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Minimum-norm solution of variationalinequality and fixed point problem inbanach spacesHabtu Zegeye a , Naseer Shahzad b & Yonghong Yao ca Department of Mathematics, University of Botswana, Gaborone,Botswanab Department of Mathematics, King Abdulaziz University, Jeddah,Saudi Arabiac Department of Mathematics, Tianjin Polytechnic University,Tianjin, ChinaPublished online: 06 Mar 2013.

To cite this article: Optimization (2013): Minimum-norm solution of variational inequality andfixed point problem in banach spaces, Optimization: A Journal of Mathematical Programming andOperations Research, DOI: 10.1080/02331934.2013.764522

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Optimization, 2013http://dx.doi.org/10.1080/02331934.2013.764522

Minimum-norm solution of variational inequality and fixed pointproblem in banach spaces

Habtu Zegeyea, Naseer Shahzadb and Yonghong Yaoc∗

aDepartment of Mathematics, University of Botswana, Gaborone, Botswana; bDepartment ofMathematics, King Abdulaziz University, Jeddah, Saudi Arabia; cDepartment of Mathematics,

Tianjin Polytechnic University, Tianjin, China

(Received 15 June 2012; final version received 24 December 2012)

We introduce an iterative process which converges strongly to a common minimum-norm solution of a variational inequality problem for an α-inverse strongly mono-tone mapping and a fixed point of relatively non-expansive mapping in Banachspaces. Our theorems improve and unify most of the results that have been provedfor this important class of non-linear operators.

Keywords: Monotone mappings; relatively asymptotically non-expansivemappings; relatively non-expansive; strong convergence; variational inequalityproblems

AMS Subject Classifications: 47H05; 47H09; 47H10; 47J05; 47J25.

1. Introduction

Let C be a non-empty subset of a real Banach space E with dual E∗. The mappingT : C → E is called Lipschitzian if there exists L > 0 such that ‖T x − T y‖ ≤ L‖x − y‖,for all x, y ∈ C. If L = 1, then T is called non-expansive.

A mapping A from C into E∗ is called α−inverse strongly monotone if there exists apositive real number α such that

〈x − y, Ax − Ay〉 ≥ α||Ax − Ay||2, for all x, y ∈ C . (1)

If A is α−inverse strongly monotone, then it is Lipschitz continuous with constant 1α

. A issaid to be monotone if for each x, y ∈ D(A), the following inequality holds:

〈x − y, Ax − Ay〉 ≥ 0. (2)

Let A : C → E∗ be a monotone mapping. The variational inequality problem [12,16]is formulated as finding

a point u ∈ C such that 〈v − u, Au〉 ≥ 0, for all v ∈ C . (3)

*Corresponding author. Email: yaoyonghong@yahoo.cn

© 2013 Taylor & Francis

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We denote the set of solutions of the variational inequality problem by V I (C, A), that is,V I (C, A) = {u ∈ C : 〈v − u, Au〉 ≥ 0, for all v ∈ C}.

Variational inequality problems have wide applications in the fields of optimizationand control, economics and transportation equilibrium, engineering science, the problem offinding a point u ∈ C satisfying Au = 0, etc. Variational inequalities were initially studiedby Stampacchia [12,16] and ever since, considerable research efforts have been devoted toiterative methods for approximating solutions of variational inequality problems when A isan α−inverse strongly monotone (see, e.g. [10,11,16,35,36] and the references containedtherein).

It is well known that if C is a non-empty closed and convex subset of a Hilbert spaceH the metric projection PC : H → C is non-expansive. This fact actually characterizesHilbert spaces and consequently, it is not available in more general Banach spaces. Recently,Alber [1] introduced a generalized projection operator �C in a Banach space E which is ananalogue of the metric projection in Hilbert spaces. Let E be a smooth Banach space withdual E∗ and φ : E × E → R defined by

φ(x, y) = ||x ||2 − 2〈x, J y〉 + ||y||2,∀x, y ∈ E,

where J is the normalized duality mapping.

A normalized duality mapping J from E into 2E∗is defined by

J x := { f ∗ ∈ E∗ : 〈x, f ∗〉 = ||x ||2 = || f ∗||2},where 〈., .〉 denotes the generalized duality pairing. It is well known (see [25]) that E issmooth if and only if J is single valued and if E is uniformly smooth then J is uniformlycontinuous on bounded subsets of E . Moreover, if E is a reflexive and strictly convexBanach space with a strictly convex dual, then J−1 is duality mapping from E∗ into Ewhich is single valued, one-to-one., surjective and J J−1 = IE∗ and J−1 J = IE (see [25]).

Following Alber [1], the generalized projection �C : E → C , is a mapping that assignsto an arbitrary point x ∈ E the minimum point of the functional φ(y, x), that is, �C x = x̄,

where x̄ is the solution to the following minimization problem:

φ(x̄, x) = infy∈C

φ(y, x).

It follows from the definition of the function φ that

(||y|| − ||x ||)2 ≤ φ(y, x) ≤ (||y|| + ||x ||)2,∀x, y ∈ E .

If E is a Hilbert space, then φ(y, x) = ||y − x ||2 and �C = PC is the metric projection ofH onto C.

Let C be a subset of a smooth real Banach space E and let T be a mapping from Cinto itself. We denote by F(T ) the fixed points set of T . A point p in C is said to bean asymptotic fixed point of T (see [19]) if C contains a sequence {xn} which convergesweakly to p such that limn→∞ ||xn − T xn|| = 0. The set of asymptotic fixed points of T

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will be denoted by F̂(T ). A mapping T from C into itself is called relatively non-expansiveif (R1) F(T ) �= ∅; (R2) φ(p, T x) ≤ φ(p, x) for x ∈ C and (R3) F(T ) = F̂(T ). T iscalled relatively quasi-non-expansive if F(T ) �= ∅ and φ(p, T x) ≤ φ(p, x) for all x ∈ C ,and p ∈ F(T ). T is called closed if xn → x and T xn → y, then T x = y.

Many authors have considered the problem of finding a common element of the fixedpoints set of relatively non-expansive mapping and the solution set of variational inequalityproblem for γ−inverse monotone mapping (see, e.g. [13,24,27,35,36]).

