 Research
 Open Access
 Published:
Projection algorithms for treating asymptotically quasiϕnonexpansive mappings in the intermediate sense
Journal of Inequalities and Applications volume 2013, Article number: 265 (2013)
Abstract
In this paper, we investigate the fixed point problem of asymptotically quasiϕnonexpansive mappings in the intermediate sense based on a projection algorithm. Strong convergence of the proposed algorithm is obtained in a reflexive, strictly convex, and smooth Banach space.
MSC:47H09, 47J25.
1 Introduction
Fixed point theory of nonlinear mapping is a popular research topic of common interest in two areas of nonlinear analysis and optimization. Over the last 60 years or so, the theory of fixed points has been revealed as a very powerful and important tool in the study of nonlinear phenomena. In particular, fixed point techniques have been applied in such diverse fields as biology, chemistry, economics, engineering, game theory, and physics; see [1–11] and the references therein. There are many results on the existence of fixed points of nonlinear mappings. 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 algorithm 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 [1, 2] for more details and the references therein.
For iterative algorithms, the oldest and simplest one is the Picard iterative algorithm. It is known that T, where T stands for a contractive mapping, enjoys a unique fixed point, and the sequence generated by the Picard iterative algorithm can converge to the unique fixed point. However, for more general nonexpansive mappings, the Picard iterative algorithm fails to converge to fixed points of nonexpansive mappings even if they enjoy fixed points. The KrasnoselskiiMann iterative algorithm has been studied for approximating fixed points of nonexpansive mappings and their extensions. However, the KrasnoselskiiMann iterative algorithm is weak convergence for nonexpansive mappings only; see [12]. In many disciplines, problems arise 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 so that the energy \parallel {x}_{n}x\parallel of the error between the iterate {x}_{n} and the solution x eventually becomes arbitrarily small. Strong convergence of iterative sequences properties has a direct impact when the process is executed directly in the underlying infinite dimensional space.
Projection methods which were first introduced by Haugazeau [13] have been considered for the approximation of fixed points of nonlinear mappings. The advantage of projection methods is that strong convergence of iterative sequences can be guaranteed without any compact assumptions.
The purpose of this paper is to investigate a projection algorithm for asymptotically quasiϕnonexpansive mappings in the intermediate sense. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a projection algorithm is investigated. Strong convergence of the purposed algorithm is obtained in a reflexive, strictly convex, and smooth Banach space. Some deduced results are also obtained.
2 Preliminaries
Let E be a real Banach space, C be a nonempty subset of E and T:C\to C be a nonlinear mapping. In this paper, we use F(T) to denote the fixed point set of T. The mapping T is said to be asymptotically regular on C if for any bounded subset K of C,
The mapping T is said to be closed if for any sequence \{{x}_{n}\}\subset C such that {lim}_{n\to \mathrm{\infty}}{x}_{n}={x}_{0} and {lim}_{n\to \mathrm{\infty}}T{x}_{n}={y}_{0}, then T{x}_{0}={y}_{0}. A point x\in C is a fixed point of T provided Tx=x. In this paper, we use → and ⇀ to denote the strong convergence and weak convergence, respectively.
Recall that the mapping T is said to be nonexpansive iff
T is said to be quasinonexpansive iff F(T)\ne \mathrm{\varnothing}, and
T is said to be asymptotically nonexpansive iff there exists a sequence \{{k}_{n}\}\subset [1,\mathrm{\infty}) with {k}_{n}\to 1 as n\to \mathrm{\infty} such that
T is said to be asymptotically quasinonexpansive iff F(T)\ne \mathrm{\varnothing} and there exists a sequence \{{k}_{n}\}\subset [1,\mathrm{\infty}) with {k}_{n}\to 1 as n\to \mathrm{\infty} such that
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [14] in 1972. In uniformly convex Banach spaces, they proved that if C is nonempty bounded closed and convex, then every asymptotically nonexpansive selfmapping T on C has a fixed point. Further, the fixed point set of T is closed and convex. Since 1972, a host of authors have studied the weak and strong convergence of iterative algorithms for such a class of mappings.
T is said to be asymptotically nonexpansive in the intermediate sense iff it is continuous and the following inequality holds:
T is said to be asymptotically quasinonexpansive in the intermediate sense iff F(T)\ne \mathrm{\varnothing} and the following inequality holds:
The class of the mappings which are asymptotically nonexpansive in the intermediate sense was considered by Bruck et al. [15]. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense may not be Lipschitz continuous. However, asymptotically nonexpansive mappings are Lipschitz continuous.
