approximation of minimum-norm fixed point of total asymptotically nonexpansive mapping

Afr. Mat.DOI 10.1007/s13370-014-0240-4

Approximation of minimum-norm fixed point of totalasymptotically nonexpansive mapping

E. U. Ofoedu · A. C. Nnubia

Received: 5 September 2013 / Accepted: 14 February 2014© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2014

Abstract In this paper, strong convergence theorem for approximation of minimum normfixed point of total asymptotically nonexpansive mapping in real Hilbert spaces are obtained.As applications, iterative methods for approximation of minimum norm fixed point of contin-uous pseudocontractive mapping, approximation of solutions of classical equilibrium prob-lems and approximation of solutions of convex minimization problems are proposed. Fur-thermore, iterative method for approximation of of common minimum norm fixed point offinite family of total asymptotically nonexpansive mapping is proposed. Our theorems unifyand complement many recently annouced results.

Keywords Asymptotically nonexpansive mappings · Total asymptoticallyquasi-nonexpansive mappings · Smooth real Banach spaces

Mathematics Subject Classification (2000) 47H06 · 47H09 · 47J05 · 47J25

1 Introduction

Let K be a nonempty subset of a real Hilbert space H . A mapping T : K → K is callednonexpansive if and only if for all x, y ∈ K , we have that

‖T x − T y‖ ≤ ‖x − y‖. (1)

The mapping T is called asymptotically nonexpansive mapping if and only if there existsa sequence {μn}n≥1 ⊂ [0,+∞), with limn→∞ μn = 0 such that for all x, y ∈ K ,

‖T n x − T n y‖ ≤ (1 + μn)‖x − y‖, ∀ n ∈ N. (2)

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk[11] as a generalisation of nonexpansive mappings. As further generalisation of class of non-

E. U. Ofoedu (B) · A. C. NnubiaDepartment of Mathematics, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State, Nigeriae-mail: [email protected]

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E. U. Ofoedu, A. C. Nnubia

expansive mappings, Alber et al. [2] introduced the class of total asymptotically nonexpansivemappings, where a mapping T : K −→ K is called total asymptotically nonexpansive if andonly if there exist two sequences {μn}n≥1, {ηn}n≥1 ⊂ [0,+∞), with limn→∞ μn = 0 =limn→∞ ηn and nondecreasing continous functionφ : [0,+∞) −→ [0,+∞)with φ(0) = 0such that for all x, y ∈ K ,

‖T n x − T n y‖ ≤ ‖x − y‖ + μnφ(‖x − y‖)+ ηn, n ≥ 1 (3)

Observe that if φ(t) = 0 ∀ t ∈ [0,+∞), then equation (3) becomes

‖T n x − T n y‖ ≤ ‖x − y‖ + ηn n ≥ 1, (4)

so that if K is bounded and T m is continuous for some integer m ≥ 1, then the mappingT is of asymptotically nonexpansive type. The class of asymptotically nonexpansive typemappings includes the class of mappings which are asymptotically nonexpansive in theintermediate sense and the class of nearly asymptotically nonexpansive mappings. Theseclasses of mappings had been studied extensively by several authors (see e.g. [9,11,14,17,26]).

If φ(t) = t ∀ t ∈ [0,+∞), then Eq. (3) becomes

‖T n x − T n y‖ ≤ (1 + μn)‖x − y‖ + ηn n ≥ 1 (5)

In addition, if ηn = 0 for all n ∈ N, then we easily see that every asymptotically nonex-pansive mapping is total asymptotically nonexpansive.

The following example shows that the class of total asymptotically nonexpansive mappingsproperly includes the class of asymptotically nonexpansive mappings.

Example 1 Let E = R × l1 be endowed with the norm ‖.‖E = |.|R + ‖.‖�1 . Let K be asubset of E defined by K := [0, 1] × B, where B is a closed unit ball of �1. ∀ u ∈ [0, 1] andx = (x1, x2, x3, . . .) ∈ B define T : K −→ K by

T (u, x) =⎧⎨

⎩

(13 ,

(0, |x1|2

3 , x23 ,

x33 ,

x43 , . . .

))if u ∈

[0, 1

3

]

(0,

(0, |x1|2

3 , x23 ,

x33 ,

x43 . . .

))if u ∈

[13 , 1

] (6)

We can easily check that T given by (6) is not continous and thus cannot be asymp-totically nonexpansive (since every asymptotically nonexpansive mapping is uniformily L-Lipschitizian, so Lipschitz and every Lipschitz mapping is continuous). Next, let {bn}n≥1

be a sequence of real numbers such that b1 = 13 and limn→∞ bn = 0. Observe that for all

(u, x), (v, y) ∈ K ,

‖T (u, x)− T (v, y)‖E ≤ |u − v| + b1 + 1

3max{|x1| + |y1|, 1}‖x − y‖�1 .

Moreover, we can equally check easily that for all n ≥ 2 and for all (u, x), (v, y) ∈ K

T n(u, x) =(1

3,(

0, 0, 0 . . . 0,︸︷︷︸

n−times

|x1|23n

,x2

3n,

x3

3n,

x4

3n, . . .

))

and

‖T n(u, x)− T n(v, y)‖E ≤ 1

3nmax{|x1| + |y1|, 1}‖x − y‖�1 .

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Total asymptotically nonexpansive mappings

So, for all n ≥ 1,

‖T n(u, x)− T n(v, y)‖E ≤ |u − v| + ‖x − y‖�1

+ 2

3n

[|u − v| + ‖x − y‖�1

]+ bn . (7)

Thus, with φ : [0,+∞) −→ [0,+∞) defined by φ(t) = 2t, μn = 13n for all n ≥ 1 and

{bn}n≥1 any null sequence with b1 = 13 , we obtain from (7) that

‖T n(u, x)− T n(v, y)‖E ≤ ‖(u, x)− (v, y)‖E + μnφ(‖(u, x)− (v, y)‖E )+ bn

Hence, the mapping T defined by (6) is total asymptotically nonexpansive but not asymp-totically nonexpansive.

