Convergence of algorithms for fixed points of generalized asymptotically quasi-ϕ-nonexpansive mappings with applications

In this article, strong convergence of Krasnoselski-Mann iterative sequences and Halpern iterative sequences are investigated based on hybrid projection methods. Strong convergence theorems for common fixed points of a family of generalized asymptotically quasi-ϕ-nonexpansive mappings are established in the framework of Banach spaces.

Mathematics Subject Classification 2000: 47H09; 47J05; 47J25

Fixed point theory as an important branch of nonlinear analysis theory has been applied in the study of nonlinear phenomena. During the four decades, many famous existence theorems of fixed points were established; see, for example, [1-5]. However, from the standpoint of real world applications it is not only to know the existence of fixed points of nonlinear mappings, but also to be able to construct an iterative process to approximate their fixed points. The computation of fixed points is important in the study of many real world problems, including inverse problems; for instance, it is not hard to show that the split feasibility problem and the convex feasibility problem in signal processing and image reconstruction can both be formulated as a problem of finding fixed points of certain operators, respectively (see [6,7] for more details and the references therein).

Recently, the study of the convergence of various iterative processes for solving various nonlinear mathematical models forms the major part of numerical mathematics. Among these iterative processes, Krasnoselski-Mann iterative process and Halpern iterative process are popular and hot. Let C be a nonempty, closed, and convex subset of a underlying space X, and T : C → C a mapping. Halpern iterative process generates a sequence {x_n} in the following manner:

where x₀ is an initial and u is a fixed element in C. Krasnoselski-Mann iterative process generates a sequence {x_n} in the following manner:

It is known that Algorithm (1.2) only has weak convergence even for nonexpansive mappings in infinite-dimensional Hilbert spaces (see [8] for more details and the reference therein). In many disciplines, including economics [9], image recovery [10], quantum physics [11], and control theory [12], problems arises in infinite dimension spaces. In such problems, strong convergence (norm convergence) is often much more desirable than weak convergence, for it translates the physically tangible property that the energy ∥x_n - x∥ of the error between the iterate x_n and the solution x eventually becomes arbitrarily small. The important of strong convergence is also underlined in [13], where a convex function f is minimized via the proximal-point algorithm: it is shown that the rate of convergence of the value sequence {f(x_n)} is better when {x_n} converges strongly that it converges weakly. Such properties have a direct impact when the process is executed directly in the underlying infinite dimensional space. To improve the weak convergence of Krasnoselski-Mann iterative process, so called hybrid projections have been considered (see [14-25] for more details and the references therein).

Algorithm (1.1) was initially introduced in [26]; for more details see the references therein. In [26], Halpern showed that the following conditions

(C1) lim_n→∞, α_n = 0;

(C2)

are necessary in the sense that if Algorithm (1.1) is strongly convergent for all nonempty, closed, and convex subsets of a Hilbert space H and all nonexpansive mappings on C, then the sequence {x_n} must satisfy conditions (C1), and (C2). Due to the restriction of (C2), Algorithm (1.1) is widely believed to have slow convergence though the rate of convergence has not be determined. Thus to improve the rate of convergence of algorithm (1.1), one can not rely only on the process itself; instead, some additional step of iteration should be taken; see [27-30] and the references therein. One of the purposes of this article is to show algorithm (1.1) is strong convergence under (C1) only with the help of projections.

The purposes of this article is to study Algorithms (1.1) and (1.2) with the help of additional metric projections for the new mapping. The organization of this article is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, Algorithms (1.1) and (1.2) are studied with the help of projections. Two main strong convergence theorems are established in a reflexive, strictly convex, and smooth Banach space such that both E and E* have Kadec-Klee property. In Section 4, applications of the main results are provided.

Let H be a real Hilbert space, C a nonempty subset of H, and T : C → C a mapping. The symbol F(T) stands for the fixed point set of T. Recall the following. T is said to be nonexpansive if

T is said to be quasi-nonexpansive if , and

T is said to be asymptotically nonexpansive if there exists a sequence {μ_n} ⊂ [0, ∞) with μ_n → 0 as n→∞ such that

It is easy to see that a nonexpansive mapping is an asymptotically nonexpansive mapping with the sequence {1}. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [2]. Since 1972, a host of authors have studied the convergence of iterative algorithms for such a class of mappings.

T is said to be asymptotically quasi-nonexpansive if , and there exists a sequence {μ_n} ⊂ [0, ∞) with μ_n → 0 as n→∞ such that

It is easy to see that a quasi-nonexpansive mapping is an asymptotically quasi-nonexpansive mapping with the sequence {1}.

