# An Iteration to a Common Point of Solution of Variational Inequality and Fixed Point-Problems in Banach Spaces

Authors: H. Zegeye; N. Shahzad
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce an iterative process which converges strongly to a common point of solution of variational inequality problem for a monotone mapping and fixed point of uniformly Lipschitzian relatively asymptotically nonexpansive mapping in Banach spaces. As a consequence, we provide a scheme that converges strongly to a common zero of finite family of monotone mappings under suitable conditions. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.


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## Body

## 1. Introduction

LetE be a smooth Banach space. Throughout this paper, we denote by ϕ:E×E→ℝ the function defined by
(1.1)ϕ(y,x)=‖y‖2-2〈y,Jx〉+‖x‖2,forx,y∈E,
which was studied by Alber [1], Kamimura and Takahashi [2], and Reich [3], where J is the normalized duality mapping from E to 2E* defined by
(1.2)Jx∶={f*∈E*:〈x,f*〉=‖x‖2=‖f*‖2},
where 〈·,·〉 denotes the duality pairing. It is well known that if E is smooth, then J is single-valued, and, if E has uniformly Gâteaux differentiable norm, then J is uniformly continuous on bounded subsets of E. Moreover, if E is a reflexive and strictly convex Banach space with a strictly convex dual, then J-1 is single valued, one-to-one, surjective, and it is the duality mapping from E* into E, and thus JJ-1=IE* and J-1J=IE (see [4]).It is obvious from the definition of the functionϕ that
(1.3)(‖x‖-‖y‖)2≤ϕ(x,y)≤(‖x‖+‖y‖)2forx,y∈E,
and, in a Hilbert space H, (1.1) reduces to ϕ(x,y)=∥x-y∥2, for x,y∈H.LetE be a reflexive, strictly convex, and smooth Banach space, and let C be a nonempty closed and convex subset of E. The generalized projection mapping, introduced by Alber [1], is a mapping ΠC:E→C that assigns an arbitrary point x∈E to the minimizer, x̅, of ϕ(·,x) over C, that is, ΠCx=x̅, where x̅ is the solution to the minimization problem
(1.4)ϕ(x̅,x)=min⁡{ϕ(y,x),y∈C}.
Let E be a real Banach space with dual E*. A mapping A:D(A)⊂E→E* is said to be monotoneif, for each x,y∈D(A), the following inequality holds:
(1.5)〈x-y,Ax-Ay〉≥0.A is said to be γ-inverse strongly monotone if there exists positive real number γ such that
(1.6)〈x-y,Ax-Ay〉≥γ‖Ax-Ay‖2,∀x,y∈K.
If A is γ-inverse strongly monotone, then it is Lipschitz continuous with constant 1/γ, that is, ∥Ax-Ay∥≤(1/γ)∥x-y∥, for all x,y∈D(A), and it is called strongly monotone if there exists k>0 such that, for all x,y∈D(A),
(1.7)〈x-y,Ax-Ay〉≥k‖x-y‖2.
Clearly, the class of monotone mappings include the class of strongly monotone and γ-inverse strongly monotone mappings.Suppose thatA is monotone mapping from C into E*. The variational inequality problem is formulated as finding a point u∈C such that 〈v-u,Au〉≥0, for all v∈C. The set of solutions of the variational inequality problems is denoted by VI(C,A).The notion of monotone mappings was introduced by Zarantonello [5], Minty [6], and Kacurovskii [7] in Hilbert spaces. Monotonicity conditions in the context of variational methods for nonlinear operator equations were also used by Vainberg and Kachurovisky [8]. Variational inequalities were initially studied by Stampacchia [9, 10] and ever since have been widely studied in general Banach spaces (see, e.g., [2, 11–13]). Such a problem is connected with the convex minimization problem, the complementarity problem, the problem of finding point u∈C satisfying 0∈Au.IfE=H, a Hilbert space, one method of solving a point u∈VI(C,A) is the projection algorithm which starts with any point x1=x∈C and updates iteratively as xn+1 according to the formula
(1.8)xn+1=PC(xn-αnAxn),foranyn≥1,
where PC is the metric projection from H onto C and {αn} is a sequence of positive real numbers. In the case that A is γ-inverse strongly monotone, Iiduka et al. [14] proved that the sequence {xn} generated by (3.35) converges weakly to some element of VI(C,A).In the case thatE is a 2-uniformly convex and uniformly smooth Banach space, Iiduka and Takahashi [15] introduced the following iteration scheme for finding a solution of the variational inequality problem for an inverse strongly monotone operator A:
(1.9)xn+1=ΠCJ-1(Jxn-αnAxn),foranyn≥1,
where ΠC is the generalized projection from E onto C, J is the normalized duality mapping from E into E*, and {αn} is a sequence of positive real numbers. They proved that the sequence {xn} generated by (1.9) converges weakly to some element of VI(C,A) provided that A satisfies ∥Ax∥≤∥Ax-Ap∥, for x∈C and p∈VI(C,A).It is worth to mention that the convergence isweak convergence.To obtain strong convergence, whenE=H, a Hilbert space and A is γ-inverse strongly monotone; Iiduka et al. [14] studied the following iterative scheme:
(1.10)x0∈C,chosenarbitrary,yn=PC(xn-αnAxn),Cn={z∈C:‖yn-z‖≤‖xn-z‖},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qn(x0),n≥1,forn≥1,
where {αn} is a sequence in [0,2γ]. They proved that the sequence {xn} generated by (1.10) converges strongly to PVI(C,A)(x0), where PVI(C,A) is the metric projection from H onto VI(C,A) provided that A satisfies ∥Ax∥≤∥Ax-Ap∥, for x∈C and p∈VI(C,A).In the case thatE is 2-uniformly convex and uniformly smooth Banach space, Iiduka and Takahashi [11] studied the following iterative scheme for a variational inequality problem for γ-inverse strongly monotone mapping:
(1.11)x0∈K,chosenarbitrary,yn=ΠCJ-1(Jxn-αnAxn),Cn={z∈E:ϕ(z,yn)≤ϕ(z,xn)},Qn={z∈E:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qn(x0),n≥1,forn≥1,
where ΠCn∩Qn is the generalized projection from E onto Cn∩Qn, J is the normalized duality mapping from E into E*, and {αn} is a positive real sequence satisfying certain condition. Then, they proved that the sequence {xn} converges strongly to an element of VI(C,A) provided that VI(C,A)≠∅ and A satisfies ∥Ax∥≤∥Ax-Ap∥ for all x∈C and p∈VI(C,A).Remark 1.1.