Recently, we notice that it is quite often to seek a particular solution of a given non-linearproblem, in particular, the minimum-norm solution. In an abstract way, we may formulatesuch problems as follows:

find x∗ ∈ C such that ||x∗|| := minx∈C

||x ||, (4)

A typical example is the split feasibility problem (SFP), formulated as finding a pointx∗ with the property that

x∗ ∈ K and Ax∗ ∈ Q, (5)

where K and Q are non-empty closed convex subsets of the infinite dimension real Hilbertspaces H1 and H2, respectively, and A is bounded linear mapping from H1 to H2. Equa-tion (5) models many applied problems arising from image reconstructions and learningtheory (see, for example [31]). Some works in the finite-dimensional setting with relevantprojection methods for solving image recovery problems can be found in [4,34]. Definingthe proximity function f by

f (x) := 1

2||Ax − PQ Ax ||2

we consider the convex optimization problem

minx∈K

f (x) := minx∈K

1

2||Ax − PQ Ax ||2. (6)

It is clear that x∗ is a solution to the SPF (5) if and only if x∗ ∈ K and Ax∗ − PQ Ax∗ = 0which is the minimum-norm solution of the minimization problem (6).

We may also see applications of the minimum-norm solution in solving the minimumFrobenius norm residual problems (see, e.g. [9]).

Let A : C → E∗ be an α−inverse strongly monotone mapping and T : C → C berelatively non-expansive mapping with F := V I (C, A) ∩ F(T ) �= ∅. Then, motivated bythe above SPF, we study the general case of finding the minimum-norm point of F , thatis, we find a common solution of variational inequality problem for an α−inverse stronglymonotone and a fixed point of relatively non-expansive mapping T satisfying

x∗ ∈ F such that x∗ = �F (0), (7)

where �F is the generalized projection from E onto F . If E = H , a real Hilbert space,then (7) reduces to x∗ ∈ F such that x∗ = PF (0), where PF is the metric projection from

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E onto F . In this paper, minimum-norm of a solution set F will mean a point x∗ ∈ F suchthat x∗ = �F (0).

Let T be a non-expansive self-mapping on a closed convex subset K of a Banach spaceE . Many problems in non-linear analysis can be reformulated as a problem of finding afixed point of a non-expansive and its generalizations. Halpern [8] was the first who intro-duced the following explicit iteration scheme for a non-expansive mapping T which wasreferred to as Halpern iteration: for u, x0 ∈ K , αn ∈ [0, 1],

xn+1 = αnu + (1 − αn)T xn . (8)

Convergence of this two schemes have been studied by many researchers with different addi-tional conditions on {αn}. Iterative methods for approximating fixed points of non-expansiveand/or a solution of an α−inverse strongly monotone mappings and its generalizations havebeen studied by various authors (see, e.g. Bruck [3], Dhompongsa et al. [6], Reich [20],Song-Xu [22], Takahashi-Ueda [26], Suzuki [23], Yao and Shahzad [32], Yao et al. [33] andmany others).

However, in all the above results, the methods can be used to find the minimum-normfixed point x∗ of T if 0 ∈ K . If, however, 0 /∈ K any of the methods above fails to providethe minimum-norm fixed point of T .

In connection with the iterative approximation of the minimum-norm fixed point of T ,Yang et al. [31] introduced an explicit scheme given by

xn+1 = βT xn + (1 − β)PK [(1 − αn)xn], n ≥ 1,

under appropriate conditions on {αn} and β, which converges strongly to the minimum-norm fixed point of T .

More recently, Cai et al. [5] have also shown that the explicit scheme xn+1 = (1 −αn)(λT xn + (1 − λ)xn), n ≥ 1, converge strongly to the minimum-norm solution of non-expansive mappings T , under appropriate conditions on λ and {αn}. However, we observethat the domain of T considered is closed convex containing 0 and hence reduces to theresults of [8].

Let C be a non-empty, closed and convex subset of a uniformly smooth and 2-uniformlyconvex real Banach space E . Let A : C → E∗ be a α-inverse strongly monotone mappingand T : C → C be an relatively non-expansive mapping.

It is our purpose in this paper to introduce an iterative scheme (see, (11)) which convergesstrongly, as n → ∞, to p = �F (0), where �F is the generalized projection from E ontoF . Our theorems improve Theorem 3.2 of Yang et al. [31] to Banach spaces.

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2. Preliminaries

Let E be a normed linear space with dim E ≥ 2. The modulus of smoothness of E is thefunction ρE : [0,∞) → [0,∞) defined by

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, ∀ τ > 0 and E is called uniformly smoothif and only if limt→0+ ρE (t)

t = 0.

The modulus of convexity of E is the function δE : (0, 2] → [0, 1] defined by

δE (ε) := inf

{1 −

∥∥∥∥ x + y

2

∥∥∥∥ : ‖x‖ = ‖y‖ = 1; ε = ‖x − y‖}

.

E is called uniformly convex if and only if δE (ε) > 0, for every ε ∈ (0, 2]. Let p > 1. ThenE is said to be p−uniformly convex if there exists a constant c > 0 such that δ(ε) ≥ cε p,for all ε ∈ [0, 2]. Observe that every p-uniformly convex space is uniformly convex.

It is well known (see for example [30]) that

L p ( l p) or W pm is

{p − uni f ormly convex i f p ≥ 2,

2 − uni f ormly convex i f 1 < p ≤ 2.

In the sequel, we shall need the following result:

Lemma 2.1 [30] Let E be a 2−uniformly convex Banach space. Then, for all x, y ∈ E,we have

||x − y|| ≤ 2

c2||J x − J y||, (9)

where J is the normalized duality mapping of E and 0 < c ≤ 1.

In the sequel, we shall make use of the following lemmas.

Lemma 2.2 [37] Let C be a non-empty closed and convex subset of a real reflexive, strictlyconvex, and smooth Banach space E. If A : C → E∗ is continuous monotone mapping,then V I (C, A) is closed and convex.

Proposition 2.3 [18] Let E be a strictly convex and smooth Banach space, let C be aclosed convex subset of E, and let T be a relatively non-expansive mapping from C intoitself. Then F(T ) is closed and convex.

Lemma 2.4 [1] Let K be a non-empty closed and convex subset of a real reflexive, strictlyconvex, and smooth Banach space E and let x ∈ E. Then ∀y ∈ K ,

φ(y,�K x) + φ(�K x, x) ≤ φ(y, x).