T is said to be generalized asymptotically nonexpansive if there exist two nonnegative sequences {μ_n} ⊂ [0, ∞) with μ_n → 0, and {ξ_n} ⊂ [0, ∞) with ξ_n → 0 as n→∞ such that

T is said to be generalized asymptotically quasi-nonexpansive if , and there exist two nonnegative sequences {μ_n} ⊂ [0, ∞) with μ_n → 0, and {ξ_n} ⊂ [0,∞) with ξ_n → 0 as n→∞ such that

The class of generalized asymptotically (quasi)-nonexpansive has been considered by Shahzad and Zegeye [31] (see also Agarwal et al. [32]). It is easy to see that the class of generalized asymptotically (quasi)-nonexpansive include the class of asymptotically (quasi)-nonexpansive as a special case.

In what follows, we always assume that E is a Banach space with the dual space E*. Let C be a nonempty, closed, and convex subset of E. We use the symbol J to stand for the normalized duality mapping from E to defined by

where 〈⋅, ⋅〉 denotes the generalized duality pairing of elements between E, and E*. It is well known that if E* is strictly convex, then J is single valued; if E* is reflexive, and smooth, then J is single valued, and demicontinuous (see [33] for more details and the references therein).

It is also well known that if D is a nonempty, closed, and convex subset of a Hilbert space H, and P_C : H → D is the metric projection from H onto D, then P_D is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [34] introduced a generalized projection operator in Banach spaces which is an analogue of the metric projection in Hilbert spaces.

Let U_E = {x ∈ E : ∥x∥ = 1} be the unit sphere of E. E is said to be strictly convex if for all x, y ∈ U_E with x ≠ y. It is said to be uniformly convex if for any ϵ ∈ (0, 2] there exists δ > 0 such that for any x, y ∈ U_E,

It is known that a uniformly convex Banach space is reflexive and strictly convex. E is said to be smooth provided exists for all x, y ∈ U_E. It is also said to be uniformly smooth if the limit is attained uniformly for all x, y ∈ U_E.

E is said to enjoy Kadec-Klee property 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→∞. For more details on Kadec-Klee property, the readers can refer to [35] and the references therein. It is well known that if E is a uniformly convex Banach spaces, then E enjoys Kadec-Klee property.

Let E be a smooth Banach space. Consider the functional defined by

Notice that, in a Hilbert space H, (2.1) is reduced to ϕ(x, y) = ║x-y║² for all x,y ∈ H. The generalized projection Π_C : E → C is a mapping that assigns to an arbitrary point x ∈ E, the minimum point of the functional ϕ(x, y); that is, , where is the solution to the following minimization problem:

The existence, and uniqueness of the operator Π_C follow from the properties of the functional ϕ(x, y), and the strict monotonicity of the mapping J (see, for example, [33,36]). In Hilbert spaces, Π_C = P_C. It is obvious from the definition of the function ϕ that

and

Remark 2.1. If E is a reflexive, strictly convex, and smooth Banach space, then, for all x,y ∈ E, 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 (2.2), we have ║x║ = ║y║. This implies that 〈x, Jy〉 = ║x║² = ║Jy║². From the definition of J, we see that Jx = Jy. It follows that x = y; see [33,36] for more details.

Next, we recall the following.

(1) A point p in C is said to be an asymptotic fixed point of T [37] if C contains a sequence {x_n} which converges weakly to p such that lim_n→∞ ║x_n - Tx_n║ = 0. The set of asymptotic fixed points of T will be denoted by .

(2) T is said to be relatively nonexpansive if

(3) T is said to be relatively asymptotically nonexpansive if

where {μ_n} ⊂ [0, ∞) is a sequence such that μ_n → 0 as n→∞.

Remark 2.2. The class of relatively asymptotically nonexpansive mappings was first considered in Su and Qin [38] (see also, Agarwal et al. [39], and Qin et al. [40]).

(4) T is said to be quasi-ϕ-nonexpansive if

(5) T is said to be asymptotically quasi-ϕ-nonexpansive if there exists a sequence {μ_n} ⊂ [0, ∞) with μ_n → 0 as n→∞ such that

Remark 2.3. The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings were first considered in Zhou et al. [24] (see also Qin and Agarwal [18], Qin et al. [20], Qin et al. [21], Qin et al. [41]).

Remark 2.4. The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are more general than the class of relatively nonex-pansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasi-ϕ-nonexpansive mappings and asymptotically quasi-ϕ-nonexpansive do not require .

Remark 2.5. The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are generalizations of the class of quasi-nonexpansive mappings and the class of asymptotically quasi-nonexpansive mappings in Banach spaces.

In this article, we introduce and consider the following new nonlinear mapping: generalized asymptotically quasi-ϕ-nonexpansive mappings.

(6) T is said to be an generalized asymptotically quasi-ϕ-nonexpansive mapping if , and there exist two nonnegative sequences {μ_n} ⊂ [0, ∞) with μ_n → 0, and {ξ_n} ⊂ [0, ∞) with ξ_n → 0 as n→∞ such that

Remark 2.6. The class of generalized asymptotically quasi-ϕ-nonexpansive mappings is a generalization of the class of generalized asymptotically quasi-nonexpansive mappings in the framework of Banach spaces.