A point x0 ∈ K is called a fixed point of a mapping T : K −→ K if and only if T x0 = x0.

We denote the set of fixed points of T by Fix(T ), that is, Fix(T ) = {x ∈ K : T x = x}. Apoint x∗ ∈ K is called a minimum norm fixed point of T if and only if x∗ ∈ Fix(T ) and‖x∗‖ = min{‖x‖ : x ∈ Fix(T )}.

Let D1 and D2 be nonempty closed convex subsets of real Hilbert spaces H1 and H2,respectively. The split feasibility problem is formulated as finding a point x satisfying

x ∈ D1 such that Ax ∈ D2, (8)

where A is bounded linear operator from HI into H2. A split feasibility problem in finitedimensional Hilbert spaces was first studied by Censor and Elfving [8] for modeling inverseproblems which arise in medical image reconstruction, image restoration and radiation ther-apy treatment planning (see e.g., [6–8]), It is clear that x ∈ D1 is a solution of the splitfeasibility problem (8) if and only if Ax − PD2 Ax = 0, where PD2 is the metric projectionfrom H2 onto D2. Consider the minimization problem:

find x∗ ∈ D1 such that1

2‖Ax∗ − PD2 Ax∗‖2 = min

x∈D1

1

2‖Ax − PD2 Ax‖2, (9)

then x∗ is a solution of (8) if and only if x∗ solves the minimization problem (9) with theminimum equal to zero. Suppose that problem (8) has solution and let � denote the (closedconvex) set of solutions of (8) (or equivalently, solution of 9), then � is a singleton if andonly if it is a set of solutions of the following variational inequality problem:

find x ∈ D1 such that 〈A∗(I − PD2)Ax, y − x〉 ≥ 0 ∀ y ∈ D2, (10)

where A∗ is the adjoint of the linear operator A. Moreover, problem (10) can be rewritten as

find x ∈ D1 such that 〈x − r A∗(I − PD2)Ax − x, y − x〉 ≤ 0 ∀ y ∈ K , (11)

where r > 0 is any positive scalar. Using the nature of projection, (11) is equivalent to thefixed point equation

x = PD1(x − r A∗(I − PD2)Ax) (12)

Thus, finding a solution of split feasibility problem (9) is equivalent to finding the minimum-norm fixed point of the mapping x −→ PD1(x − r A∗(I − PD2)Ax).

Motivated by the above split feasibility problem, we study the general case of findingthe minimum norm fixed point of total asymptotically nonexpansive mapping using approx-imation methods. Approximation of solutions of equations involving nonexpansive map-pings and their generalization by iterative methods has been of increasing research interestfor numerous mathematicians in recent years. One of the first results of this nature was

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obtained by Browder [5] for nonexpansive self mappings in Hilbert spaces. Suppose K isa closed convex nonempty subset of a real Hilbert space H . Browder [5] studied the pathu ∈ K , xt = tu + (1 − t)T xt , t ∈ (0, 1), where T : K −→ K is a nonexpansive map-ping. In [5], Browder proved that limt→0 xt exists and limt→0 xt ∈ Fix(T ). The result wasextended by Reich [20] to uniformly smooth real Banach spaces. Reich [20] proved, in fact,that limt→0 xt is a sunny nonexpansive retraction of K onto Fix(T ). In [12], Halpern studiedthe convergence of the explicit iteration method defined from x1 ∈ K by

xn+1 = αnu + (1 − αn)T xn; n ≥ 1 (13)

in the frame work of real Hilbert spaces. Under appropriate conditions on the iterative para-meter αn , it had been shown by Halpern [12], Lions [15], Wittmann [22] and Bauschke [3]that the sequence {xn} generated by (13) converges strongly to a fixed point of T nearest tou, that is, PFix(T )u, Browder and Halpern iterative methods had motivated different iterativemethods for approximation of fixed points of asymptotically nonexpansive mappings. In thisregard, Lim and Xu [14] introduced and studied the following implicit iterative method forasymptotically nonexpansive mapping T

zn = αnu + (1 − αn)Tnzn; n ≥ 1. (14)

They showed that the sequence {zn}n≥1 generated by (14) converges strongly to a fixed pointof T in the frame work of uniformly smooth real Banach spaces under suitable conditions onthe iterative parameters. In [10], Chidume et al. proved the strong convergence of the explicititerative method generated from x1, u ∈ K by

xn+1 = αnu + (1 − αn)Tn xn n ≥ 1, (15)

where limn→∞ αn = 0,∑∞

n=0 αn = +∞ and T is asymptotically nonexpansive.Alber et al. [1], obtained strong convergence of (15) for a total asymptotically nonexpan-

sive self mapping T on K in the setting of smooth reflexive real Banach space with weaklysequentially continuous duality mapping.

We note immediately that in Hilbert spaces the methods studied above are used to approx-imate the fixed point of T which is closest to the point u ∈ K . These methods can be usedto find the minimum norm fixed point of T if 0 ∈ K . If, however, 0 ∈ K , the above methodsmay fail to provide the minimum norm fixed point of T . In connection with the iterativeapproximation of minimum norm fixed point of the mapping T , Yang et al. [24] introducedan explicit iterative method generated from x1 ∈ K by

xn+1 = βnT xn + (1 − βn)PK [(1 − αn)xn], n ≥ 1. (16)

They proved under appropriate conditions on {αn}n≥1 and {βn}n≥1 that the sequence {xn}n≥1

converges strongly to the minimum norm fixed point of T in Hilbert spaces.In [23], Yao and Xu proved that the explicit iterative method generated from x1 ∈ K

defined byxn+1 = PK [(1 − αn)T xn], n ≥ 1 (17)

converges strongly to the minimum norm fixed point of nonexpansive mapping T : K −→ Kprovided that {αn}n≥1 satisfies appropriate conditions. Recently, Zegeye and Shahzad [26]proved that the iterative method generated from arbitrary x1 ∈ K by

yn = PK [(1 − αn)xn],xn+1 = βn xn + (1 − βn)T

n yn, n ≥ 1 (18)

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converges strongly to minimum norm fixed point of asymptotically nonexpansive self mapT on K .

