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Relaxed Halperntype iteration method for countable families of totally quasiϕasymptotically nonexpansive mappings
Journal of Inequalities and Applications volume 2013, Article number: 367 (2013)
Abstract
Based on an original idea, namely, a specific way of choosing the indexes of involved mappings, we propose a relaxed Halperntype iterative algorithm for approximating some common fixed point of a kind of nonlinear mappings and obtain a strong convergence theorem under suitable conditions. Since the involved mappings need no assumption of being uniformly totally quasiϕasymptotically nonexpansive, and there is no need to compute projections onto intersections of countably many closed and convex sets, the results improve those of the authors with related interest.
MSC:47H09, 47H10, 47J25.
1 Introduction
Throughout this paper, we assume that E is a real Banach space with its dual {E}^{\ast}, C is a nonempty closed convex subset of E, and J:E\to {2}^{{E}^{\ast}} is the normalized duality mapping defined by
In the sequel, we use F(T) to denote the set of fixed points of a mapping T.
Definition 1.1 [1]
A mapping T:C\to C is said to be totally quasiϕasymptotically nonexpansive if F(T)\ne \mathrm{\varnothing} and there exist nonnegative real sequences \{{\nu}_{n}\}, \{{\mu}_{n}\} with {\nu}_{n},{\mu}_{n}\to 0 (as n\to \mathrm{\infty}) and a strictly increasing continuous function \zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} with \zeta (0)=0 such that
where \varphi :E\times E\to {\mathbb{R}}^{+} denotes the Lyapunov functional defined by
It is obvious from the definition of ϕ that
Definition 1.2 [1]

(1)
A countable family of mappings \{{T}_{i}\}:C\to C is said to be totally uniformly quasiϕasymptotically nonexpansive if F:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\ne \mathrm{\varnothing} and there exist nonnegative real sequences \{{\nu}_{n}\}, \{{\mu}_{n}\} with {\nu}_{n},{\mu}_{n}\to 0 (as n\to \mathrm{\infty}) and a strictly increasing continuous function \zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} with \zeta (0)=0 such that
\varphi (p,{T}_{i}^{n}x)\le \varphi (p,x)+{\nu}_{n}\zeta (\varphi (p,x))+{\mu}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall}n\ge 1,i\ge 1,x\in C,p\in F(T).(1.4) 
(2)
A mapping T:C\to C is said to be uniformly LLipschitz continuous if there exists a constant L>0 such that
\parallel {T}^{n}x{T}^{n}y\parallel \le L\parallel xy\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}n\ge 1,x,y\in C.(1.5)
In 2012, Chang et al. [1] used the following modified Halperntype iteration algorithm for totally quasiϕasymptotically nonexpansive mappings to have the strong convergence under a limit condition only in the framework of Banach spaces.
where \{{T}_{i}\}:C\to C is a countable family of closed and uniformly totally quasiϕasymptotically nonexpansive mappings; and {\xi}_{n}={\nu}_{n}{sup}_{p\in F}\zeta (\varphi (p,{x}_{n}))+{\mu}_{n}, {\mathrm{\Pi}}_{{C}_{n+1}} is the generalized projection (see (2.1)) of E onto {C}_{n+1}. Their results extended and improved the corresponding results of Qin et al. [2, 3], Wang et al. [4], MartinezYanes and Xu [5] and others.
However, it is obviously a very strong condition that the involved mappings are assumed to be uniformly totally quasiϕasymptotically nonexpansive. Additionally, the assumption conditions imposed on all {C}_{n} (n\ge 2) are not weak at all since each one is in fact an intersection of countably many sets. This fact makes the projection very hard to compute, and therefore the method proposed in their paper does not seem to be valuable in practice.
Inspired and motivated by the studies mentioned above, in this article, we introduce a relaxed iterative algorithm for approximating some common fixed point of a countable family of totally quasiϕasymptotically nonexpansive mappings and obtain a strong convergence theorem.
2 Preliminaries
Following Alber [6], the generalized projection {\mathrm{\Pi}}_{C}:E\to C is defined by
Lemma 2.1 [6]
Let E be a smooth, strictly convex and reflexive Banach space, and let C be a nonempty closed convex subset of E. Then the following conclusions hold:

(1)
\varphi (x,{\mathrm{\Pi}}_{C}y)+\varphi ({\mathrm{\Pi}}_{C}y,y)\le \varphi (x,y) for all x\in C and y\in E;

(2)
If x\in E and z\in C, then z={\mathrm{\Pi}}_{C}x\iff \u3008zy,JxJz\u3009\ge 0, \mathrm{\forall}y\in C;

(3)
For x,y\in E, \varphi (x,y)=0 if and only if x=y.
Remark 2.2 The following basic properties for a Banach space E can be found in Cioranescu [7].

