Abstract
We construct a relaxed hybrid shrinking iteration algorithm for approximating common fixed points of a countable family of totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings. A strong convergence theorem for solving generalized mixed equilibrium problems is established in the framework of Banach spaces under relaxed conditions. Since there is no need to impose a uniformity assumption on the involved mappings and no need to compute complex series in the iteration process, the results improve those of the authors with related interests.
Similar content being viewed by others
1 Introduction
Throughout this paper we assume that E is a real Banach space with its dual \(E^{*}\), C is a nonempty closed convex subset of E and \(J : E\rightarrow2^{E^{*}}\) 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]
(1) A multi-valued mapping \(T: C\rightarrow2^{C}\) is said to be totally quasi-ϕ-asymptotically nonexpansive, if \(F(T)\neq\emptyset\) and there exist nonnegative real sequences \(\{\nu_{n}\}\), \(\{\mu_{n}\}\) with \(\nu_{n},\mu_{n}\rightarrow0\) (as \(n\rightarrow\infty\)) and a strictly increasing continuous function \(\zeta:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}^{+}\cup\{0\}\) with \(\zeta(0)=0\) such that
where \(\phi:E\times E\rightarrow\mathbb{R}^{+}\cup\{0\}\) denotes the Lyapunov functional defined by
It is obvious from the definition of ϕ that
and
(2) A countable family of multi-valued mappings \(\{T_{i}\}:C\rightarrow C\) said to be uniformly totally quasi-ϕ-asymptotically nonexpansive, if \(F:=\bigcap^{\infty}_{i=1}F(T_{i})\neq\emptyset\) and there exist nonnegative real sequences \(\{\nu_{n}\}\), \(\{\mu_{n}\}\) with \(\nu_{n},\mu_{n}\rightarrow0\) (as \(n\rightarrow\infty\)) and a strictly increasing continuous function \(\zeta:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}^{+}\cup\{0\}\) with \(\zeta(0)=0\) such that
(3) A totally quasi-ϕ-asymptotically nonexpansive multi-valued mapping \(T: C\rightarrow2^{C}\) is said to be uniformly L-Lipschitz continuous, if there exists a constant \(L>0\) such that
Let \(\theta:C\times C\rightarrow\mathbb{R}\) be a bifunction, \(\psi:C\rightarrow\mathbb{R}\) a real valued function and \(A:C\rightarrow E^{*}\) a nonlinear mapping. The so-called generalized mixed equilibrium problem GMEP is to find an \(u\in C\) such that
whose set of solutions is denoted by Ω.
In 2012, Chang et al. [1] used the following hybrid shrinking iteration algorithm finding a common element of the set of solutions for a GMEP, the set of solutions for variational inequality problems, and the set of common fixed points for a countable family of multi-valued total quasi-ϕ-asymptotically nonexpansive mappings in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property:
where \(\{T_{i}\}:C\rightarrow2^{C}\) is a countable family of closed and uniformly totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings; \(w_{n,i}\in T^{n}_{i}x_{n}\), \(\forall n\geq1\), \(i\geq1\), \(\xi_{n}:=\nu_{n}\sup_{p\in F}\zeta(\phi(p,x_{n}))+\mu_{n}\), \(\Pi_{C_{n+1}}\) is the generalized projection (see (2.1)) of E onto \(C_{n+1}\). Their results not only generalized the corresponding results of [2–19] from single-valued mappings to multi-valued mappings, but they also improved and extended the main results of Homaeipour and Razani [20].
However, it is obviously a quite strong condition that the involved multi-valued mappings are assumed to be uniformly \((\{\nu_{n}\},\{\mu_{n}\},\zeta)\)-totally quasi-ϕ-asymptotically nonexpansive. In addition, the accurate computation of the series \(\sum^{\infty}_{i=1}\beta_{n,i}Jw_{n,i}\) at each step of the iteration process is not easily attainable, which leads to gradually increasing errors.
