n →0$, $μ n →0$ as $n→∞$ and a strictly increasing continuous function $ψ: R + → R +$ with $ψ(0)=0$. Assume that S is uniformly asymptotically regular on C with $μ 1 =0$ and $F:=F(S)∩EP(f)∩ A − 1 0≠∅$. For arbitrary $x 1 ∈C$, $C 1 =C$, generate a sequence ${ x n }$ by

${ z n = J − 1 ( J x n − λ n A x n ) , u n = T r n z n , y n = J − 1 ( α n J x n + β n J w n + γ n J u n ) , w n ∈ S n x n , C n + 1 = { v ∈ C n : ϕ ( v , y n ) ≤ ϕ ( v , z n ) ≤ ϕ ( v , x n ) + K n } , x n + 1 = Π C n + 1 x 1 , n ∈ N ,$
(3.1)

where $K n = ν n sup q ∈ F ψ(ϕ(q, x n ))+ μ n$. Assume that the control sequences ${ α n }$, ${ β n }$, ${ γ n }$, ${ λ n }$ and ${ r n }$ satisfy the following conditions:

1. 1.

${ α n }$, ${ β n }$ and ${ γ n }$ are sequences in $(0,1)$ such that $α n + β n + γ n =1$, $lim inf n → ∞ α n β n >0$,

2. 2.

${ λ n }⊂[a,b]$ for some a, b with $0 and $1 c$ is the 2-uniformly convex constant of E,

3. 3.

${ r n }⊂[d,∞)$ for some $d>0$,

then ${ x n }$ converges strongly to $Π F x 1$.

Proof We will show that $C n$ is closed and convex for all $n∈N$. Since $C 1 =C$ is closed and convex. Suppose that $C n$ is closed and convex for all $n∈N$. For any $v∈ C n$, we know that $ϕ(v, y n )≤ϕ(v, x n )+ K n$ is equivalent to

$2〈v,J x n −J y n 〉≤ ∥ x n ∥ 2 − ∥ y n ∥ 2 + K n .$

That is, $C n + 1$ is closed and convex, hence $C n$ is closed and convex for all $n∈N$.

We show by induction that $F⊂ C n$ for all $n∈N$. It is obvious that $F⊂C= C 1$. Suppose that $F⊂ C n$ where $n∈N$. Let $q∈F$, we have

$ϕ ( q , z n ) = ϕ ( q , J − 1 ( J x n − λ n A x n ) ) = V ( q , J x n − λ n A x n ) ≤ V ( q , ( J x n − λ n A x n ) + λ n A x n ) − 2 〈 J − 1 ( J x n − λ n A x n ) − q , λ n A x n 〉 = V ( q , J x n ) − 2 λ n 〈 J − 1 ( J x n − λ n A x n ) − q , A x n 〉 = ϕ ( q , x n ) − 2 λ n 〈 x n − q , A x n 〉 + 2 〈 J − 1 ( J x n − λ n A x n ) − x n , − λ n A x n 〉 .$
(3.2)

Since A is an α-inverse-strongly monotone mapping, we get

$− 2 λ n 〈 x n − q , A x n 〉 = − 2 λ n 〈 x n − q , A x n − A q 〉 − 2 λ n 〈 x n − q , A q 〉 ≤ − 2 λ n 〈 x n − q , A x n − A q 〉 = − 2 α λ n ∥ A x n − A q ∥ 2 .$
(3.3)

It follows from Lemma 2.17 that

$2 〈 J − 1 ( J x n − λ n A x n ) − x n , − λ n A x n 〉 = 2 〈 J − 1 ( J x n − λ n A x n ) − J − 1 ( J x n ) , − λ n A x n 〉 ≤ 2 ∥ J − 1 ( J x n − λ n A x n ) − J − 1 ( J x n ) ∥ ∥ λ n A x n ∥ ≤ 4 c 2 ∥ J J − 1 ( J x n − λ n A x n ) − J J − 1 ( J x n ) ∥ ∥ λ n A x n ∥ = 4 c 2 ∥ J x n − λ n A x n − J x n ∥ ∥ λ n A x n ∥ = 4 c 2 ∥ λ n A x n ∥ 2 = 4 c 2 λ n 2 ∥ A x n ∥ 2 ≤ 4 c 2 λ n 2 ∥ A x n − A q ∥ 2 .$
(3.4)

