The purpose of this paper is to introduce the modified Halpern-type iterative method by the generalized f-projection operator for finding a common solution of fixed-point problem of a totally quasi-ϕ-asymptotically nonexpansive mapping and a system of equilibrium problems in a uniform smooth and strictly convex Banach space with the Kadec-Klee property. Consequently, we prove the strong convergence for a common solution of above two sets. Our result presented in this paper generalize and improve the result of Chang et al., (2012), and some others.

1. Introduction

In 1953, Mann [1] introduced the following iteration process which is now known as Mann′s iteration:

()

where T is nonexpansive, the initial guess element x₁ ∈ C is arbitrary, and {α_n} is a sequence in [0, 1]. Mann iteration has been extensively investigated for nonexpansive mappings. In an infinite-dimensional Hilbert space, Mann iteration can conclude only weak conviergence (see [2, 3]).

Later, in 1967, Halpern [4] considered the following algorithm:

()

where T is nonexpansive. He proved the strong convergence theorem of {x_n} to a fixed point of T under some control condition {α_n}. Many authors improved and studied the result of Halpern [4] such as Qin et al. [5], Wang et al. [6], and reference therein.

In 2008-2009, Takahashi and Zembayashi [7, 8] studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of the Banach spaces.

On the other hand, Li et al. [9] introduced the following hybrid iterative scheme for approximation fixed points of relatively nonexpansive mapping using the generalized f-projection operator in a uniformly smooth real Banach space which is also uniformly convex. They obtained strong convergence theorem for finding an element in the fixed point set of T.

Recently, Ofoedu and Shehu [10] extended algorithm of Li et al. [9] to prove a strong convergence theorem for a common solution of a system of equilibrium problem and the set of common fixed points of a pair of relatively quasi-nonexpansive mappings in the Banach spaces by using generalized f-projection operator. Chang et al. [11] extended and improved Qin and Su [12] to obtain a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem, the set of solutions for a variational inequality problem, and the set of common fixed points for a pair of relatively nonexpansive mappings in a Banach space.

Very recently, Chang et al. [13] extended the results of Qin et al. [5] and Wang et al. [6] to consider a modification to the Halpern-type iteration algorithm for a total quasi-ϕ-asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of Banach spaces.

The purpose of this paper is to be motivated and inspired by the works mentioned above, we introduce a modified Halpern-type iterative method by using the new hybrid projection algorithm of the generalized f-projection operator for solving the common solution of fixed point for totally quasi-ϕ-asymptoically nonexpansive mappings and the system of equilibrium problems in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. The results presented in this paper improve and extend the corresponding ones announced by many others.

2. Preliminaries and Definitions

Let E be a real Banach space with dual E^*, and let C be a nonempty closed and convex subset of E. Let {θ_i} _i∈Γ : C × C → ℝ be a bifunction, where Γ is an arbitrary index set. The system of equilibrium problems is to find x ∈ C such that

()

If Γ is a singleton, then problem (2.1) reduces to the equilibrium problem, which is to find x ∈ C such that

()

A mapping T from C into itself is said to be nonexpansive if

()

T is said to be asymptotically nonexpansive if there exists a sequence {k_n}⊂[1, ∞) with k_n → 1 as n → ∞ such that

()

T is said to be total asymptotically nonexpansive if there exist nonnegative real sequences ν_n, μ_n with ν_n → 0, μ_n → 0 as n → ∞ and a strictly increasing continuous function φ : ℝ⁺ → ℝ⁺ with φ(0) = 0 such that

()

A point x ∈ C is a fixed point of T provided Tx = x. Denote by F(T) the fixed point set of T; that is, F(T) = {x ∈ C : Tx = x}. A point p in C is called an asymptotic fixed point of T if C contains a sequence {x_n} which converges weakly to p such that lim _n→∞∥x_n − Tx_n∥ = 0. The asymptotic fixed point set of T is denoted by

The normalized duality mapping

is defined by J(x) = {x^* ∈ E^* : 〈x, x^*〉 = ∥x∥², ∥x^*∥ = ∥x∥}. If E is a Hilbert space, then J = I, where I is the identity mapping. Consider the functional defined by

()

where J is the normalized duality mapping and 〈·, ·〉 denote the duality pairing of E and E^*.

