Skip to main content

Advertisement

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

Article metrics

  • 1124 Accesses

  • 1 Citations

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

AxAy,xy0,x,yC;

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

AxAy,xyα A x A y 2 ,x,yC;

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

AxAyLxy,x,yC.

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

AxAy 1 α xy,x,yC.

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 xE such that

0Ax,
(1.1)

a point xE is called a zero point of A. Denote by A 1 0 the set of all points xE such that 0Ax. 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,yC.
(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 xE satisfying Ax=0.

The value of x E at xE 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 xE. 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

ϕ(y,x)= y 2 2y,Jx+ x 2 for x,yE,
(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,yE.
(1.4)

Alber [1] introduced that the generalized projection Π C :EC is a map that assigns to an arbitrary point xE 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 [2] 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 =xC and

x n + 1 = Π C J 1 (J x n λ n A x n ),n1,
(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 [3] 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, [46].

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

ϕ(Tx,Ty)ϕ(x,y),x,yC.

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

ϕ(p,Tx)ϕ(p,x),xC,pF(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 ,n1,xC,pF(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 qC, n1 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 pC 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 xC, pF(S),

ϕ(p, w n )ϕ(p,x)+ ν n φ ( ϕ ( p , x ) ) + μ n ,n1, 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 wSx.

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. [9] 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×CR be a bifunction, the equilibrium problem is to find xC such that

f(x,y)0,yC.
(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, [1221].

For a mapping A:C E , let f(x,y)=Ax,yx for all x,yC. Then xEP(f) if and only if Tx,yx0 for all yC; 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,yE with x=y=1 and xy. Let U={xE: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,yU. It is also called uniformly smooth if the limit exists uniformly for all x,yU. 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 p2. 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 xE 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 [22] 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 [24] and Xu [25])

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

xy 2 c 2 JxJy,

where J is the normalized duality mapping of E and 0<c1.

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

Lemma 2.4 (Beauzamy [24] and Zalinescu [26])

If E is a p-uniformly convex Banach space, and let p be a given real number with p2, then, for all x,yE, J x J p (x) and J y J p (y),

xy, 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 [27])

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 [1])

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

x 0 y,JxJ x 0 0,yC.

Lemma 2.7 (Alber [1])

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

ϕ(y, Π C x)+ϕ( Π C x,x)ϕ(y,x),yC.

Lemma 2.8 (Chang et al. [9])

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×CR, let us assume that f satisfies the following conditions:

  1. (A1)

    f(x,x)=0 for all xC;

  2. (A2)

    f is monotone, i.e., f(x,y)+f(y,x)0 for all x,yC;

  3. (A3)

    for each x,y,zC,

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

    for each xC, yf(x,y) is convex and lower semi-continuous.

Lemma 2.9 (Blum and Oettli [28])

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 xE. Then there exists zC such that

f(z,y)+ 1 r yz,JzJx0,yC.

Lemma 2.10 (Takahashi and Zembayashi [11])

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 xE, define a mapping T r :EC 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 [29], that is, for all x,yE,

    T r x T r y,J T r xJ T r y T r x T r y,JxJy;
  3. (3)

    F( T r )=EP(f);

  4. (4)

    EP(f) is closed and convex.

Lemma 2.11 (Takahashi and Zembayashi [11])

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 xE and qF( 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,yC, the mapping h of [0,1] into E , defined by h(t)=A(tx+(1t)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 vC, that is,

N C (v)= { x E : v y , x 0 , y C } .
(2.1)

Theorem 2.12 (Rockafellar [30])

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 BE× 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 [31])

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 [31])

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 [1]:

V ( x , x ) = x 2 2 x , x + x 2 ,xE, x E ,
(2.4)

that is, V(x, x )=ϕ(x, J 1 ( x )).

Lemma 2.15 (Alber [1])

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 ) ,xE, x , y E .

Lemma 2.16 (Beauzamy [24] and Xu [25])

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

xy 2 c 2 JxJy,

where J is the normalized duality mapping of E and 0<c1.

Lemma 2.17 (Cho et al. [32])

Let E be a uniformly convex Banach space, and let B r (0)={xE:xr} 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 [33])

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 xE and y n A x n with y n y E , then xD(A) and yAx.

