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Strong convergence theorems of the HalpernMann’s mixed iteration for a totally quasiϕasymptotically nonexpansive nonself multivalued mapping in Banach spaces
Journal of Inequalities and Applications volume 2014, Article number: 225 (2014)
Abstract
In this paper, we introduce a class of totally quasiϕasymptotically nonexpansive nonself multivalued mapping to modify the HalpernManntype iteration algorithm for a totally quasiϕasymptotically nonexpansive nonself multivalued mapping, which has the strong convergence under a limit condition only in the framework of Banach spaces. Our results are applied to study the approximation problem of solution to a system of equilibrium problems. Also, the results presented in the paper improve and extend the corresponding results of Chang et al. (Appl. Math. Comput. 218:78647870, 2012) and others.
1 Introduction and preliminaries
A Banach space X is said to be strictly convex if \parallel \frac{x+y}{2}\parallel \le 1 for all x,y\in X with \parallel x\parallel =\parallel y\parallel =1 and x\ne y. A Banach space 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}\},\{{y}_{n}\}\subset X with \parallel {x}_{n}\parallel =\parallel {y}_{n}\parallel =1 and {lim}_{n\to \mathrm{\infty}}\parallel \frac{{x}_{n}+{y}_{n}}{2}\parallel =0.
The norm of Banach space X is said to be Gâteaux differentiable, if, for each x,y\in S(x), the limit
exists, where S(x)=\{x:\parallel x\parallel =1,x\in X\}. In this case, X is said to be smooth. The norm of Banach space X is said to be Fréchet differentiable, if, for each x\in S(x), the limit (1.1) is attained uniformly for y\in S(x) and the norm is uniformly Fréchet differentiable if the limit (1.1) is attained uniformly for x,y\in S(x). In this case, X is said to be uniformly smooth.
Let D be a nonempty closed subset of a real Banach space X. A mapping T:D\to D is said to be nonexpansive if \parallel TxTy\parallel \le \parallel xy\parallel for all x,y\in D. An element p\in D is called a fixed point of a nonself multivalued mapping T:D\to X if p\in Tp. The set of fixed points of T is represented by F(T).
A subset D of X is said to be retract of X, if there exists a continuous mapping P:X\to D such that Px=x, for all x\in X. It is well known that every nonempty, closed, convex subset of a uniformly convex Banach space X is a retract of X. A mapping P:X\to D is said to be a retraction, if {P}^{2}=P. It follows that if a mapping P is a retraction, then Py=y for all y in the range of P. A mapping P:X\to D is said to be a nonexpansive retraction, if it is nonexpansive and it is a retraction from X to D.
Assume that X is a real Banach space with the dual {X}^{\ast}, D is a nonempty, closed, convex subset of X. We also denote by J the normalized duality mapping from X to {2}^{{X}^{\ast}} which is defined by
where \u3008\cdot ,\cdot \u3009 denotes the generalized duality pairing.
Next we assume that X is a smooth, strictly convex and reflexive Banach space and D is a nonempty, closed, convex subset of X. In the sequel, we always use \varphi :X\times X\to {R}^{+} to denote the Lyapunov functional defined by
It is obvious from the definition of the function ϕ that
and
for all \lambda \in [0,1] and x,y,z\in X.
Following Alber [2], the generalized projection {\mathrm{\Pi}}_{D}:X\to D is defined by
Lemma 1.1 (see [3])
Let X be a uniformly convex and smooth Banach space and let \{{x}_{n}\} and \{{y}_{n}\} be two sequences of X such that \{{x}_{n}\} and \{{y}_{n}\} is bounded, if \varphi ({x}_{n},{y}_{n})\to 0, then \parallel {x}_{n}{y}_{n}\parallel \to 0.
Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping.
In the sequel, we denote the strong convergence and weak convergence of the sequence \{{x}_{n}\} by {x}_{n}\to x and {x}_{n}\rightharpoonup x, respectively.
Lemma 1.2 (see [2])
Let X be a smooth, strictly convex, and reflexive Banach space and D be a nonempty, closed, convex subset of X. Then the following conclusions hold:

(a)
\varphi (x,y)=0 if and only if x=y;

(b)
\varphi (x,{\mathrm{\Pi}}_{D}y)+\varphi ({\mathrm{\Pi}}_{D}y,y)\le \varphi (x,y), \mathrm{\forall}x,y\in D;

(c)
if x\in X and z\in D, then z={\mathrm{\Pi}}_{D}x if and only if \u3008zy,JxJz\u3009\ge 0, \mathrm{\forall}y\in D.
Remark 1.1 (see [4])
Let {\mathrm{\Pi}}_{D} be the generalized projection from a smooth, reflexive and strictly convex Banach space X onto a nonempty, closed, convex subset D of X. Then {\mathrm{\Pi}}_{D} is a closed and quasiϕnonexpansive from X onto D.
Remark 1.2 (see [4])
If H is a real Hilbert space, then \varphi (x,y)={\parallel xy\parallel}^{2}, and {\mathrm{\Pi}}_{D} is the metric projection of H onto D.
Definition 1.1 Let P:X\to D be the nonexpansive retraction.