Inspired and motivated by the study mentioned above, in this paper, we use a relaxed hybrid iteration algorithm for approximating common fixed points of a countable family of multi-valued totally quasi-ϕ-asymptotically nonexpansive mappings and obtain a strong convergence theorem under some suitable conditions. The results improve those of Chang et al. [1].
2 Preliminaries
We say that a Banach space E is strictly convex if the following implication holds for \(x, y\in E\):
E is also said to be uniformly convex if for any \(\epsilon>0\), there exists \(\delta>0\) such that
It is well known that if E is a uniformly convex Banach space, then E is reflexive and strictly convex. A Banach space E is said to be smooth if
exists for each \(x, y\in S(E) := \{x \in E : \|x\|= 1\}\). E is said to be uniformly smooth if the limit (2.3) is attained uniformly for \(x,y \in S(E)\).
Following Alber [21], the generalized projection \(\Pi_{C}:E\rightarrow C\) is defined by
Lemma 2.1
[21]
Let E be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of E. Then the following conclusions hold:
-
(1)
\(\phi(x,\Pi_{C}y)+\phi(\Pi_{C}y,y)\leq\phi(x,y)\) for all \(x\in C\) and \(y\in E\);
-
(2)
If \(x\in E\) and \(z\in C\), then \(z=\Pi_{C}x\Leftrightarrow\langle z-y,Jx-Jz\rangle\geq0\), \(\forall y\in C\);
-
(3)
For \(x,y\in E\), \(\phi(x,y)=0\) if and only if \(x=y\).
Remark 2.2
The following basic properties for a Banach space E can be found in Cioranescu [22].
-
(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 norm-weak-continuous;
-
(iii)
If E is a smooth, strictly convex and reflexive Banach space, then the normalized duality mapping \(J : E\rightarrow2^{E^{*}}\) is single valued, one-to-one and onto;
-
(iv)
A Banach space E is uniformly smooth if and only if \(E^{*}\) is uniformly convex;
-
(v)
Each uniformly convex Banach space E has the Kadec-Klee property, i.e., for any sequence \(\{x_{n}\}\subset E\), if \(x_{n}\rightharpoonup x\in E\) and \(\|x_{n}\|\rightarrow\|x\|\), then \(x_{n}\rightarrow x\) as \(n\rightarrow\infty\).
Lemma 2.3
[6]
Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed convex subset of E. Let \(\{x_{n}\}\) and \(\{y_{n}\}\) be two sequences in C such that \(x_{n}\rightarrow p\) and \(\phi(x_{n},y_{n})\rightarrow0\), where ϕ is the function defined by (1.2), then \(y_{n}\rightarrow p\).
Lemma 2.4
[1]
Let E and C be the same as in Lemma 2.3. Let \(T:C\rightarrow C\) be a closed and totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings with nonnegative real sequences \(\{\nu_{n}\}\), \(\{\mu_{n}\}\) and a strictly increasing continuous function \(\zeta:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}^{+}\cup\{0\}\) such that \(\nu_{n},\mu_{n}\rightarrow0\) 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
[6]
Let E be a real uniformly convex Banach space and let \(B_{r}(0)\) be the closed ball of E with center at the origin and radius \(r>0\). Then for any for any sequence \(\{x_{i}\}\subset B_{r}(0)\) and for any sequence \(\{\lambda_{i}\}\) of positive numbers with \(\sum^{\infty}_{i=1}\lambda_{i}=1\), there exists a continuous strictly increasing convex function \(g:[0,\infty)\rightarrow[0,\infty)\) with \(g(0)=0\) such that such that for any positive integer \(i\neq1\), the following hold:
and, for all \(x\in E\),
Assume that, to obtain the solution of GMEP, the function \(\psi:C\rightarrow\mathbb{R}\) is convex and lower semi-continuous, the nonlinear mapping \(A:C\rightarrow E^{*}\) is continuous and monotone, and the bifunction \(\theta:C\times C\rightarrow\mathbb{R}\) satisfies the following conditions:
- (A1):
-
\(\theta(x,x)=0\);
- (A2):
-
θ is monotone, i.e., \(\theta(x,y)+\theta(y,x)\leq0\);
- (A3):
-
\(\lim \sup_{t\downarrow0}\theta(x+t(z-x),y)\leq\theta(x,y)\);
- (A4):
-
the mapping \(y\mapsto\theta(x,y)\) is convex and lower semicontinuous.