(7) T is said to be asymptotically regular on C if, for any bounded subset K of C,

In order to prove our main results, we also need the following lemmas:

Lemma 2.1. [34]Let C be a nonempty, closed, and convex subset of a smooth Banach space E, and x ∈ E. Then x₀ = Π_Cx if and only if

Lemma 2.2. [34]Let E be a reflexive, strictly, convex, and smooth Banach space, C a nonempty, closed, and convex subset of E, and x ∈ E. Then

Theorem 3.1. Let E be a reflexive, strictly convex, and smooth Banach space such that both E and E* have Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. Let Δ be an index set, and T_i : C → C a closed, asymptotically regular, and generalized asymptotically quasi-ϕ-nonexpansive mapping with the sequences {μ_n,i}, and {ξ_n,i}, for every i ∈ Δ. Assume that ⋂_i∈Δ F(T_i) is nonempty, and bounded. Let {x_n} be a sequence generated in the following manner:

where M_n = sup{ϕ(z, x_n) : z ∈ ⋂_i∈Δ F(T_i)}, and {α_n,i} are sequences in (0,1] such that lim inf_n→∞ α_n,i > 0. Then {x_n} converges strongly to , where stands for the generalized projection from E onto ⋂_i∈Δ F(T_i).

Proof. The proof is split into seven steps.

Step 1. Show, for every i∈Δ, that F(T_i) is closed, and convex. This proves that is well defined, for every x₀ ∈ E. On the closedness of ⋂_i∈Δ F(T_i), we can easily conclude from the closedness of T_i the desired conclusion. We only prove that ⋂_i∈Δ F(T_i) is convex. Let p_1,i,p_2,i ∈ F(T_i), and p_i = t_ip _1,i + (1 - t_i)p_2,i, where t_i ∈ (0,1), for every i ∈ Δ. We see that p_i = T_ip_i. Indeed, we see from the definition of T_i that

and

In view of (2.3), we obtain that

and

It follows from (3.1), (3.2), (3.3), and (3.4) that

and

Multiplying t_i and (1 - t_i) on the both sides of (3.5) and (3.6), respectively yields that

It follows that

In light of (2.2), we arrive at

It follows that

Since E* is reflexive, we may, without loss of generality, assume that . In view of the reflexivity of E, we have J(E) = E*. This shows that there exists an element eⁱ ∈ E such that Jeⁱ = e*^,i. It follows that

Taking lim inf_n→∞ on the both sides of the equality above, we obtain that

This implies that p_i = eⁱ, that is, Jp_i = e*^,i. It follows that . In view of Kadec-Klee property of E*, we obtain from (3.8) that

Since J^-1 : E* → E is demicontinuous, we see that . By virtue of Kadec-Klee property of E, we see from (3.7) that as n→∞. Hence

as n→∞. In view of the closedness of T_i, we can obtain that p_i ∈ F(T_i), for every i ∈ Δ. This shows, for every i ∈ Δ, that F(T_i) is convex. This proves that ⋂_i∈Δ F(T_i) is convex. This completes the proof of Step 1.

Step 2. Show that C_n is closed, and convex for all n ≥ 1. It suffices to show, for any fixed but arbitrary i ∈ Δ, that C_n,i is closed, and convex, for every n ≥ 1. This can be proved by induction on n. It is obvious that C_1,i = C is closed, and convex. Assume that C_h,i is closed, and convex for some h ≥ 1. We next prove that C_h+1,i is closed, and convex for the same h. This completes the proof that C_n is closed, and convex. The closedness of C_h+1,i is clear. We only prove the convexness. Indeed, ∀a, b ∈ C_h+1,i, we see that a,b ∈ C_h,i, and

and

Notice that (3.9), and (3.10) are equivalent to the following inequalities, respectively.

and

These imply that

Since C_h,i is convex, we see that ta + (1 - t)b ∈ C_h,i. Notice that (3.11) is equivalent to

This proves that C_h+1,i is convex. This completes the proof of Step 2.

Step 3. Show that ⋂_i∈Δ F(T_i) ⊂ C_n, for every n ≥ 1. It suffices to claim that ⋂_i∈Δ F(T_i) ⊂ C_n,i, for every n ≥ 1, and for every i ≥ Δ. Note that ⋂_i∈Δ F(T_i) ⊂ C_1,i = C. Suppose that ⋂_i∈Δ F(T_i) ⊂ C_h,i for some h, and for every i ∈ Δ. Then, for all w ∈ ⋂_i∈Δ F(T_i) ⊂ C_h,i, we have

where .This shows that w ∈ C_h+1,i.This implies that ⋂_i∈Δ F(T_i) ⊂ C_n, for every n ≥ 1. This completes the proof of Step 3.