Motivated by the results of these authors, it is our aim in this paper to prove strongconvergence theorem for minimum norm fixed point of total asymptotically nonexpansivemapping. Our theorems generalize and unify the corresponding results of Zegeye and Shahzad[26], Yao and Xu [23], Yang et al. [24].

The paper is organized as follows: In the preliminary section (Sect. 2), we present severalLemmas that will aid the proof of our main Theorem. Our main results are discussed in Sect.3 of the paper. Section 4 took care of application of our main theorem to approximation offxed point of continuous pseudocontractive mappings, while Sect. 5 dealt with applicationto approximation of solutions of classical equilibrium problems. Application of our theoremto approximation of solutions of convex optimization problem is discussed in Sect. 6. Insection seven which is the last section of the paper, we proposed an iterative scheme forapproximation of common minimum norm fixed point of finite family of total asymptoticallynonexpansive mappings; and made our concluding remark.

2 Preliminaries

We shall make use of the following lemmas.

Lemma 2 Let H be a real Hilbert space, then for all x, y ∈ H the following inequalityholds.

‖x + y‖2 ≤ ‖x‖2 + 2〈y, x + y〉Lemma 3 For any x, y, z in a real Hilbert space H and a real number λ ∈ [0, 1],

‖λx + (1 − λ)y − z‖2 = λ‖x − z‖2 + (1 − λ)‖y − z‖2 − λ(1 − λ)‖x − y‖2.

Lemma 4 [21] Let K be a closed convex nonempty subset of a real Hilbert space H. Letx ∈ H, then x0 = PK x if and only if

〈z − x0, x − x0〉 ≤ 0 ∀ z ∈ K

Let T : K −→ K be a mapping and I be the identity mapping of K , we say that (I − T )is demiclosed at zero if and only if for any sequence {xn}n≥1 in K such that xn convergesweakly to x and xn − T xn → 0, as n → ∞, we have that x = T x .

Lemma 5 (see Corollary 2.6 of [1]) Let E be a reflexive Banach space with weakly con-tinuous normalised duality mapping. Let K be a closed convex subset of E and let T bea uniformly continuous total asymptotically nonexpansive mapping from K into itself withbounded orbit, then (I − T ) is demiclosed at zero.

Lemma 6 (See e.g. [1]) Let {an}n≥1 be a sequence of nonegative real numbers satisfyingthe following relation:

an+1 ≤ an − αnan + δn, n ≥ 1.

Suppose that for n ≥ 1, δnαn

≤ c1 and αn ≤ α (for some α, c1 > 0), then an ≤ max{a1, (1 +α)c1}. Moreover, if

∑∞n=0 αn = ∞ and δn,= o(αn), then limn→∞ an = 0.

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Lemma 7 (Mainge [16]) Let {n} be sequence of real numbers that does not decrease atinfinity in the sense that there exists a subsequence {n j } of {n} which satisfiesn j < n j+1∀ j ∈ N. Define τ : N\{1, 2, 3 . . . , n0 − 1} (for some n0 ∈ N sufficiently

large) by

τ(n) = max{k ≤ n : k < k+1},then the following hold: (i) the set {k ≤ n : k < k+1} is not empty; (ii) τ(n) ≤ τ(n + 1)and τ(n) → ∞ as n → ∞ (iii) τ(n) ≤ τ(n+1) and n ≤ τ(n+1)∀ n ∈ N.

3 Main results

Proposition 8 Let K be a closed convex nonempty subset of a real Hilbert space H and letT : K → K be a continuous total asymptotically nonexpansive mapping with function φ :[0,+∞) −→ [0,+∞) satisfying φ(t) ≤ M0t ∀ t > M1 for some constants M0,M1 > 0,then Fix(T ) is closed and convex.

Proof If Fix(T ) = ∅, then we are done. Suppose Fix(T ) = ∅, then it is easy to see thatcontinuity of T implies that Fix(T ) is closed. Next, we show that Fix(T ) is convex. Fort ∈ [0, 1] and x, y ∈ Fix(T ), put z := t x + (1 − t)y, we show that z = T z. Since φis continuous, it follows that it attains its maximum (say M) in the interval [0.M1]. Thiscombined with the fact that φ(t) ≤ M0t whenever t > M1 give φ(t) ≤ M + M0t ∀ t ∈[0,∞). Using this and Lemma 3, we obtain that

‖T nz − z‖2 = ‖t (T nz − x)+ (1 − t)(T nz − y)‖2

= t‖T nz − x‖2 + (1 − t)‖T nz − y‖2 − t (1 − t)‖x − y‖2

≤ t(‖z − x‖ + μnφ(‖z − x‖)+ ηn

)2

+ (1 − t)(‖z − y‖ + μnφ(‖z − y‖)+ ηn

)2 − t (1 − t)‖x − y‖2

≤ t(‖z − x‖ + μn(M + M0‖z − x‖)+ ηn

)2

+ (1 − t)(‖z − y‖ + μn(M + M0‖z − y‖)+ ηn

)2 − t (1 − t)‖x − y‖2

= t((1 + μn M0)‖z − x‖ + (μn M + ηn)

)2

(1 − t)((1 + μn M0)‖z − y‖ + (μn M + ηn)

)2 − t (1 − t)‖x − y‖2

= t (1 − t)2(1 + μn M0)2‖x − y‖2 + (1 − t)t2(1 + μn M0)

2‖x − y‖2

− t (1 − t)‖x − y‖2 + 2t (1 + μn M0)(μn M + ηn)‖z − x‖+ t (μn M + ηn)

2 + 2(1 − t)(1 + μn M0)(μn M + ηn)‖z − y‖+ (1 − t)(μn M + ηn)

2

= t (1 − t)((1 + μn M0)

2 − 1)‖x − y‖2

+ 2(1 + μn M0)(μn M + ηn)[t‖z − x‖ + (1 − t)‖z − y‖

]

+ (μn M + ηn)2.