Lemma 2.6
[16]
Let E be a smooth, strictly convex, and reflexive Banach space, and C be a nonempty closed convex subset of E. Let \(A:C\rightarrow E^{*}\) be a continuous and monotone mapping, \(\psi:C\rightarrow\mathbb{R}\) a lower semi-continuous and convex function, and \(\theta:C\times C\rightarrow\mathbb{R}\) a bifunction satisfying the conditions (A1)-(A4). Let \(r>0\) and \(x\in E\). Then the following hold:
-
(1)
There exists an \(u\in C\) such that
$$\theta(u,y)+\langle Au,y-u\rangle+\psi(y)-\psi(u)+\frac{1}{r}\langle y-u ,Ju-Jx\rangle\geq0,\quad\forall y\in C. $$ -
(2)
A mapping \(\kappa_{r}:C\rightarrow C\) is defined by
$$\kappa_{r}(x)= \biggl\{ u\in C:\theta(u,y)+\langle Au,y-u\rangle+ \psi(y)-\psi(u)+\frac{1}{r}\langle y-u ,Ju-Jx\rangle\geq0 \biggr\} . $$
Then the mapping \(\kappa_{r}\) has the following properties:
-
(i)
\(\kappa_{r}\) is single-valued;
-
(ii)
\(\kappa_{r}\) a firmly nonexpansive-type mapping, i.e.,
$$\langle\kappa_{r}z-\kappa_{r}y,J\kappa_{r}z-J \kappa_{r}y\rangle\leq\langle \kappa_{r}z- \kappa_{r}y,Jz-Jy\rangle; $$ -
(iii)
\(F(\kappa_{r})=\Omega=\tilde{F}(\kappa_{r})\);
-
(iv)
Ω is a closed convex set of C;
-
(v)
\(\phi(p,\kappa_{r}z)+\phi(\kappa_{r}z,z)\leq\phi(p,z)\), \(\forall p\in F(\kappa_{r})\), \(z\in E\),
where \(\tilde{F}(\kappa_{r})\) denotes the set of asymptotic fixed points of \(\kappa_{r}\), i.e.,
Lemma 2.7
[23]
The unique solutions to the positive integer equation
are
where \([x]\) denotes the maximal integer that is not larger than x.
3 Main results
Recall that a multi-valued mapping \(T:C\rightarrow2^{C}\) is said to be closed, if for any sequence \(\{x_{n}\}\subset C\) with \(x_{n}\rightarrow x\) and \(w_{n}\in Tx_{n}\) with \(w_{n}\rightarrow y\) as \(n\rightarrow\infty\), then \(y\in Tx\).
Theorem 3.1
Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and C a nonempty closed convex subset of E. Let \(\theta:C\times C\rightarrow\mathbb{R}\) be a bifunction satisfying the conditions (A1)-(A4), \(A:C\rightarrow E^{*}\) a continuous and monotone mapping, and \(\psi:C\rightarrow\mathbb{R}\) a lower semi-continuous and convex function. Let \(\{T_{i}\}:C\rightarrow2^{C}\) be a countable family of closed and totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings with nonnegative real sequences \(\{\nu^{(i)}_{n}\}\), \(\{\mu^{(i)}_{n}\}\) satisfying \(\nu^{(i)}_{n}\rightarrow0\) and \(\mu^{(i)}_{n}\rightarrow0\) (as \(n\rightarrow\infty\) and for each \(i\geq1\)) and a strictly increasing and continuous function \(\zeta:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}^{+}\cup\{0\}\) satisfying condition (1.1) and each \(T_{i}\) is uniformly \(L_{i}\)-Lipschitz continuous with \(\mu^{(i)}_{1}=0\). Let \(\{\alpha_{i}\}\) be a sequence in \([0, 1)\) and \(\{\beta_{i}\}\) be a sequence in \((0, 1)\). Let \(\{x_{n}\}\) be the sequence generated by
where \(w^{(i_{n})}_{m_{n}}\in T^{m_{n}}_{i_{n}}x_{n}\), \(\forall n\geq1\), \(\xi_{n}:=\nu^{(i_{n})}_{m_{n}}\sup_{p\in F}\zeta_{i_{n}}(\phi(p,x_{n}))+\mu^{(i_{n})}_{m_{n}}\), \(\Pi_{C_{n+1}}\) is the generalized projection of E onto \(C_{n+1}\); and \(i_{n}\) and \(m_{n}\) are the solutions to the positive integer equation: \(n=i_{n}+\frac{(m_{n}-1)m_{n}}{2} \) (\(m_{n}\geq i_{n}\), \(n=1,2,\ldots\)), that is, for each \(n\geq1\), there exist unique \(i_{n}\) and \(m_{n}\) such that
If \(G:=F\cap\Omega\neq\emptyset\) and \(F:=\bigcap^{\infty}_{i=1}F(T_{i})\) is bounded, then \(\{x_{n}\}\) converges strongly to \(\Pi_{G}x_{1}\).