We remark that the computation ofxn+1 in Algorithms (1.10) and (1.11) is not simple because of the involvement of computation of Cn+1 from Cn and Qn for each n≥1.LetT be a mapping from C into itself. We denote by F(T) the fixed points set of T. A point p in C is said to be an asymptotic fixed point ofT (see [3]) if C contains a sequence {xn} which converges weakly to p such that limn→∞∥xn-Txn∥=0. The set of asymptotic fixed points of T will be denoted by F̂(T). A mapping T from C into itself is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for each x,y∈C and is called relatively nonexpansive if (R1) F(T)≠∅; (R2) ϕ(p,Tx)≤ϕ(p,x) for x∈C and (R3) F(T)=F̂(T). T is called relatively quasi-nonexpansive if F(T)≠∅ and ϕ(p,Tx)≤ϕ(p,x) for all x∈C, and p∈F(T).A mappingT from C into itself is said to be asymptotically nonexpansiveif there exists {kn}⊂[1,∞) such that kn→1 and ∥Tnx-Tny∥≤kn∥x-y∥ for each x,y∈C and is called relatively asymptotically nonexpansive if there exists {kn}⊂[1,∞) such that (N1) F(T)≠∅; (N2) ϕ(p,Tnx)≤knϕ(p,x) for x∈C and p∈F(T), and (N3) F(T)=F̂(T), where kn→1 as n→∞. A-self mapping on C is called uniformlyL-Lipschitzianif there exists L>0 such that ∥Tnx-Tny∥≤L∥x-y∥ for all x,y∈C. T is called closed if xn→x and Txn→y, then Tx=y.Clearly, we note that the class of relatively nonexpansive mappings is contained in a class of relatively asymptotically nonexpansive mappings but the converse is not true. Now, we give an example of relatively asymptotically nonexpansive mapping which is not relatively nonexpansive.Example 1.2 (see [16]).
LetX=lp, where 1<p<∞, and C={x=(x1,x2,…)∈X;xn≥0}. Then C is closed and convex subset of X. Note that C is not bounded. Obviously, X is uniformly convex and uniformly smooth. Let {λn} and{λ̅n} be sequences of real numbers satisfying the following properties:(i)
0<λn<1, λ̅n>1, λn↑1, and λ̅n↓1,(ii)
λn+1λ̅n=1 and λj+1λ̅n+j<1 for all n and j (e.g., λn=1-1/(n+1), λ̅n=1+1/(n+1)). Then, the map T:C→C defined by
(1.12)Tx∶=(0,λ̅1|sin⁡x1|,λ2x2,λ̅2x3,λ3x4,λ̅3x5,…),for all x=(x1,x2,…)∈C, is uniformly Lipschitzian which is relatively asymptotically nonexpansive but not relatively nonexpansive (see [16] for the details). Note also that F(T)={0}.In 2005, Matsushita and Takahashi [17] proposed the following hybrid iteration method with generalized projection for relatively nonexpansive mapping T in a Banach space E:
(1.13)x0∈C,chosenarbitrary,yn=J-1(αnJxn+(1-αn)JTxn),Cn={z∈C:ϕ(z,yn)≤ϕ(z,xn)},Qn={z∈C;〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qn(x0),n≥1.