Replacing (3.2) by (3.3) and (3.4), we get

$ϕ ( q , z n ) ≤ ϕ ( q , x n ) − 2 α λ n ∥ A x n − A q ∥ 2 + 4 c 2 λ n 2 ∥ A x n − A q ∥ 2 = ϕ ( q , x n ) + 2 λ n ( 2 c 2 λ n − α ) ∥ A x n − A q ∥ 2 ≤ ϕ ( q , x n ) .$
(3.5)

From Lemma 2.11, we know that

$ϕ(q, u n )=ϕ(q, T r n z n )≤ϕ(q, z n )≤ϕ(q, x n ).$
(3.6)

Since S is a total quasi-ϕ-asymptotically nonexpansive multi-valued mapping and $w n ∈ S n x n$, it follows that

$ϕ ( q , y n ) = ϕ ( q , J − 1 ( α n J x n + β n J w n + γ n J u n ) ) = ∥ q ∥ 2 − 2 〈 q , α n J x n + β n J w n + γ n J u n 〉 + ∥ α n J x n + β n J w n + γ n J u n ∥ 2 ≤ α n ϕ ( q , x n ) + β n ϕ ( q , w n ) + γ n ϕ ( q , u n ) ≤ α n ϕ ( q , x n ) + β n ϕ ( q , x n ) + β n ν n ψ ( ϕ ( q , x n ) ) + β n μ n + γ n ϕ ( q , u n ) ≤ α n ϕ ( q , x n ) + β n ϕ ( q , x n ) + ν n sup q ∈ F ψ ( ϕ ( q , x n ) ) + μ n + γ n ϕ ( q , u n ) = α n ϕ ( q , x n ) + β n ϕ ( q , x n ) + γ n ϕ ( q , u n ) + K n ≤ α n ϕ ( q , x n ) + β n ϕ ( q , x n ) + γ n ϕ ( q , T r n z n ) + K n ≤ α n ϕ ( q , x n ) + β n ϕ ( q , x n ) + γ n ϕ ( q , z n ) + K n ≤ α n ϕ ( q , x n ) + β n ϕ ( q , x n ) + γ n ϕ ( q , x n ) + K n ≤ ϕ ( q , x n ) + K n ,$
(3.7)

where $K n = ν n sup q ∈ F ψ(ϕ(q, x n ))+ μ n$.

This shows that $q∈ C n + 1$, which implies that $F⊂ C n + 1$. Hence $F⊂ C n$ for all $n∈N$ and the sequence ${ x n }$ is well defined.

From the definition of $C n + 1$ with $x n = Π C n x 1$ and $x n + 1 = Π C n + 1 x 1 ∈ C n + 1 ⊂ C n$, it follows that

$ϕ( x n , x 1 )≤ϕ( x n + 1 , x 1 ),∀n≥1,$
(3.8)

that is, ${ϕ( x n , x 1 )}$ is nondecreasing. By Lemma 2.7, we get

$ϕ ( x n , x 1 ) = ϕ ( Π C n x 1 , x 1 ) ≤ ϕ ( q , x 1 ) − ϕ ( q , x n ) ≤ ϕ ( q , x 1 ) , ∀ q ∈ F .$
(3.9)

This implies that ${ϕ( x n , x 1 )}$ is bounded and so $lim n → ∞ ϕ( x n , x 1 )$ exists. In particular, by (1.4), the sequence ${ ( ∥ x n ∥ − ∥ x 1 ∥ ) 2 }$ is bounded. This implies ${ x n }$ is also bounded. So, we have ${ u n }$, ${ z n }$ and ${ y n }$ are also bounded.