Proof
Two functions \(\tau:C\times C\rightarrow\mathbb{R}\) and \(\kappa_{r}:C\rightarrow C\) are defined by
By Lemma 2.6, we know that the function τ satisfies the conditions (A1)-(A4) and \(\kappa_{r}\) has the properties (i)-(v). Therefore, (3.1) can be rewritten as
We divide the proof into several steps.
(I) F and \(C_{n}\) (\(\forall n\geq1\)) 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 \(n\geq2\). In view of the definition of ϕ we have
where \(\varphi(v)=2\langle v,Jx_{n}-Jy_{n}\rangle\) and \(a=\|x_{n}\|^{2}-\|y_{n}\|^{2}+\xi_{n}\). This shows that \(C_{n+1}\) is closed and convex.
(II) G is a subset of \(\bigcap^{\infty}_{n=1}C_{n}\).
It is obvious that \(G\subset C_{1}\). Suppose that \(G\subset C_{n}\) for some \(n\geq2\). Since \(u_{n}=\kappa_{r_{n}}y_{n}\), by Lemma 2.6, it is easily shown that \(\kappa_{r_{n}}\) is quasi-ϕ-nonexpansive. Hence, for any \(p\in G\subset C_{n}\), it follows from (1.4) that
Furthermore, it follows from Lemma 2.5 that for any \(p\in G\subset C_{n}\), \(w^{(i_{n})}_{m_{n}} \in T^{m_{n}}_{i_{n}}x_{n}\), we have
Substituting (3.4) into (3.3) and simplifying it, we have
This implies that \(p\in C_{n+1}\), and so \(G\subset C_{n+1}\).
(III) \(x_{n}\rightarrow x^{*}\in C\) as \(n\rightarrow\infty\).