They proved that, if the sequence {αn} is bounded above from one, then the sequence {xn} generated by (1.13) converges strongly to ΠF(T)x0.Recently, many authors have considered the problem of finding a common element of the fixed-point set of relatively nonexpansive mapping and the solution set of variational inequality problem forγ-inverse monotone mapping (see, e.g., [12, 13, 18–20]).In [21], Iiduka and Takahashi studied the following iterative scheme for a common point of solution of a variational inequality problem for γ-inverse strongly monotone mapping A and fixed point of nonexpansive mapping T in a Hilbert space H:
(1.14)x1=x∈C,xn+1=αnx+(1-αn)SPC(xn-λnAxn),n≥1,
where {αn} is sequences satisfying certain condition. They proved that the sequence {xn} converges strongly to an element of F∶=F(S)∩VI(C,A) provided that F≠∅.In the case thatE is a Banach space more general than Hilbert spaces, Zegeye et al. [12] studied the following iterative scheme for a common point of solution of a variational inequality problem for γ-inverse strongly monotone mapping A and fixed point of a closed relatively quasi-nonexpansive mapping T in a 2-uniformly convex and uniformly smooth Banach space E:
(1.15)C1=C,chosenarbitrary,zn=ΠC(xn-λnAxn),yn=J-1(βJxn+(1-β)JTzn),Cn+1={z∈Cn:ϕ(z,yn)≤ϕ(z,xn)},xn+1=ΠCn+1(x0),n≥1,
where {λn} is sequences satisfying certain condition. They proved that the sequence {xn} converges strongly to an element of F:=F(S)∩VI(C,A)≠∅ provided that F≠∅ and A satisfies ∥Ax∥≤∥Ax-Ap∥ for all x∈C and p∈F.Furthermore, Zegeye and Shahzad [22] studied the following iterative scheme for common point of solution of a variational inequality problem for γ-inverse strongly monotone mapping A and fixed point of a relatively asymptotically nonexpansive mapping on a closed convex and bounded set C which is a subset of a real Hilbert space H:
(1.16)C1=C,chosenarbitrary,zn=PC(xn-λnAxn),yn=αnxn+(1-αn)Snzn,Cn+1={z∈Cn:‖z-un‖2≤‖z-xn‖2+θn},xn+1=PCn+1(x0),n≥1,
where PCn is the metric projection from H into Cn and θn=(1-αn)(kn2-1)(diam(C))2 and {αn},{λn} are sequences satisfying certain condition. Then, they proved that the sequence {xn} converges strongly to an element of F∶=F(S)∩VI(C,A)≠∅ provided that F≠∅ and A satisfies ∥Ax∥≤∥Ax-Ap∥ for all x∈C and p∈F.Remark 1.3.
We again remark that the computation ofxn+1 in Algorithms (1.13), (1.15), and (1.16) is not simple because of the involvement of computation of Cn+1 from Cn for each n≥1.
It is our purpose in this paper to introduce an iterative scheme{xn} which converges strongly to a common point of solution of variational inequality problem for a monotone operator A:C→E* satisfying appropriate conditions, for some nonempty closed convex subset C of a Banach space E and fixed points of uniformly L-Lipschitzian relatively asymptotically nonexpansive mapping in Banach spaces. As a consequence, we provide a scheme which converges strongly to a common zero of finite family of monotone mappings. Our scheme does not involve computation of Cn+1 from Cn or Qn, for each n≥1. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.


## 2. Preliminaries

LetE be a normed linear space with dimE≥2. The modulus of smoothness of E is the function ρE:[0,∞)→[0,∞) defined by
(2.1)ρE(τ)∶=sup⁡{‖x+y‖+‖x-y‖2-1:‖x‖=1;‖y‖=τ}.
The space E is said to be smooth if ρE(τ)>0, for all τ>0, and E is called uniformly smooth if and only if limt→0+(ρE(t)/t)=0.Themodulus of convexity of E is the function δE:(0,2]→[0,1] defined by
(2.2)δE(ϵ)∶=inf⁡{1-‖x+y2‖:‖x‖=‖y‖=1;ϵ=‖x-y‖}.E is called uniformly convex if and only if δE(ϵ)>0, for every ϵ∈(0,2].In the sequel, we will need the following results.Lemma 2.1 (see [23]).
LetC be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space  E. If  A:C→E* is continuous monotone mapping, then  VI(C,A) is closed and convex.Lemma 2.2.
LetC be a closed convex subset of a uniformly convex and smooth Banach space  E, and let  S be continuous relatively asymptotically nonexpansive mapping from  C into itself. Then,  F(S) is closed and convex.Lemma 2.3 (see [1]).
LetK be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space  E, and let  x∈E. Then, for all  y∈K,
(2.3)ϕ(y,ΠKx)+ϕ(ΠKx,x)≤ϕ(y,x).Lemma 2.4 (see [2]).
LetE be a real smooth and uniformly convex Banach space, and let  {xn} and  {yn} be two sequences of  E. If either  {xn} or  {yn} is bounded and  ϕ(xn,yn)→0 as  n→∞, then  xn-yn→0, as  n→∞.We make use of the functionV:E×E*→ℝ defined by
(2.4)V(x,x*)=‖x‖2-2〈x,x*〉+‖x‖2,∀x∈E,x*∈E,
studied by Alber [1]. That is, V(x,y)=ϕ(x,J-1x*) for all x∈E and x*∈E*. We know the following lemma.Lemma 2.5 (see [1]).
LetE be reflexive strictly convex and smooth Banach space with  E* as its dual. Then,
(2.5)V(x,x*)+2〈J-1x*-x,y*〉≤V(x,x*+y*),
for all x∈E and x*,y*∈E*.Lemma 2.6 (see [1]).
LetC be a convex subset of a real smooth Banach space  E. Let  x∈E. Then  x0=ΠCx if and only if
(2.6)〈z-x0,Jx-Jx0〉≤0,∀z∈C.Lemma 2.7 (see [12]).