Since $x m = Π C m x 1 ∈ C m ⊂ C n$ for all $m,n≥1$ with $m>n$, by Lemma 2.7, we have

$ϕ ( x m , x n ) = ϕ ( x m , Π C n x 1 ) ≤ ϕ ( x m , x 1 ) − ϕ ( Π C n x 1 , x 1 ) = ϕ ( x m , x 1 ) − ϕ ( x n , x 1 ) ,$

taking $m,n→∞$, we have $ϕ( x m , x n )→0$. This implies that ${ x n }$ is a Cauchy sequence. From Lemma 2.5, it follows that $∥ x n − x m ∥→0$ and ${ x n }$ is a Cauchy sequence. By the completeness of E and the closedness of C, we can assume that there exists $p∈C$ such that

$lim n → ∞ x n =p,$
(3.10)

we also get that

$lim n → ∞ K n = lim n → ∞ ν n sup q ∈ F ψ ( ϕ ( q , x n ) ) + μ n =0.$
(3.11)

Next, we show that $p∈F:=F(S)∩ A − 1 0∩EP(f)$.

(a) We show that $p∈F(S)$. By the definition of $Π C n x 1$, we have

$ϕ ( x n + 1 , x n ) = ϕ ( x n + 1 , Π C n x 1 ) ≤ ϕ ( x n + 1 , x 1 ) − ϕ ( Π C n x 1 , x 1 ) = ϕ ( x n + 1 , x 1 ) − ϕ ( x n , x 1 ) .$

Since $lim n → ∞ ϕ( x n , x 1 )$ exists, we get

$lim n → ∞ ϕ( x n + 1 , x n )=0.$
(3.12)

It follows from Lemma 2.5 that

$lim n → ∞ ∥ x n + 1 − x n ∥=0.$
(3.13)

From the definition of $C n + 1$ and $x n + 1 = Π C n + 1 x 1 ∈ C n + 1 ⊂ C n$, we have $ϕ( x n + 1 , y n )≤ϕ( x n + 1 , x n )+ K n →0$ as $n→∞$. By Lemma 2.5, it follows that

$lim n → ∞ ∥ x n + 1 − y n ∥=0.$
(3.14)

From $lim n → ∞ x n =p$, we also have

$lim n → ∞ y n =p.$
(3.15)

By using the triangle inequality, we get $∥ x n − y n ∥≤∥ x n − x n + 1 ∥+∥ x n + 1 − y n ∥→0$ as $n→∞$. Since J is uniformly norm-to-norm continuous, we obtain $∥J x n −J y n ∥→0$ as $n→∞$. On the other hand, we note that

$ϕ ( q , x n ) − ϕ ( q , y n ) = ∥ x n ∥ 2 − ∥ y n ∥ 2 − 2 〈 q , J x n − J y n 〉 ≤ ∥ x n − y n ∥ ( ∥ x n + y n ∥ ) + 2 ∥ q ∥ ∥ J x n − J y n ∥ .$

In view of $∥ x n − y n ∥→0$ and $∥J x n −J y n ∥→0$ as $n→∞$, we obtain that

(3.16)

From Lemma 2.17, we have

$ϕ ( q , y n ) = ϕ ( q , J − 1 [ α n J x n + β n J w n + γ n J u n ] ) ≤ ∥ q ∥ 2 − 2 〈 q , α n J x n + β n J w n + γ n J u n 〉 + ∥ α n J x n + β n J w n + γ n J u n ∥ 2 − α n β n g ( ∥ J x n − J w n ∥ ) = α n ϕ ( q , x n ) + β n ϕ ( q , w n ) + γ n ϕ ( q , u n ) − α n β n g ( ∥ J x n − J w n ∥ ) ≤ ϕ ( q , x n ) + K n − α n β n g ( ∥ J x n − J w n ∥ ) .$
(3.17)

It follows from $lim inf n → ∞ α n β n >0$, (3.16), (3.11) and the property of g that

$lim n → ∞ ∥J x n −J w n ∥=0.$

Since $J − 1$ is uniformly norm-to-norm continuous, we obtain

$lim n → ∞ ∥ x n − w n ∥=0.$
(3.18)

From (3.10) it follows that

$lim n → ∞ ∥ w n −p∥=0.$
(3.19)

For $w n ∈ S n x n$, generate a sequence ${ s n }$ by

$s 2 ∈ S w 1 ⊂ S 2 x 1 , s 3 ∈ S w 2 ⊂ S 3 x 2 , s 4 ∈ S w 3 ⊂ S 4 x 3 , ⋮ s n + 1 ∈ S w n ⊂ S n + 1 x n .$

On the other hand, we have $∥ s n + 1 −p∥≤∥ s n + 1 − w n ∥+∥ w n −p∥$. Since S is uniformly asymptotically regular, it follows that

$lim n → ∞ ∥ s n + 1 −p∥=0,$
(3.20)

we have

$lim n → ∞ ∥ S n + 1 x n − p ∥ =0,$
(3.21)

that is, $S S n x n →p$ as $n→∞$. From the closedness of S, we have $p∈F(S)$.