Lemma 2.5 [11] Let E 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, asn → ∞, then xn − yn → 0, as n → ∞.

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We make use of the function V : E × E∗ → R defined by

V (x, x∗) = ||x ||2 − 2〈x, x∗〉 + ||x∗||2, for all x ∈ E and x∗ ∈ E∗,

studied by Alber [1]. That is, V (x, x∗) = φ(x, J−1x∗) for all x ∈ E and x∗ ∈ E∗. Weknow the following lemma.

Lemma 2.6 [1] Let E be reflexive strictly convex and smooth Banach space with E∗ asits dual. Then

V (x, x∗) + 2〈J−1x∗ − x, y∗〉 ≤ V (x, x∗ + y∗),for all x ∈ E and x∗, y∗ ∈ E∗.

Lemma 2.7 [1] Let C be a convex subset of a real smooth Banach space E. Let x ∈ E.Then x0 = �C x if and only if

〈z − x0, J x − J x0〉 ≤ 0,∀z ∈ C.

Lemma 2.8 [35] Let E be a uniformly convex Banach space and BR(0) be a closed ball ofE. Then, there exists a continuous strictly increasing convex function g : [0,∞) → [0,∞)

with g(0) = 0 such that

||αx + (1 − α)y||2 ≤ α||x ||2 + (1 − α)||y||2 − α(1 − α)g(||x − y||),for α ∈ (0, 1) and for x, y ∈ BR(0) := {x ∈ E : ||x || ≤ R}.Lemma 2.9 [29] Let {an} be a sequence of nonnegative real numbers satisfying thefollowing relation:

an+1 ≤ (1 − αn)an + αnδn, n ≥ n0,

where {αn} ⊂ (0, 1) and {δn} ⊂ R satisfying the following conditions: limn→∞αn = 0,

∑∞n=1

αn = ∞, and limn→∞ δn ≤ 0. Then, limn→∞ an = 0.

Lemma 2.10 [17] Let {an} be sequences of real numbers such that there exists a subse-quence {ni } of {n} such that ani < ani +1 for all i ∈ N. Then there exists a nondecreasingsequence {mk} ⊂ N such that mk → ∞ and the following properties are satisfied by all(sufficiently large) numbers k ∈ N:

amk ≤ amk+1 and ak ≤ amk+1.

In fact, mk = max{ j ≤ k : a j < a j+1}.

3. Main result

We note that, as it is mentioned in [38], if C is a subset of a real Banach space E andA : C → E∗ is a monotone mapping satisfying ||Ax || ≤ ||Ax − Ap||, ∀x ∈ C andp ∈ V I (C, A), then

V I (C, A) = A−1(0) = {p ∈ C : Ap = 0}. (10)

Theorem 3.1 Let C be a non-empty, closed and convex subset of a smooth and 2-uniformly convex real Banach space E. Let A : C → E∗ be an α-inverse strongly monotone

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mapping satisfying ||Ax || ≤ ||Ax − Ap||, ∀x ∈ C and p ∈ V I (C, A), and let T : C → Cbe an relatively non-expansive mapping. Assume that F := V I (C, A)∩ F(T ) is non-empty.Let {xn} be a sequence generated by

⎧⎨⎩

x1 ∈ C, chosen arbitrarily,yn = �C

[(1 − αn)J−1(J xn − λn Axn)

],

xn+1 = �C J−1(βn J yn + (1 − βn)J T yn), n ≥ 1,

(11)

where αn ∈ (0, 1) such that limn→∞ αn = 0,∑∞

n=1 αn = ∞, {βn} ⊂ [c, d] ⊂ (0, 1) and

{λn} is a sequence in [a, b] for some real numbers a, b such that 0 < a ≤ λn ≤ b < c2α2 , for

1c a 2-uniformly convex constant of E. Then {xn} converges strongly to the minimum-normpoint of F.

Proof Let p := �F 0 and wn = J−1(J xn −λn Axn). From (11), Lemma 2.4 and propertyof φ we get that

φ(p, yn) = φ(p,�C[(1 − αn)wn

]) ≤ φ(p, (1 − αn)wn)

= φ(p, J−1(αn J0 + (1 − αn)Jwn)

= ||p||2 − 2〈p, αn J0 + (1 − αn)Jwn〉 + ||αn J0 + (1 − αn)Jwn||2≤ ||p||2 − 2αn〈p, J0〉 − 2(1 − αn)〈p, Jwn〉

+ αn||J0||2 + (1 − αn)||Jwn||2= αnφ(p, 0) + (1 − αn)φ(p, wn) (12)

But, by Lemma 2.6 we get that

φ(p, wn) = φ(p, J−1(J xn − λn Axn)) = V (p, J xn − λn Axn)

≤ V (p, (J xn − λn Axn) + λn Axn〉−2〈J−1(J xn − λn Axn) − p, λn Axn〉

= V (p, J xn) − 2λn〈J−1(J xn − λn Axn) − p, Axn〉= φ(p, xn) − 2λn〈xn − p, Axn〉 − 2λn〈J−1(J xn − λn Axn) − xn, Axn〉(13)

Thus, since p ∈ F implies, by (10), that p ∈ A−1(0), and A is an α−inverse stronglymonotone we have from (13) and Lemma 2.1 that

φ(p, wn) ≤ φ(p, xn) − 2λn〈xn − p, Axn − Ap〉− 2λn〈J−1(J xn − λn Axn) − xn, Axn〉

≤ φ(p, xn) − 2λnα||Axn||2+ 2λn||J−1(J xn − λn Axn) − J−1(J xn)||||Axn||

≤ φ(p, xn) − 2λnα||Axn||2 + 4

c2λ2

n||Axn||2

= φ(p, xn) + 2λn

(2

c2λn − α

)||Axn||2 ≤ φ(p, xn). (14)

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Therefore, from (12), (14) and λn < c2

2 α we get that

φ(p, yn) ≤ αnφ(p, 0) + (1 − αn)φ(p, xn)

+ 2(1 − αn)λn

(2

c2λn − α

)||Axn||2 (15)

≤ αnφ(p, 0) + (1 − αn)φ(p, xn). (16)

Furthermore, from (11), Lemma 2.4 and relatively non-expansiveness of T , property of φ

and (16) we get that

φ(p, xn+1) = φ(p,�C J−1(βn J yn + (1 − βn)J T yn)

≤ φ(p, J−1(βn J yn + (1 − βn)J T yn)

≤ βnφ(p, yn) + (1 − βn)φ(p, T yn))

= βnφ(p, yn) + (1 − βn)φ(p, yn) ≤ φ(p, yn)

≤ αnφ(p, 0) + (1 − αn)φ(p, xn).