LetE be a uniformly convex Banach space and  BR(0) a closed ball of  E. Then, there exists a continuous strictly increasing convex function  g:[0,∞)→[0,∞) with  g(0)=0 such that
(2.7)‖α1x1+α2x2+α3x3‖2≤αi‖x1‖2+α2‖x2‖2+α3‖x3‖2-αiαjg(‖xi-xj‖),
for αi∈(0,1) such that α1+α2+α3=1 and xi∈BR(0)∶={x∈E:∥x∥≤R}, for i=1,2,3.LetE be a smooth and strictly convex Banach space, C a nonempty closed convex subset of E, and A:C→E* a monotone operator satisfying
(2.8)D(A)⊆C⊆J-1(∩r>0R(J+rA)),
for r>0. Then, we can define the resolvent Qr:C→D(A) of A by
(2.9)Qrx∶={z∈D(A):Jx∈Jz+rAz},∀x∈C.
In other words, Qrx=(J+rA)-1Jx for x∈C. We know that Qrx is single-valued mapping from C into D(A), for all x∈C and r>0 and F(Qr)=A-1(0), where F(Qr) is the set of fixed points of Qr (see, [4]).Lemma 2.8 (see [24]).
LetE be a smooth and strictly convex Banach space,  C a nonempty closed convex subset of  E, and  A⊂E×E* a monotone operator satisfying (2.8) and  A-1(0) is nonempty. Let  Qr be the resolvent of  A. Then, for each  r>0,
(2.10)ϕ(u,Qrx)+ϕ(Qrx,x)≤ϕ(u,x)
for all u∈A-1(0) and x∈C, that is, Qr is relatively nonexpansive.Lemma 2.9 (see [25]).
Let{an} be a sequence of nonnegative real numbers satisfying the following relation:
(2.11)an+1≤(1-βn)an+βnδn,n≥n0,
where {βn}⊂(0,1) and {δn}⊂R satisfying the following conditions: limn→∞βn=0,∑n=1∞βn=∞, and limsupn→∞δn≤0. Then, limn→∞an=0.Lemma 2.10 (see [26]).
Let{an} be sequences of real numbers such that there exists a subsequence  {ni} of  {n} such that  ani<ani+1 for all  i∈N. Then, there exists a nondecreasing sequence  {mk}⊂N such that  mk→∞, and the following properties are satisfied by all (sufficiently large) numbers  k∈N:
(2.12)amk≤amk+1,ak≤amk+1.
In fact, mk=max{j≤k:aj<aj+1}.


## 3. Main Result

We note that, as it is mentioned in [27], if C is a subset of a real Banach space E and A:C→E* is a mapping satisfying ∥Ax∥≤∥Ax-Ap∥,for all x∈C and p∈VI(C,A), then
(3.1)VI⁡(C,A)=A-1(0)={p∈C:Ap=0}.In fact, clearly,A-1(0)⊆VI(C,A). Now, we show that VI(C,A)⊆A-1(0). Let p∈VI(C,A), then we have by hypothesis that ∥Ap∥≤∥Ap-Ap∥=0 which implies that p∈A-1(0). Hence, VI(C,A)⊆A-1(0). Therefore, VI(C,A)=A-1(0). Now we prove the main theorem of our paper.Theorem 3.1.
LetC be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space E. Let A:C→E* be a monotone mapping satisfying (2.8) and ∥Ax∥≤∥Ax-Ap∥,for all x∈C and p∈VI(C,A). Let T:C→C be a uniformly L-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {kn}. Assume that F∶=VI(C,A)∩F(S) is nonempty. Let Qr be the resolvent of A and {xn} a sequence generated by
(3.2)x0=w∈C,chosenarbitrarily,yn=ΠCJ-1(αnJw+(1-αn)Jxn),xn+1=J-1(βnJxn+γnJTnyn+θnJQryn),
where αn∈(0,1) such that limn→∞αn=0,limn→∞((kn-1)/αn)=0,∑n=1∞αn=∞, {βn},{γn},{θn}⊂[c,d]⊂(0,1) such that βn+γn+θn=1. Then, {xn} converges strongly to an element of F.Proof.
Letp∶=ΠFw. Then, from (3.2), Lemma 2.3, and property of ϕ, we get that
(3.3)ϕ(p,yn)=ϕ(p,ΠCJ-1(αnJw+(1-αn)Jxn))≤ϕ(p,J-1(αnJw+(1-αn)Jxn))=‖p‖2-2〈p,αnJw+(1-αn)Jxn〉+‖αnJw+(1-αn)Jxn‖2≤‖p‖2-2αn〈p,Jw〉-2(1-αn)〈p,Jxn〉+αn‖Jw‖2+(1-αn)‖Jxn‖2=αnϕ(p,w)+(1-αn)ϕ(p,xn).