Step 4. Show that {x_n} is bounded. In view of , we see that

Since ⋂_i∈Δ F(T_i) ⊂ C_n, we arrive at

It follows from Lemma 2.2 that

This implies that the sequence {ϕ(x_n, x₀)} is bounded. It follows from (2.2) that the sequence {x_n} is also bounded. This completes the proof of Step 4.

Step 5. Show that , where is some point in C as n→∞. Since {x_n} is bounded, and the space is reflexive, we may assume that . Since C_n is closed, and convex, we see that . On the other hand, we see from the weakly lower semicontinuity of the norm that

which implies that as n→∞. Hence, as n→∞. In view of Kadec-Klee property of E, we see that as n→∞. This completes the proof of Step 5.

Step 6. Show that . In view of construction of , we arrive at

Since , and , we arrive at ϕ(x_n, x₀) ≤ ϕ(x_n+1,x₀), ∀n ≥ 1. This shows that {ϕ(x_n, x₀)} is nondecreasing. It follows from the boundedness that lim_n→∞ ϕ(x, x₀) exists. It follows that

Since , we arrive at

This in turn implies from (3.13) that

In view of (2.2), we see that

This in turn implies that

It follows that

This implies that {Jy_n,i} is bounded. Note that both E and E* are reflexive. We may assume that Jy_n,i ⇀ y*^,i ∈ E*, for every i ∈ Δ. In view of the reflexivity of E, we see that J(E) = E*. This shows that there exists an element yⁱ ∈ E such that Jyⁱ = y*^,i. It follows that

Taking lim inf_n→∞ on the both sides of the equality above yields that

That is, , which in turn implies that , for every i ∈ Δ. It follows that , for every i ∈ Δ. Since E* enjoys Kadec-Klee property, we obtain from (3.15) that

Notice that

It follows that

Notice from (ϒ) that

In view of the assumption that lim inf_n→∞ α_n,i > 0, we arrive at

Notice that

This implies from (3.17) that

The demi-continuity of J^-1 : E* → E implies that , for every i ∈ Δ. Note that

In view of (3.18), we see that , for every i ∈ Δ as n→∞. Since E enjoy Kadec-Klee property, we obtain that

Notice that

It follows from the asymptotic regularity of T_i, and (3.19) that

that is, as n→∞. It follows from the closedness of T_i that , for every i ∈ Δ. This completes the proof of Step 6.

Step 7. Show that . Letting n→∞ in (3.12), we arrive at

It follows from Lemma 2.1 that . This completes the proof of Step 7. The proof of Theorem 3.1 is completed.

Remark 3.2. Comparing Theorem 3.1 with Theorem 2.1 in Qin et al. [21], we have the following:

(a) extend the mapping from the class of asymptotically quasi-ϕ-nonexpansive mappings to the class of generalized asymptotically quasi-ϕ-nonexpansive mappings;

(b) extend the mapping from a single mapping to a family of mappings;

(c) extend the space from a uniformly smooth, and strictly convex Banach space which also enjoys the Kadec-Klee property to a reflexive, strictly convex, and smooth Banach space such that both E and E* have Kadec-Klee property.

Remark 3.3. Strictly convex, reflexive, and smooth Musielak-Orlicz spaces satisfy the restrictions imposed on the framework of the spaces [35], while, in general, these spaces need not to be uniformly convex or uniformly smooth.

For a single mapping, we can easily conclude the following.

Corollary 3.4. Let E be a reflexive, strictly convex, and smooth Banach space such that both E and E* have Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. Let T : C → C be a closed, asymptotically regular, and generalized asymptotically quasi-ϕ-nonexpansive mapping with the sequences {μ_n}, and {ξ_n}. Assume that F(T) is nonempty, and bounded. Let {x_n} be a sequence generated in the following manner:

where M_n = sup{ϕ(z, x_n) : z ∈ F(T)}, and {α_n} is a sequence in (0,1] such that lim inf_n→∞ α_n > 0. Then {x_n} converges strongly to Π_F(T)x₀, where Π_F(T) stands for the generalized projection from E onto F(T).

If α_n = 1, then Theorem 3.1 is reduced to the following.

Corollary 3.5. Let E be a reflexive, strictly convex, and smooth Banach space such that both E and E* have Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. Let Δ be an index set, and T_i : C → C a closed, asymptotically regular, and generalized asymptotically quasi-ϕ-nonexpansive mapping with the sequences {μ_n,i}, and {ξ_n,i}, for every i ∈ Δ. Assume that ⋂_i∈Δ F(T_i) is nonempty, and bounded. Let {x_n} be a sequence generated in the following manner:

where M_n = sup{ϕ(z, x_n) : z ∈ ⋂_i∈Δ F(T_i)}, and {α_n,i} are sequences in (0,1] such that lim inf_n→∞ α_n,i > 0. Then {x_n} converges strongly to , where stands for the generalized projection from E onto ⋂_i∈Δ F(T_i).