One of classical iterations is the Halpern iteration [16] which generates a sequence in the following manner:
where \{{\alpha}_{n}\} is a sequence in the interval (0,1) and u\in C is a fixed element.
Since 1967, the Halpern iteration has been studied extensively by many authors. It is well known that the following two restrictions:
(C1) {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0;
(C2) {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}
are necessary in the sense that if the Halbern iterative sequence is strongly convergent for all nonexpansive selfmappings defined on C. To improve the rate of convergence of the Halbern iterative sequence, we cannot rely only on the iteration itself. Hybrid projection methods recently have been applied to solve the problem.
MartinezYanes and Xu [17] considered the hybrid projection algorithm for a nonexpansive mapping in a Hilbert space. Strong convergence theorems are established under the condition (C1) only imposed on the control sequence. To be more precise, they proved the following theorem.
Theorem 2.1 Let H be a real Hilbert space, C be a closed convex subset of H and T:C\to C be a nonexpansive mapping such that F(T)\ne \mathrm{\varnothing}. Assume that \{{\alpha}_{n}\}\subset (0,1) is such that {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0. Then the sequence \{{x}_{n}\} defined by
converges strongly to {P}_{F(T)}{x}_{0}.
Let E be a Banach space with the dual {E}^{\ast}. We denote by J the normalized duality mapping from E to {2}^{{E}^{\ast}} defined by
where \u3008\cdot ,\cdot \u3009 denotes the generalized duality pairing.
A Banach space E is said to be strictly convex if \parallel \frac{x+y}{2}\parallel <1 for all x,y\in E with \parallel x\parallel =\parallel y\parallel =1 and x\ne y. It is said to be uniformly convex if {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{y}_{n}\parallel =0 for any two sequences \{{x}_{n}\} and \{{y}_{n}\} in E such that \parallel {x}_{n}\parallel =\parallel {y}_{n}\parallel =1 and {lim}_{n\to \mathrm{\infty}}\parallel \frac{{x}_{n}+{y}_{n}}{2}\parallel =1. Let {U}_{E}=\{x\in E:\parallel x\parallel =1\} be the unit sphere of E. Then the Banach space E is said to be smooth provided
exists for each x,y\in {U}_{E}. It is also said to be uniformly smooth if the above limit is attained uniformly for x,y\in {U}_{E}. It is well known that if E is uniformly smooth, then J is uniformly normtonorm continuous on each bounded subset of E. It is also well known that E is uniformly smooth if and only if {E}^{\ast} is uniformly convex.
Recall that a Banach space E enjoys the KadecKlee property if for any sequence \{{x}_{n}\}\subset E, and x\in E with {x}_{n}\rightharpoonup x, and \parallel {x}_{n}\parallel \to \parallel x\parallel, then \parallel {x}_{n}x\parallel \to 0 as n\to \mathrm{\infty}. For more details on the KadecKlee property, the readers can refer to [18] and the references therein. It is well known that if E is a uniformly convex Banach space, then E enjoys the KadecKlee property.
As we all know, if C is a nonempty closed convex subset of a Hilbert space H and {P}_{C}:H\to C is the metric projection of H onto C, then {P}_{C} is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently, it is not available in more general Banach spaces. In this connection, Alber [19] recently introduced a generalized projection operator {\mathrm{\Pi}}_{C} in a Banach space E which is an analogue of the metric projection in Hilbert spaces.
Next, we assume that E is a smooth Banach space. Consider the functional defined by
Observe that, in a Hilbert space H, (2.1) is reduced to \varphi (x,y)={\parallel xy\parallel}^{2}, x,y\in H. The generalized projection {\mathrm{\Pi}}_{C}:E\to C is a map that assigns to an arbitrary point x\in E the minimum point of the functional \varphi (x,y), that is, {\mathrm{\Pi}}_{C}x=\overline{x}, where \overline{x} is the solution to the minimization problem
the existence and uniqueness of the operator {\mathrm{\Pi}}_{C} follow from the properties of the functional \varphi (x,y) and strict monotonicity of the mapping J; for more details, see [18] and [19] and the references therein. In Hilbert spaces, {\mathrm{\Pi}}_{C}={P}_{C}. It is obvious from the definition of a function ϕ that
and
Remark 2.2 If E is a reflexive, strictly convex, and smooth Banach space, then for x,y\in E, \varphi (x,y)=0 if and only if x=y. It is sufficient to show that if \varphi (x,y)=0, then x=y. From (2.2), we have \parallel x\parallel =\parallel y\parallel. This implies that \u3008x,Jy\u3009={\parallel x\parallel}^{2}={\parallel Jy\parallel}^{2}. From the definition of J, we have Jx=Jy. Therefore, we have x=y; for more details, see [18] and [19] and the references therein.