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Thus, limn→∞ ‖z − T nz‖ = 0, which implies that limn→∞ T nz = z. Now, by continuityof T , we obtain that z = limn→∞ T nz = limn→∞ T (T n−1z) = T (limn→∞ T n−1z) = T z.Hence, z ∈ Fix(T ), that is, Fix(T ) is convex.

Theorem 9 Let K be a closed convex nonempty subset of a real Hilbert space H andlet T : K → K be a uniformly continuous total asymptotically nonexpansive mappingwith function φ : [0,+∞) −→ [0,+∞) satisfying φ(t) ≤ M0t ∀ t > M1 for someconstants M0,M1 > 0. Suppose that Fix(T ) = ∅ and let {xn}n≥1 be a sequence generatediteratively from x1 ∈ K by

yn = PK [(1 − αn)xn],xn+1 = (1 − βn)xn + βnT n yn, n ≥ 1 (19)

where {αn}n≥1, {βn}n≥1 are sequences in (0, 1) satisfying the following conditions:∑∞

n=1 αn = ∞, limn→∞ αn = 0, limn→∞ α−1n μn = 0 = limn→∞ α−1

n ηn and 0 <

ζ0 < βn < ε0 < 1 ∀ n ≥ 1 (for some ζ0, ε0 ∈ (0, 1)), then {xn}n≥1 converges strongly toPFix(T )(0) which is a minimum norm fixed point of T .

Proof Let x∗ = PFix(T )(0) (observe that well definedness of x∗ follows from Proposition8), then from (19) and hypothesis on T, we have that

‖yn − x∗‖ = ‖PK [(1 − αn)xn] − PK x∗‖≤ ‖(1 − αn)xn − x∗‖= ‖αn(0 − x∗)+ (1 − αn)(xn − x∗)‖ (20)

≤ αn‖x∗‖ + (1 − αn)‖xn − x∗‖and

‖xn+1 − x∗‖ = ‖(1 − βn)xn + βnT n yn − x∗‖≤ (1 − βn)‖xn − x∗‖ + βn

[‖yn − x∗‖ + μnφ(‖yn − x∗‖)+ ηn

](21)

Since φ is continuous, it follows that φ attains its maximum (say M) on the interval [0,M1];moreover, φ(t) ≤ M0t whenever t > M1. Thus,

φ(t) ≤ M + M0t ∀ t ∈ [0,+∞). (22)

Using (20) and (22), we obtain from (21) that

‖xn+1 − x∗‖ ≤ (1 − βn)‖xn − x∗‖ + βn

[‖yn − x∗‖ + μn

(M + M0‖yn − x∗‖

)+ ηn

]

≤ (1 − βn)‖xn − x∗‖ + βn

{αn‖x∗‖ + (1 − αn)‖xn − x∗‖

+μn

[M + M0

(αn‖x∗‖ + (1 − αn)‖xn − x∗‖

)]+ ηn

}

=[1 − αnβn + (1 − αn)βnμn M0

]‖xn − x∗‖ + δn, (23)

where δn = (αnβn + αnβnμn M0)‖x∗‖ + βnμn M + βnηn . Since limn→∞ α−1n μn = 0, we

may assume without loss of generality that there exists k0 ∈ (0, 1) and M2 > 0 such thatα−1

n μn <1−k0

(1−αn)M0and δn

αnβn< M2. Thus, we obtain from (23) that

‖xn+1 − x∗‖ ≤ ‖xn − x∗‖ − k0αnβn‖xn − x∗‖ + δn (24)

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So, applying Lemma 6 on (24), we obtain that

‖xn − x∗‖ ≤ max{‖x1 − x∗‖, (1 + k0)M2}.Therefore, {xn}n≥1 is bounded and by (20) we obtain that {yn}n≥1 is bounded. Moreover,using Lemma 2, we obtain that

‖yn − x∗‖2 = ‖PK [(1 − αn)xn] − PK x∗‖2

≤ ‖(1 − αn)xn − x∗‖2

= ‖αn(0 − x∗)+ (1 − αn)(xn − x∗)‖2

≤ (1 − αn)2‖xn − x∗‖2 − 2αn(1 − αn)〈x∗, xn − x∗〉 + 2α2

n‖x∗‖2;and using Lemma 3, we obtain that

‖xn+1 − x∗‖2 = ‖(1 − βn)xn + βnT n yn − x∗‖2

= (1 − βn)‖xn − x∗‖2 + βn‖T n yn − x∗‖2 − βn(1 − βn)‖xn − T n yn‖2

≤ (1 − βn)‖xn − x∗‖2 + βn

[‖yn − x∗‖ + μn

(M + M0‖yn − x∗‖

)+ ηn

]2

−βn(1 − βn)‖xn − T n yn‖2

= (1 − βn)‖xn − x∗‖2 + βn

[(1 + μn M0)‖yn − x∗‖ + μn M + ηn

]2


= (1 − βn)‖xn − x∗‖2 + βn(1 + μn M0)2‖yn − x∗‖2

+βn(μn M + ηn)[2(1 + μn M0)‖yn − x∗‖2 + μn M + ηn

]


≤ (1 − βn)‖xn − x∗‖2 + βn(1 + μn M0)2[(1 − αn)‖xn − x∗‖2 (25)

−2αn(1 − αn)〈x∗, xn − x∗〉 + 2α2n‖x∗‖2

]

+βn(μn M + ηn)[2(1 + μn M0)‖yn − x∗‖2 + μn M + ηn

]


From (25), we have that

‖xn+1 − x∗‖2 ≤ ‖xn − x∗‖2 − γn‖xn − x∗‖2 − 2γn(1 − αn)〈x∗, xn − x∗〉 + θn

−βn(1 − βn)‖xn − T n yn‖2 (26)

where γn = βnαn(1+μn M0)2 and θn = βn

[(μn M +ηn)

(2(1+μn M0) supn≥1 ‖yn −x∗‖2 +

μn M + ηn

)+ (2μn M0 + μ2

n M20 ) supn≥1 ‖xn − x∗‖2

]+ 2α2

nβn(1 + μn M0)2‖x∗‖2.