(i)
If E is uniformly smooth, then J is uniformly continuous on each bounded subset of E;

(ii)
If E is reflexive and strictly convex, then {J}^{1} is normweakcontinuous;

(iii)
If E is a smooth, strictly convex and reflexive Banach space, then the normalized duality mapping J:E\to {2}^{{E}^{\ast}} is singlevalued, onetoone and onto;

(iv)
A Banach space E is uniformly smooth if and only if {E}^{\ast} is uniformly convex;

(v)
Each uniformly convex Banach space E has the KadecKlee property, i.e., for any sequence \{{x}_{n}\}\subset E, if {x}_{n}\rightharpoonup x\in E and \parallel {x}_{n}\parallel \to \parallel x\parallel, then {x}_{n}\to x as n\to \mathrm{\infty}.
Lemma 2.3 [1]
Let E be a real uniformly smooth and strictly convex Banach space with the KadecKlee property, and let C be a nonempty closed convex subset of E. Let \{{x}_{n}\} and \{{y}_{n}\} be two sequences in C such that {x}_{n}\to p and \varphi ({x}_{n},{y}_{n})\to 0, where ϕ is the function defined by (1.2), then {y}_{n}\to p.
Lemma 2.4 [1]
Let E and C be the same as in Lemma 2.3. Let T:C\to C be a closed and totally quasiϕasymptotically nonexpansive mapping with nonnegative real sequences \{{\nu}_{n}\}, \{{\mu}_{n}\} and a strictly increasing continuous function \zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} such that {\nu}_{n},{\mu}_{n}\to 0 and \zeta (0)=0. If {\mu}_{1}=0, then the fixed point set F(T) of T is a closed and convex subset of C.
Lemma 2.5 [8]
The unique solutions to the positive integer equation
are
where [x] denotes the maximal integer that is not larger than x.
3 Main results
Theorem 3.1 Let E be a real uniformly smooth and strictly convex Banach space with the KadecKlee property, let C be a nonempty closed convex subset of E, and let {T}_{i}:C\to C, i=1,2,\dots , be a countable family of closed and totally quasiϕasymptotically nonexpansive mappings with nonnegative real sequences \{{\nu}_{n}^{(i)}\}, \{{\mu}_{n}^{(i)}\} satisfying {\nu}_{n}^{(i)}\to 0 and {\mu}_{n}^{(i)}\to 0 (as n\to \mathrm{\infty} and for each i\ge 1) and a strictly increasing and continuous function \zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} satisfying condition (1.1) and each {T}_{i} is uniformly {L}_{i}Lipschitz continuous. Let \{{\alpha}_{n}\} be a sequence in [0,1] with {\alpha}_{n}\to 0. Let \{{x}_{n}\} be the sequence generated by
where {\xi}_{n}:={\nu}_{{m}_{n}}^{({i}_{n})}{sup}_{p\in F}\zeta (\varphi (p,{x}_{n}))+{\mu}_{{m}_{n}}^{({i}_{n})}, {\mathrm{\Pi}}_{{C}_{n+1}} is the generalized projection of E onto {C}_{n+1}, and {i}_{n} and {m}_{n} are the solutions to the positive integer equation: n=i+\frac{(m1)m}{2} (m\ge i, n=1,2,\dots), that is, for each n\ge 1, there exist unique {i}_{n} and {m}_{n} such that
If F:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}) is bounded and {\mu}_{1}^{(i)}=0 for each i\ge 1, then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F}{x}_{1}.
Proof We divide the proof into several steps.