Let C be a nonempty closed convex subset of E and let T be a mapping from C into itself. A point p in C is said to be an asymptotic fixed point of T [20] if C contains a sequence \{{x}_{n}\} which converges weakly to p such that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0. The set of asymptotic fixed points of T will be denoted by \tilde{F}(T). A mapping T from C into itself is said to be relatively nonexpansive [21] if \tilde{F}(T)=F(T)\ne \mathrm{\varnothing} and \varphi (p,Tx)\le \varphi (p,x) for all x\in C and p\in F(T). The mapping T is said to be relatively asymptotically nonexpansive [22, 23] if \tilde{F}(T)=F(T)\ne \mathrm{\varnothing} and there exists a sequence \{{k}_{n}\}\subset [1,\mathrm{\infty}) with {k}_{n}\to 1 as n\to \mathrm{\infty} such that \varphi (p,Tx)\le {k}_{n}\varphi (p,x) for all x\in C, p\in F(T) and n\ge 1.
The mapping T is said to be quasiϕnonexpansive [24] if F(T)\ne \mathrm{\varnothing} and \varphi (p,Tx)\le \varphi (p,x) for all x\in C and p\in F(T). T is said to be asymptotically quasiϕnonexpansive [25–27] if F(T)\ne \mathrm{\varnothing} and there exists a sequence \{{k}_{n}\}\subset [0,\mathrm{\infty}) with {k}_{n}\to 1 as n\to \mathrm{\infty} such that \varphi (p,Tx)\le {k}_{n}\varphi (p,x) for all x\in C, p\in F(T) and n\ge 1.
Remark 2.3 The class of asymptotically quasiϕnonexpansive mappings is more general than the class of relatively asymptotically nonexpansive mappings which requires the restriction F(T)=\tilde{F}(T).
In 2007, Qin and Su [28] extended the results of MartinezYanes and Xu [17] from Hilbert spaces to Banach spaces. To be more precise, they established the following theorem.
Theorem 2.4 Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E and let T:C\to C be a relatively nonexpansive mapping. Assume that \{{\alpha}_{n}\} is a sequence in (0,1) such that {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0. Define a sequence \{{x}_{n}\} in C in the following manner:
If F(T) is nonempty, then \{{x}_{n}\} converges to {\mathrm{\Pi}}_{F(T)}{x}_{0}.
T is said to be an asymptotically quasiϕnonexpansive in the intermediate sense [29] if F(T)\ne \mathrm{\varnothing} and
Put
It follows that {\xi}_{n}\to 0 as n\to \mathrm{\infty}. Then (2.4) is reduced to the following:
Remark 2.5 The class of asymptotically quasiϕnonexpansive mappings in the intermediate sense is a generalization of the class of asymptotically quasinonexpansive mappings in the intermediate sense in the framework of Banach spaces.
Recently, many authors investigated fixed point problems of asymptotically quasiϕnonexpansive mappings and asymptotically quasiϕnonexpansive mappings in the intermediate sense based on projection algorithms; for more details, see [30–36]. In this paper, we investigate the fixed point problems of asymptotically quasiϕnonexpansive mappings in the intermediate sense based on a projection algorithm. Strong convergence of the proposed algorithm is obtained in a reflexive, strictly convex, and smooth Banach space such that both E and {E}^{\ast} have the KadecKlee property. The results presented in this paper mainly improve the corresponding results in Cho et al. [30].
In order to prove our main results, we need the following lemmas.