(1)
A nonself multivalued mapping T:D\to X is said to be quasiϕnonexpansive, if F(T)\ne \mathrm{\Phi}, and
\varphi (p,{z}_{n})\le \varphi (p,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,p\in F(T),{z}_{n}\in T{(PT)}^{n1}x,\mathrm{\forall}n\ge 1;(1.7) 
(2)
A nonself multivalued mapping T:D\to X is said to be quasiϕasymptotically nonexpansive, if F(T)\ne \mathrm{\Phi} and there exists a real sequence {k}_{n}\subset [1,+\mathrm{\infty}), {k}_{n}\to 1 (as n\to \mathrm{\infty}) such that
\varphi (p,{z}_{n})\le {k}_{n}\varphi (p,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,p\in F(T),{z}_{n}\in T{(PT)}^{n1}x,\mathrm{\forall}n\ge 1;(1.8) 
(3)
A nonself multivalued mapping T:D\to X is said to be totally quasiϕasymptotically nonexpansive, if F(T)\ne \mathrm{\Phi} and there exist nonnegative real sequences \{{v}_{n}\}, \{{\mu}_{n}\}, with {v}_{n},{\mu}_{n}\to 0 (as n\to \mathrm{\infty}) and a strictly increasing continuous function \zeta :{R}^{+}\to {R}^{+} with \zeta (0)=0 such that
\begin{array}{r}\varphi (p,{z}_{n})\le \varphi (p,x)+{v}_{n}\zeta [\varphi (p,x)]+{\mu}_{n},\\ \phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,p\in F(T),{z}_{n}\in T{(PT)}^{n1}x,\mathrm{\forall}n\ge 1.\end{array}(1.9)
Remark 1.3 From the definitions, it is obvious that a quasiϕnonexpansive nonself multivalued mapping is a quasiϕasymptotically nonexpansive nonself multivalued mapping, and a quasiϕasymptotically nonexpansive nonself multivalued mapping is a totally quasiϕasymptotically nonexpansive nonself multivalued mapping, but the converse is not true.
Now, we give an example of totally quasiϕasymptotically nonexpansive nonself multivalued mapping.
Example 1.1 (see [4])
Let D be a unit ball in a real Hilbert space {l}^{2} and let T:D\to {l}^{2} be a nonself multivalued mapping defined by
where \{{a}_{i}\} is a sequence in (0,1) such that {\prod}_{i=2}^{\mathrm{\infty}}{a}_{i}=\frac{1}{2}.
It is proved in [5] that

(i)
\parallel TxTy\parallel \le 2\parallel xy\parallel, \mathrm{\forall}x,y\in D;