Thus, by induction,

φ(p, xn+1) ≤ max{φ(p, 0), φ(p, x1)},∀n ≥ 1,

which implies that {xn} and hence {yn} is bounded. Now let zn = (1−αn)wn = J−1(αn J0+(1 − αn)Jwn). Then, since αn → 0, as n → ∞, we obtain that

zn − wn → 0, as n → ∞. (17)

Now, using Lemma 2.4, Lemma 2.6 and property of φ we obtain that

φ(p, yn) ≤ φ(p, zn) = V (p, J zn)

≤ V (p, J zn − αn(J0 − J p)) − 2〈zn − p,−αn(J0 − J p)〉= φ(p, J−1(αn J p + (1 − αn)Jwn) + 2αn〈zn − p,−J p〉≤ αnφ(p, p) + (1 − αn)φ(p, wn) + 2αn〈zn − p,−J p〉= (1 − αn)φ(p, wn) + 2αn〈zn − p,−J p〉≤ (1 − αn)φ(p, xn) + 2αn〈zn − p,−J p〉. (18)

Furthermore, from (11), Lemma 2.8, relatively non-expansiveness of T and (15) we havethat

φ(p, xn+1) = φ(p,�C J−1(βn J yn + (1 − βn)J T yn))

≤ βnφ(p, yn) + (1 − βn)φ(p, T yn)

− (1 − βn)βng(||J yn − J T yn||)≤ βnφ(p, yn) + (1 − βn)φ(p, yn)

− (1 − βn)βng(||J yn − J T yn||)≤ φ(p, yn) − (1 − βn)βng(||J yn − J T yn||)≤ αnφ(p, 0) + (1 − αn)φ(p, xn) + 2(1 − αn)λn

(2

c2λn − α

)||Axn||2

− (1 − βn)βng(||J yn − J T yn||). (19)

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Moreover, from (11) and (18) we obtain that

φ(p, xn+1) = φ(p,�C J−1(βn J yn + (1 − βn)J T yn))

≤ βnφ(p, yn) + (1 − βn)φ(p, T yn)

≤ βnφ(p, yn) + (1 − βn)φ(p, yn) ≤ φ(p, yn)

≤ (1 − αn)φ(p, xn) + 2αn〈zn − p,−J p〉. (20)

Now, we consider two cases:

Case 1 Suppose that there exists n0 ∈ N such that {φ(p, xn)} is non-increasing forall n ≥ n0. Then we get that {φ(p, xn)} is convergent. Now, from (19) we have that2λn(1 − αn)(α − 2

c2 λn)||Axn||2 → 0 and (1 − βn)βng(||J yn − J T yn|| → 0, as n → ∞.

These imply, by the property of g and the fact that λn < c2

2 α, that

Axn → 0 and J yn − J T yn → 0, as n → ∞, (21)

and hence, since J−1 is uniformly continuous on bounded sets we obtain that

yn − T yn → 0, as n → ∞. (22)

Furthermore, since yn = �C zn , the fact that αn → 0, as n → ∞, and property of φ implythat

φ(wn, yn) = φ(wn,�C zn) ≤ φ(wn, zn)

≤ φ(wn, J−1(αn J0 + (1 − αn)Jwn)

≤ αnφ(wn, 0) + (1 − αn)φ(wn, wn)

≤ αnφ(wn, 0) + (1 − αn)φ(wn, wn) → 0, as n → ∞, (23)

and hence

wn − yn → 0, as n → ∞. (24)

Since {zn} is bounded and E is reflexive, we choose a subsequence {zni } of {zn} such thatzni ⇀ z and limn→∞〈zn − p,−J p〉 = limi→∞〈zni − p,−J p〉. Then, from (24) and (17)we get that

yni ⇀ z, wni ⇀ z, as i → ∞. (25)

Thus, since T satisfies condition (R3) we obtain from (22) and (25) that z ∈ F(T ).

Next, we show that z ∈ V I (C, A). From Lemma 2.6, we have that

φ(xn, wn) = φ(xn, J−1(J xn − λn Axn) ≤ V (xn, J xn − λn Axn)

≤ V (xn, (J xn − λn Axn) + λn Axn) − 2〈J−1(J xn − λn Axn) − xn, λn Axn〉= φ(xn, xn) + 2λn〈J−1(J xn − λn Axn) − xn,−An xn〉= 2λn〈J−1(J xn − λn Axn) − xn,−An xn〉≤ 2λn||J−1(J xn − λn Axn) − J−1 J xn||.||Axn|| ≤ 4

c2λ2

n||Axn||2,

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then, using (21) we obtain that

φ(xn, wn) → 0, as n → ∞, (26)

which implies by Lemma 2.5 that

xn − wn → 0, as n → ∞, (27)

and hence from (17) and (27) we have that xni ⇀ z. Now, since A is α-inverse stronglymonotone, we have

α||Axni − Az||2 ≤ 〈xni − z, Axni − Az〉 → 0, as i → ∞, (28)

which implies that Axni → Az and hence since Axn → 0, we have Az = 0. Therefore,z ∈ A−1(0) and hence by (10) we have that z ∈ V I (C, A).

Thus, from the above discussions we obtain that z ∈ F := F(T )∩V I (C, A). Therefore,by Lemma 2.7, we immediately obtain that limn→∞〈zn − p,−J p〉 = limi→∞〈zni −p,−J p〉 = 〈z − p,−J p〉 ≤ 0. It follows from Lemma 2.9 and (20) that φ(p, xn) → 0, asn → ∞. Consequently, xn → p.

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

φ(p, xni ) < φ(p, xni +1)

for all i ∈ N. Then, by Lemma 2.10, there exist a non-decreasing sequence {mk} ⊂ N suchthat mk → ∞, φ(p, xmk ) ≤ φ(p, xmk+1) and φ(p, xk) ≤ φ(p, xmk+1) for all k ∈ N. Thenfrom (19) and the fact that αn → 0 we have

Axmk → 0 and g(||J ymk − J T ymk ||) → 0, as k → ∞.