Lemma 2.6 [18]
Let C be a nonempty, closed, and convex subset of a smooth Banach space E, and x\in E. Then {x}_{0}={\mathrm{\Pi}}_{C}x if and only if
Lemma 2.7 [18]
Let E be a reflexive, strictly, convex, and smooth Banach space, let C be a nonempty, closed, and convex subset of E, and x\in E. Then
3 Main results
Theorem 3.1 Let E be a reflexive, strictly convex, and smooth Banach space such that both E and {E}^{\ast} have the KadecKlee property. Let C be a nonempty closed and convex subset of E. Let T:C\to C be a closed and asymptotically quasiϕnonexpansive mapping in the intermediate sense. Assume that T is asymptotically regular on C and F(T) is nonempty and bounded. Let \{{x}_{n}\} be a sequence generated in the following manner:
where
If the sequence \{{\alpha}_{n}\} satisfies the restriction {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, then the sequence \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F(T)}{x}_{1}, where {\mathrm{\Pi}}_{F(T)} is the generalized projection from C onto F(T).
Proof First, we show that F(T) is closed and convex. Since T is closed, we can easily conclude that F(T) is also closed. This proves that F(T) is closed. Next, we prove the convexness of F(T). Let {p}_{1},{p}_{2}\in F(T), and p=t{p}_{1}+(1t){p}_{2}, where t\in (0,1). We see that p=Tp. Indeed, we see from the definition of T that
and
In view of (2.2), we find that
and
Combining (3.1), (3.2), (3.3) with (3.4) yields that
and
Multiplying t and (1t) on the both sides of (3.5) and (3.6), respectively, yields that
In light of (2.3), we arrive at
It follows that
Since {E}^{\ast} is reflexive, we may, without loss of generality, assume that J({T}^{n}p)\rightharpoonup {e}^{\ast}\in {E}^{\ast}. In view of the reflexivity of E, we have J(E)={E}^{\ast}. This shows that there exists an element e\in E such that Je={e}^{\ast}. It follows that
Taking {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}} on the both sides of the equality above, we obtain that
This implies that p=e, that is, Jp={e}^{\ast}. It follows that J({T}^{n}p)\rightharpoonup Jp\in {E}^{\ast}. In view of the KadecKlee property of {E}^{\ast}, we obtain from (3.8) that {lim}_{n\to \mathrm{\infty}}\parallel J({T}^{n}p)Jp\parallel =0. Since {J}^{1}:{E}^{\ast}\to E is demicontinuous, we see that {T}^{n}p\rightharpoonup p. By virtue of the KadecKlee property of E, we see from (3.7) that {T}^{n}p\to p as n\to \mathrm{\infty}. Hence T{T}^{n}p={T}^{n+1}p\to p as n\to \mathrm{\infty}. In view of the closedness of T, we can obtain that p\in F(T). This shows that F(T) is convex. This completes the proof that F(T) is convex and closed. This means that {\mathrm{\Pi}}_{F(T)}x is well defined for any x\in C. Next, we show that {C}_{n} is closed and convex. It is obvious that {C}_{1}=C is closed and convex. Suppose that {C}_{h} is closed and convex for some h\in \mathbb{N}. For z\in {C}_{h}, we see that
is equivalent to
It is not hard to see that {C}_{h+1} is closed and convex. Then, for each n\ge 1, {C}_{n} is closed and convex. This shows that {\mathrm{\Pi}}_{{C}_{n+1}}{x}_{1} is well defined.