(I)
F and {C}_{n} (\mathrm{\forall}n\ge 1) both are closed and convex subsets in C.
In fact, it follows from Lemma 2.4 that each F({T}_{i}) is a closed and convex subset of C, so is F. In addition, with {C}_{1} (=C) being closed and convex, we may assume that {C}_{n} is closed and convex for some n\ge 2. In view of the definition of ϕ, we have that
where \phi (z)=2{\alpha}_{n}\u3008z,J{x}_{1}\u3009+2(1{\alpha}_{n})\u3008z,J{x}_{n}\u30092\u3008z,J{y}_{n}\u3009 and a={\alpha}_{n}{\parallel {x}_{1}\parallel}^{2}+(1{\alpha}_{n}){\parallel {x}_{n}\parallel}^{2}{\parallel {y}_{n}\parallel}^{2}+{\xi}_{n}. This shows that {C}_{n+1} is closed and convex.

(II)
F is a subset of {\bigcap}_{n=1}^{\mathrm{\infty}}{C}_{n}.
It is obvious that F\subset {C}_{1}. Suppose that F\subset {C}_{n} for some n\ge 2. Then, for any p\in F\subset {C}_{n}, we have
This implies that p\in {C}_{n+1}, and so F\subset {C}_{n+1}.

(III)
{x}_{n}\to {x}^{\ast}\in C as n\to \mathrm{\infty}.
In fact, since {x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{1}, from Lemma 2.1(2) we have \u3008{x}_{n}y,J{x}_{1}J{x}_{n}\u3009\ge 0, \mathrm{\forall}y\in {C}_{n}. Again since F\subset {\bigcap}_{n=1}^{\mathrm{\infty}}{C}_{n}, we have \u3008{x}_{n}p,J{x}_{1}J{x}_{n}\u3009\ge 0, \mathrm{\forall}p\in F. It follows from Lemma 2.1(1) that for each p\in F and for each n\ge 1,
which implies that \{\varphi ({x}_{n},{x}_{1})\} is bounded, so is \{{x}_{n}\}. Since for all n\ge 1, {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 have \varphi ({x}_{n},{x}_{1})\le \varphi ({x}_{n+1},{x}_{1}). This implies that \{\varphi ({x}_{n},{x}_{1})\} is nondecreasing, hence the limit
Since E is reflexive, there exists a subsequence \{{x}_{{n}_{i}}\} of \{{x}_{n}\} such that {x}_{{n}_{i}}\rightharpoonup {x}^{\ast}\in C as i\to \mathrm{\infty}. Since {C}_{n} is closed and convex and {C}_{n+1}\subset {C}_{n}, this implies that {C}_{n} is weakly closed and {x}^{\ast}\in {C}_{n} for each n\ge 1. In view of {x}_{{n}_{i}}={\mathrm{\Pi}}_{{C}_{{n}_{i}}}{x}_{1}, we have
Since the norm \parallel \cdot \parallel is weakly lower semicontinuous, we have
and so
This implies that {lim}_{i\to \mathrm{\infty}}\varphi ({x}_{{n}_{i}},{x}_{1})=\varphi ({x}^{\ast},{x}_{1}), and so \parallel {x}_{{n}_{i}}\parallel \to \parallel {x}^{\ast}\parallel as i\to \mathrm{\infty}. Since {x}_{{n}_{i}}\rightharpoonup {x}^{\ast}, by virtue of the KadecKlee property of E, we obtain that
Since \{\varphi ({x}_{n},{x}_{1})\} is convergent, this, together with {lim}_{i\to \mathrm{\infty}}\varphi ({x}_{{n}_{i}},{x}_{1})=\varphi ({x}^{\ast},{x}_{1}), shows that {lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1})=\varphi ({x}^{\ast},{x}_{1}). If there exists some subsequence \{{x}_{{n}_{j}}\} of \{{x}_{n}\} such that {x}_{{n}_{j}}\to y as j\to \mathrm{\infty}, then from Lemma 2.1(1) we have that
that is, {x}^{\ast}=y and so