(ii)
\parallel {T}^{n}x{T}^{n}y\parallel \le 2{\prod}_{j=2}^{n}{a}_{j}, \mathrm{\forall}x,y\in D, n\ge 2.
Let \sqrt{{k}_{1}}=2, \sqrt{{k}_{n}}=2{\prod}_{j=2}^{n}{a}_{j}, n\ge 2. Then {lim}_{n\to \mathrm{\infty}}{k}_{n}=1. Letting {\nu}_{n}={k}_{n}1 (n\ge 2), \zeta (t)=t (t\ge 0) and \{{\mu}_{n}\} be a nonnegative real sequence with {\mu}_{n}\to 0, then from (i) and (ii) we have
Since D is a unit ball in a real Hilbert space {l}^{2}, it follows from Remark 1.2 that \varphi (x,y)={\parallel xy\parallel}^{2}, \mathrm{\forall}x,y\in D. The inequality above can be written as
Again since 0\in D and 0\in F(T), this implies that F(T)\ne \mathrm{\Phi}. From the inequality above, we get
where P is the nonexpansive retraction. This shows that the mapping T defined above is a totally quasiϕasymptotically nonexpansive nonself multivalued mapping.
Lemma 1.3 Let X be a smooth, strictly convex and reflexive Banach space and D be a nonempty, closed, convex subset of X. Let T:D\to X be a totally quasiϕasymptotically nonexpansive nonself multivalued mapping with {\mu}_{1}=0. Then F(T) is a closed and convex subset of D.
Proof Let \{{x}_{n}\} be a sequence in F(T) such that {x}_{n}\to p. Since T is a totally quasiϕasymptotically nonexpansive nonself multivalued mapping, we have
Therefore,
By Lemma 1.2, we obtain p=z\in Tp. So we have p\in F(T). This implies that F(T) is closed.
Let p,q\in F(T) and t\in (0,1), and put w=tp+(1t)q. We prove that w\in F(T). Indeed, in view of the definition of ϕ, let \{{u}_{n}\} be a sequence generated by {u}_{1}\in Tw,{u}_{2}\in T(PT)w,{u}_{3}\in T{(PT)}^{2}w,\dots ,{u}_{n}\in T{(PT)}^{n1}w\subset TP{u}_{n1}, we have
Since
Substituting (1.11) into (1.10) and simplifying it, we have
Hence, {u}_{n}\to w holds, which yields {u}_{n+1}\to w. Since TP is closed and {u}_{n+1}\in T{(PT)}^{n}w\subset TP{u}_{n}, we have w\in TPw. It follows from w\in D that w\in Tw, i.e., w\in F(T). This implies that F(T) is convex. This completes the proof of Lemma 1.3. □
Definition 1.2 (see [1])
A nonself mapping T:D\to X is said to be uniformly LLipschitz continuous, if there exists a constant L>0, such that
Definition 1.3 A nonself multivalued mapping T:D\to X is said to be uniformly LLipschitz continuous, if there exists a constant L>0, such that
where d(\cdot ,\cdot ) is Hausdorff metric.
Strong and weak convergence of asymptotically nonexpansive self or nonself mappings, relatively nonexpansive, quasiϕnonexpansive and quasiϕasymptotically nonexpansive self or nonself mappings have been considered extensively by several authors in the setting of Hilbert or Banach spaces (see [1–4, 6–24]). In recent years, by hybrid projection methods, strong and weak convergence problems for totally quasiϕ quasiϕasymptotically nonexpansive nonself and multivalued mapping, respectively, was also studied by Kim et al. (see [6, 7]), Li et al. (see [8]), Chang et al. (see [9]) and Yang et al. (see [10]).
Inspired by specialists above, the purpose of this paper is to modify the HalpernMann’s mixed type iteration algorithm for a totally quasiϕasymptotically nonexpansive nonself multivalued mapping, which has the strong convergence under a limit condition only in the framework of Banach spaces. As an application, we utilize our results to study the approximation problem of solution to a system of equilibrium problems. The results presented in the paper improve and extend the corresponding results of Chang et al. [1, 11–13], Hao et al. [14], Guo et al. [15], Yildirim et al. [16], Thianwan [17], Nilsrakoo et al. [18], Pathak et al. [19], Qin et al. [20], Su et al. [21], Wang [22, 23], Yang et al. [24] and others.
2 Main results
Theorem 2.1 Let X be a real uniformly smooth and uniformly convex Banach space, D be a nonempty, closed, convex subset of X. Let P:X\to D be the nonexpansive retraction. Let T:D\to X be a totally quasiϕasymptotically nonexpansive nonself multivalued mapping with sequence \{{v}_{n}\}, \{{\mu}_{n}\} ({\mu}_{1}=0), with {v}_{n},{\mu}_{n}\to 0 (as n\to \mathrm{\infty}) and a strictly increasing continuous function \zeta :{R}^{+}\to {R}^{+} with \zeta (0)=0 such that T is uniformly LLipschitz continuous. Let \{{\alpha}_{n}\} be a sequence in [0,1] and \{{\beta}_{n}\} be a sequence in (0,1) satisfying the following conditions:

(i)
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0;

(ii)
0<{lim}_{n\to \mathrm{\infty}}inf{\beta}_{n}\le {lim}_{n\to \mathrm{\infty}}sup{\beta}_{n}<1.
Let \{{x}_{n}\} be a sequence generated by
where {\xi}_{n}={v}_{n}{sup}_{p\in F(T)}\zeta (\varphi (p,{x}_{n}))+{\mu}_{n}, {\mathrm{\Pi}}_{{D}_{n+1}} is the generalized projection of X onto {D}_{n+1}. If F(T)\ne \mathrm{\varnothing}, then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F(T)}{x}_{1}.
Proof (I) First, we prove that {D}_{n} are closed and convex subsets in D.
In fact, by Lemma 1.3, F(T) is closed and convex in D. By the assumption, {D}_{1}=D is closed and convex. Suppose that {D}_{n} is closed and convex for some n\ge 1. In view of the definition of ϕ, we have
This shows that {D}_{n+1} is closed and convex. The conclusions are proved.