If E is a Hilbert space, then ϕ(y, x) = ∥y − x∥². It is obvious from the definition of ϕ that

()

A mapping T from C into itself is said to be ϕ-nonexpansive [14, 15] if

()

T is said to be quasi-ϕ-nonexpansive [14, 15] if F(T) ≠ ∅ and

()

T is said to be asymptotically ϕ-nonexpansive [15] if there exists a sequence {k_n}⊂[0, ∞) with k_n → 1 as n → ∞ such that

()

T is said to be quasi-ϕ-asymptotically nonexpansive [15] if F(T) ≠ ∅ and there exists a sequence {k_n}⊂[0, ∞) with k_n → 1 as n → ∞ such that

()

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

()

A mapping T from C into itself is said to be closed if for any sequence {x_n} ⊂ C such that lim _n→∞ x_n = x₀ and lim _n→∞ Tx_n = y₀, then Tx₀ = y₀.

Alber [16] introduced the generalized projection Π_C : E → C is a map that assigns to an arbitrary point x ∈ E the minimum point of the functional ϕ(x, y); that is,

, where

is the solution of the minimization problem:

()

The existence and uniqueness of the operator Π_C follows from the properties of the functional ϕ(y, x) and the strict monotonicity of the mapping J (see, e.g., [16–20]). If E is a Hilbert space, then ϕ(x, y) = ∥x − y∥² and Π_C becomes the metric projection P_C : H → C. If C is a nonempty, closed, and convex subset of a Hilbert space H, then P_C is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently, it is not available in more general Banach spaces. Later, Wu and Huang [21] introduced a new generalized f-projection operator in the Banach space. They extended the definition of the generalized projection operators and proved some properties of the generalized f-projection operator. Next, we recall the concept of the generalized f-projection operator. Let G : C × E^* → ℝ ∪ {+∞} be a functional defined by

()

where y ∈ C, ϖ ∈ E^*, ρ is positive number, and f : C → ℝ ∪ {+∞} is proper, convex, and lower semicontinuous. From the definition of G, Wu and Huang [21] proved the following properties:

(1)
G(y, ϖ) is convex and continuous with respect to ϖ when y is fixed;
(2)
G(y, ϖ) is convex and lower semicontinuous with respect to y when ϖ is fixed.

Definition 2.1. Let E be a real Banach space with its dual E^*. Let C be a nonempty, closed, and convex subset of E. We say that is a generalized f-projection operator if

()

A Banach space E with norm ∥·∥ is called strictly convex if ∥(x + y)/2∥<1 for all x, y ∈ E with ∥x∥ = ∥y∥ = 1 and x ≠ y. Let U = {x ∈ E : ∥x∥ = 1} be the unit sphere of E. A Banach space E is called smooth if the limit lim _t→0 ((∥x + ty∥−∥x∥)/t) exists for each x, y ∈ U. It is also called uniformly smooth if the limit exists uniformly for all x, y ∈ U. The modulus of smoothness of E is the function ρ_E : [0, ∞)→[0, ∞) defined by ρ_E(t) = sup {(∥x + y∥+∥x − y∥)/2 − 1 : ∥x∥ = 1, ∥y∥≤t}. The modulus of convexity of E (see [22]) is the function δ_E : [0,2]→[0,1] defined by δ_E(ɛ) = inf {1 − ∥(x + y)/2∥:x, y ∈ E, ∥x∥ = ∥y∥ = 1, ∥x − y∥≥ɛ}. In this paper we denote the strong convergence and weak convergence of a sequence {x_n} by x_n → x and x_n⇀x, respectively.

Remark 2.2. The basic properties of E, E^*, J, and J⁻¹ (see [18]) are as follows.

(i)
If E is an arbitrary Banach space, then J is monotone and bounded.
(ii)
If E is a strictly convex, then J is strictly monotone.
(iii)
If E is a smooth, then J is single valued and semicontinuous.
(iv)
If E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.
(v)
If E is reflexive smooth and strictly convex, then the normalized duality mapping J is single valued, one-to-one, and onto.
(vi)
If E is a reflexive strictly convex and smooth Banach space and J is the duality mapping from E into E^*, then J⁻¹ is also single valued, bijective, and is also the duality mapping from E^* into E, and thus and J⁻¹J = I_E.
(vii)
If E is uniformly smooth, then E is smooth and reflexive.
(viii)
E is uniformly smooth if and only if E^* is uniformly convex.
(ix)
If E is a reflexive and strictly convex Banach space, then J⁻¹ is norm-weak*-continuous.