(b) We show that $p∈ A − 1 0$.

From the definition of $C n + 1$ and $x n + 1 = Π C n + 1 x 1 ∈ C n + 1 ⊂ C n$, we have $ϕ( x n + 1 , z n )≤ϕ( x n + 1 , x n )+ K n →0$ as $n→∞$. By Lemma 2.5, it follows that $lim n → ∞ ∥ x n + 1 − z n ∥=0$. By the triangle inequality, we get $∥ x n − z n ∥≤∥ x n − x n + 1 ∥+∥ x n + 1 − z n ∥→0$ as $n→∞$. From $lim n → ∞ ∥ z n − x n ∥=0$ and from (3.10), it follows that

$lim n → ∞ z n =p.$
(3.22)

Since J is uniformly norm-to-norm continuous, we also have

$lim n → ∞ ∥J z n −J x n ∥=0.$
(3.23)

Hence, from the definition of the sequence ${ z n }$, it follows that

$∥A x n ∥= ∥ J z n − J x n ∥ λ n .$
(3.24)

From (3.23) and the definition of the sequence ${ λ n }$, we have

$lim n → ∞ ∥A x n ∥=0,$
(3.25)

that is,

$lim n → ∞ A x n =0.$
(3.26)

Since A is Lipschitz continuous, it follows from (3.10) that

$Ap=0.$
(3.27)

Again, since A is Lipschitz continuous and monotone so it is maximal monotone. It follows from Lemma 2.18 that $p∈ A − 1 0$.

(c) We show that $p∈EP(f)$.

From $x n , y n →0$ and $K n →0$ as $n→∞$ and applying (3.7) for any $q∈F$, we get $lim n → ∞ ϕ(q, u n )→ϕ(q,p)$, it follows that

$ϕ ( u n , x n ) = ϕ ( T r n , x n ) ≤ ϕ ( q , x n ) − ϕ ( q , T r n x n ) = ϕ ( q , x n ) − ϕ ( q , u n ) .$

Taking limit as $n→∞$ on the both sides of the inequality, we have $lim n → ∞ ϕ( u n , x n )=0$. From Lemma 2.5, it follows that

$lim n → ∞ ∥ u n − x n ∥=0$
(3.28)

and

$lim n → ∞ u n =p.$
(3.29)

Since J is uniformly norm-to-norm continuous on bounded subsets of E, we obtain

$lim n → ∞ ∥J u n −J z n ∥=0.$

Since $r n >0$ for all $n≥1$, we have $∥ J u n − J z n ∥ r n →0$ as $n→∞$ and

$f( u n ,y)+ 1 r n 〈y− u n ,J u n −J z n 〉≥0,∀y∈C.$

From (A2), the fact that

$∥ y − u n ∥ ∥ J u n − J z n ∥ r n ≥ 1 r n 〈 y − u n , J u n − J z n 〉 ≥ − f ( u n , y ) ≥ f ( y , u n ) , ∀ y ∈ C ,$

taking the limit as $n→∞$ in the above inequality and from the fact that $u n →p$ as $n→∞$, it follows that $f(y,p)≤0$ for all $y∈C$. For any $0, define $y t =ty+(1−t)p$. Then $y t ∈C$, which implies that $f( y t ,p)≤0$. Thus it follows from (A1) that

$0=f( y t , y t )≤tf( y t ,y)+(1−t)θ( y t ,p)≤tf( y t ,y),$

and so $f( y t ,y)≥0$. From (A3) we have $f(p,y)≥0$ for all $y∈C$ and so $p∈EP(f)$. Hence, by (a), (b) and (c), that is, $p∈F(S)∩ A − 1 0∩EP(f)$.