Two cases arise:

Case 1: Suppose {‖xn − x∗‖}n≥1 is nonincreasing for n ≥ n0, for some n0 ∈ N, thatis, suppose that ‖xn+1 − x∗‖ ≤ ‖xn − x∗‖ ∀ n ≥ n0. Thus, limn→∞ ‖xn − x∗‖ exists andlimn→∞(‖xn+1−x∗‖−‖xn−x∗‖) = 0.Moreover, using the fact that 0 < ζ0 < βn < ε0 < 1,we obtain from (26) that limn→∞ ‖xn − T n yn‖ = 0.

Next, we observe that

‖xn+1 − xn‖ = βn‖T n yn − xn‖ (27)

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and

‖yn − xn‖ = ‖PK [(1 − αn)xn] − PK xn‖ ≤ αn‖xn‖. (28)

Furthermore,

‖yn+1 − yn‖ ≤ ‖yn+1 − xn+1‖ + ‖xn+1 − xn‖ + ‖xn − yn‖ (29)

and‖yn − T n yn‖ ≤ ‖yn − xn‖ + ‖xn − T n yn‖. (30)

So, using the fact that αn → 0 and ‖xn − T n yn‖ → 0 as n → ∞,we obtain from (27), (28),(29) and (30) that

limn→∞ ‖xn+1 − xn‖ = lim

n→∞ ‖yn − xn‖= lim

n→∞ ‖yn+1 − yn‖ = limn→∞ ‖yn − T n yn‖ = 0. (31)

Moreover, we obtain that

‖yn − T yn‖ ≤ ‖yn − yn+1‖ + ‖yn+1 − T n+1 yn+1‖+‖T n+1 yn+1 − T n+1 yn‖ + ‖T n+1 yn − T yn‖

≤ ‖yn − yn+1‖ + ‖yn+1 − T n+1 yn+1‖ + ‖yn − yn+1‖+μn+1φ(‖yn − yn+1‖)+ ηn+1 + ‖T (T n)yn − T yn‖.

This gives,

‖yn − T yn‖ ≤ ‖yn − yn+1‖ + ‖yn+1 − T n+1 yn+1‖ + ‖yn − yn+1‖+μn+1

(M + M0‖yn − yn+1‖

)+ ηn+1 + ‖T (T n)yn − T yn‖

≤ ‖yn+1 − T n+1 yn+1‖ + (2 + μn+1 M + μn+1 M0)‖yn − yn+1‖+ηn+1 + ‖T (T n)yn − T yn‖ (32)

Using (31) and uniform continuity of the mapping T , we obtain from (32) that

limn→∞ ‖yn − T yn‖ = 0. (33)

But

‖xn − T xn‖ ≤ ‖xn − yn‖ + ‖yn − T yn‖ + ‖T yn − T xn‖ for all n ∈ N. (34)

So, using (31), uniform continuity of T and (33), we obtain from (34) that

limn→∞ ‖xn − T xn‖ = 0. (35)

Now, let {xnk }k≥1 be a subsequence of {xn}n≥1 such that

lim infn→∞ 〈x∗, xn − x∗〉 = lim

k→∞〈x∗, xnk − x∗〉, (36)

then (since {xnk }k≥1 is a bounded sequence in H and H is a reflexive real Banach space),there exists a subsequence {xnk j

} j≥1 of {xnk }k≥1 such that {xnk j} j≥1 converges weakly to

some z ∈ H . Thus, (36) gives

lim infn→∞ 〈x∗, xn − x∗〉 = lim

k→∞〈x∗, xnk − x∗〉 = limj→∞〈x∗, xnk j

− x∗〉. (37)

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Furthermore, since by (35), lim j→∞ ‖xnk j− T xnk j

‖ = 0 and by Lemma 5, (I − T ) is

demiclosed at zero, we obtain that z ∈ Fix(T ). So, using (37) and the fact that x∗ =PFix(T )(0), we obtain from Lemma 4 that

lim supn→∞

(− 〈x∗, xn − x∗〉

)= − lim inf

n→∞ 〈x∗, xn − x∗〉= − lim

j→∞〈x∗, xnk j− x∗〉 = −〈x∗, z − x∗〉 ≤ 0. (38)

Therefore, defining

ζn := max{0,−〈x∗, xn − x∗〉}, (39)

then it is easy to see that limn→∞ ζn = 0. Moreover, we obtain from (26) (using (39)) that

‖xn+1 − x∗‖2 ≤ (1 − γn)‖xn − x∗‖2 − 2γn(1 − αn)〈x∗, xn − x∗〉 + θn

≤ (1 − γn)‖xn − x∗‖2 + 2γn(1 − αn)ζn + θn

= (1 − γn)‖xn − x∗‖2 + σn, (40)

where σn = 2γn(1 − αn)ζn + θn . Conditions on our iterative parameters easily give σn =o(γn). Hence, we obtain from (40) using Lemma 6 that {xn}n≥1 converges strongly to x∗ =PF (0).

Case 2: Set n := ‖xn − x∗‖ for all n ∈ N. Suppose that there exists a subsequence {n j } j≥1

of {n}n≥1 such that n j < n j+1 for all j ∈ N. Let τ : N → N be defined for all n ≥ n0

(for some n0 ∈ N sufficiently large) by

τ(n) = max{k ∈ N : k ≤ n, k < k+1}.Then, by Lemma 7, we obtain that {τ(n)}n≥1 is a non-decreasing sequence such thatlimn→∞ τ(n) = +∞ and τ(n) ≤ τ(n+1) for all n ≥ n0. That is, ‖xτ(n) − x∗‖ ≤‖xτ(n+1) − x∗‖ for all n ≥ n0. In other words, {‖xτ(n) − x∗‖}n≥1 is non-decreasing. So,limn→∞ ‖xτ(n) − x∗‖ exists. Moreover, we obtain from (26) that

γτ(n)‖xτ(n) − x∗‖2 ≤ ‖xτ(n) − x∗‖2 − ‖xτ(n+1) − x∗‖2

−2γτ(n)〈x∗, xτ(n) − x∗〉 + θτ(n) ∀ n ∈ N. (41)

Inequality (41) implies that

‖xτ(n) − x∗‖2 ≤ −2〈x∗, xτ(n) − x∗〉 + θτ(n)

γτ(n)∀ n ∈ N. (42)

Observe that following the argument of case 1, we obtain that limn→∞ ‖xτ(n+1) − xτ(n)‖ =limn→∞ ‖xτ(n) − T xτ(n)‖ = 0 and lim supn→∞(−〈x∗, xτ(n) − x∗〉) ≤ 0. Thus, settingζτ(n) := max{0,−〈x∗, xτ(n) − x∗〉}, we obtain that ζτ(n) → 0 as n → ∞. Furthermore, it

could be seen easily that from conditions on our iterative parameters, we obatin θτ(n)γτ(n)

→ 0as n → ∞. So, we obtain from (42) that

‖xτ(n) − x∗‖2 ≤ 2ζτ(n) + θτ(n)

γτ(n)∀ n ∈ N.