Remark 2.3. If E is a reflexive, strictly convex, and smooth Banach space, then ϕ(x, y) = 0, if and only if x = y. It is sufficient to show that if ϕ(x, y) = 0 then x = y. From (2.6), we have ∥x∥ = ∥y∥. This implies that 〈x, Jy〉 = ∥x∥² = ∥Jy∥². From the definition of J, one has Jx = Jy. Therefore, we have x = y (see [18, 20, 23] for more details).

Recall that a Banach space E has the Kadec-Klee property [18, 20, 24], if for any sequence {x_n} ⊂ E and x ∈ E with x_n⇀x and ∥x_n∥→∥x∥, then ∥x_n − x∥→0 as n → ∞. It is well known that if E is a uniformly convex Banach space, then E has the Kadec-Klee property.

We also need the following lemmas for the proof of our main results.

Lemma 2.4 (see Change et al. [25].)Let C be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let T : C → C be a closed and total quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequence ν_n and μ_n with ν_n → 0, μ_n → 0 as n → ∞ and a strictly increasing continuous function ζ : ℝ⁺ → ℝ⁺ with ζ(0) = 0. If μ₁ = 0, then the fixed point set F(T) is a closed convex subset of C.

Lemma 2.5 (see Wu and Hung [21].)Let E be a real reflexive Banach space with its dual E^* and C a nonempty, closed, and convex subset of E. The following statement hold:

(1)
is a nonempty, closed and convex subset of C for all ϖ ∈ E^*;
(2)
if E is smooth, then for all ϖ ∈ E^*, if and only if
()
(3)
if E is strictly convex and f : C → ℝ ∪ {+∞} is positive homogeneous (i.e., f(tx) = tf(x) for all t > 0 such that tx ∈ C where x ∈ C), then is single-valued mapping.

Lemma 2.6 (see Fan et al. [26].)Let E be a real reflexive Banach space with its dual E^* and C be a nonempty, closed and convex subset of E. If E is strictly convex, then is single valued.

Recall that J is single-valued mapping when E is a smooth Banach space. There exists a unique element ϖ ∈ E^* such that ϖ = Jx where x ∈ E. This substitution in (2.14) gives

()

Now we consider the second generalized f projection operator in Banach space (see [9]).

Definition 2.7. Let E be a real smooth Banach space, and let C be a nonempty, closed, and convex subset of E. We say that is generalized f-projection operator if

()

Lemma 2.8 (see Deimling [27].)Let E be a Banach space, and let f : E → ℝ ∪ {+∞} be a lower semicontinuous convex function. Then there exist x^* ∈ E^* and α ∈ ℝ such that

()

Lemma 2.9 (see Li et al. [9].)Let E be a reflexive smooth Banach space, and let C be a nonempty, closed, and convex subset of E. The following statements hold:

(1)
is nonempty, closed and convex subset of C for all x ∈ E;
(2)
for all x ∈ E, if and only if
()
(3)
if E is strictly convex, then is single-valued mapping.

Lemma 2.10 (see Li et al. [9].)Let E be a real reflexive smooth Banach space, let C be a nonempty, closed, and convex subset of E, x ∈ E, and let . Then

()

Remark 2.11. Let E be a uniformly convex and uniformly smooth Banach space and f(x) = 0 for all x ∈ E, then Lemma 2.10 reduces to the property of the generalized projection operator considered by Alber [16].

If f(y) ≥ 0 for all y ∈ C and f(0) = 0, then the definition of totally quasi-ϕ-asymptotically nonexpansive T is equivalent to if F(T) ≠ ∅, and there exist nonnegative real sequences ν_n, μ_n with ν_n → 0, μ_n → 0 as n → ∞ and a strictly increasing continuous function ζ : ℝ⁺ → ℝ⁺ with ζ(0) = 0 such that

()

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

(A1)
θ(x, x) = 0 for all x ∈ C;
(A2)
θ is monotone; that is, θ(x, y) + θ(y, x) ≤ 0 for all x, y ∈ C;
(A3)
for each x, y, z ∈ C,
()
(A4)
for each x ∈ C, y ↦ θ(x, y) is convex and lower semicontinuous.

For example, let A be a continuous and monotone operator of C into E^* and define

()

Then, θ satisfies (A1)–(A4). The following result is in Blum and Oettli [28].