(IV)
{x}^{\ast} is a member of F.
Set {\mathcal{K}}_{i}=\{k\in \mathbb{N}:k=i+\frac{(m1)m}{2},m\ge i,m\in \mathbb{N}\} for each i\ge 1. For example, by Lemma 2.5 and the definition of {\mathcal{K}}_{1}, we have {\mathcal{K}}_{1}=\{1,2,4,7,11,16,\dots \} and {i}_{1}={i}_{2}={i}_{4}={i}_{7}={i}_{11}={i}_{16}=\cdots =1. Then we have
Note that {\{{m}_{k}\}}_{k\in {\mathcal{K}}_{i}}=\{i,i+1,i+2,\dots \}, i.e., {m}_{k}\uparrow \mathrm{\infty} as {\mathcal{K}}_{i}\ni k\to \mathrm{\infty}. It follows from (3.3) and (3.4) that
Since {x}_{n+1}\in {C}_{n+1}, it follows from (3.1), (3.3) and (3.5) that
as {\mathcal{K}}_{i}\ni k\to \mathrm{\infty}. Since {x}_{k}\to {x}^{\ast} as {\mathcal{K}}_{i}\ni k\to \mathrm{\infty}, it follows from (3.6) and Lemma 2.3 that
Note that {T}_{{i}_{k}}^{{m}_{k}}={T}_{i}^{{m}_{k}} whenever k\in {\mathcal{K}}_{i} for each i\ge 1. Since {\{{x}_{k}\}}_{k\in {\mathcal{K}}_{i}} is bounded, so is {\{{T}_{i}^{{m}_{k}}{x}_{k}\}}_{k\in {\mathcal{K}}_{i}}. In view of {\alpha}_{k}\to 0, hence from (3.1) we have that
In addition, J{y}_{k}\to J{x}^{\ast} implies that {lim}_{{\mathcal{K}}_{i}\ni k\to \mathrm{\infty}}J{T}_{i}^{{m}_{k}}=J{x}^{\ast}. Remark 2.2(ii) yields that, as {\mathcal{K}}_{i}\ni k\to \mathrm{\infty},
Again, since for each i\ge 1, as {\mathcal{K}}_{i}\ni k\to \mathrm{\infty},
this, together with (3.10) and the KadecKlee property of E, shows that
On the other hand, by the assumptions that for each i\ge 1, {T}_{i} is uniformly {L}_{i}Lipschitz continuous, and noting that {m}_{k+1}1={m}_{k} for all k\in {\mathcal{K}}_{i}, we then have
From (3.11) and {x}_{k}\to {x}^{\ast} ({\mathcal{K}}_{i}\ni k\to \mathrm{\infty}), we have that {lim}_{{\mathcal{K}}_{i}\ni k\to \mathrm{\infty}}\parallel {T}_{i}^{{m}_{k+1}}{x}_{k}{T}_{i}^{{m}_{k}}{x}_{k}\parallel =0 and {lim}_{{\mathcal{K}}_{i}\ni k\to \mathrm{\infty}}{T}_{i}^{{m}_{k+1}}{x}_{k}={x}^{\ast}, i.e., {lim}_{{\mathcal{K}}_{i}\ni k\to \mathrm{\infty}}{T}_{i}({T}_{i}^{{m}_{k+1}1}{x}_{k})={x}^{\ast}. It then follows that, for each i\ge 1,
In view of the closedness of {T}_{i}, it follows from (3.11) that {T}_{i}{x}^{\ast}={x}^{\ast}, namely, for each i\ge 1, {x}^{\ast}\in F({T}_{i}) and hence {x}^{\ast}\in F.