Finally, we show that $p= Π F x 1$. From $x n = Π C n x 1$, we have $〈J x 1 −J x n , x n −z〉≥0$ for all $z∈ C n$. Since $F⊂ C n$, we also have

$〈J x 1 −J x n , x n − p ˆ 〉≥0,∀ p ˆ ∈F.$

Taking limit $n→∞$, we obtain

$〈J x 1 −Jp,p− p ˆ 〉≥0,∀ p ˆ ∈F.$

By Lemma 2.6, we can conclude that $p= Π F x 1$ and $x n →p$ as $n→∞$. The proof is completed. □

Next, we define $z n = Π C J − 1 (J x n − λ n A x n )$ and assume that $∥Ay∥≤∥Ay−Au∥$ for all $y∈C$ and $u∈VI(A,C)≠∅$. We can prove the strong convergence theorem for finding the set of solutions of the variational inequality problem in a real uniformly smooth and 2-uniformly convex Banach space.

Remark 3.2 (Qin et al. )

Let $Π C$ be the generalized projection from a smooth strictly convex and reflexive Banach space E onto a nonempty closed convex subset C of E. Then $Π C$ is a closed quasi-ϕ-nonexpansive mapping from E onto C with $F( Π C )=C$.

Corollary 3.3 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from $C×C$ to satisfying conditions (A1)-(A4), and let A be an α-inverse-strongly monotone mapping of C into $E ∗$ satisfying $∥Ay∥≤∥Ay−Au∥$ for all $y∈C$ and $u∈VI(A,C)≠∅$. Let $S:C→ 2 C$ be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences $ν n$, $μ n$ with $ν n →0$, $μ n →0$ as $n→∞$ and a strictly increasing continuous function $ψ: R + → R +$ with $ψ(0)=0$. Assume that S is uniformly asymptotically regular on C with $μ 1 =0$ and $F:=F(S)∩EP(f)∩VI(A,C)≠∅$. For arbitrary $x 1 ∈C$, $C 1 =C$, generate a sequence ${ x n }$ by

${ z n = Π C J − 1 ( J x n − λ n A x n ) , u n = T r n x n , y n = J − 1 ( α n J x n + β n J w n + γ n J u n ) , w n ∈ S n x n , C n + 1 = { v ∈ C n : ϕ ( v , y n ) ≤ ϕ ( v , z n ) ≤ ϕ ( v , x n ) + K n } , x n + 1 = Π C n + 1 x 1 , n ∈ N ,$
(3.30)

where $K n = ν n sup q ∈ F ψ(ϕ(q, x n ))+ μ n$. Assume that the control sequences ${ α n }$, ${ β n }$, ${ γ n }$, ${ λ n }$ and ${ r n }$ satisfy the following conditions:

1. 1.

${ α n }$, ${ β n }$ and ${ γ n }$ are sequences in $(0,1)$ such that $α n + β n + γ n =1$, $lim inf n → ∞ α n β n >0$,

2. 2.

${ λ n }⊂[a,b]$ for some a, b with $0 and $1 c$ is the 2-uniformly convex constant of E,

3. 3.

${ r n }⊂[d,∞)$ for some $d>0$,

then ${ x n }$ converges strongly to $Π F x 1$.

Proof For $q∈F$ and $Π C$ is quasi-ϕ-nonexpansive mapping, we have

$ϕ(q, z n )=ϕ ( q , Π C J − 1 ( J x n − λ n A x n ) ) ≤ϕ ( q , J − 1 ( J x n − λ n A x n ) ) .$

So, we can show that $p∈VI(A,C)$.

Define $B⊂E× E ∗$ by Theorem 2.14, B is maximal monotone and $B − 1 0=VI(A,C)$. Let $(z,w)∈G(B)$. Since $w∈Bz=Az+ N C (z)$, we get $w−Az∈ N C (z)$.