Now, from (3.2) and relatively asymptotically nonexpansiveness of T, relatively nonexpansiveness of Qr, property of ϕ, and (3.3), we get that
(3.4)ϕ(p,xn+1)=ϕ(p,J-1(βnJxn+γnJTnyn+θnJQryn))≤βnϕ(p,xn)+γnϕ(p,Tnyn)+θnϕ(p,Qryn)≤βnϕ(p,xn)+γnknϕ(p,yn)+θnϕ(p,yn)≤βnϕ(p,xn)+(γnkn+θn)ϕ(p,yn)≤βnϕ(p,xn)+(γnkn+θn)[αnϕ(p,w)+(1-αn)ϕ(p,xn)]≤(γnkn+θn)αnϕ(p,w)+[βn+(γnkn+θn)(1-αn)]ϕ(p,xn)≤(γnkn+θn)αnϕ(p,w)+[(1-αn(γnkn+θn))+γn(kn-1)]×ϕ(p,xn)≤cnϕ(p,w)+[1-(1-ϵ)cn]ϕ(p,xn),
where cn=αn(γnkn+θn), since there exists N0>0 such that γn(kn-1)/αn≤ϵ(γnkn+θn) for all n≥N0 and for some ϵ>0 satisfying (1-ϵ)cn≤1. Thus, by induction,
(3.5)ϕ(p,xn+1)≤max⁡⁡{ϕ(p,x0),(1-ϵ)-1ϕ(p,w)},∀n≥N0
which implies that {xn}, and hence {yn} is bounded. Now, let zn=J-1(αnJw+(1-αn)Jxn). Then we have that yn=ΠCzn. Using Lemmas 2.3, 2.5, and property of ϕ, we obtain that
(3.6)ϕ(p,yn)≤ϕ(p,zn)=V(p,Jzn)≤V(p,Jzn-αn(Jw-Jp))-2〈zn-p,-αn(Jw-Jp)〉=ϕ(p,J-1(αnJp+(1-αn)Jwn))+2αn〈zn-p,Jw-Jp〉≤αnϕ(p,p)+(1-αn)ϕ(p,wn)+2αn〈zn-p,Jw-Jp〉=(1-αn)ϕ(p,wn)+2αn〈zn-p,Jw-Jp〉≤(1-αn)ϕ(p,xn)+2αn〈zn-p,Jw-Jp〉.
Furthermore, from (3.2), Lemma 2.7, relatively asymptotically nonexpansiveness of T, relatively nonexpansiveness of Qr, and (3.6), we have that
(3.7)ϕ(p,xn+1)=ϕ(p,J-1(βnJxn+γnJTnyn+θnJQryn))=‖p‖2-〈p,βnJxn+γnJTnyn+θnJQryn〉+‖βnJxn+γnJTnyn+θnJQryn‖2(3.8)≤‖p‖2-2βn〈p,Jxn〉-2γn〈p,JTnyn〉-2θn〈p,JQryn〉+βn‖Jxn‖2+γn‖JTnyn‖2+θn‖JQryn‖2-γnβng(‖Jxn-JTnyn‖)≤βnϕ(p,xn)+γnϕ(p,Tnyn)+θnϕ(p,Qryn)-γnβng(‖Jxn-JTnyn‖)≤βnϕ(p,xn)+(γn+θn)ϕ(p,yn)+γn(kn-1)ϕ(p,yn)-γnβng(‖Jxn-JTnyn‖)≤βnϕ(p,xn)+(γn+θn)[(1-αn)ϕ(p,xn)+2αn〈zn-p,Jw-Jp〉]+γn(kn-1)ϕ(p,yn)-γnβng(‖Jxn-JTnyn‖)≤[βn+(γn+θn)(1-αn)]ϕ(p,xn)+2αn(γn+θn)〈zn-p,Jw-Jp〉+γn(kn-1)ϕ(p,yn)-γnβng(‖Jxn-JTnyn‖)≤(1-δn)ϕ(p,xn)+2δn〈zn-p,Jw-Jp〉+(kn-1)M-γnβng(‖Jxn-JTnyn‖)(3.9)≤(1-δn)ϕ(p,xn)+2δn〈zn-p,Jw-Jp〉+(kn-1)M,
for some M>0, where δn=(γn+θn)αn.
Similarly, from (3.7), we obtain that
(*)ϕ(p,xn+1)=ϕ(p,J-1(βnJxn+γnJTnyn+θnJQryn))≤(1-δn)ϕ(p,xn)+2δ〈zn-p,Jw-Jp〉+(kn-1)M-θnβng(‖Jxn-JQryn‖)≤(1-δn)ϕ(p,xn)+2δn〈zn-p,Jw-Jp〉+(kn-1)M,
for some M>0. Note that {δn} satisfies that limn→∞δn=0 and ∑δn=∞.
Now, the rest of the proof is divided into two parts.
Case 1.
Suppose that there existsn0∈N>N0 such that {ϕ(p,xn)} is nonincreasing for all n≥n0. In this situation, {ϕ(p,xn)} is then convergent. Then, from (3.8) and (*), we have that
(3.10)γnβng(‖Jxn-JTnyn‖)⟶0,θnβng(‖Jxn-JQryn‖)⟶0,
which implies, by the property of g, that
(3.11)Jxn-JTnyn⟶0,Jxn-JQryn⟶0,asn⟶∞,
and, hence, since J-1 is uniformly continuous on bounded sets, we obtain that
(3.12)xn-Tnyn⟶0,xn-Qryn⟶0,asn⟶∞.
Furthermore, Lemma 2.3, property of ϕ, and the fact that αn→0 as n→∞ imply that
(3.13)ϕ(xn,yn)=ϕ(xn,ΠCzn)≤ϕ(xn,zn)=ϕ(xn,J-1(αnJw+(1-αn)Jxn)≤αnϕ(xn,w)+(1-αn)ϕ(xn,xn)≤αnϕ(xn,w)+(1-αn)ϕ(xn,xn)⟶0,asn⟶∞,
and hence
(3.14)xn-yn⟶0,xn-zn⟶0,asn⟶∞.