Thus, using the same proof as in Case 1, we obtain that ymk − T ymk → 0, ymk −wmk → 0,wmk − xmk → 0, as k → ∞ and hence we obtain that

lim supk→∞

〈zmk − p,−J p〉 ≤ 0. (29)

Then from (20) we have that

φ(p, xmk+1) ≤ (1 − αmk )φ(p, xmk ) + 2αmk 〈zmk − p,−J p〉. (30)

Since φ(p, xmk ) ≤ φ(p, xmk+1), (30) implies that

αmk φ(p, xmk ) ≤ φ(p, xmk ) − φ(p, xmk+1) + 2αmk 〈zmk − p,−J p〉≤ 2αmk 〈zmk − p,−J p〉.

In particular, since αmk > 0, we get

φ(p, xmk ) ≤ 2〈zmk − p,−J p〉.Then, using (29) we obtain that φ(p, xmk ) → 0, as k → ∞. This together with (30) givesφ(p, xmk+1) → 0, as k → ∞. But φ(p, xk) ≤ φ(p, xmk+1), for all k ∈ N , thus we obtain

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that xk → p. Therefore, from the above two cases, we can conclude that {xn} convergesstrongly to p, the minimum-norm point of F , and the proof is complete. �

We note that Theorem 3.1 could be relaxed to continuous monotone mappings withoutthe requirement that A satisfies ||Ax || ≤ ||Ax − Ap||, ∀x ∈ C and p ∈ V I (C, A) in moregeneral Banach spaces. In fact, we have the following.

Let C be a non-empty, closed and convex subset of a smooth, strictly convex andreflexive real Banach space E with dual E∗. Let A : C → E∗, be a continuous monotonemapping. For the rest of this paper, Frn x is a mapping defined as follows: For x ∈ E , letFrn : E → C be given by

Frn x := {z ∈ C : 〈y − z, Az〉 + 1

rn〈y − z, J z − J x〉 ≥ 0,∀ y ∈ C},

where {rn}n∈N ⊂ [c1,∞) for some c1 > 0.

Theorem 3.2 Let C be a non-empty, closed and convex subset of a a uniformly smoothand uniformly convex real Banach space E. Let A : C → E∗ be a continuously monotonemapping and let T : C → C be an relatively non-expansive mapping. Assume that F :=V I (C, A) ∩ F(T ) is non-empty. Let {xn} be a sequence generated by⎧⎨

⎩x1 ∈ C, chosen arbitrarily,yn = �C

[(1 − αn)Frn xn],

xn+1 = �C J−1(βn J yn + (1 − βn)J T yn), n ≥ 1,

(31)

where αn ∈ (0, 1) such that limn→∞ αn = 0,∑∞

n=1 αn = ∞, {βn} ⊂ [c, d] ⊂ (0, 1). Then{xn} converges strongly to the minimum-norm point of F.

Proof Following the method of proof of Theorem 3.1, replacing ‘J−1(J xn − λn Axn)’with ‘wn := Frn xn’, we obtain the required result provided that z ∈ V I (C, A). In fact, wecan show that z ∈ V I (C, A). First, we note that the method in the proof of Theorem 3.1and Lemma 2.3 of [38] give that

φ(p, xn+1) ≤ αnφ(p, 0) + (1 − αn)φ(p, xn) − (1 − αn)φ(wn, xn)

− (1 − βn)βng(||J yn − J T yn||).Again using similar argument we get that φ(wn, xn) → 0, as n → ∞, and hence

wn − xn → 0, as n → ∞. (32)

Now, from the definition of wn we have that

〈y − wn, Awn〉 +⟨y − wn,

Jwn − J xn

rn

⟩≥ 0, ∀ y ∈ C. (33)

and hence

〈y − wni , Auni 〉 +⟨y − wni ,

Jwni − J xni

rni

⟩≥ 0, ∀ y ∈ C. (34)

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Set vt = t y + (1 − t)z, for all t ∈ (0, 1] and y ∈ C . Consequently, we get that vt ∈ C .Now, from (34) it follows that

〈vt − wni , Avt 〉 ≥ 〈vt − wni , Avt 〉 − 〈vt − wni , Awni 〉 −⟨vt − wni ,

Jwni − J xni

rni

⟩

= 〈vt − wni , Avt − Awni 〉 −⟨vt − uni ,

Jwni − J xni

rni

⟩.

But, from (32) we have thatJwni − J xni

rni

→ 0, as i → ∞, and the monotonicity of A

implies that 〈vt − wni , Avt − Awni 〉 ≥ 0. Thus, it follows that

0 ≤ limi→∞〈vt − wni , Avt 〉 = 〈vt − z, Avt 〉,

and hence

〈y − z, Avt 〉 ≥ 0, ∀ y ∈ C.

If t → 0, the continuity of A implies that

〈y − z, Az〉 ≥ 0, ∀ y ∈ C.

This implies that z ∈ V I (C, A). �

We now give a convergence theorem to a common solution of monotone mapping anda fixed point of relatively non-expansive mapping in Banach spaces.

Corollary 3.3 Let C be a non-empty, closed and convex subset of a smooth and 2-uniformly convex real Banach space E. Let A : C → E∗ be an α-inverse strongly monotonemapping with A−1(0) closed and convex and let T : C → C be an relatively non-expansivemapping. Assume that F := A−1(0)∩ F(T ) is non-empty. Let {xn} be a sequence generatedby ⎧⎨

⎩x1 ∈ C chosen arbitrarily,yn = �C

[(1 − αn)J−1(J xn − λn Axn)

],

xn+1 = �C J−1(βn J yn + (1 − βn)J T yn), n ≥ 1,

(35)

where αn ∈ (0, 1) such that limn→∞ αn = 0,∑∞

n=1 αn = ∞, {βn} ⊂ [c, d] ⊂ (0, 1) and

{λn} is a sequence in [a, b] for some real numbers a, b such that 0 < a ≤ λn ≤ b < c2α2 , for

1c a 2-uniformly convex constant of E. Then {xn} converges strongly to the minimum-normpoint of F.

If in Theorem 3.1, we assume that A ≡ 0, then the assumption that E be 2-uniformlyconvex may be relaxed. In fact, we have the following corollary.