From $z n ∈C$, we have

$〈z− z n ,w−Az〉≥0.$
(3.31)

On the other hand, since $z n = Π C J − 1 (J x n − λ n A x n )$. Then, by Lemma 2.6, we have

$〈 z − z n , J z n − ( J x n − λ n A x n ) 〉 ≥0,$

and thus

$〈 z − z n , J x n − J z n λ n − A x n 〉 ≤0.$
(3.32)

It follows from (3.31) and (3.32) that

$〈 z − z n , w 〉 ≥ 〈 z − z n , A z 〉 ≥ 〈 z − z n , A z 〉 + 〈 z − z n , J x n − J z n λ n − A x n 〉 = 〈 z − z n , A z − A x n 〉 + 〈 z − z n , J x n − J z n λ n 〉 = 〈 z − z n , A z − A z n 〉 + 〈 z − z n , A z n − A x n 〉 + 〈 z − z n , J x n − J z n λ n 〉 ≥ − ∥ z − z n ∥ ∥ z n − x n ∥ α − ∥ z − z n ∥ ∥ J x n − J z n ∥ a ≥ − M ( ∥ z n − x n ∥ α + ∥ J x n − J z n ∥ a ) ,$

where $M= sup n ≥ 1 ∥z− z n ∥$. From $∥ x n − z n ∥→0$ as $n→∞$ and (3.23), taking $lim n → ∞$ on the both sides of the equality above, we have $〈z−p,w〉≥0$. By the maximality of B, we have $p∈ B − 1 0$, that is, $p∈VI(A,C)$. From Theorem 3.1, we have $p∈F(S)∩EP(f)∩VI(A,C)$. The proof is completed. □

Let A be a strongly monotone mapping with constant k, Lipschitz with constant $L>0$, that is,

$∥Ax−Ay∥≤L∥x−y∥,∀x,y∈D(A),$

which implies that

$1 L ∥Ax−Ay∥≤∥x−y∥,∀x,y∈D(A).$

It follows that

$〈Ax−Ay,x−y〉≥k ∥ x − y ∥ 2 ≥ k L ∥ A x − A y ∥ 2$

hence A is α-inverse-strongly monotone with $α= k L$. Therefore, we have the following corollaries.

Corollary 3.4 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from $C×C$ to satisfying conditions (A1)-(A4), and let $A:E→ E ∗$ be a strongly monotone mapping with constant k, Lipschitz with constant $L>0$. Let $S:C→ 2 C$ be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences $ν n$, $μ n$ with $ν n →0$, $μ n →0$ as $n→∞$ and a strictly increasing continuous function $ψ: R + → R +$ with $ψ(0)=0$. Assume that S is uniformly asymptotically regular on C with $μ 1 =0$ and $F:=F(S)∩EP(f)∩ A − 1 0≠∅$. For arbitrary $x 1 ∈C$, $C 1 =C$, a sequence ${ x n }$ is generated by

${ z n = J − 1 ( J x n − λ n A x n ) , u n = T r n z n , y n = J − 1 ( α n J x n + β n J w n + γ n J u n ) , w n ∈ S n x n , C n + 1 = { v ∈ C n : ϕ ( v , y n ) ≤ ϕ ( v , z n ) ≤ ϕ ( v , x n ) + K n } , x n + 1 = Π C n + 1 x 1 , n ∈ N ,$
(3.33)

where $K n = ν n sup q ∈ F ψ(ϕ(q, x n ))+ μ n$. Assume that the control sequences ${ α n }$, ${ β n }$, ${ γ n }$, ${ λ n }$ and ${ r n }$ satisfy the following conditions:

1. 1.

${ α n }$, ${ β n }$ and ${ γ n }$ are sequences in $(0,1)$ such that $α n + β n + γ n =1$, $lim inf n → ∞ α n β n >0$,

2. 2.

${ λ n }⊂[a,b]$ for some a, b with $0 and $1 c$ is the 2-uniformly convex constant of E,

3. 3.

${ r n }⊂[d,∞)$ for some $d>0$,

then ${ x n }$ converges strongly to $Π F x 1$.

Corollary 3.5 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from $C×C$ to satisfying conditions (A1)-(A4), and let $A:C→ E ∗$ be a strongly monotone mapping with constant k,Lipschitz with constant $L>0$ satisfying $∥Ay∥≤∥Ay−Au∥$ for all $y∈C$ and $u∈VI(A,C)≠∅$. Let $S:C→ 2 C$ be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences $ν n$, $μ n$ with $ν n →0$, $μ n →0$ as $n→∞$ and a strictly increasing continuous function $ψ: R + → R +$ with $ψ(0)=0$. Assume that S is uniformly asymptotically regular on C with $μ 1 =0$ and $F:=F(S)∩EP(f)∩VI(A,C)≠∅$. For arbitrary $x 1 ∈C$, $C 1 =C$, generate a sequence ${ x n }$ by