Therefore, from (3.12) and (3.14), we obtain that
(3.15)yn-zn⟶0,yn-Tnyn⟶0,yn-Qryn⟶0,asn⟶∞.
But observe that from (3.2) and (3.11), we have
(3.16)‖Jxn+1-Jxn‖≤γn‖JTnyn-Jxn‖+θn‖JQryn-Jxn‖⟶0,
as n→∞. Thus, as J-1 is uniformly continuous on bounded sets, we have that xn+1-xn→0 which implies from (3.14) that xn+1-yn→0, as n→∞, and that
(3.17)‖yn+1-yn‖≤‖yn+1-xn+1‖+‖xn+1-yn‖⟶0,asn⟶∞.
Furthermore, since
(3.18)‖yn-Tyn‖≤‖yn-Tnyn‖+‖Tnyn-Tn+1yn‖+‖Tn+1yn-Tyn‖,≤‖yn-Tnyn‖+‖Tn+1yn-Tn+1yn+1‖+‖Tn+1yn+1-yn+1‖+‖yn+1-yn‖+‖yn-Tnyn‖+‖T(Tnyn)-Tyn‖≤‖yn-Tnyn‖+L‖yn-yn+1‖+‖Tn+1yn+1-yn+1‖+‖yn+1-yn‖+‖yn-Tnyn‖+‖T(Tnyn)-Tyn‖≤2‖yn-Tnyn‖+(1+L)‖yn-yn+1‖+‖Tn+1yn+1-yn+1‖+‖T(Tnyn)-Tyn‖,
we have from (3.17), (3.15), and uniform continuity of T that
(3.19)‖yn-Tyn‖⟶0,asn⟶∞.
Since {zn} is bounded and E is reflexive, we choose a subsequence {zni} of {zn} such that zni⇀z and limsupn→∞〈zn-p,Jw-Jp〉=limi→∞〈zni-p,Jw-Jp〉. Then, from (3.14) and (3.15) we get that
(3.20)yni⇀z,xni⇀z,asi⟶∞.
Thus, since T satisfies condition (N3), we obtain from (3.19) that z∈F(T) and the fact that Qr is relatively nonexpansive and yni⇀z implies that z∈F(Qr)=A-1(0), and, hence, using (3.1), we obtain that z∈VI(C,A).
Therefore, from the above discussions, we obtain thatz∈F:=F(T)∩VI(C,A). Hence, by Lemma 2.6, we immediately obtain that limsupn→∞〈zn-p,Jw-Jp〉=limi→∞〈zni-p,Jw-Jp〉=〈z-p,Jw-Jp〉≤0. It follows from Lemma 2.9 and (3.9) that ϕ(p,xn)→0, as n→∞. Consequently, xn→p.Case 2.
Suppose that there exists a subsequence{ni} of {n} such that
(3.21)ϕ(p,xni)<ϕ(p,xni+1)
for all i∈N. Then, by Lemma 2.10, there exists a nondecreasing sequence {mk}⊂N such that mk→∞, ϕ(p,xmk)≤ϕ(p,xmk+1) and ϕ(p,xk)≤ϕ(p,xmk+1) for all k∈N. Then, from (3.8), (*) and the fact δn→0, we have
(3.22)g(‖Jxmk-JTmkymk‖)⟶0,g(‖Jxmk-JQrymk‖)⟶0,ask⟶∞.
Thus, using the same proof as in Case 1, we obtain that ymk-Tymk→0, ymk-Qrymk→0, as k→∞, and, hence, we obtain that
(3.23)limsup⁡k→∞〈zmk-p,Jw-Jp〉≤0.
Then, from (3.9), we have that
(3.24)ϕ(p,xmk+1)≤(1-δmk)ϕ(p,xmk)+2δmk〈zmk-p,Jw-Jp〉+(kmk-1)M.
Since ϕ(p,xmk)≤ϕ(p,xmk+1), (3.24) implies that
(3.25)δmkϕ(p,xmk)≤ϕ(p,xmk)-ϕ(p,xmk+1)+2δmk〈zmk-p,Jw-Jp〉+(kmk-1)M≤2δmk〈zmk-p,Jw-Jp〉+(kmk-1)M.
In particular, since δmk>0, we get
(3.26)ϕ(p,xmk)≤2〈zmk-p,Jw-Jp〉+(kmk-1)δmkM.
Then, from (3.23) and the fact that (kmk-1)/δmk→0, we obtain ϕ(p,xmk)→0, as k→∞. This together with (3.24) gives ϕ(p,xmk+1)→0, as k→∞. But ϕ(p,xk)≤ϕ(p,xmk+1), for all k∈N, thus we obtain that xk→p. Therefore, from the above two cases, we can conclude that {xn} converges strongly to p and the proof is complete.If, in Theorem3.1, we assume that T is relatively nonexpansive, we get the following corollary.Corollary 3.2.