In the framework of Hilbert spaces, Theorem 3.1 is reduced to the following.

Corollary 3.6. Let C be a nonempty, closed, and convex subset of a Hilbert space E. Let Δ be an index set, and T_i : C → C a closed, asymptotically regular, and generalized asymptotically quasi-nonexpansive mapping with the sequences {μ_n,i}, and {ξ_n,i}, for every i ∈ Δ. Assume that ⋂_i∈Δ F(T_i) is nonempty, and bounded. Let {x_n} be a sequence generated in the following manner:

where M_n = sup{║z - x_n║² : z ∈ ⋂_i∈Δ F(T_i)}, and {α_n,i} are sequences in (0,1] such that lim inf_n→∞ α_n,i > 0. Then {x_n} converges strongly to , where stands for the metric projection from E onto ⋂_i∈Δ F(T_i).

For a single mapping, we can easily conclude the following.

Corollary 3.7. Let C be a nonempty, closed, and convex subset of a Hilbert space E. Let T : C → C be a closed, asymptotically regular, and generalized asymptotically quasi-nonexpansive mapping with the sequences {μ_n}, and {ξ_n}. Assume that F(T) is nonempty, and bounded. Let {x_n} be a sequence generated in the following manner:

where M_n = sup{║z - x_n║² : z ∈ ⋂_i∈Δ F(T_i)}, and {α_n} is a sequence in (0,1] such that lim inf_n→∞ α_n > 0. Then {x_n} converges strongly to P_F(T)x₀, where P_F(T) stands for the metric projection from E onto F(T).

Next, we turn our attention to Algorithm (1.1).

Theorem 3.8. Let E be a reflexive, strictly convex, and smooth Banach space such that both E and E* have Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. Let Δ be an index set, and T_i : C → C a closed, asymptotically regular, and generalized asymptotically quasi-ϕ-nonexpansive mapping with the sequences {μ_n,i}, and {ξ_n,i}, for every i ∈ Δ. Assume that ⋂_i∈Δ F(T_i) is nonempty, and bounded. Let {x_n} be a sequence generated in the following manner:

where , and {α_n,i} are sequences in (0,1) such that lim_n→∞ α_n,i = 0. Assume that . Then {x_n} converges strongly to , where stands for the generalized projection from E onto ⋂_i∈Δ F(T_i).

Proof. In view of the proof of Theorem 3.1, we show the difference only. From the proof of Step 1 of Theorem 3.1, we see that ⋂_i∈Δ F(T_i) is closed, and convex.

Next, we show that C_n is closed, and convex for all n ≥ 1. It suffices to show, for any fixed but arbitrary i ∈ Δ, that C_n,i is closed, and convex, for every n ≥ 1. This can be proved by induction on n. It is obvious that C_1,i = C is closed, and convex. Assume that C_h,i is closed, and convex for some h ≥ 1. We next prove that C_h+1,i is closed, and convex for the same h. This completes the proof that C_n is closed, and convex. The closedness of C_h+1,i is clear. We only prove the convexness. Indeed, ∀a, b ∈ C_h+1,i, we see that a,b ∈ C_h,i, and

and

Notice that (3.20), and (3.21) are equivalent to the following inequalities, respectively.

and

These imply that

Since C_h,i is convex, we see that ta + (1 - t)b ∈ C_h,i. Notice that (3.22) is equivalent to

This proves that C_h+1,i is convex. This completes the proof that C_n is closed, and convex for all n ≥ 1.

Next, we show that ⋂_i∈Δ F(T_i) ⊂ C_n, for every n ≥ 1. It suffices to claim that ⋂_i∈Δ F(T_i) ⊂ C_n,i, for every n ≥ 1, and for every i ≥ Δ. Note that ⋂_i∈Δ F(T_i) ⊂ C_1,i = C. Suppose that ⋂_i∈Δ F(T_i) ⊂ C_h,i for some h, and for every i ∈ Δ. Then, for ∀w ∈ ⋂_i∈Δ F(T_i) ⊂ C_h,i, we obtain from the restriction that

where . This shows that w ∈ C_h+1,i. This implies that ⋂_i∈Δ F(T_i) ⊂ C_n, for every n ≥ 1. This completes the proof that ⋂_i∈Δ F(T_i) ⊂ C_n, for every n ≥ 1.

In the light of the proof of Step 4 of Theorem 3.1, we find that {x_n} is bounded. It follows the proof of Step 5 of Theorem 3.1 that as n → ∞. Next, we show that . In view of the proof of Step 6 of Theorem 3.1, we find that

Since , we arrive at

This in turn implies that

In view of the proof of Step 6 of Theorem 3.1, we find that

Notice from (ϒϒ) that

In view of the assumption that lim_n→∞ α_n,i = 0, ∀i ∈ Δ, we find from (3.25) that

Next, following Steps 6 and 7, we can easily conclude the desired conclusion. This completes the proof of Theorem 3.8.