${ z n = Π C J − 1 ( J x n − λ n A x n ) , u n = T r n x n , y n = J − 1 ( α n J x n + β n J w n + γ n J u n ) , w n ∈ S n x n , C n + 1 = { v ∈ C n : ϕ ( v , y n ) ≤ ϕ ( v , z n ) ≤ ϕ ( v , x n ) + K n } , x n + 1 = Π C n + 1 x 1 , n ∈ N ,$
(3.34)

where $K n = ν n sup q ∈ F ψ(ϕ(q, x n ))+ μ n$. Assume that the control sequences ${ α n }$, ${ β n }$, ${ γ n }$, ${ λ n }$ and ${ r n }$ satisfy the following conditions:

1. 1.

${ α n }$, ${ β n }$ and ${ γ n }$ are sequences in $(0,1)$ such that $α n + β n + γ n =1$, $lim inf n → ∞ α n β n >0$,

2. 2.

${ λ n }⊂[a,b]$ for some a, b with $0 and $1 c$ is the 2-uniformly convex constant of E,

3. 3.

${ r n }⊂[d,∞)$ for some $d>0$,

then ${ x n }$ converges strongly to $Π F x 1$.

Let F be a Fréchet differentiable functional in a Banach space E and F be the gradient of F, denote $( ∇ F ) − 1 0={x∈E:F(x)= min y ∈ E F(y)}$. Baillon and Haddad  proved the following lemma.

Let E be a Banach space. Let F be a continuously Fréchet differentiable convex functional on E and F be the gradient of F. If F is $1 α$-Lipschitz continuous, then F is an α-inverse strongly monotone mapping.

We replace A in Theorem 3.1 by F, then we can obtain the following corollary.

Corollary 3.7 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from $C×C$ to satisfying conditions (A1)-(A4). Let F be a continuously Fréchet differentiable convex functional on E and F be $1 α$-Lipschitz continuous. Let $S:C→ 2 C$ be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences $ν n$, $μ n$ with $ν n →0$, $μ n →0$ as $n→∞$ and a strictly increasing continuous function $ψ: R + → R +$ with $ψ(0)=0$. Assume that S is uniformly asymptotically regular on C with $μ 1 =0$ and $F:=F(S)∩F(T)∩EP(f)∩ A − 1 0≠∅$. For an initial point $x 1 ∈E$, $C 1 =C$, define the sequence ${ x n }$ by

${ z n = J − 1 ( J x n − λ n ∇ F x n ) , u n = T r n x n , y n = J − 1 ( α n J x n + β n J w n + γ n J u n ) , w n ∈ S n x n , C n + 1 = { v ∈ C n : ϕ ( v , y n ) ≤ ϕ ( v , z n ) ≤ ϕ ( v , x n ) + K n } , x n + 1 = Π C n + 1 x 1 , n ∈ N ,$
(3.35)

where $μ n =sup{ μ n S , μ n T }$, $ν n =sup{ ν n S , ν n T }$, $ψ=sup{ ψ S , ψ T }$, $k n = ν n sup q ∈ F ψ(ϕ(q, x n ))+ μ n$.

Assume that the control sequences ${ α n }$, ${ β n }$, ${ γ n }$, ${ λ n }$ and ${ r n }$ satisfy the following conditions:

1. 1.

${ α n }$, ${ β n }$ and ${ γ n }$ are sequences in $(0,1)$ such that $α n + β n + γ n =1$, $lim inf n → ∞ α n β n >0$ and $lim inf n → ∞ α n γ n >0$,

2. 2.

${ λ n }⊂[a,b]$ for some a, b with $0 and the 2-uniformly convex constant $1 c$ of E,

3. 3.

${ r n }⊂[d,∞)$ for some $d>0$,

then ${ x n }$ converges strongly to $Π F x 1$.

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## Acknowledgements

This work was supported by Thaksin University Research Fund. Moreover, the author also would like to thank Faculty of Science, Thaksin University.

## Author information

Correspondence to Siwaporn Saewan.

### Competing interests

The author declares that they have no competing interests.

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