In fact, since \(x_{n}=\Pi_{C_{n}}x_{1}\), from Lemma 2.1(2) we have \(\langle x_{n}-y,Jx_{1}-Jx_{n}\rangle\geq0\), \(\forall y\in C_{n}\). Again since \(F\subset\bigcap^{\infty}_{n=1}C_{n}\), we have \(\langle x_{n}-p,Jx_{1}-Jx_{n}\rangle\geq0\), \(\forall p\in F\). It follows from Lemma 2.1(1) that for each \(p\in F\) and for each \(n\geq1\),
which implies that \(\{\phi(x_{n},x_{1})\}\) is bounded, so is \(\{x_{n}\}\). Since for all \(n\geq1\), \(x_{n}=\Pi_{C_{n}}x_{1}\) and \(x_{n+1}=\Pi_{C_{n+1}}x_{1}\in C_{n+1}\subset C_{n}\), we have \(\phi(x_{n},x_{1})\leq\phi(x_{n+1},x_{1})\). This implies that \(\{\phi(x_{n},x_{1})\}\) is nondecreasing, hence the limit
Since E is reflexive, there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{i}}\rightharpoonup x^{*}\in C\) as \(i\rightarrow\infty\). Since \(C_{n}\) is closed and convex and \(C_{n+1} \subset C_{n}\), this implies that \(C_{n}\) is weakly closed and \(x^{*}\in C_{n}\) for each \(n\geq1\). In view of \(x_{n_{i}}=\Pi_{C_{n_{i}}}x_{1}\), we have
Since the norm \(\|\cdot\|\) is weakly lower semi-continuous, we have
and so
This implies that \(\lim_{i\rightarrow\infty}\phi(x_{n_{i}},x_{1})=\phi(x^{*},x_{1})\), and so \(\|x_{n_{i}}\|\rightarrow\|x^{*}\|\) as \(i\rightarrow\infty\). Since \(x_{n_{i}}\rightharpoonup x^{*}\), by virtue of Kadec-Klee property of E, we obtain
Since \(\{\phi(x_{n},x_{1})\}\) is convergent, this, together with \(\lim_{i\rightarrow\infty}\phi(x_{n_{i}},x_{1})=\phi(x^{*},x_{1})\), shows that \(\lim_{n\rightarrow\infty}\phi(x_{n},x_{1})=\phi(x^{*},x_{1})\). If there exists some subsequence \(\{x_{n_{j}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{j}}\rightarrow y\) as \(j\rightarrow\infty\), then from Lemma 2.1(1) we have
that is, \(x^{*}=y\) and so
(IV) \(x^{*}\) is a member of F.
Set \(\mathcal{K}_{i}= \{k\geq1:k=i_{k}+\frac{(m_{k}-1)m_{k}}{2},m_{k}\geq i_{k},m_{k}\in\mathbb{N} \}\) for each \(i\geq1\). Note that \(\nu^{(i_{k})}_{m_{k}}=\nu^{(i)}_{m_{k}}\), \(\mu^{(i_{k})}_{m_{k}}=\mu^{(i)}_{m_{k}}\), and \(\zeta_{i_{k}}=\zeta_{i}\) whenever \(k\in\mathcal{K}_{i}\) for each \(i\geq1\). For example, by Lemma 2.7 and the definition of \(\mathcal{K}_{1}\), we have \(\mathcal{K}_{1}=\{1,2,4,7,11,16,\ldots\} \) and \(i_{1}=i_{2}=i_{4}=i_{7}=i_{11}=i_{16}=\cdots=1\). Then we have
Note that \(\{m_{k}\}_{k\in\mathcal{K}_{i}}=\{i,i+1,i+2,\ldots\}\), i.e., \(m_{k}\uparrow\infty\) as \(\mathcal{K}_{i}\ni k\rightarrow\infty\). It follows from (3.6) and (3.7) that
Since \(x_{n+1}\in C_{n+1}\), it follows from (3.1), (3.6), and (3.8) that
as \(\mathcal{K}_{i}\ni k\rightarrow\infty\). Since \(x_{k}\rightarrow x^{*}\), it follows from (3.9) and Lemma 2.3 that
Note that \(w^{(i_{k})}_{m_{k}}=w^{(i)}_{m_{k}}\), \(T^{m_{k}}_{i_{k}}=T^{m_{k}}_{i}\), \(\alpha _{i_{k}}=\alpha_{i}\), and \(\beta_{i_{k}}=\beta_{i}\) whenever \(k\in\mathcal{K}_{i}\) for each \(i\geq1\). From (3.5), for any \(p\in F\) and \(w^{(i)}_{m_{k}}\in T^{m_{k}}_{i}x_{k}\), \(\forall k\in\mathcal{K}_{i}\), we have