Corollary 3.4 Let C be a non-empty, closed and convex subset of a smooth anduniformly convex real Banach space E. Let T : C → C be an relatively non-expansive

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mapping. Assume that F := F(T ) is non-empty. Let {xn} be a sequence generated by⎧⎨⎩

x1 ∈ C, chosen arbitrarily,yn = �C

[(1 − αn)xn

],

xn+1 = J−1(βn J yn + (1 − βn)J T yn), n ≥ 1,

(36)

where αn ∈ (0, 1) such that limn→∞ αn = 0,∑∞

n=1 αn = ∞, {βn} ⊂ [c, d] ⊂ (0, 1). Then{xn} converges strongly to the minimum-norm point of F.

Proof If we put A ≡ 0 in (11) then we get that yn = �[(1 − αn)xn] and (11) reducesto (36). Therefore, following the method of proof of Theorem 3.1 we obtain the requiredassertion without the assumption that E is 2-uniformly convex. �

If in Corollary 3.3, we assume that T ≡ I, identity map on C then we get the followingcorollary.

Corollary 3.5 Let C be a non-empty, closed and convex subset of a 2-uniformly convexand smooth real Banach space E. Let A : C → E∗ be an α-inverse strongly monotonemapping with A−1(0) closed and convex. Assume that F := A−1(0) is non-empty. Let {xn}be a sequence generated by{

x1 ∈ C, chosen arbitrarily,xn+1 = �C J−1

[(1 − αn)(J xn − λn Axn)

], n ≥ 1,

(37)

where αn ∈ (0, 1) such that limn→∞ αn = 0,∑∞

n=1 αn = ∞ and {λn} is a sequence in

[a, b] for some real numbers a, b such that 0 < a ≤ λn ≤ b < c2α2 , for 1

c a 2-uniformlyconvex constant of E. Then {xn} converges strongly to the minimum-norm point of F.

Proof If we put T ≡ I , identity map on C , then (35) reduces to (37). Then the conclusionfollows from Corollary 3.3. �

If E = H , a real Hilbert space, then E is 2-uniformly convex and uniformly smooth realBanach space. In this case, J = I , identity map on H and �C = PC , projection mappingfrom H onto C . Thus, the following corollaries hold.

Corollary 3.6 Let C be a non-empty, closed and convex subset of a real Hilbert spaceH. Let A : C → H be an α-inverse strongly monotone mapping satisfying ||Ax || ≤||Ax − Ap||, ∀x ∈ C and p ∈ V I (C, A). Let T : C → C be an relatively non-expansivemapping. Assume that F := V I (C, A) ∩ F(T ) is non-empty. Let {xn} be a sequencegenerated by ⎧⎨

⎩x1 ∈ C, chosen arbitrarily,yn = PC

[(1 − αn)(xn − λn Axn)

],

xn+1 = βn yn + (1 − βn)T yn, n ≥ 1,

(38)

where αn ∈ (0, 1) such that limn→∞ αn = 0,∑∞

n=1 αn = ∞, {βn} ⊂ [c, d] ⊂ (0, 1) and{λn} is a sequence in [a, b] for some real numbers a, b such that 0 < a ≤ λn ≤ b < 2α.Then {xn} converges strongly to the minimum-norm point of F.

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Corollary 3.7 Let C be a non-empty, closed and convex subset of a real Hilbert spaceH. Let A : C → H be an α-inverse strongly monotone mapping with A−1(0) closedand convex. Let T : C → C be an relatively non-expansive mapping. Assume that F :=A−1(0) ∩ F(T ) is non-empty. Let {xn} be a sequence generated by⎧⎨

⎩x1 ∈ C, chosen arbitrarily,yn = PC

[(1 − αn)(xn − λn Axn)

],

xn+1 = βn yn + (1 − βn)T yn, n ≥ 1,

(39)

where αn ∈ (0, 1) such that limn→∞ αn = 0,∑∞

n=1 αn = ∞, {βn} ⊂ [c, d] ⊂ (0, 1) and{λn} is a sequence in [a, b] for some real numbers a, b such that 0 < a ≤ λn ≤ b < 2α.Then {xn} converges strongly to the minimum-norm point of F.

Corollary 3.8 Let C be a non-empty, closed and convex subset of a real Hilbert spaceH. Let A : C → H be an α-inverse strongly monotone mapping with A−1(0) closed andconvex. Assume that F := A−1(0) is non-empty. Let {xn} be a sequence generated by{

x1 ∈ C, chosen arbitrarily,xn+1 = PC

[(1 − αn)(xn − λn Axn)

], n ≥ 1,

(40)

where αn ∈ (0, 1) such that limn→∞ αn = 0,∑∞

n=1 αn = ∞ and {λn} is a sequence in[a, b] for some real numbers a, b such that 0 < a ≤ λn ≤ b < 2α. Then {xn} convergesstrongly to the minimum-norm point of F.

4. Applications

4.1. Convexly constrained minimization problem

In this sub-section, we study the problem of finding a minimum-norm solution of a contin-uously Fréchet differentiable convex functional in a Banach space.

Consider the optimization problem

minx∈C

f (x), (41)

where f : E → R is a convex and differentiable function and ∇ f is α-inverse stronglymonotone. Assume (41) is consistent. Then, it is wellknown [4] that the gradient projectionalgorithm {xn} generated by {

x1 ∈ C, chosen arbitrarily,

xn+1 = PC [xn − r∇ f (xn)], n ≥ 1,(42)

where 0 < r < 2α, converges weakly to a solution of (41) in Hilbert spaces. In what follows,we present an algorithm with strong convergence for solving (41) in Banach spaces. Weshall make use of the following lemma.

Lemma 4.1 [2] Let E be a Banach space, let f : E → R be a continuous Fréçhet differ-entiable convex functional and ∇ f be the gradient of f . If ∇ f is 1

α-Lipschitz continuous,

then ∇ f is α−inverse strongly monotone.

Now we state the following corollary.