(II)
Next, we prove that F(T)\subset {D}_{n}, for all n\ge 1.
It is obvious that F(T)\subset {D}_{1}. Suppose that F(T)\subset {D}_{n}, {w}_{n}={J}^{1}({\beta}_{n}J{x}_{n}+(1{\beta}_{n})J{z}_{n}) and {z}_{n}\in T{(PT)}^{n1}{x}_{n}. Hence, for any u\in F(T)\subset {D}_{n}, by (1.5), we have
and
Therefore, we have
where {\xi}_{n}={v}_{n}{sup}_{p\in F(T)}\zeta (\varphi (p,{x}_{n}))+{\mu}_{n}. This shows that u\in {D}_{n+1} and so F(T)\subset {D}_{n}. The conclusion is proved.

(III)
Now we prove that \{{x}_{n}\} converges strongly to some point {p}^{\ast}.
Since {x}_{n}={\mathrm{\Pi}}_{{D}_{n}}{x}_{1}, from Lemma 1.2(c), we have
Again since F(T)\subset {D}_{n}, we have
It follows from Lemma 1.2(b) that, for each u\in F(T) and for each n\ge 1,
Therefore, \{\varphi ({x}_{n},{x}_{1})\} is bounded, and so is \{{x}_{n}\}. Since {x}_{n}={\mathrm{\Pi}}_{{D}_{n}}{x}_{1} and {x}_{n+1}={\mathrm{\Pi}}_{{D}_{n+1}}{x}_{1}\in {D}_{n+1}\subset {D}_{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 {lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1}) exists.
By the construction of \{{D}_{n}\}, for any m\ge n, we have {D}_{m}\subset {D}_{n} and {x}_{m}={\mathrm{\Pi}}_{{D}_{m}}{x}_{1}\in {D}_{n}. This shows that
It follows from Lemma 1.1 that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{m}{x}_{n}\parallel =0. Hence \{{x}_{n}\} is a Cauchy sequence in D. Since D is complete, without loss of generality, we can assume that {lim}_{n\to \mathrm{\infty}}{x}_{n}={p}^{\ast} (some point in D).
By the assumption, it is easy to see that

(IV)
Now we prove that {p}^{\ast}\in F(T).
Since {x}_{n+1}\in {D}_{n+1}, from (2.1) and (2.6), we have
Since {x}_{n}\to {p}^{\ast}, it follows from (2.7) and Lemma 1.1 that
Since \{{x}_{n}\} is bounded and T is a totally quasiϕasymptotically nonexpansive nonself multivalued mapping, we have
This implies that \{{z}_{n}\} is also bounded.
By condition (ii), we have
this implies that \{{w}_{n}\} is also bounded.
In view of {\alpha}_{n}\to 0, from (2.1), we have
Since {J}^{1} is uniformly continuous on each bounded subset of {X}^{\ast}, it follows from (2.8) and (2.9) that
Since J is uniformly continuous on each bounded subset of X, we have
By condition (ii), we have
Since J is uniformly continuous, this shows that
Again by the assumptions that T:D\to X be uniformly LLipschitz continuous, thus we have
We get {lim}_{n\to \mathrm{\infty}}d(T{(PT)}^{n}{x}_{n},T{(PT)}^{n1}{x}_{n})=0, since {lim}_{n\to \mathrm{\infty}}{z}_{n}={p}^{\ast} and {lim}_{n\to \mathrm{\infty}}{x}_{n}={p}^{\ast}.
In view of the continuity of TP, it yields {p}^{\ast}\in TP{p}^{\ast}. We have {p}^{\ast}\in C, which implies that {p}^{\ast}\in T{p}^{\ast}. We have {p}^{\ast}\in F(T).