LetC be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space E. Let A:C→E* be a monotone mapping satisfying (2.8) and ∥Ax∥≤∥Ax-Ap∥,for all x∈C and p∈VI(C,A). Let T:C→C be a relatively nonexpansive mapping. Assume that F∶=VI(C,A)∩F(S) is nonempty. Let {xn} be a sequence generated by
(3.27)x0=w∈C,chosenarbitrarily,yn=ΠCJ-1(αnJw+(1-αn)Jxn),xn+1=J-1(βnJxn+γnJTyn+θnJQryn),
where αn∈(0,1) such that limn→∞αn=0, ∑n=1∞αn=∞, {βn},{γn},{θn}⊂[c,d]⊂(0,1) such that βn+γn+θn=1. Then, {xn} converges strongly to an element of F.Proof.
We note that the method of proof of Theorem3.1 provides the required assertion.IfE=H, a real Hilbert space, then E is uniformly convex and uniformly smooth real Banach space. In this case, J=I, identity map on H and ΠC=PC, projection mapping from H onto C. Thus, the following corollary holds.Corollary 3.3.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let A:C→H be a monotone mapping satisfying (2.8) and ∥Ax∥≤∥Ax-Ap∥,for all x∈C and p∈VI(C,A). Let T:C→C be a uniformly L-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {kn}. Assume that F∶=VI(C,A)∩F(S) is nonempty. Let {xn} be a sequence generated by
(3.28)x0=w∈C,chosenarbitrarily,yn=PC(αnw+(1-αn)xn),xn+1=βnxn+γnTnyn+θnQryn,
where αn∈(0,1) such that limn→∞αn=0,limn→∞((kn-1)/αn)=0,∑n=1∞αn=∞, {βn},{γn},{θn}⊂[c,d]⊂(0,1) such that βn+γn+θn=1. Then, {xn} converges strongly to an element of F.Now, we state the second main theorem of our paper.Theorem 3.4.
LetC be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space E. Let A:C→E* be a monotone mapping satisfying (2.8). Let T:C→C be a uniformly L-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {kn}. Assume that F:=A-1(0)∩F(S) is nonempty. Let Qr be the resolvent of A and {xn} a sequence generated by
(3.29)x0=w∈C,chosenarbitrarily,yn=ΠCJ-1(αnJw+(1-αn)Jxn),xn+1=J-1(βnJxn+γnJTnyn+θnJQryn),
where αn∈(0,1) such that limn→∞αn=0,limn→∞((kn-1)/αn)=0, ∑n=1∞αn=∞, {βn},{γn},{θn}⊂[c,d]⊂(0,1) such that βn+γn+θn=1. Then, {xn} converges strongly to an element of F.Proof.
Similar method of proof of Theorem3.1 provides the required assertion.If, in Theorem3.4, A=0, then we have the following corollary. Similar proof of Theorem 3.1 provides the assertion.Corollary 3.5.
LetC be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space E. Let T:C→C be a uniformly L-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {kn}. Assume that F∶=F(S) is nonempty. Let {xn} be a sequence generated by
(3.30)x0=w∈C,chosenarbitrarily,yn=ΠCJ-1(αnJw+(1-αn)Jxn),xn+1=J-1(βnJxn+(1-βn)JTnyn),
where αn∈(0,1) such that limn→∞αn=0,limn→∞((kn-1)/αn)=0, ∑n=1∞αn=∞, {βn}⊂[c,d]⊂(0,1). Then, {xn} converges strongly to an element of F.If, in Theorem3.4, T=I, identity mapping on C, then we have the following corollary.Corollary 3.6.
LetC be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex real Banach space  E. Let  A:C→E* be a monotone mapping satisfying (2.8). Assume that  F:=A-1(0) is nonempty. Let  Qr be the resolvent of  A and  {xn} a sequence generated by
(3.31)x0=w∈C,chosenarbitrarily,yn=ΠCJ-1(αnJw+(1-αn)Jxn),xn+1=J-1(βnJxn+(1-βn)JQryn),
where αn∈(0,1) such that limn→∞αn=0, ∑n=1∞αn=∞, {βn}⊂[c,d]⊂(0,1). Then, {xn} converges strongly to an element of F.If, in Theorem3.4, we assume that T is relatively nonexpansive, we get the following corollary.Corollary 3.7.
LetC be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space  E. Let  A:C→E* be a monotone mapping satisfying (2.8). Let  T:C→C be a relatively nonexpansive mapping. Assume that  F∶=A-1(0)∩F(S) is nonempty. Let  {xn} be a sequence generated by
(3.32)x0=w∈C,chosenarbitrarily,yn=ΠCJ-1(αnJw+(1-αn)Jxn),xn+1=J-1(βnJxn+γnJTyn+θnJQryn),
where αn∈(0,1) such that limn→∞αn=0, ∑n=1∞αn=∞, {βn},{γn},{θn}⊂[c,d]⊂(0,1) such that βn+γn+θn=1. Then, {xn} converges strongly to an element of F.We may also get the following corollary for a common zero of monotone mappings.Corollary 3.8.