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Corollary 4.2 Let C be a non-empty, closed and convex subset of a 2-uniformly convexand smooth real Banach space E. Let f : E → R be a continuously Fréchet differentiableconvex functional such that ∇ f is 1

α−Lipschtz continuous and S := {z ∈ E : f (z) =

miny∈C f (y)} �= ∅. Let {xn} be a sequence generated by{x1 ∈ C, chosen arbitrarily,xn+1 = �C J−1

[(1 − αn)(J xn − λn∇ f (xn))

], n ≥ 1,

(43)

where αn ∈ (0, 1) such that limn→∞ αn = 0,∑∞

n=1 αn = ∞ and {λn} is a sequence in

[a, b] for some real numbers a, b such that 0 < a ≤ λn ≤ b < c2α2 , for 1

c a 2-uniformlyconvex constant of E. Then {xn} converges strongly to a minimum-norm of the minimization(41).

Proof From Lemma 4.1 we have that ∇ f is an α−inverse strongly monotone mappingfrom C into E∗. Then the conclusion follows from Corollary 3.5. �

4.2. Convexly constrained linear inverse problem

Consider the convexly constrained linear inverse problem (cf. [7]){Ax = b,

x ∈ C,(44)

where H1 and H2 are real Hilbert spaces and A : H1 → H2 is a bounded linear mappingand b ∈ H2. To solve (44), we consider the following convexly constrained minimizationproblem

minx∈C

f (x) := minx∈C

1

2||Ax − b||2. (45)

In general, every solution to (44) is a solution to (45). However, a solution to the Equation(45) may not necessarily satisfy (44). Moreover, if a solution of the Equation (44) is non-empty then it follows from Lemma 4.2 of [28] that

C ∩ (� f )−1(0) �= ∅.

It is well-known that the projected Landweber method (see, [15]) given by{x1 ∈ C, chosen arbitrarily,

xn+1 = PC [xn − λA∗(Axn − b)], n ≥ 1,(46)

where 0 < λ < 2α with α = 1||A||2 converges weakly to a solution of (44). In what follows,

we present an algorithm with strong convergence for solving (44).

Corollary 4.3 Assume that the system (44) is consistent, where C is a closed and convexsubset of H1. Then, the sequence {xn} generated by{

x1 ∈ C, chosen arbitrarily,xn+1 = PC

[(1 − αn)(xn − λn A∗(Axn − b))

], n ≥ 1,

(47)

where αn ∈ (0, 1) such that limn→∞ αn = 0,∑∞

n=1 αn = ∞ and {λn} is a sequence in[a, b] for some real numbers a, b such that 0 < a ≤ λn ≤ b < 2

||A||2 . Then {xn} convergesstrongly to a solution of (44).

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Proof Let f be defined by (45). Then from [4] we have that � f (x) = A∗(Ax−b), x ∈ H1which is α−strongly monotone with α = 1

||A||2 . Thus, by Corollary 3.8 we obtain therequired assertion. �Example Now we provide examples of α-inverse strongly monotone mapping A andrelatively non-expansive mapping T with F := V I (C, A) ∩ F(T ) �= ∅. Let E = R withstandard topology and C = [−1, 1] ⊂ R. Let A : C → R be given by

Ax :={

0, −1 ≤ x ≤ 0,13 x2, 0 < x ≤ 1; (48)

and T : C → C be given by

T x :={

x, −1 ≤ x ≤ 12 ,

x − (x − 12 )2, 1

2 < x ≤ 1.(49)

Now, first we show that A is α-inverse strongly monotone with α = 12 . Let C1 = [−1, 0]

and C2 = (0, 1] then we have the following cases:

Case 1 If x, y ∈ C1 then clearly A is α-inverse strongly monotone.

Case 2 If x, y ∈ C2 then, since x + y ≤ 2, we get that

〈Ax − Ay, x − y〉 = 1

3(x2 − y2)(x − y) = 1

3(x − y)2(x + y)

= 1

62(x − y)2(x + y) ≥ 1

6(x − y)2(x + y)2

≥ 1

2

∣∣∣∣1

3(x2 − y2)

∣∣∣∣2

≥ 1

2|Ax − Ay|2.

Case 3 If x ∈ C1 and y ∈ C2, since y − x ≥ y2

6 , we have that

〈Ax − Ay, x − y〉 = 1

3y2(y − x) ≥ 1

3y2

(y2

6

)

= 1

2

(1

3y2

)2

= 1

2|Ax − Ay|2.

Therefore, from the above three cases we get that A is α-inverse strongly monotone mapping.

Next, we show that T is non-expansive. But by mean value theorem, we have that forx, y ∈ [−1, 1] there exists c ∈ (−1, 1) such that T (x) − T (y) = T ′(c)(x − y) and hence,since sup{T ′x : x ∈ (−1, 1)} ≤ 1, we get that |T x − T y| ≤ |x − y|, Thus, T is non-expansive and hence relatively non-expansive.

Furthermore, we observe that V I (C, A) = [−1, 0] and F(T ) = [−1, 12 ] and hence

V I (C, A)∩F(T ) = [−1, 0]. In addition, we see that A satisfies |Ax | ≤ |Ax−Ap|, ∀x ∈ Cand p ∈ V I (C, A).

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Now, taking αn = 1n+1 , λn = 1

10 + 1n+4 and βn := 1

3 + 1n+3 , for all n ≥ 1, then by

Corollary 3.6 we obtain that the sequence generated by x1 ∈ C, yn = PC[(1 − αn)(xn −

λn Axn)], xn+1 = βn yn + (1 − βn)T yn, n ≥ 1, converges strongly to the zero, a common

minimum-norm point of V I (C, A) and F(T ).

Remark 4.4 Corollary 3.4 improves Theorem 3.2 of Yang et al. [31] to a more generalclass of relatively non-expansive mappings in Banach spaces more general than Hilbertspaces.

Remark 4.5 We note that in the iterations schemes of our Theorems and Corollaries wehave used metric projection in Hilbert spaces and generalized projection, in the sense ofAlber [1], which plays the role of metric projection, in Banach spaces. Thus, it is worth tomention that whether we can remove the projections from those iterative schemes or not isan open problem.

Remark 4.6 It is worth to mention that we can have corollaries to Theorem 3.2 similar tocorollaries of Theorem 3.1 stated above.

AcknowledgementsThe authors would like to thank the referee for valuable suggestions which helped to improve thismanuscript.

References

[1] Ya. Alber, Metric and generalized projection operators in Banach spaces: Properties andApplications, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type(A.G. Kartsatos Ed.), Lecture notes in Pure and Appl. Math., Vol. 178, Dekker, New York, 1996,pp. 15–50.