(V)
{x}^{\ast}={\mathrm{\Pi}}_{F}{x}_{1}, and so {x}_{n}\to {\mathrm{\Pi}}_{F}{x}_{1} as n\to \mathrm{\infty}.
Put u={\mathrm{\Pi}}_{F}{x}_{1}. Since u\in F\subset {C}_{n} and {x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{1}, we have \varphi ({x}_{n},{x}_{1})\le \varphi (u,{x}_{1}), \mathrm{\forall}n\ge 1. Then
which implies that {x}^{\ast}=u since u={\mathrm{\Pi}}_{F}{x}_{1}, and hence {x}_{n}\to {x}^{\ast}={\mathrm{\Pi}}_{F}{x}_{1}. This completes the proof. □
Remark 3.2 Note that algorithm (3.1) just depends on the projection onto a single closed and convex set for each fixed n. An example [9] of how to compute such a projection is given as follows.
Dykstra’s algorithm Let {\mathrm{\Omega}}_{1},{\mathrm{\Omega}}_{2},\dots ,{\mathrm{\Omega}}_{p} be closed and convex subsets of {\mathbb{R}}^{n}. For any i=1,2,\dots ,p and {x}^{0}\in {\mathbb{R}}^{n}, the sequences \{{x}_{i}^{k}\} are defined by the following recursive formulae:
for k=1,2,\dots with initial values {x}_{p}^{0}={x}^{0} and {y}_{i}^{0}=0 for i=1,2,\dots ,p. If \mathrm{\Omega}:={\bigcap}_{i=1}^{p}{\mathrm{\Omega}}_{i}\ne \mathrm{\varnothing}, then \{{x}_{i}^{k}\} converges to {x}^{\ast}={P}_{\mathrm{\Omega}}({x}^{0}), where {P}_{\mathrm{\Omega}}(x):=arg{inf}_{y\in \mathrm{\Omega}}{\parallel yx\parallel}^{2}, \mathrm{\forall}x\in {\mathbb{R}}^{n}.
We now give a nontrivial example of the calculation of common fixed points for specific mappings.
Example 3.3 Let E={\mathbb{R}}^{1} with the standard norm \parallel \cdot \parallel =\cdot  and C=[0,1]. Let \{{T}_{i}\}:C\to C be a sequence of nonexpansive mappings defined by {T}_{i}x=\frac{{x}^{i}}{i}. Consider the following iteration sequence generated by
where \{{\alpha}_{n}\}=\{\frac{2}{3}\frac{1}{4n}\}, \{{\beta}_{n}\}=\{\frac{4}{5}\frac{1}{2n}\} and {\mathrm{\Pi}}_{{C}_{n+1}}(x):=arg{inf}_{y\in {C}_{n+1}}yx. Note that J=I and \varphi (x,y)={xy}^{2} for all x,y\in E since E is a Hilbert space. Moreover, it is not difficult to obtain that {C}_{n+1}=[0,\frac{{x}_{n}+{y}_{n}}{2}] for all n\ge 1. Then (3.16) is reduced to
where {i}_{n} is the solution to the positive integer equation: n=i+\frac{(m1)m}{2} (m\ge i, n=1,2,\dots). It is clear that \{{T}_{i}\} is a sequence of closed and totally quasiϕasymptotically nonexpansive mappings with a common fixed point zero. It then can be shown by a similar way of Theorem 3.1 that \{{x}_{n}\} converges strongly to zero. The numerical experiment outcome obtained by using MATLAB 7.10.0.499 shows that as {x}_{1}=1, the computations of {x}_{100}, {x}_{200}, {x}_{300} and {x}_{400} are 0.025141746, 0.00078472044, 0.000025282198 and 0.00000082531714, respectively. This example illustrates the effectiveness of the introduced algorithm for countable families of totally quasiϕasymptotically nonexpansive mappings.
4 Applications
The socalled convex feasibility problem for a family of mappings {\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}} is to find a point in the nonempty intersection {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}), which exactly illustrates the importance of finding common fixed points of infinite families. The following example also clarifies the same thing.
Example 4.1 Let E be a smooth, strictly convex and reflexive Banach space, let C be a nonempty and closed convex subset of E, and let {\{{f}_{i}\}}_{i=1}^{\mathrm{\infty}}:C\times C\to \mathbb{R} be a sequence of bifunctions satisfying the conditions: for each i\ge 1,
(A_{1}) {f}_{i}(x,x)=0;
(A_{2}) {f}_{i} is monotone, i.e., {f}_{i}(x,y)+{f}_{i}(y,x)\le 0;
(A_{3}) lim{sup}_{t\downarrow 0}{f}_{i}(x+t(zx),y)\le {f}_{i}(x,y);
(A_{4}) the mapping y\mapsto {f}_{i}(x,y) is convex and lower semicontinuous.
A system of equilibrium problems for {\{{f}_{i}\}}_{i=1}^{\mathrm{\infty}} is to find an {x}^{\ast}\in C such that
whose set of common solutions is denoted by EP:={\bigcap}_{i=1}^{\mathrm{\infty}}EP({f}_{i}), where EP({f}_{i}) denotes the set of solutions to the equilibrium problem for {f}_{i} (i=1,2,\dots). It is shown in [[10], Theorem 4.3] that such a system of problems can be reduced to the approximation of some fixed point of a sequence of nonlinear mappings.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant No. 11061037).
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Deng, WQ. Relaxed Halperntype iteration method for countable families of totally quasiϕasymptotically nonexpansive mappings. J Inequal Appl 2013, 367 (2013). https://doi.org/10.1186/1029242X2013367
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DOI: https://doi.org/10.1186/1029242X2013367