Remark 3.9. In view of the mappings, and the framework of the spaces, we see that Theorem 3.8 can be viewed as a generalization of the corresponding results announced in Cho et al. [27], Qin et al. [28], and Qin and Su [29].

For a single mapping, we obtain from Theorem 3.8 the following.

Corollary 3.10. Let E be a reflexive, strictly convex, and smooth Banach space such that both E and E* have Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. Let T : C → C a closed, asymptotically regular, and generalized asymptotically quasi-ϕ-nonexpansive mapping with the sequences {μ_n}, and {ξ_n}. Assume that F(T) is nonempty, and bounded. Let {x_n} be a sequence generated in the following manner:

where M = sup_z∈F(T){ϕ(z,x₁)}, and {α_n} is a sequence in (0,1) such that lim_n→∞ α_n = 0. Assume that . Then {x_n} converges strongly to Π_F(T)x₁, where Π_F(T) stands for the generalized projection from E onto F(T).

In the framework of Hilbert spaces, Theorem 3.8 is reduced to the following.

Corollary 3.11. Let C be a nonempty, closed, and convex subset of a Hilbert space E. Let Δ be an index set, and T_i : C → C a closed, asymptotically regular, and generalized asymptotically quasi-nonexpansive mapping with the sequences {μ_n,i}, and {ξ_n,i}, for every i ∈ Δ. Assume that ⋂_i∈Δ F(T_i) is nonempty, and bounded. Let {x_n} be a sequence generated in the following manner:

where , and {α_n,i} are sequences in (0,1) such that lim_n→∞ α_n,i = 0. Assume that .Then {x_n} converges strongly to , where stands for the metric projection from E onto ⋂_i∈Δ F(T_i).

Remark 3.12. Comparing with Theorem 3.1 in Martinez-Yanes and Xu [30], we have the following:

(a) improve the mapping from nonexpansive mappings to asymptotically quasi-nonexpansive mappings;

(b) improve the mapping from a single mapping to a family of mappings;

(b) the hybrid projection in Corollary 3.1 is different with the one in [30].

For a single mapping, we obtain from Corollary 3.11 the following.

Corollary 3.13. Let C be a nonempty, closed, and convex subset of a Hilbert space E. Let T : C → C a closed, asymptotically regular, and generalized asymptotically quasi-nonexpansive mapping with the sequences {μ_n}, and {ξ_n}. Assume that F(T) is nonempty, and bounded. Let {x_n} be a sequence generated in the following manner:

where M = sup_z∈F(T){║z- x₁║²}, and {α_n} is a sequence in (0,1) such that lim_n→∞ α_n = 0. Assume that . Then {x_n} converges strongly to P_F(T)x₁, where P_F(T) stands for the metric projection from E onto F(T).

First, we consider the problem of approximating a common minimizer of a family of proper, lower semicontinuous, and convex functionals.

Let E be a Banach space with the dual E*. For a proper lower semicontinuous convex function f : E → (-∞,∞], the subdifferential mapping ∂f ⊂ E × E* of f is defined by

Rockafellar [42] proved that ∂f is a maximal monotone operator. It is easy to verify that 0 ∈ ∂f(v) if and only if f(v) = min_x∈E f(x).

Theorem 4.1. Let E be a reflexive, strictly convex, and smooth Banach space such that both E and E* have Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. Let Δ be an index set, and f_i : C → C a proper, lower semicontinuous, and convex functionals, for every i ∈ Δ. Assume that ⋂_i∈Δ (∂f_i)^-1(0) is nonempty. Let {x_n} be a sequence generated in the following manner:

where r_i > 0, ∀i ∈ Δ, and {α_n,i} are sequences in (0,1] such that lim inf_n→∞ α_n,i > 0.Then {x_n} converges strongly to , where stands for the generalized projection from E onto ⋂_i∈Δ (∂f_i)^-1(0).

Proof. For each r_i > 0, and x ∈ E, we see that there exists a unique such that , where . Notice that

is equivalent to

This shows that z_n,i = (J + r_i∂f_i)^-1Jx_n. In view of the Example 2.3 in Qin et al. [41], we find that (J + r_i∂f_i)^-1J is closed quasi-ϕ-nonexpansive with F((J+ r_i∂f_i)^-1 J) = (∂f_i)^-1(0).

Following the proof of Theorem 3.1, we can immediately conclude the desired conclusion.

Theorem 4.2. Let E be a reflexive, strictly convex, and smooth Banach space such that both E and E* have Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. Let Δ be an index set, and f_i : C → C a proper, lower semicontinuous, and convex functionals, for every i ∈ Δ. Assume that ⋂_i∈Δ(∂f_i)^-1(0) is nonempty. Let {x_n} be a sequence generated in the following manner:

where r_i > 0, and {α_n,i} are sequences in (0,1) such that lim_n→∞ α_n,i = 0. Then {x_n} converges strongly to , where stands for the generalized projection from E onton ⋂_i∈Δ (∂f_i)^-1(0).