Next, we prove that \mathcal{F}\subset {C}_{n}. F(T)\subset {C}_{1}=C is obvious. Suppose that F(T)\subset {C}_{h} for some h\in \mathbb{N}. Then, for \mathrm{\forall}w\in F(T)\subset {C}_{h}, we have
This shows that w\in {C}_{h+1}. This implies that \mathcal{F}\subset {C}_{n}. In view of {x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{1}, we see that
Since F(T)\subset {C}_{n}, we arrive at
It follows from Lemma 2.7 that
This implies that the sequence \{\varphi ({x}_{n},{x}_{0})\} is bounded. It follows from (2.2) that the sequence \{{x}_{n}\} is also bounded. Now, we are in a position to show that {x}_{n}\to \overline{x}, where \overline{x}\in F(T) as n\to \mathrm{\infty}. Since \{{x}_{n}\} is bounded, and the space is reflexive, we may assume that {x}_{n}\rightharpoonup \overline{x}. Since {C}_{n} is closed and convex, we see that \overline{x}\in {C}_{n}. On the other hand, we see from the weakly lower semicontinuity of the norm that
which implies that {lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1})=\varphi (\overline{x},{x}_{1}). Hence, we have {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}\parallel =\parallel \overline{x}\parallel. In view of the KadecKlee property of E, we see that {x}_{n}\to \overline{x} as n\to \mathrm{\infty}. Next, we show that \overline{x}\in F(T). Since {x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{1} and {x}_{n+1}={\mathrm{\Pi}}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}\subset {C}_{n}, we find that \varphi ({x}_{n},{x}_{1})\le \varphi ({x}_{n+1},{x}_{1}). This shows that \{\varphi ({x}_{n},{x}_{1})\} is nondecreasing. It follows from the boundedness that {lim}_{n\to \mathrm{\infty}}\varphi ({x}_{,}{x}_{1}) exists. In view of construction of {x}_{n+1}={\mathrm{\Pi}}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}\subset {C}_{n}, we arrive at
It follows that
Since {x}_{n+1}={\mathrm{\Pi}}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}, we arrive at
This in turn implies from (3.10) that
In view of (2.2), we see that
This in turn implies that
It follows that
This implies that \{J{y}_{n}\} is bounded. Note that both E and {E}^{\ast} are reflexive. We may assume that J{y}_{n}\rightharpoonup {y}^{\ast}\in {E}^{\ast}. In view of the reflexivity of E, we see that J(E)={E}^{\ast}. This shows that there exists an element y\in E such that Jy={y}^{\ast}. It follows that
Taking {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}} on the both sides of the equality above yields that
That is, \overline{x}=y, which in turn implies that {y}^{\ast}=J\overline{x}. It follows that J{y}_{n}\rightharpoonup J\overline{x}\in {E}^{\ast}. Since {E}^{\ast} enjoys the KadecKlee property, we obtain from (3.12) that
Notice that
It follows that
In view of
we find from {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0 that
Notice that
This implies from (3.14) that
The demicontinuity of {J}^{1}:{E}^{\ast}\to E implies that {T}^{n}{x}_{n}\rightharpoonup \overline{x}. Note that
In view of (3.15), we see that {lim}_{n\to \mathrm{\infty}}\parallel {T}^{n}{x}_{n}\parallel =\parallel \overline{x}\parallel. Since E has the KadecKlee property, we find that
Notice that
In view of the asymptotic regularity of T, we find from (3.16) that
that is, T{T}^{n}{x}_{n}\overline{x}\to 0 as n\to \mathrm{\infty}. It follows from the closedness of T that T\overline{x}=\overline{x}.
Finally, we show that \overline{x}={\mathrm{\Pi}}_{F(T)}{x}_{1}. Letting n\to \mathrm{\infty} in (3.9), we arrive at
It follows from Lemma 2.6 that \overline{x}={\mathrm{\Pi}}_{F(T)}{x}_{1}. This completes the proof of the theorem. □
Remark 3.2 Since every uniformly smooth and uniformly convex Banach space is a reflexive, strictly convex, and smooth Banach space such that both E and {E}^{\ast} have the KadecKlee property, we find that Theorem 3.1 is still valid in the framework of uniformly smooth and uniformly convex Banach spaces.
If E is a Hilbert space, then we have the following result.
Corollary 3.3 Let E be a Hilbert space. Let C be a nonempty closed and convex subset of E. Let T:C\to C be a closed quasinonexpansive mapping in the intermediate sense. Assume that T is asymptotically regular on C and F(T) is nonempty and bounded. Let \{{x}_{n}\} be a sequence generated in the following manner:
where
If the sequence \{{\alpha}_{n}\} satisfies the restriction {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, then the sequence \{{x}_{n}\} converges strongly to {P}_{F(T)}{x}_{1}, where {P}_{F(T)} is the metric projection from C onto F(T).
If T is quasiϕnonexpansive, then we have the following result.
Corollary 3.4 Let E be a reflexive, strictly convex, and smooth Banach space such that both E and {E}^{\ast} have the KadecKlee property. Let C be a nonempty closed and convex subset of E. Let T:C\to C be a closed quasiϕnonexpansive mapping with a nonempty fixed point set. Let \{{x}_{n}\} be a sequence generated in the following manner:
If the sequence \{{\alpha}_{n}\} satisfies the restriction {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, then the sequence \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F(T)}{x}_{1}, where {\mathrm{\Pi}}_{F(T)} is the generalized projection from C onto F(T).