[2] J. B. Baillon, G. Haddad, Quelques propriétés des opérateurs angle-bornés et n-cycliquementmonotones, Israel J. Math., 26 (1997), φ(wn, xn) → 0, as n → ∞, and hence137-150.

[3] RE. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banachspaces, Israel J Math. 32, (1979), 107–116.

[4] Byrne C. A unified treatment of some iterative algorithms in signal processing and imagereconstruction. Inverse Probl. 2004;20:103–120.

[5] Y. Cai, Y. Tang and L. Liu, Iterative algorithms for minimum-norm fixed point of non-expansivemapping in Hilbert space, Fixed Point Theory andApplications, 2012, (2012), doi:10.1186/1687-1812-2012-49.

[6] Dhompongsa S, Takahashi W, Yingtaweesittikul H. Weak Convergence Theorems forEquilibrium Problems With Nonlinear Operators in Hilbert Spaces. Fixed Point Theory.2011;12:309–320.

[7] Eicke B. Iteration methods for convexly constrained ill-posed problems in Hilbert space. Numer.Funct. Anal. Optim. 1992;13:413–429.

[8] Halpern B. Fixed points of nonexpansive maps. Bull. Amer. Math. Soc. 1967;73:957–961.[9] J-J Hou, Z-Y. Peng and X-L Zhang, An iterative method for the least squares symmetric solution

of matrix equation AX B = C , Numer Algor, (2006) 42: 181–192.[10] 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.

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 20:

35 0

6 Se

ptem

ber

2013

18 H. Zegeye et al.

[11] Kamimura S, Takahashi W. Strong convergence of proximal-type algorithm in a Banach space.SIAM J. Optim. 2002;13:938–945.

[12] Kinderlehrer D, Stampaccia G. An iteration to variational inequalities and their applications.New York: Academic Press; 1990.

[13] P. Kumam, A hybrid approximation method for equilibrium and fixed point problemsfor a monotone mapping and a nonexpansive, Nonlinear Anal.: Hybrid syst, (2008)doi:10.1016/j.nahs.2008.09.017.

[14] Landweber L. An iterative formula for Fredholm integral equations of the first kind. Amer. J.Math. 1951;73:615–624.

[15] Engl HW, Hanke M, Neubauer A. Regularization of Inverse Problems. Dordrecht: KluwerAcademic Publishers Group; 1996.

[16] Lions JL, Stampacchia G. Variational inequalities. Comm. Pure Appl. Math. 1967;20:493–517.[17] Maing’e PE. Strong convergence of projected subgradient methods for nonsmooth and non-

strictly convex minimization. Set-Valued Anal. 2008;16:899–912.[18] Matsushita SY, Takahashi W. A strong convergence theorem for relatively nonexpansive

mappings in a Banach space. J. Approx. Theory. 2005;134:257–266.[19] S. Reich,Aweak convergence theorem for the alternating method with Bergman distance, Theory

and Applications of Nonlinear Operators of Accretive and Monotone Type (A.G. Kartsatos Ed.),Lecture notes in Pure and Appl. Math., Vol. 178, Dekker, New York, 1996, pp. 313–318.

[20] Reich S. Approximating zeros of accretive operators. Proc Amer Math Soc. 1975;51:381–384.[21] Reich S. Strong convergence theorems for resolvents of accretive operators in Banach spaces. J

Math Anal Appl. 1980;75:287–292.[22] Song Y, Xu S. Strong convergence theorems for nonexpansive semigroup in Banach spaces. J

Math Anal Appl. 2008;338:152–161.[23] Suzuki T. Strong convergence theorems for infinite families of nonexpansive mappings in general

Banach spaces. Fixed Point Theory Appl. 2005;2005(1):103–123.[24] Tada A, Takahashi W. Weak and strong convergence theorems for nonexpansive mappings and

equilbrium problems. J. Optim. Theory Appl. 2007;133:359–370.[25] Takahashi W. Nonlinear Functional Analysis (Japanese). Tokyo: Kindikagaku; 1988.[26] Takahashi W, Ueda Y. On Reich’s strong convergence theorems for resolvents of accretive

operators. J. Math. Anal. Appl. 1984;104:546–553.[27] 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.[28] F. Wang and H. K. Xu, Approximating curve and strong convergence of the CQ algorithm for

the SPF, J. Inequal. Appl., 2010 (2010), Article ID 102085.[29] Xu HK. Another control condition in an iterative method for nonexpansive mappings. Bull.

Austral. Math. Soc. 2002;65:109–113.[30] Xu HK. Inequalities in Banach spaces with applications. Nonlinear Anal. 1991;16:1127–1138.[31] X.Yang, Y-C Liou and Y. Yao, Finding Minimum Norm Fixed Point of Nonexpansive Mappings

and Applications, Mathematical Problems in Engineering V. 2011, Article ID 106450, 13 pagesdoi:10.1155/2011/106450.

[32] Y. Yao and N. Shahzad, New methods with perturbations for non-expansive mappings in Hilbertspaces, Fixed Point Theory Appl. 2011, 2011:79, 9 pp.

[33] Yao Y, Liou YC, Shahzad N. Iterative methods for nonexpansive mappings in Banach spaces.Numer. Funct. Anal. Optim. 2011;32:583–592.

[34] D. Youla, Mathematical theory of image restoration by the method of convex projections, ImageRecovery Theory and Applications ed H Stark (Orlando: Academic) (1987) pp 29–77.

[35] Zegeye H, Ofoedu EU, Shahzad N. Convergence theorems for equilibrium problem, variotionalinequality problem and countably infinite relatively quasi-nonexpansive mappings. AppliedMath. Comput. 2010;216:3439–3449.

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nloa

ded

by [

Uni

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ity o

f R

egin

a] a

t 20:

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2013

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[36] Zegeye H, Shahzad N. Strong convergence for monotone mappings and relatively weaknonexpansive mappings. Nonlinear Anal. 2009;70:2707–2716.

[37] Zegeye H, Shahzad N. A hybrid scheme for finite families of equilibrium, variational inequalityand fixed point problems. Nonlinear Anal. 2011;74:263–272.

[38] Zegeye H, Shahzad N. Approximating common solution of variational inequality problems fortwo monotone mappings in Banach spaces. Optim. Lett. 2011;5:691–704.

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 20:

35 0

6 Se

ptem

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