This inequality, therefore, implies that limn→∞ ‖xτ(n) − x∗‖ = 0. But Lemma 7 furthergives that n ≤ τ(n+1) for all n ∈ N. This implies that ‖xn − x∗‖ ≤ ‖xτ(n+1) − x∗‖ for alln ∈ N. Thus, we obtain by sandwich theorem that limn→∞ ‖xn − x∗‖ = 0. Hence, {xn}n≥1

converges strongly to x∗ = PFix(T )(0), the minimum norm fixed point of the mapping T .This completes that proof.

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The folowing corollaries easily follow from Theorem 9 above:

Corollary 10 Let K be a closed convex nonempty subset of a real Hilbert space H and letT be asymptotically nonexpansive mapping from K into itself with sequences {μn}n≥1 ⊂[0,+∞). Suppose that Fix(T ) = ∅ and let {xn}n≥1 be a sequence generated iterativelyfrom x1 ∈ K by

yn = PK [(1 − αn)xn],xn+1 = (1 − βn)xn + βnT n yn, n ≥ 1, (43)


n=1 αn = ∞, limn→∞ αn = 0, limn→∞ α−1n μn = 0 and 0 < ζ < βn < ε < 1 ∀ n ≥

1, then {xn}n≥1 converges strongly to PFix(T )(0) which is a minimum norm fixed point of T .

Corollary 11 Let K be a closed convex nonempty subset of a real Hilbert space H and letT be a nonexpansive mapping from K into itself. Suppose that Fix(T ) = ∅ and let {xn}n≥1

be a sequence generated iteratively from x1 ∈ K by

yn = PK [(1 − αn)xn],xn+1 = (1 − βn)xn + βnT yn, n ≥ 1, (44)


n=1 αn = ∞, limn→∞ αn = 0, and 0 < ζ < βn < ε < 1 ∀ n ≥ 1, then {xn}n≥1

converges strongly to PFix(T )(0) which is a minimum norm fixed point of T .

4 Application to approximation of fixed points of continuous pseudocontractivemappings

The most important generalization of the class of nonexpansive mappings is, perhaps, theclass of pseudocontractive mappings. These mappings are intimately connected with theimportant class of nonlinear monotone operators. This connection will be made precise inwhat follows.

A mapping T ′ with domain D(T ′), and range R(T ′), in real Hilbert space H is calledpseudocontractive if and only if for all x, y ∈ D(T ′), the following inequality holds:

‖x − y‖ ≤ ‖(1 + r)(x − y)− r(T ′x − T ′y)‖ (45)

for all r > 0. Inequality (45) could be shown to be equivalent to the following inequality:

for any x, y ∈ D(T ′), 〈T ′x − T ′y, x − y〉 ≤ ‖x − y‖2. (46)

It now follows trivially from (46) that every nonexpansive mapping is pseudocontractive. Wenote immediately that the class of pseudocontractive mappings is larger than that of nonex-pansive mappings. For examples of pseudocontractive mappings which are not nonexpansive,the reader may see [9].

To see the connection between the class of pseudocontractive mappings and the monotonemappings, we give the following definition: a mapping A with domain, D(A), and range,R(A), in H is called monotone if and only if for all x, y ∈ D(A), the following inequalityis satisfied:

‖x − y‖ ≤ ‖x − y + r(Ax − Ay)‖ (47)

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for all r > 0; or equivalently,

for any x, y ∈ D(A), 〈Ax − Ay, x − y〉 ≥ 0. (48)

It is easy to see from inequalities (45) and (47) that an operator A is monotone if and onlyif the mapping T ′ := (I − A) is pseudocontractive. Consequently, the fixed point theory forpseudocontractive mappings is intimately connected with the mapping theory of monotoneoperators. For the importance of monotone operators and their connections with evolutionequations, the reader may consult [9,19].

Due to the above connection, fixed point theory of pseudocontractive mappings became aflourishing area of intensive research for several authors. Recently, Zegeye [27] establishedthe following Lemmas:

Lemma 12 (See Zegeye [27]) Let K be a nonempty closed convex subset of a real Hilbertspace H. Let T ′ : K → H be a continuous pseudocontractive mapping, then for all r > 0and x ∈ H, there exists z ∈ K such that

〈y − z, T ′z〉 − 1

r〈y − z, (1 + r)z − x〉 ≤ 0 ∀ y ∈ K .

Lemma 13 (See Zegeye [27]) Let K be a nonempty closed convex subset of a real Hilbertspace H. Let T ′ : K → K be a continuous pseudocontractive mapping, then for all r > 0and x ∈ H, define a mapping Fr : H → K by

Fr (x) ={

z ∈ K : 〈y − z, T ′z〉 − 1

r〈y − z, (1 + r)z − x〉 ≤ 0 ∀ y ∈ K

}

,

then the following hold:

(1) Fr is single-valued;(2) Fr is firmly nonexpansive type mapping, i.e., for all x, y ∈ H,

‖Fr (x)− Fr (y)‖2 ≤ 〈Fr (x)− Fr (y), x − y〉;(3) Fix(Fr ) is closed and convex; and Fix(Fr ) = Fix(T ′) for all r > 0.