Next, we consider an equilibrium problem based on the projection algorithm. Find p\in C such that
We use EP(f) to denote the solution set of the equilibrium problem (3.17). That is,
Given a mapping Q:C\to {E}^{\ast}, let
Then p\in EP(f) if and only if p is a solution of the following variational inequality. Find p such that
For studying the equilibrium problem (3.17), let us assume that f satisfies the following conditions:
(A1) f(x,x)=0, \mathrm{\forall}x\in C;
(A2) f is monotone, i.e., f(x,y)+f(y,x)\le 0, \mathrm{\forall}x,y\in C;
(A3)
(A4) for each x\in C, y\mapsto f(x,y) is convex and weakly lower semicontinuous.
Lemma 3.5 Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Let f be a bifunction from C\times C to ℝ satisfying (A1)(A4). Let r>0 and x\in E. Then

(a)
[37]. There exists z\in C such that
f(z,y)+\frac{1}{r}\u3008yz,JzJx\u3009\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in C; 
(b)
[24, 38]. Define a mapping {T}_{r}:E\to C by
{S}_{r}x=\{z\in C:f(z,y)+\frac{1}{r}\u3008yz,JzJx\u3009,\mathrm{\forall}y\in C\}.
Then the following conclusions hold:

(1)
{S}_{r} is singlevalued;

(2)
{S}_{r} is a firmly nonexpansivetype mapping, i.e., for all x,y\in E,
\u3008{S}_{r}x{S}_{r}y,J{S}_{r}xJ{S}_{r}y\u3009\le \u3008{S}_{r}x{S}_{r}y,JxJy\u3009 
(3)
F({S}_{r})=EP(f);

(4)
{S}_{r} is quasiϕnonexpansive;

(5)
\varphi (q,{S}_{r}x)+\varphi ({S}_{r}x,x)\le \varphi (q,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}q\in F({S}_{r});

(6)
EP(f) is closed and convex.
Corollary 3.6 Let E be a reflexive, strictly convex, and smooth Banach space such that both E and {E}^{\ast} have the KadecKlee property. Let C be a nonempty closed and convex subset of E. Let f be a bifunction from C\times C to ℝ satisfying (A1)(A4) with a nonempty solution set. Let r be a positive real number. Let \{{x}_{n}\} be a sequence generated in the following manner:
where {z}_{n} is such that
If the sequence \{{\alpha}_{n}\} satisfies the restriction {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0, then the sequence \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{EP(f)}{x}_{1}, where {\mathrm{\Pi}}_{EP(f)} is the generalized projection from C onto EP(f).
Proof Put {z}_{n}={S}_{r}{x}_{n}. In view of Lemma 3.5, we find that {S}_{r} is quasiϕnonexpansive which is an asymptotically quasiϕnonexpansive. We immediately find from Theorem 3.1 the desired conclusion. □
References
Byrne C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 2008, 20: 103–120.
Vanderluge A: Optical Signal Processing. Wiley, New York; 2005.
Qin X, Chang SS, Kang SM: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 2010, 11: 2963–2972. 10.1016/j.nonrwa.2009.10.017
Luo H, Wang Y: Iterative approximation for the common solutions of a infinite variational inequality system for inversestrongly accretive mappings. J. Math. Comput. Sci. 2012, 2: 1660–1670.
Zegeye H, Shahzad N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory 2012, 2: 374–397.
Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011, 24: 224–228. 10.1016/j.aml.2010.09.008
Ye J, Huang J: Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasinonexpansive mappings in Banach spaces. J. Math. Comput. Sci. 2011, 1: 1–18.
Cho SY, Kang SM, Qin X: Hybrid projection algorithms for treating common fixed points of a family of demicontinuous pseudocontractions. Appl. Math. Lett. 2012, 25: 854–857. 10.1016/j.aml.2011.10.031
Noor MA, Noor KI, Waseem M: Decomposition method for solving system of linear equations. Eng. Math. Lett. 2013, 2: 34–41.
Cho SY, Kang SM, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607–1618.