Lemma 2.12 (see Blum and Oettli [28].)Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let θ be a bifunction from C × C to ℝ satisfying (A1)–(A4), and let r > 0 and x ∈ E. Then, there exists z ∈ C such that

()

Lemma 2.13 (see Takahashi and Zembayashi [8].)Let C be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space E, and let θ be a bifunction from C × C to ℝ satisfying conditions (A1)–(A4). For all r > 0 and x ∈ E, define a mapping as follows:

()

Then the following hold:

(1)
is single-valued;
(2)
is a firmly nonexpansive-type mapping [29]; that is, for all x, y ∈ E,
()
(3)
;
(4)
EP (θ) is closed and convex.

Lemma 2.14 (see Takahashi and Zembayashi [8].)Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let θ be a bifunction from C × C to ℝ satisfying (A1)–(A4), and let r > 0. Then, for x ∈ E and ,

()

3. Main Result

Theorem 3.1. Let C be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. For each j = 1,2, …, m, let θ_j be a bifunction from C × C to ℝ which satisfies conditions (A1)–(A4). Let S : C → C be a closed totally quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences ν_n, μ_n with ν_n → 0, μ_n → 0 as n → ∞, and a strictly increasing continuous function ψ : ℝ⁺ → ℝ⁺ with ψ(0) = 0. Let f : E → ℝ be a convex and lower semicontinuous function with C ⊂ int (D(f)) such that f(x) ≥ 0 for all x ∈ C and f(0) = 0. Assume that . For an initial point x₁ ∈ E and C₁ = C, one define the sequence {x_n} by

()

where {α_n} is a sequence in [0,1], ζ_n = ν_n sup _q∈ℱ ψ(G(q, x_n)) + μ_n and {r_j,n}⊂[d, ∞) for some d > 0. If

, then {x_n} converges strongly to

Proof. We split the proof into four steps.

Step 1. First, we show that C_n is closed and convex for all n ∈ ℕ.

Clearly C₁ = C is closed and convex. Suppose that C_n is closed and convex for all n ∈ ℕ. For any υ ∈ C_n, we know that G(υ, Jz_n) ≤ G(υ, Jx_n) + ζ_n is equivalent to

()

So, C_n+1 is closed and convex. Hence by induction C_n is closed and convex for all n ≥ 1.

Step 2. We will show that the sequence {x_n} is well defined.

We will show by induction that ℱ ⊂ C_n for all n ∈ ℕ. It is obvious that ℱ ⊂ C₁=C. Suppose that ℱ ⊂ C_n for some n ∈ ℕ. Let q ∈ ℱ, put for all j = 1,2, 3, …, m, , we have that

()

From (3.3) and S which is a totally quasi-ϕ asymptotically nonexpansive mappings, it follows that

()

This shows that q ∈ C_n+1 which implies that ℱ⊂C_n+1, and hence, ℱ⊂C_n for all n ∈ ℕ. and the sequence {x_n} is well defined. From , we see that

()

Since

for each n ∈ ℕ, we arrive at

()

Hence, the sequence {x_n} is well defined.

Step 3. We will show that .

Let f : E → ℝ is convex and lower semicontinuous function, follows from Lemma 2.8, there exist x^* ∈ E^* and α ∈ ℝ such that

()

Since x_n ∈ C_n ⊂ E, it follows that

()

For q ∈ ℱ and

, we have

()

This shows that {x_n} is bounded and so is {G(x_n, Jx₁)}. From the fact that

and

, it follows from Lemma 2.10 that

()

That is, {G(x_n, Jx₁)} is nondecreasing. Hence, we obtain that lim _n→∞G(x_n, Jx₁) exists. Taking n → ∞, we obtain

()

Since E is reflexive, {x_n} is bounded, and C_n is closed and convex for all n ∈ ℕ. Without loss of generality, we can assume that x_n⇀p ∈ C_n. From the fact that

, we get that

()

Since f is convex and lower semicontinuous, we have

()

By (3.12) and (3.13), we get

()

That is, lim _n→∞ G(x_n, Jx₁) = G(p, Jx₁); this implies that ∥x_n∥→∥p∥; by virtue of the Kadec-Klee property of E, we obtain that

()

We also have

()

From (3.15), we get that

()

(a) We show that .