Proof. We easily find from Theorems 3.8 and 4.1 the conclusion.

Second, we consider the problem of approximating a solution of a family of variational inequalities.

Let C be a nonempty, closed, and convex subset of a Banach space E. Let E* be the dual space of E. let A : C → E* be a single valued monotone operator which is hemicontinuous; that is, continuous along each line segment in C with respect to the weak* topology of E*.

Consider the following variational inequality problem of finding a point x ∈ C such that

In this chapter, we use VI(C, A) to denote the solution set of the variational inequality involving A. The symbol N_C(x) stand for the normal cone for C at a point x ∈ C; that is,

Theorem 4.3. Let E be a reflexive, strictly convex, and smooth Banach space such that both E and E* have Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. Let Δ be an index set, and A_i : C → E* a single valued, monotone and hemicontinuous operator. Assume that ⋂_i∈Δ VI(C, A_i) is not empty. Let {x_n} be a sequence generated in the following manner:

where {α_n,i} are sequences in (0,1] such that lim inf_n→∞ α_n,i > 0. Then {x_n} converges strongly to , where stands for the generalized projection from E onto ⋂_i∈Δ VI(C,A_i).

Proof. Define a mapping T_i ⊂ E × E* by

By Rockafellar [42], we know that T_i is maximal monotone, and . For each r_i > 0, and x ∈ E, we see that there exists a unique such that , where . Notice that

which is equivalent to

that is,

This implies that . In view of the Example 2.3 in Qin et al. [41], we find that is closed quasi-ϕ-nonexpansive with .

Following the proof of Theorem 3.1, we can immediately conclude the desired conclusion.

Theorem 4.4. Let E be a reflexive, strictly convex, and smooth Banach space such that both E and E* have Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. Let Δ be an index set, and A_i : C → E* a single valued, monotone and hemicontinuous operator. Assume that ⋂_i∈Δ VI(C, A_i) is not empty. Let {x_n} be a sequence generated in the following manner:

where r_i > 0, and {α_n,i} are sequences in (0,1) such that lim_n→∞ α_n,i = 0. Then {x_n} converges strongly to , where stands for the generalized projection from E onto ⋂_i∈Δ VI(C, A_i).

Proof. We easily find from Theorems 3.8 and 4.3 the conclusion.

The authors declare that they have no competing interests.

All authors contribute equally and significantly in writing this paper. All authors read and approved the final manuscript.

The authors are grateful to the referees for their valuable comments and suggestions which improve the contents of the article.

1. Browder, FE: Nonexpansive nonlinear operators in a Banach space. Proc Natl Acad Sci USA. 54, 1041–1044 (1965). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

2. Goebel, K, Kirk, WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc Amer Math Soc. 35, 171–174 (1972). Publisher Full Text

3. Kirk, WA: A fixed point theorem for mappings which do not increase distances. Amer Math Monthly. 72, 1004–1006 (1965). Publisher Full Text

4. Kirk, WA: Fixed point theorems for non-Lipschitzian mappings of as ymptotically non-expansive type. Ireal J Math. 17, 339–346 (1974)

5. Xu, HK: Existence and convergence for fixed points of mappings of asymptotically nonexpansive type. Nonlinear Anal. 16, 1139–1146 (1991). Publisher Full Text

6. Vanderlugt, A: Optical Signal Processing. John Wiley & Sons, New York (1992)

7. Byrne, C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems. 20, 103–120 (2008)

8. Genel, A, Lindenstruss, J: An example concerning fixed points. Israel J Math. 22, 81–86 (1975). Publisher Full Text

9. Khan, MA, Yannelis, NC: Equilibrium Theory in Infinite Dimensional Spaces. Springer-Verlag, New York (1991)

10. Combettes, PL: The convex feasibility problem in image recovery. In: Hawkes, P (eds.) Advanced in Imaging and Electron Physcis, vol. 95, pp. 155–270. Academic Press, New York (1996)

11. Dautray, R, Lions, JL: Mathematical Analysis and Numerical Methods for Science and Technology. pp. 1–6. Springer-Verlag, New York (1988)

12. Fattorini, HOL: Infinite-dimensional Optimization and Control Theory. Cambridge University Press, Cambridge (1999)

13. Güler, O: On the convergence of the proximal point algorithm for convex minimization. SIAM J Control Optim. 29, 403–409 (1991). Publisher Full Text

14. Kim, JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-ϕ-nonexpansive mappings. Fixed Point Theory Appl. 58, 2011 (2011)

15. Ye, J, Huang, J: Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. J Math Comput Sci. 1, 1–18 (2011)

16. Matsushita, SY, Takahashi, W: A strong convergence theorem for relatively nonexpan-sive mappings in a Banach space. J Approx Theory. 134, 257–266 (2005). Publisher Full Text