Remark 14 We observe that Lemmas 12 and 13 hold in particular for r = 1. Thus, if T ′ acontinuous pseudocontractive mapping and we define

F1x = {z ∈ K : 〈y − z, T ′z〉 − 〈y − z, 2z − x〉 ≤ 0 ∀ y ∈ K }, (49)

then F1 satisfies the conditions of Lemma 13. Hence, we easily see that F1 is nonexpansiveand Fix(F1) = Fix(T ′). Thus, we have the following theorem:

Theorem 15 Let K be a closed convex nonempty subset of a real Hilbert space, H and let T ′be a pseudocotractive mapping from K into itself. Suppose that Fix(T ) = ∅ and let {xn}n≥1

be a sequence generated iteratively from x1 ∈ K by

yn = PK [(1 − αn)xn],xn+1 = (1 − βn)xn + βn F1(yn), n ≥ 1 (50)



converges strongly to PF(T ′)(0) which is a minimum norm fixed point of T ′.

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5 Application to approximation of solutions of classical equilibrium problems

Let K be a closed convex nonempty subset of a real Hilbert space H . Let f : K × K → R

be a bifunction The classical equilibrium problem (abbreviated E P) for f is to find u∗ ∈ Ksuch that

f (u∗, y) ≥ 0 ∀ y ∈ K . (51)

The set of solutions of classical equilibrium problem is denoted by E P( f ), where

E P( f ) = {u ∈ K : f (u, y) ≥ 0 ∀ y ∈ K }.The classical equilibrium problem (E P) includes as special cases the monotone inclu-

sion problems, saddle point problems, variational inequality problems, minimization prob-lems, optimization problems, vector equilibrium problems, Nash equilibria in noncoopera-tive games. Furthermore, there are several other problems, for example, the complementarityproblems and fixed point problems, which can also be written in the form of the classical equi-librium problem. In other words, the classical equilibrium problem is a unifying model forseveral problems arising from engineering, physics, statistics, computer science, optimiza-tion theory, operations research, economics and countless other fields. For the past 20 yearsor so, many existence results have been published for various equilibrium problems (see e.g.,[4,13,25]).

In the sequel, we shall require that the bifunction f : K × K → R satisfies the followingconditions:

(A1) f (x, x) = 0 ∀ x ∈ K ;(A2) f is monotone, in the sense that f (x, y)+ f (y, x) ≤ 0 for all x, y ∈ K ;(A3) lim supt→0+ f

(t z + (1 − t)x, y

) ≤ f (x, y) for all x, y, z ∈ K ;(A4) the function y �→ f (x, y) is convex and lower semicontinuous for all x ∈ K .

Lemma 16 (Compare with Lemma 2.4 of [13]) Let K be a closed convex nonempty subset ofa real Hilbert space H. Let f : K × K → R be a bifunction satisfying conditions (A1)–(A4).Then, for all r > 0 and x ∈ H there exists u ∈ K such that

f (u, y)+ 1

r〈y − u, u − x〉 ≥ 0 ∀ y ∈ K .

Moreover, if for all x ∈ H we define a mapping Gr : H → 2K by

Gr (x) ={

u ∈ K : f (u, y)+ 1

r〈y − u, u − x〉 ≥ 0, ∀ y ∈ K

},

then the following hold:

(1) Gr is single-valued for all r > 0;(2) Gr is firmly nonexpansive, that is, for all x, z ∈ H,

‖Gr (x)− Gr (z)‖2 ≤ 〈Gr (x)− Gr (z), x − z〉;(3) Fix(Gr ) = E P( f ) for all r > 0;(4) E P( f ) is closed and convex.

Remark 17 We observe that Lemma 16 holds in particular for r = 1. Thus, if we define

G1(x) ={

u ∈ K : f (u, y)+ 〈y − u, u − x〉 ≥ 0, ∀ y ∈ K}, (52)

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then G1 satisfies the conditions of Lemma 16. Hence, we easily see that G1 is nonexpansiveand Fix(G1) = E P( f ). Thus, we have the following theorem:

Theorem 18 Let K be a closed convex nonempty subset of a real Hilbert space H and letf : K × K → R be a bifunction satisfying conditions (A1)–(A4). Suppose that E P( f ) = ∅and suppose that {xn}n≥1 is a sequence generated iteratively from x1 ∈ K by

yn = PK [(1 − αn)xn],xn+1 = (1 − βn)xn + βnG1(yn), n ≥ 1 (53)



converges strongly to PE P( f )(0) which is a minimum norm fixed point of T .

Remark 19 Several authors (see e.g. [13,18] and references therein) have studied the fol-lowing problem: Let K be a closed convex nonempty subset of a real Hilbert space H . Letf : K × K → R be a bifunction and � : K → R ∪ {+∞} be a proper extended real valuedfunction, where R denotes the set of real numbers. Let� : K → H be a nonlinear monotonemapping. The generalized mixed equilibrium problem (abbreviated G M E P) for f, � and� is to find u∗ ∈ K such that

f (u∗, y)+�(y)−�(u∗)+ 〈�u∗, y − u∗〉 ≥ 0 ∀ y ∈ K . (54)

The set of solutions for G M E P (54) is denoted by

G M E P( f,�,�) = {u ∈ K : f (u, y)+�(y)−�(u)+ 〈�u, y − u〉 ≥ 0 ∀ y ∈ K }.These authors always claim that if � ≡ 0 ≡ � in (54), then (54) reduces to the classical

equilibrium problem (abbreviated E P), that is, the problem of finding u∗ ∈ K such that

f (u∗, y) ≥ 0 ∀ y ∈ K (55)

and G M E P( f, 0, 0) is denoted by E P( f ), where

E P( f ) = {u ∈ K : f (u, y) ≥ 0 ∀ y ∈ K }.If f ≡ 0 ≡ � in (54), then G M E P (54) reduces to the classical variational inequalityproblem and G M E P(0, 0,�) is denoted by V I (�, K ), where

V I (�, K ) = {u ∈ K : 〈�u, y − u〉 ≥ 0 ∀ y ∈ K }.If f ≡ 0 ≡ �, then G M E P (54) reduces to the following minimization problem:

find u∗ ∈ K such that �(y) ≥ �(u∗) ∀ y ∈ K ;and G M E P(0,�, 0) is denoted by Argmin(�), where

Argmin(�) = {u ∈ K : �(u) ≤ �(y) ∀ y ∈ K }.If � ≡ 0, then (54) becomes the mixed equilibrium problem (abbreviated M E P) andG M E P( f,�, 0) is denoted by M E P( f,�), where

M E P( f,�) = {u ∈ K : f (u, y)+�(y)−�(u) ≥ 0 ∀ y ∈ K }.If� ≡ 0, then (54) reduces to the generalized equilibrium problem (abbreviated, G E P) andG M E P( f, 0,�) is denoted by G E P( f,�), where

G E P( f,�) = {u ∈ K : f (u, y)+ 〈�u, y − u〉 ≥ 0 ∀ y ∈ K }.