Since and the definition of C_n+1, we have

()

is equivalent to

()

From (3.11), (3.15), and (3.17), it follows that

()

From (2.7), we have

()

Since ∥x_n+1∥→∥p∥, we have

()

It follow that

()

That is, {∥Ju_n∥} is bounded in E^* and E^* is reflexive; we assume that Ju_n⇀u^* ∈ E^*. In view of J(E) = E^*, there exists u ∈ E such that Ju = u^*. It follows that

()

Taking liminf _n→∞ on both sides of the equality above and ∥·∥ is the weak lower semicontinuous, it yields that

()

That is, p = u, which implies that u^* = Jp. It follows that Ju_n⇀Jp ∈ E^*. From (3.23) and the Kadec-Klee property of E^* we have Ju_n → Jp as n → ∞. Note that J⁻¹ : E^* → E is norm-weak ^*-continuous; that is, u_n⇀p. From (3.22) and the Kadec-Klee property of E, we have

()

For q ∈ F ⊂ C_n, by nonexpansiveness, we observe that

()

By Lemma 2.14, we have for j = 1,2, 3, …, m

()

Since x_n, u_n → p as n → ∞, we get

as n → ∞, for j = 1,2, 3, …, m. From (2.7), it follow that

()

Since ∥x_n∥→∥p∥, we also have

()

Since

is bounded and E is reflexive, without loss of generality we assume that

. We know that C_n is closed and convex for each n ≥ 1 it is obvious that h ∈ C_n. Again since

()

taking liminf _n→∞ on the both sides of equality above, we have

()

That is, h = p, for all j = 1,2, 3, …, m; it follow that

()

from (3.30), (3.33), and the Kadec-Klee property, it follows that

()

By using triangle inequality, we have

()

Since

as n → ∞, we have

()

Again by using triangle inequality, we have

()

From (3.36), we also have

()

Since J is uniformly norm-to-norm continuous, we obtain

()

From r_j,n > 0, we have

as n → ∞ for all j = 1,2, 3, …, m, and

()

By (A2), that

()

and

as n → ∞, we get θ_j(y, p) ≤ 0, for all y ∈ C. For 0 < t < 1, define y_t = ty + (1 − t)p, then y_t ∈ C which imply that θ_j(y_t, p) ≤ 0. From (A1), we obtain that

()

We have that θ_j(y_t, y) ≥ 0. From (A3), we have θ_j(p, y) ≥ 0, for all y ∈ C and j = 1,2, 3, …, m. That is, p ∈ EP (θ_j), for all j = 1,2, 3, …, m. This imply that

(b) We show that p ∈ F(S).

Since and the definition of C_n+1, we have

()

is equivalent to

()

Following (3.11), (3.15), and (3.17), we get that

()

From (2.7), we also have

()

It follows that

()

This implies that {∥Jz_n∥} is bounded in E^*. Since E is reflexive and E^* is also reflexive, we can assume that Jz_n⇀z^* ∈ E^*. In view of the reflexive of E, we see that J(E) = E^*. There exists z ∈ E such that Jz = z^*. It follows that

()

Taking liminf _n→∞ on both sides of the equality above and in view of the weak lower semicontinuity of norm ∥·∥, it yields that

()

That is p = z, which implies that z^* = Jp. It follows that Jz_n⇀Jp ∈ E^*.From (3.47) and the Kadec-Klee property of E^* we have Jz_n → Jp as n → ∞. Since J⁻¹ : E^* → E is norm-weak ^*-continuous,z_n⇀p as n → ∞. From (3.46) and the Kadec-Klee property of E, we have

()

Since {x_n} is bounded, then a mapping S is also bounded. From the condition lim _n→∞ α_n = 0, we have that

()

From (3.47), we get

()

Since J⁻¹ : E^* → E is norm-weak*-continuous,

()

On the other hand, we observe that

()

In view of (3.52), we obtain ∥Sⁿu_n∥→∥p∥. Since E has the Kadee-Klee property, we get

()

From Sⁿu_n → p, we get Sⁿ⁺¹u_n → p; that is, SSⁿu_n → p. In view of closeness of S, we have Sp = p. This implies that p ∈ F(S). From (a) and (b), it follows that

Step 4. We will show that .

Since ℱ is closed and convex set from Lemma 2.9, we have which is single valued, denoted by υ. By definition and v ∈ ℱ ⊂ C_n, we also have

()

By the definition of G and f, we know that, for each given x, G(ξ, Jx) is convex and lower semicontinuous with respect to ξ. So

()

From the definition of

and since p ∈ ℱ, we conclude that

and x_n → p as n → ∞. The proof is completed.