that is,
This, together with assumption conditions imposed on the sequence \(\{\alpha_{i}\}\) and \(\{\beta_{i}\}\), shows that \(\lim_{\mathcal{K}_{i}\ni k\rightarrow\infty}g (\Vert Jx_{k}-Jw^{(i)}_{m_{k}}\Vert )=0\). In view of property of g, we have
In addition, \(Jx_{k}\rightarrow Jx^{*}\) implies that \(\lim_{\mathcal{K}_{i}\ni k\rightarrow\infty}Jw^{(i)}_{m_{k}}=Jx^{*}\). From Remark 2.2(ii) it yields, as \(\mathcal{K}_{i}\ni k\rightarrow\infty\),
Again, since, for each \(i\geq1\), as \(\mathcal{K}_{i}\ni k\rightarrow\infty\),
this, together with (3.11) and the Kadec-Klee property of E, shows that
For each \(i\geq1\), we now consider the sequence \(\{s^{(i)}_{m_{k}} \}_{k\in\mathcal{K}_{i}}\) generated by
By the assumptions that for each \(i\geq1\), \(T_{i}\) is uniformly \(L_{i}\)-Lipschitz continuous. Noting again that \(\{m_{k}\}_{k\in\mathcal{K}_{i}}=\{i,i+1,i+2,\ldots\}\), i.e., \(m_{k+1}-1=m_{k}\) for all \({k\in\mathcal{K}_{i}}\), we then have
From (3.12) and \(x_{k}\rightarrow x^{*}\) we have \(\lim_{\mathcal{K}_{i}\ni k\rightarrow\infty} \Vert s^{(i)}_{m_{k+1}}-w^{(i)}_{m_{k}}\Vert =0\) and
In view of the closedness of \(T_{i}\), it follows from (3.12) and (3.13) that \(x^{*}\in T_{i}x^{*}\) for each \(i\geq1\), namely \(x^{*}\in F\).
(V) \(x^{*}\) is also a member of G.
Since \(x_{n+1}=\Pi_{C_{n+1}}x_{1}\), it follows from (3.1) and (3.6) that
as \(\mathcal{K}_{i}\ni k\rightarrow\infty\). Since \(x_{k}\rightarrow x^{*}\), by virtue of Lemma 2.1 we have
This, together with (3.10), shows that \(\lim_{\mathcal{K}_{i}\ni k\rightarrow\infty}\|u_{k}-y_{k}\|=0\) and \(\lim_{\mathcal{K}_{i}\ni k\rightarrow\infty}\|Ju_{k}-Jy_{k}\|=0\). By the assumption that \(\{r_{k}\}_{k\in\mathcal{K}_{i}}\subset[a,\infty)\) for some \(a>0\), we have
Since \(\tau(u_{k},y)+\frac{1}{r_{k}}\langle y-u_{k},Ju_{k}-Jy_{k}\rangle\geq0\), \(\forall y\in C\), by condition (A1), we have
By the assumption that the mapping \(y\mapsto\tau(x,y)\) is convex and lower semi-continuous, letting \(\mathcal{K}_{i}\ni k\rightarrow\infty\) in (3.18), from (3.16) and (3.17), we have \(\tau(y,x^{*})\leq0\), \(\forall y\in C\).
For any \(t\in(0,1]\) and any \(y\in C\), set \(y_{t}=ty+(1-t)x^{*}\). Then \(\tau(y_{t},x^{*})\leq0\) since \(y_{t}\in C\). By condition (A1) and (A4), we have
Dividing both sides of the above equation by t, we have \(\tau(y_{t},y)\geq0\), \(\forall y\in C\). Letting \(t\downarrow0\), from condition (A3), we have \(\tau(x^{*},y)\geq0\), \(\forall y\in C\), i.e., \(x^{*}\in\Omega\) and so \(x^{*}\in G\).
(VI) \(x^{*}=\Pi_{G}x_{1}\), and so \(x_{n}\rightarrow\Pi_{G}x_{1}\) as \(n\rightarrow\infty\).
Put \(u=\Pi_{G}x_{1}\). Since \(u\in G\subset C_{n}\) and \(x_{n}=\Pi_{C_{n}}x_{1}\), we have \(\phi(x_{n},x_{1})\leq\phi(u,x_{1})\), \(\forall n\geq1\). Then
which implies that \(x^{*}=u\) since \(u=\Pi_{Gx_{1}}\), and hence \(x_{n}\rightarrow x^{*}=\Pi_{F}x_{1}\) as \(n\rightarrow\infty\). This completes the proof. □
A numerical result is given as follows.