LetC be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space  E. Let  A,B:C→E* be monotone mappings satisfying (2.8). Suppose that  T1=(J+rA)-1J and  T2=(J+rB)-1J. Assume that  F:=A-1(0)∩B-1(0) is nonempty. Let  {xn} be a sequence generated by
(3.33)x0=w∈C,chosenarbitrarily,yn=ΠCJ-1(αnJw+(1-αn)Jxn),xn+1=J-1(βnJxn+γnJT1yn+θnJT2yn),
where αn∈(0,1) such that limn→∞αn=0, ∑n=1∞αn=∞, {βn},{γn},{θn}⊂[c,d]⊂(0,1) such that βn+γn+θn=1. Then, {xn} converges strongly to an element of F.Proof.
Clearly, from Lemma2.8, we know that T1 and T2 are relatively nonexpansive mappings. We also have that F(T1)=A-1(0) and F(T2)=B-1(0). Thus, the conclusion follows from Corollary 3.7.Remark 3.9.
We remark that from Corollary3.8 the scheme converges strongly to a common zero of two monotone operators. We may also have the following theorem for a common zero of finite family of monotone mappings.Theorem 3.10.
LetC be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space  E. Let  Ai:C→E*,  i=1,2,…,N be monotone mappings satisfying (2.8). Suppose that  Ti=(J+rAi)-1J, and assume that  F:=∩i=1NAi-1(0) is nonempty. Let  {xn} be a sequence generated by
(3.34)x0=w∈C,chosenarbitrarily,yn=ΠCJ-1(αnJw+(1-αn)Jxn),xn+1=J-1(βn,0Jxn+βn,1JT1yn+⋯+βn,NJTNyn),
where αn∈(0,1) such that limn→∞αn=0, ∑n=1∞αn=∞, {βn,i}⊂[c,d]⊂(0,1), for i=0,1,2,…,N, such that ∑i=0Nβn,i=1. Then, {xn} converges strongly to an element of F.A monotone mappingA:C→E* is said to be maximal monotoneif its graph is not properly contained in the graph of any monotone operator. We know that if A is maximal monotone operator, then A-1(0) is closed and convex: see [4] for more details. The following Lemma is well known.Lemma 3.11 (see [28]).
LetE be a smooth and strictly convex and reflexive Banach space, let  C be a nonempty closed convex subset of  E, and let  A:C→E* be a monotone operator. Then  A is maximal if and only if  R(J+rA)=E* for all  r>0.We note from the above lemma that ifA is maximal then it satisfies condition (2.8) and hence we have the following corollary.Corollary 3.12.
LetC be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space E. Let A:C→E* be a maximal monotone mapping. Let T:C→C be a uniformly L-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {kn}. Assume that F:=A-1(0)∩F(S) is nonempty. Let {xn} be a sequence generated by
(3.35)x0=w∈C,chosenarbitrarily,yn=ΠCJ-1(αnJw+(1-αn)Jxn),xn+1=J-1(βnJxn+γnJTnyn+θnJQryn),
where αn∈(0,1) such that limn→∞αn=0,limn→∞((kn-1)/αn)=0, ∑n=1∞αn=∞, {βn},{γn},{θn}⊂[c,d]⊂(0,1) such that βn+γn+θn=1. Then, {xn} converges strongly to an element of F.


## 4. Application

In this section, we study the problem of finding a minimizer of a lower semicontinuous continuously convex functional in Banach spaces.Theorem 4.1.
LetE be a uniformly convex and uniformly smooth real Banach space. Let  f,g:E→(-∞,∞) be a proper lower semicontinuous convex functions. Assume that  F∶=(∂f)-1(0)∩(∂g)-1(0) is nonempty. Let  {xn} be a sequence generated by
(4.1)x0=w∈C,chosenarbitrarily,yn=ΠCJ-1(αnJw+(1-αn)Jxn),hn=argmin{f(z)+12rϕ(z,yn),z∈E},tn=argmin{g(z)+12rϕ(z,yn),z∈E},xn+1=J-1(βnJxn+γnJhn+θnJtn),
where αn∈(0,1) such that limn→∞αn=0, ∑n=1∞αn=∞, {βn},{γn},{θn}⊂[c,d]⊂(0,1) such that βn+γn+θn=1. Then, {xn} converges strongly to an element of F.Proof.
LetA and B be operators defined by A=∂f and B=∂g and Qr=(J+rA)-1J, QrB=(J+rB)-1J for all r>0. Then, by Rockafellar [29], A and B are maximal monotone mappings. We also have that
(4.2)hn=QrAy=argmin{f(z)+12rϕ(z,y),z∈E},tn=QrBy=argmin{g(z)+12rϕ(z,y),z∈E},
for all y∈E and r>0. Furthermore, we have that F(QrA)=A-1(0) and F(QrB)=B-1(0). Thus, by Corollary 3.8, we obtain the desired result.Remark 4.2.
Consider the following.(1)
Theorem3.1 improves and extends the corresponding results of Zegeye et al. [12] and Zegeye and Shahzad [22] in the sense that either our scheme does not require computation of Cn+1 for each n≥1 or the space considered is more general.(2)
Corollary3.5 improves the corresponding results of Nakajo and Takahashi [30] and Matsushita and Takahashi [17] in the sense that either our scheme does not require computation of Cn+1 for each n≥1 or the class of mappings considered in our corollary is more general.(3)
Corollary3.6 improves the corresponding results of Iiduka and Takahashi [11], Iiduka et al. [14], and Alber [1] in the sense that our scheme does not require computation of Cn+1 for each n≥1 or the class of mappings considered in our corollary is more general.



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