Setting ν_n ≡ 0 and μ_n ≡ 0 in Theorem 3.1, then we have the following corollary.

Corollary 3.2. Let C be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. For each j = 1,2, …, m, let θ_j be a bifunction from C × C to ℝ which satisfies conditions (A1)–(A4). Let S : C → C be a closed and quasi-ϕ-asymptotically nonexpansive mappings, and let f : E → ℝ be a convex and lower semicontinuous function with C ⊂ int (D(f)) such that f(x) ≥ 0 for all x ∈ C and f(0) = 0. Assume that . For an initial point x₁ ∈ E and C₁ = C, we define the sequence {x_n} by

()

where {α_n} is a sequence in [0,1], ζ_n = ν_n sup _q∈ℱ ψ(G(q, x_n)) + μ_n, and {r_j,n}⊂[d, ∞) for some d > 0. If lim _n→∞α_n = 0, then {x_n} converges strongly to

Let E be a real Banach space, and let C be a nonempty closed convex subset of E. Given a mapping A : C → E^*, let θ(x, y) = 〈Ax, y − x〉 for all x, y ∈ C. Then x^* ∈ EP (θ) if and only if 〈Ax^*, y − x^*〉≥0 for all y ∈ C; that is, x^* is a solution of the classical variational inequality problem. The set of this solution is denoted by VI (A, C). For each r > 0 and x ∈ E, we define the mapping

()

Hence, we obtain the following corollary.

Corollary 3.3. Let C be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. For each j = 1,2, …, m, let{A_j} be a continuous monotone mapping of C into E^*. Let S : C → C be a closed totally quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences ν_n, μ_n with ν_n → 0, μ_n → 0 as n → ∞ and a strictly increasing continuous function ψ : ℝ⁺ → ℝ⁺ with ψ(0) = 0, and let f : E → ℝ be a convex and lower semicontinuous function with C ⊂ int (D(f)) such that f(x) ≥ 0 for all x ∈ C and f(0) = 0. Assume that . For an initial point x₁ ∈ E and C₁ = C, one defines the sequence {x_n} by

()

where ζ_n = ν_n sup _q∈ℱ ψ(G(q, x_n)) + μ_n, {α_n} is a sequence in [0,1], and {r_j,n}⊂[d, ∞) for some d > 0. If lim _n→∞α_n = 0, then {x_n} converges strongly to

If f(x) = 0 for all x ∈ E, we have G(ξ, Jx) = ϕ(ξ, x) and . From Theorem 3.1, we obtain the following corollary.

Corollary 3.4. Let C be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. For each j = 1,2, …, m, let θ_j be a bifunction from C × C to ℝ which satisfies conditions (A1)–(A4). Let S : C → C be a closed totally quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences ν_n, μ_n with ν_n → 0, μ_n → 0 as n → ∞ and a strictly increasing continuous function ψ : ℝ⁺ → ℝ⁺ with ψ(0) = 0. Assume that . For an initial point x₁ ∈ E and C₁ = C, we define the sequence {x_n} by

()

where {α_n} is a sequence in [0,1], ζ_n = ν_n sup _q∈ℱ ψ(G(q, x_n)) + μ_n, and {r_j,n}⊂[d, ∞) for some d > 0. If lim _n→∞α_n = 0, then {x_n} converges strongly to Π_ℱx₁.

Remark 3.5. Our main result extends and improves the result of Chang et al. [13] in the following sense.

(i)
From the algorithm we used new method replace by the generalized f-projection method which is more general than generalized projection.
(ii)
For the problem, we extend the result to a common problem of fixed point problems and equilibrium problems.

Acknowledgments

The authors would like to thank The National Research Council of Thailand (NRCT) and Faculty of Science, King Mongkut′s University of Technology Thonburi (Grant NRCT-2555). Furthermore, the authors would like to express their thanks to the referees for their helpful comments.