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If f ≡ 0, then G M E P (54) reduces to the generalized variational inequality problem(abbreviated GV I P) and G M E P(0,�,�) is denoted by GV I P(�,�, K ), where

GV I P(�,�, K ) = {u ∈ K : �(y)−�(u)+ 〈�u, y − u〉 ≥ 0 ∀ y ∈ K }.It is worthy to note that if we define : K × K → R by

(x, y) = f (x, y)+�(y)−�(x)+ 〈�x, y − x〉,then it could be easily checked that is a bi-function and satisfies properties (A1)–(A4).Thus, the so called generalized mixed equilibrium problem reduces to the classical equilib-rium problem for the by function . Thus, consideration of the so called generalized mixedequilibrium problem in place of the classical equilibrium problem studied in this sectionleads to no further generalization.

6 Applications to convex optimization

Let us look at the problem of minimizing a continuously Frechet-differentiable convex func-tional with minimum norm in Hilbert spaces.

Let K be a closed convex subset of a real Hilbert space H , Consider the minimizationproblem given by

minx∈K

φ(x) (56)

where φ is a Frechet-differentiable convex functional. Let � the solution set of (56) benonempty. It is known that a point z ∈ K is a solution of (56) if and only if the followingoptimality condition holds:

z ∈ K , 〈∇φ(z), x − z〉 ≥ 0, x ∈ K , (57)

where ∇ is the gradient of φ at x ∈ K . It is also known that the optimality condition (57) isequivalent to the following fixed point problem:

z = Tγ (z), where Tγ := PK (I − γ∇φ). (58)

for all γ > 0 So we have the following corollary deduced from theorem 9.

Theorem 20 Let H be a real Hilbert space, let K be a closed convex nonempty subsetof H. Let ψ be a continuously Frechet-differentiable convex functional on K such thatTγ := PK (I −γ∇ψ) be a uniformly continuous total asymptotically nonexpansive mappingfrom K into itself with sequences {μn}n≥1, {ηn}n≥1 ⊂ [0,+∞) such that limn→∞ μn = 0 =limn→∞ ηn, and with function φ : [0,+∞) −→ [0,+∞) satisfying φ(t) ≤ M0t ∀ t > M1,for some constants M0,M1 > 0. Suppose that Fix(T ) = ∅ and let {xn}n≥1 be a sequencegenerated iteratively from x1 ∈ K by

yn = PK [(1 − αn)xn],xn+1 = (1 − βn)xn + βn[PK (I − γ∇ψ)]n yn, n ≥ 1 (59)


n=1 αn = ∞, limn→∞ αn = 0, limn→∞ α−1n μn = 0 = limn→∞ α−1

n ηn and 0 < ζ <

βn < ε < 1 ∀ n ≥ 1, then {xn}n≥1 converges strongly to the minimum norm solution of theminimization problem (56).

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7 Iterative scheme for approximation of common minimum norm fixed point of finitefamily of total asymptotically nonexpansive mappings

For approximation of common minimum norm fixed point of finite family {Ti }mi=1 of total

asymptotically nonexpansive mappings, we propose the following: Let {xn} generated itera-tively from x1,∈ K by

yn = PK [(1 − αn)xn],

xn+1 = β0,n xn +m∑

i=1

βi,nT ni yn, n ≥ 1 (60)

Infact we state the following theorem.

Theorem 21 Let K be a closed convex nonempty subset of a real Hilbert space, H and let{Ti }m

i=1 be a finite family uniformly continuous total asymptotically nonexpansive mappingfrom K into itself with sequences {μi,n}n≥1, {ηi,n}n≥1 ⊂ [0,+∞) such that limn→∞ μi,n =0 = limn→∞ ηi,n for each i = 1, 2, . . . N, and with function φi : [0,+∞) −→ [0,+∞)

satisfying φi (t) ≤ Mi,0t ∀ t > Mi,1, for some constants Mi,0,Mi,1 > 0, i = 1, 2, . . . .Suppose that � = ⋂m

i=1 Fix(Ti ) = ∅ and let {xn}n≥1 be a sequence generated iterativelyfrom x1 ∈ K by (60) where {αn}n≥1, {βi,n}n≥1 are sequences in (0, 1) satisfying the followingconditions:

∑∞n=1 αn = ∞, limn→∞ αn = 0, limn→∞ α−1

n μi,n = 0 = limn→∞ α−1n ηi,n, 0 < ζ <

βi,n < ε0 < 1 ∀ n ≥ 1, i = 1, 2, . . . N, then {xn}n≥1 converges strongly to P�(0) which isa minimum norm fixed point of T .

Remark 22 We skip the proof of Theorem 21 because the method of proof of this theoremis basically a slight modification the methods of proof of Theorem 9 of this paper combinedwith method of proof of Theorem 3.1 of Zegeye and Shahzad [26], using Lemma 2.2 of[26] in addition. Our Theorems generalize, improve and unify most of the results that havebeen announced for the class of asymptotically nonexpansive mappings of which the resultsobtained in [26] are examples. It is worthy to note that replica of the theorems obtained inSects. 4, 5 and 6 of this paper are obtainable for finite families of corresponding problems.Applications of our theorems are therefore of independent interest.

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