17. Plubtieng, S, Ungchittrakool, K: Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space. J Approx Theory. 149, 103–115 (2007). Publisher Full Text

18. Qin, X, Agarwal, RP: Shrinking projection methods for a pair of asymptotically quasi-ϕ-nonexpansive mappings. Numer Funct Anal Optim. 31, 1072–1089 (2010). Publisher Full Text

19. Qin, X, Cho, YJ, Cho, SY, Kang, SM: Strong convergence theorems of common fixed points for a family of quasi-ϕ-nonexpansive mappings. Fixed Point Theory Appl. 2010, 11 Article ID 754320 (2010)

20. Qin, X, Cho, SY, Kang, SM: On hybrid projection methods for asymptotically quasi-ϕ-nonexpansive mappings. Appl Math Comput. 215, 3874–3883 (2010). Publisher Full Text

21. Qin, X, Huang, S, Wang, T: On the convergence of hybrid projection algorithms for asymptotically quasi-ϕ-nonexpansive mappings. Comput Math Appl. 61, 851–859 (2011). Publisher Full Text

22. Su, Y, Wang, Z, Xu, H: Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings. Nonlinear Anal. 71, 5616–5628 (2009). Publisher Full Text

23. Su, Y, Qin, X: Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators. Nonlinear Anal. 68, 3657–3664 (2008). Publisher Full Text

24. Zhou, H, Gao, G, Tan, B: Convergence theorems of a modified hybrid algorithm for a family of quasi-ϕ-asymptotically nonexpansive mappings. J Appl Math Comput. 32, 453–464 (2010). Publisher Full Text

25. Zhou, H, Gao, X: A strong convergence theorem for a family of quasi-ϕ-nonexpansive mappings in a Banach space. Fixed Point Theory Appl. 2009, 12 Article ID 351265 (2009)

26. Halpern, B: Fixed points of nonexpanding maps. Bull Amer Math Soc. 73, 957–961 (1967). Publisher Full Text

27. Cho, YJ, Qin, X, Kang, SM: Strong convergence of the modified Halpern-type iterative algorithms in Banach spaces. An Ştiinţ Univ Ovidius Constanta Ser Mat. 17, 51–68 (2009). PubMed Abstract

28. Qin, X, Cho, YJ, Kang, SM, Zhou, H: Convergence of a modified Halpern-type iteration algorithm for quasi-ϕ-nonexpansive mappings. Appl Math Lett. 22, 1051–1055 (2009). Publisher Full Text

29. Qin, X, Su, Y: Strong convergence theorem for relatively nonexpansive mappings in a Banach space. Nonlinear Anal. 67, 1958–1965 (2007). Publisher Full Text

30. Martinez-Yanes, C, Xu, HK: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal. 64, 2400–2411 (2006). Publisher Full Text

31. Shzhzad, N, Zegeye, H: Strong convergence of an implicit iteration process for a finite family of genrealized asymptotically quasi-nonexpansive maps. Appl Math Comput. 189, 1058–1065 (2007). Publisher Full Text

32. Agarwal, RP, Qin, X, Kang, SM: An implicit iterative algorithm with errors for two families of generalized asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2011, 58 (2011)

33. Cioranescu, I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer, Dordrecht (1990)

34. Alber, YaI: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, AG (eds.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15–50. Marcel Dekker, New York (1996)

35. Hudzik, H, Kowalewski, W, Lewicki, G: Approximative compactness and full rotundity in Musielak-Orlicz spaces and Lorentz-Orlicz spaces. Zeitschrift Analysis Anwendungen. 25, 163–192 (2006)

36. Takahashi, W: Nonlinear Functional Analysis. Yokohama-Publishers, Yokohama (2000)

37. Reich, S: A weak convergence theorem for the alternating method with Bregman distance. In: Kartsatos AG (ed.) Theory and Applications of Nonlinear Operatorsof Accretive and Monotone Type, Marcel Dekker, New York (1996)

38. Su, Y, Qin, X: Strong convergence of modified Ishikawa iterations for nonlinear mappings. Proc Indian Acad Sci Math Sci. 117, 97–107 (2007). Publisher Full Text

39. Agarwal, RP, Cho, YJ, Qin, X: Generalized projection algorithms for nonlinear operators. Numer Funct Anal Optim. 28, 1197–1215 (2007). Publisher Full Text

40. Qin, X, Su, Y, Wu, C, Liu, K: Strong convergence theorems for nonlinear operators in Banach spacess. Commun Appl Nonlinear Anal. 14, 35–50 (2007)

41. Qin, X, Cho, YJ, Kang, SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J Comput Appl Math. 225, 20–30 (2009). Publisher Full Text

42. Rockafellar, RT: Characterization of the subdifferentials of convex functions. Pacific J Math. 17, 497–510 (1966)