References

1 Mann W. R., Mean value methods in iteration, Proceedings of the American Mathematical Society. (1953) 4, 506–510, 0054846.
10.1090/S0002-9939-1953-0054846-3
Web of Science® Google Scholar
2 Bauschke H. H., Matoušková E., and Reich S., Projection and proximal point methods: convergence results and counterexamples, Nonlinear Analysis. Theory, Methods & Applications. (2004) 56, no. 5, 715–738, https://doi.org/10.1016/j.na.2003.10.010, 2036787.
10.1016/j.na.2003.10.010
Web of Science® Google Scholar
3 Genel A. and Lindenstrauss J., An example concerning fixed points, Israel Journal of Mathematics. (1975) 22, no. 1, 81–86, 0390847.
10.1007/BF02757276
Web of Science® Google Scholar
4 Halpern B., Fixed points of nonexpanding maps, Bulletin of the American Mathematical Society. (1967) 73, 957–961, 0218938.
10.1090/S0002-9904-1967-11864-0
Web of Science® Google Scholar
5 Qin X. L., Cho Y. J., Kang S. M., and Zhou H. Y., Convergence of a modified Halpern-type iteration algorithm for quasi-ϕ-nonexpansive mappings, Applied Mathematics Letters. (2009) 22, no. 7, 1051–1055, https://doi.org/10.1016/j.aml.2009.01.015, 2522998.
10.1016/j.aml.2009.01.015
Web of Science® Google Scholar
6 Wang Z. M., Su Y. F., Wang D. X., and Dong Y. C., A modified Halpern-type iteration algorithm for a family of hemi-relatively nonexpansive mappings and systems of equilibrium problems in Banach spaces, Journal of Computational and Applied Mathematics. (2011) 235, no. 8, 2364–2371, https://doi.org/10.1016/j.cam.2010.10.036, 2763149.
10.1016/j.cam.2010.10.036
Web of Science® Google Scholar
7 Takahashi W. and Zembayashi K., Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings, Fixed Point Theory and Applications. (2008) 2008, 11, 528476, 2395311.
10.1155/2008/528476
Google Scholar
8 Takahashi W. and Zembayashi K., Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Analysis. (2009) 70, no. 1, 45–57, https://doi.org/10.1016/j.na.2007.11.031, 2468217.
10.1016/j.na.2007.11.031
Web of Science® Google Scholar
9 Li X., Huang N., and O′Regan D., Strong convergence theorems for relatively nonexpansive mappings in Banach spaces with applications, Computers & Mathematics with Applications. (2010) 60, no. 5, 1322–1331, https://doi.org/10.1016/j.camwa.2010.06.013, 2672931.
10.1016/j.camwa.2010.06.013
Web of Science® Google Scholar
10 Ofoedu E. U. and Shehu Y., Convergence analysis for finite family of relatively quasi nonexpansive mappings and systems of equilibrium problems, Applied Mathematics and Computation. (2011) 217, no. 22, 9142–9150, https://doi.org/10.1016/j.amc.2011.03.147, 2803977.
10.1016/j.amc.2011.03.147
Web of Science® Google Scholar
11 Chang S.-S., Lee H. W. J., and Chan C. K., A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces, with applications, Nonlinear Analysis. Theory, Methods & Applications. (2010) 73, no. 7, 2260–2270, https://doi.org/10.1016/j.na.2010.06.006, 2674202.
10.1016/j.na.2010.06.006
Web of Science® Google Scholar
12 Qin X. and Su Y., Strong convergence theorems for relatively nonexpansive mappings in a Banach space, Nonlinear Analysis. Theory, Methods & Applications. (2007) 67, no. 6, 1958–1965, https://doi.org/10.1016/j.na.2006.08.021, 2326043.
10.1016/j.na.2006.08.021
Web of Science® Google Scholar
13 Chang S.-S., Joseph Lee H. W., Chan C. K., and Yang L., Approximation theorems for total quasi-ϕ-asymptotically nonexpansive mappings with applications, Applied Mathematics and Computation. (2012) 218, no. 11, 6489–6497, https://doi.org/10.1016/j.amc.2011.08.036, 2838194.
10.1016/j.amc.2011.12.019
Web of Science® Google Scholar
14 Qin X., Cho Y. J., and Kang S. M., Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, Journal of Computational and Applied Mathematics. (2009) 225, no. 1, 20–30, https://doi.org/10.1016/j.cam.2008.06.011, 2490167.
10.1016/j.cam.2008.06.011
Web of Science® Google Scholar