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<a<b< c 2 α 2 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 nN. Since C 1 =C is closed and convex. Suppose that C n is closed and convex for all nN. For any v C n , we know that ϕ(v, y n )ϕ(v, x n )+ K n is equivalent to

2v,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 nN.

We show by induction that F C n for all nN. It is obvious that FC= C 1 . Suppose that F C n where nN. Let qF, 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 nN 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 ),n1,
(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,n1 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 pC 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 pF:=F(S) A 1 0EP(f).

(a) We show that pF(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

ϕ(q, x n )ϕ(q, y n )0as n.
(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 pF(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 pEP(f).

From x n , y n 0 and K n 0 as n and applying (3.7) for any qF, 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 n1, 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,yC.

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 yC. For any 0<t<1, define y t =ty+(1t)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)+(1t)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 yC and so pEP(f). Hence, by (a), (b) and (c), that is, pF(S) A 1 0EP(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 z0 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 AyAyAu for all yC and uVI(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. [7])

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 AyAyAu for all yC and uVI(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<a<b< c 2 α 2 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 qF 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 pVI(A,C).

Define BE× E by Theorem 2.14, B is maximal monotone and B 1 0=VI(A,C). Let (z,w)G(B). Since wBz=Az+ N C (z), we get wAz N C (z).

From z n C, we have

z z n ,wAz0.
(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 zp,w0. By the maximality of B, we have p B 1 0, that is, pVI(A,C). From Theorem 3.1, we have pF(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,

AxAyLxy,x,yD(A),

which implies that

1 L AxAyxy,x,yD(A).

It follows that

AxAy,xyk 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<a<b< c 2 k 2 L 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 AyAyAu for all yC and uVI(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<a<b< c 2 k 2 L 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={xE:F(x)= min y E F(y)}. Baillon and Haddad [34] proved the following lemma.

Lemma 3.6 (Baillon and Haddad [34])

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<a<b< c 2 α 2 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 .

References

  1. 1.

    Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos A. Dekker, New York; 1996:15–50.

  2. 2.

    Iiduka H, Takahashi W: Weak convergence of a projection algorithm for variational inequalities in a Banach space. J. Math. Anal. Appl. 2008, 339: 668–679. 10.1016/j.jmaa.2007.07.019

  3. 3.

    Iiduka H, Takahashi W: Strong convergence studied by a hybrid type method for monotone operators in a Banach space. Nonlinear Anal. 2008, 68: 3679–3688. 10.1016/j.na.2007.04.010

  4. 4.

    Ceng LC, Latif A, Yao JC: On solutions of system of variational inequalities and fixed point problems in Banach spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 176 10.1186/1687-1812-2013-176

  5. 5.

    Kangtunyakarn A: A new mapping for finding a common element of the sets of fixed points of two finite families of nonexpansive and strictly pseudo-contractive mappings and two sets of variational inequalities in uniformly convex and 2-smooth Banach spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 157 10.1186/1687-1812-2013-157

  6. 6.

    Kassay G, Reich S, Sabach S: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 2011, 21: 1319–1344. 10.1137/110820002

  7. 7.

    Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011

  8. 8.

    Zhou H, Gao G, Tan B: Convergence theorems of a modified hybrid algorithm for a family of quasi- ϕ -asymptotically nonexpansive mappings. J. Appl. Math. Comput. 2010, 32(2):453–464. 10.1007/s12190-009-0263-4

  9. 9.

    Chang S-S, Wang L, Tang Y-K, Zhao Y-H, Ma Z-L: Strong convergence theorems of nonlinear operator equations for countable family of multi-valued total quasi- ϕ -asymptotically nonexpansive mappings with applications. Fixed Point Theory Appl. 2012., 2012: Article ID 69 10.1186/1687-1812-2012-69

  10. 10.

    Takahashi W, Zembayashi K: Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings. Fixed Point Theory Appl. 2008., 2008: Article ID 528476

  11. 11.

    Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 2009, 70: 45–57. 10.1016/j.na.2007.11.031

  12. 12.

    Cholamjiak P: Strong convergence of projection algorithms for a family of relatively quasi-nonexpansive mappings and an equilibrium problem in Banach spaces. Optimization 2011, 60(4):495–507. 10.1080/02331930903477192

  13. 13.

    Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.

  14. 14.

    Iusema AN, Sosa W: Iterative algorithms for equilibrium problems. Optimization 2003, 52(3):301–316. 10.1080/0233193031000120039

  15. 15.

    Saewan S, Kumam P, Wattanawitoon K: Convergence theorem based on a new hybrid projection method for finding a common solution of generalized equilibrium and variational inequality problems in Banach spaces. Abstr. Appl. Anal. 2010., 2010: Article ID 734126

  16. 16.

    Ceng LC, Al-Homidan S, Ansari QH, Yao J-C: An iterative scheme for equilibrium problems and fixed points problems of strict pseudo-contraction mappings. J. Comput. Appl. Math. 2009, 223(2):967–974. 10.1016/j.cam.2008.03.032

  17. 17.

    Zeng L-C, Ansari QH, Shyu DS, Yao J-C: Strong and weak convergence theorems for common solutions of generalized equilibrium problems and zeros of maximal monotone operators. Fixed Point Theory Appl. 2010., 2010: Article ID 590278

  18. 18.

    Ceng LC, Ansari QH, Yao JC: Strong and weak convergence theorems for asymptotically strict pseudocontractive mappings in intermediate sense. J. Nonlinear Convex Anal. 2010, 11(2):283–308.

  19. 19.

    Ceng L-C, Ansari QH, Yao J-C: Hybrid proximal-type and hybrid shrinking projection algorithms for equilibrium problems, maximal monotone operators and relatively nonexpansive mappings. Numer. Funct. Anal. Optim. 2010, 31(7):763–797. 10.1080/01630563.2010.496697

  20. 20.

    Zeng LC, Al-Homidan S, Ansari QH: Hybrid proximal-type algorithms for generalized equilibrium problems, maximal monotone operators and relatively nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 973028

  21. 21.

    Zeng L-C, Ansari QH, Schaible S, Yao J-C: Hybrid viscosity approximate method for zeros of m -accretive operators in Banach spaces. Numer. Funct. Anal. Optim. 2012, 33(2):142–165. 10.1080/01630563.2011.594197

  22. 22.

    Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht; 1990.

  23. 23.

    Reich S: Geometry of Banach spaces, duality mappings and nonlinear problems. Bull. Am. Math. Soc. 1992, 26: 367–370. 10.1090/S0273-0979-1992-00287-2

  24. 24.

    Beauzamy B: Introduction to Banach Spaces and Their Geometry. 2nd edition. North-Holland, Amsterdam; 1985.

  25. 25.

    Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-K

  26. 26.

    Zalinescu C: On uniformly convex functions. J. Math. Anal. Appl. 1983, 95: 344–374. 10.1016/0022-247X(83)90112-9

  27. 27.

    Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 2002, 13: 938–945. 10.1137/S105262340139611X

  28. 28.

    Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.

  29. 29.

    Kohsaka F, Takahashi W: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaced. SIAM J. Optim. 2008, 19(2):824–835. 10.1137/070688717

  30. 30.

    Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 1970, 149: 75–88. 10.1090/S0002-9947-1970-0282272-5

  31. 31.

    Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama; 2000.

  32. 32.

    Cho YJ, Zhou HY, Guo G: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. 2004, 47: 707–717. 10.1016/S0898-1221(04)90058-2

  33. 33.

    Pascali D, Sburlan S: Nonlinear Mappings of Monotone Type. Editura Academiae Bucaresti, Romania; 1978.

  34. 34.

    Baillon JB, Haddad G: Quelques propriétés des opérateurs andle-bornés et n -cycliquement monotones. Isr. J. Math. 1977, 26: 137–150. 10.1007/BF03007664

Download references

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.

Additional information

Competing interests

The author declares that they have no competing interests.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Saewan, S. Strong convergence theorem for total quasi-ϕ-asymptotically nonexpansive mappings in a Banach space. Fixed Point Theory Appl 2013, 297 (2013) doi:10.1186/1687-1812-2013-297

Download citation

Keywords

  • total quasi-ϕ-asymptotically nonexpansive multi-valued mappings
  • hybrid scheme
  • equilibrium problem
  • variational inequality problems
  • inverse-strongly monotone operator