(V)
Finally, we prove that {p}^{\ast}={\mathrm{\Pi}}_{F(T)}{x}_{1} and so {x}_{n}\to {\mathrm{\Pi}}_{F(T)}{x}_{1}={p}^{\ast}.
Let w={\mathrm{\Pi}}_{F(T)}{x}_{1}. Since w\in F(T)\subset {D}_{n} and {x}_{n}={\mathrm{\Pi}}_{{D}_{n}}{x}_{1}, we have \varphi ({x}_{n},{x}_{1})\le \varphi (w,{x}_{1}). This implies that
which yields {p}^{\ast}=w={\mathrm{\Pi}}_{F(T)}{x}_{1}. Therefore, {x}_{n}\to {\mathrm{\Pi}}_{F(T)}{x}_{1}. The proof of Theorem 3.1 is completed. □
By Remark 1.3, the following corollary is obtained.
Corollary 2.1 Let X, D, \{{\alpha}_{n}\}, \{{\beta}_{n}\} be the same as in Theorem 2.1. Let T:D\to X be a quasiϕasymptotically nonexpansive nonself multivalued mapping with sequence {k}_{n}\subset [1,+\mathrm{\infty}), {k}_{n}\to 1, T:D\to X be uniformly LLipschitz continuous.
Suppose \{{x}_{n}\} be a sequence generated by
where {\xi}_{n}=({k}_{n}1){sup}_{p\in F(T)}\varphi (p,{x}_{n}), {\mathrm{\Pi}}_{{D}_{n+1}} is the generalized projection of X onto {D}_{n+1}. If F(T)\ne \mathrm{\varnothing}, then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F(T)}{x}_{1}.
Corollary 2.2 Let X, D, \{{\alpha}_{n}\}, \{{\beta}_{n}\} be the same as in Theorem 2.1. Let T:D\to X be a quasiϕnonexpansive nonself multivalued mapping, T:D\to X be uniformly LLipschitz continuous.
Suppose \{{x}_{n}\} is a sequence generated by
where {\xi}_{n}=({k}_{n}1){sup}_{p\in F(T)}\varphi (p,{x}_{n}), {\mathrm{\Pi}}_{{D}_{n+1}} is the generalized projection of X onto {D}_{n+1}. If F(T)\ne \mathrm{\varnothing}, then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F(T)}{x}_{1}.
3 Application
First, we present an example of a quasiϕnonexpansive nonself multivalued mapping.
Example 3.1 (see [4])
Let H be a real Hilbert space, D be a nonempty closed and convex subset of H and f:D\times D\to R be a bifunction satisfying the conditions: (A1) f(x,x)=0, \mathrm{\forall}x\in D; (A2) f(x,y)+f(y,x)\le 0, \mathrm{\forall}x,y\in D; (A3) for each x,y,z\in D, {lim}_{t\to 0}f(tz+(1t)x,y)\le f(x,y); (A4) for each given x\in D, the function y\mapsto f(x,y) is convex and lower semicontinuous. The ‘socalled’ equilibrium problem for f is to find a {x}^{\ast}\in D such that f({x}^{\ast},y)\ge 0, \mathrm{\forall}y\in D. The set of its solutions is denoted by EP(f).
Let r>0, x\in H and define a mapping {T}_{r}:D\to D\subset H as follows:
Then (1) {T}_{r} is singlevalued, so z={T}_{r}(x); (2) {T}_{r} is a relatively nonexpansive nonself mapping, therefore it is a closed quasiϕnonexpansive nonself mapping; (3) F({T}_{r})=EP(f) and F({T}_{r}) is a nonempty and closed convex subset of D; (4) {T}_{r}:D\to D is a nonexpansive. Since F({T}_{r}) nonempty, it is a quasiϕnonexpansive nonself mapping from D to H, where \varphi (x,y)={\parallel xy\parallel}^{2}, x,y\in H.
In this section we utilize Corollary 2.1 to study a modified Halpern iterative algorithm for a system of equilibrium problems. We have the following result.
Theorem 3.1 Let H be a real Hilbert space, D be a nonempty closed and convex subset of H, \{{\alpha}_{n}\}, \{{\beta}_{n}\} be the same as in Theorem 2.1. Let f:D\times D\to R be a bifunction satisfying conditions (A1)(A4) as given in Example 3.1. Let {T}_{r}:D\to D\subset H be mapping defined by (3.1), i.e.,
Let \{{x}_{n}\} be the sequence generated by
If F({T}_{r})\ne \mathrm{\varnothing}, then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F({T}_{r})}{x}_{1}, which is a common solution of the system of equilibrium problems for f.
Proof In Example 3.1, we have pointed out that {u}_{n}={T}_{r}({x}_{n}), F({T}_{r})=EP(f) is nonempty and convex, {T}_{r} is a quasiϕnonexpansive nonself mapping. Since F({T}_{r}) is nonempty, and so {T}_{r} is a quasiϕnonexpansive mapping and {T}_{r} is uniformly 1Lipschitzian mapping. Hence (3.1) can be rewritten as follows:
Therefore, the conclusion of Theorem 3.1 can be obtained from Corollary 2.1. □
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Bo, L.H., Yi, L. Strong convergence theorems of the HalpernMann’s mixed iteration for a totally quasiϕasymptotically nonexpansive nonself multivalued mapping in Banach spaces. J Inequal Appl 2014, 225 (2014). https://doi.org/10.1186/1029242X2014225
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DOI: https://doi.org/10.1186/1029242X2014225