15 Zhou H., Gao G., and Tan B., Convergence theorems of a modified hybrid algorithm for a family of quasi-ϕ-asymptotically nonexpansive mappings, Journal of Applied Mathematics and Computing. (2010) 32, no. 2, 453–464, https://doi.org/10.1007/s12190%2D009%2D0263%2D4, 2595857.
10.1007/s12190-009-0263-4
Google Scholar
16 Alber Y. I., Metric and generalized projection operators in Banach spaces: properties and applications, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, 1996, 178, Dekker, New York, NY, USA, 15–50, 1386667.
Google Scholar
17 Alcprimeber Ya. I. and Reich S., An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamerican Mathematical Journal. (1994) 4, no. 2, 39–54, 1274188.
Google Scholar
18 Cioranescu I., Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, 1990, 62, Kluwer, Dordrecht, Germany, 1079061.
10.1007/978-94-009-2121-4
Google Scholar
19 Kamimura S. and Takahashi W., Strong convergence of a proximal-type algorithm in a Banach space, SIAM Journal on Optimization. (2002) 13, no. 3, 938–945, https://doi.org/10.1137/S105262340139611X, 1972223.
10.1137/S105262340139611X
Web of Science® Google Scholar
20 Takahashi W., Nonlinear Functional Analysis, 2000, Yokohama, 1864294.
Google Scholar
21 Wu K. Q. and Huang N. J., The generalized f-projection operator with an application, Bulletin of the Australian Mathematical Society. (2006) 73, no. 2, 307–317, https://doi.org/10.1017/S0004972700038892, 2217950.
10.1017/S0004972700038892
Web of Science® Google Scholar
22 Goebel K. and Reich S., Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, 1984, 83, Marcel Dekker, New York, NY, USA, 744194.
Google Scholar
23 Reich S., Book review: geometry of banach spaces, duality mappings and nonlinear problems, American Mathematical Society. (1992) 26, no. 2, 367–370, https://doi.org/10.1090/S0273%2D0979%2D1992%2D00287%2D2, 1567976.
10.1090/S0273-0979-1992-00287-2
Google Scholar
24 Hudzik H., Kowalewski W., and Lewicki G., Approximate compactness and full rotundity in Musielak-Orlicz spaces and Lorentz-Orlicz spaces, Zeitschrift für Analysis und ihre Anwendungen. (2006) 25, no. 2, 163–192, https://doi.org/10.4171/ZAA/1283, 2229444.
10.4171/zaa/1283
Web of Science® Google Scholar
25 Chang S.-S., Joseph Lee H. W., Chan C. K., and Zhang W. B., Approximation theorems for total quasi-ϕ-asymptotically nonexpansive mappings with applications, Applied Mathematics and Computation. (2011) 218, no. 6, 2921–2931, https://doi.org/10.1016/j.amc.2011.08.036, 2838194.
10.1016/j.amc.2011.08.036
Web of Science® Google Scholar
26 Fan J. L., Liu X., and Li J., Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces, Nonlinear Analysis. Theory, Methods & Applications. (2009) 70, no. 11, 3997–4007, https://doi.org/10.1016/j.na.2008.08.008, 2515316.
10.1016/j.na.2008.08.008
Web of Science® Google Scholar
27 Deimling K., Nonlinear Functional Analysis, 1985, Springer, Berlin, Germany, 787404.
10.1007/978-3-662-00547-7
Google Scholar
28 Blum E. and Oettli W., From optimization and variational inequalities to equilibrium problems, The Mathematics Student. (1994) 63, no. 1–4, 123–145, 1292380.
Google Scholar
29 Kohsaka F. and Takahashi W., Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM Journal on Optimization. (2008) 19, no. 2, 824–835, https://doi.org/10.1137/070688717, 2448915.
10.1137/070688717
Web of Science® Google Scholar

Citing Literature

All articles

A Modified Halpern-TypeIterative Method of a System of Equilibrium Problems and a Fixed Point for a Totally Quasi-ϕ-Asymptotically Nonexpansive Mapping in a Banach Space

Abstract

1. Introduction

2. Preliminaries and Definitions

3. Main Result

Acknowledgments

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

A Modified Halpern-TypeIterative Method of a System of Equilibrium Problems and a Fixed Point for a Totally Quasi-ϕ-Asymptotically Nonexpansive Mapping in a Banach Space

Abstract

1. Introduction

2. Preliminaries and Definitions

3. Main Result

Acknowledgments

References

Citing Literature

References

Related

Information