Qin X, Cho SY, Kang SM: Strong convergence of shrinking projection methods for quasi ϕ nonexpansive mappings and equilibrium problems. J. Comput. Appl. Math. 2010, 234: 750–760. 10.1016/j.cam.2010.01.015
Genel A, Lindenstrass J: An example concerning fixed points. Isr. J. Math. 1975, 22: 81–86. 10.1007/BF02757276
Haugazeau, Y: Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes. Ph.D. thesis, Université de Paris (1968)
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1972, 35: 171–174. 10.1090/S00029939197202985003
Bruck RE, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloq. Math. 1993, 65: 169–179.
Halpern B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S000299041967118640
MartinezYanes C, Xu HK: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal. 2006, 64: 2400–2411. 10.1016/j.na.2005.08.018
Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht; 1990.
Alber YaI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos AG. Dekker, New York; 1996.
Reich S: A weak convergence theorem for the alternating method with Bregman distance. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos AG. Dekker, New York; 1996.
Butnariu D, Reich S, Zaslavski AJ: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. J. Appl. Anal. 2001, 7: 151–174.
Agarwal RP, Cho YJ, Qin X: Generalized projection algorithms for nonlinear operators. Numer. Funct. Anal. Optim. 2007, 28: 1197–1215. 10.1080/01630560701766627
Qin X, Su Y, Wu C, Liu K: Strong convergence theorems for nonlinear operators in Banach spaces. Commun. Appl. Nonlinear Anal. 2007, 14: 35–50.
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. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011
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. 2010, 32: 453–464. 10.1007/s1219000902634
Qin X, Cho SY, Kang SM: On hybrid projection methods for asymptotically quasi ϕ nonexpansive mappings. Appl. Math. Comput. 2010, 215: 3874–3883. 10.1016/j.amc.2009.11.031
Qin X, Agarwal RP: Shrinking projection methods for a pair of asymptotically quasi ϕ nonexpansive mappings. Numer. Funct. Anal. Optim. 2010, 31: 1072–1089. 10.1080/01630563.2010.501643
Qin X, Su Y: Strong convergence theorem for relatively nonexpansive mappings in a Banach space. Nonlinear Anal. 2007, 67: 1958–1965. 10.1016/j.na.2006.08.021
Qin X, Wang L: On asymptotically quasi ϕ nonexpansive mappings in the intermediate sense. Abstr. Appl. Anal. 2012., 2012: Article ID 636217
Cho YJ, Qin X, Kang SM: Strong convergence of the modified Halperntype iterative algorithms in Banach spaces. An. Univ. “Ovidius” Constanţa, Ser. Mat. 2009, 17: 51–68.
Chen JW, Cho YJ, Wan Z: Shrinking projection algorithms for equilibrium problems with a bifunction defined on the dual space of a Banach space. Fixed Point Theory Appl. 2011., 2011: Article ID 91
Qin X, Cho YJ, Kang SM, Zhou H: Convergence of a modified Halperntype iteration algorithm for quasi ϕ nonexpansive mappings. Appl. Math. Lett. 2009, 22: 1051–1055. 10.1016/j.aml.2009.01.015
Chang SS, Lee HWJ, Chan CK, Yang L: Approximation theorems for total quasi ϕ asymptotically nonexpansive mappings with applications. Appl. Math. Comput. 2011, 218: 2921–2931. 10.1016/j.amc.2011.08.036
Wei L, Cho YJ, Zhou HY: A strong convergence theorem for common fixed points of two relatively nonexpansive mappings and its applications. J. Appl. Math. Comput. 2009, 29: 95–103. 10.1007/s121900080092x
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. 2011., 2011: Article ID 10
Qin X, Huang S: On the convergence of hybrid projection algorithms for asymptotically quasi ϕ nonexpansive mappings. Comput. Math. Appl. 2011, 61: 851–859. 10.1016/j.camwa.2010.12.033
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.
Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 2009, 70: 45–57. 10.1016/j.na.2007.11.031
Acknowledgements
The authors are grateful to the editor and the reviewers for their useful suggestions which improved the contents of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Hecai, Y., Aichao, L. Projection algorithms for treating asymptotically quasiϕnonexpansive mappings in the intermediate sense. J Inequal Appl 2013, 265 (2013). https://doi.org/10.1186/1029242X2013265
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X2013265
Keywords
 asymptotically quasiϕnonexpansive mapping in the intermediate sense
 relatively nonexpansive mapping
 generalized projection
 fixed point