Example 3.2
Let \(E=\mathbb{R}^{1}\) with the standard norm \(\|\cdot\|=|\cdot|\) and \(C=[0,1]\). Let \(\{T_{i}\}^{\infty}_{i=1}:C\rightarrow2^{C}\) be a sequence of multi-valued nonlinear mappings defined by
Consider the following iteration sequence generated by
where \(w_{i}:=\frac{x^{i+1}}{i+1}\in T_{i}x\), \(\{\alpha_{n}\}=\{\frac{2}{3}-\frac{1}{4n}\}\), \(\{\beta_{n}\}=\{\frac{4}{5}-\frac{1}{2n}\}\), and \(\Pi_{C_{n+1}}(x):=\arg\inf_{y\in C_{n+1}}|y-x|\). Note that \(J=I\) and \(\phi(x,y)=|x-y|^{2}\) for all \(x,y\in E\) since E is a Hilbert space. Moreover, it is not difficult to obtain \(C_{n+1}=[0,\frac{x_{n}+y_{n}}{2}]\) for all \(n\geq1\). Then (3.20) is reduced to
where \(i_{n}\) is the solution to the positive integer equation: \(n=i_{n}+\frac{(m_{n}-1)m_{n}}{2}\) (\(m_{n}\geq i_{n}\), \(n=1,2,\ldots\)). It is clear that \(\{T_{i}\}\) is a sequence of closed and totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings with a common fixed point zero. It then can be shown by 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 0.023899039, 0.00074538945, 0.000024001481, and 0.00000078318587, respectively. This example illustrates the effectiveness of the introduced algorithm for countable families of totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings.
References
Chang, S-s, Wang, L, Tang, Y-K, Zhao, Y-H, Ma, Z-L: Strong convergence theorems of nonlinear operator equations for countable family of multi-valued total quasi-ϕ-asymptotically nonexpansive mappings with applications. Fixed Point Theory Appl. 2012, Article ID 69 (2012). doi:10.1186/1687-1812-2012-69
Chang, S-s, Joseph Lee, HW, Chan, CK, Zhang, WB: A modified Halpern-type iterative algorithm for totally quasi-ϕ-asymptotically nonexpansive mappings with applications. Appl. Math. Comput. 218, 6489-6497 (2012). doi:10.1016/j.amc.2011.12.019
Matsushita, S, Takahashi, W: A strong convergence theorem for relatively nonexpansive mappings in Banach spaces. J. Approx. Theory 134, 257-266 (2005). doi:10.1016/j.jat.2005.02.007
Plubtieng, S, Ungchittrakool, K: Hybrid iterative method for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2008, Article ID 583082 (2008). doi:10.1155/2008/58308
Chang, S-s, Joseph Lee, HW, Chan, CK: A block hybrid method for solving generalized equilibrium problems and convex feasibility problem. Adv. Comput. Math. (2011). doi:10.1007/s10444-011-9249-5
Chang, S-s, Joseph Lee, HW, Chan, CK, Yang, L: Approximation theorems for total quasi-ϕ-asymptotically nonexpansive mappings with applications. Appl. Math. Comput. 218, 2921-2931 (2011). doi:10.1016/j.amc.2011.08.036
Ceng, L-C, Guu, S-M, Hu, H-Y, Yao, J-C: Hybrid shrinking projection method for a generalized equilibrium problem, a maximal monotone operator and a countable family of relatively nonexpansive mappings. Comput. Math. Appl. 61, 2468-2479 (2011). doi:10.1016/j.camwa.2011.02.028
Su, YF, Xu, HK, Zhang, X: Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications. Nonlinear Anal. 73, 3890-3906 (2010). doi:10.1016/j.na.2010.08.021
Ofoedu, EU, Malonza, DM: Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type. Appl. Math. Comput. 217, 6019-6030 (2011). doi:10.1016/j.amc.2010.12.073
Wang, ZM, Su, YF, Wang, DX, Dong, YC: A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces. J. Comput. Appl. Math. 235, 2364-2371 (2011). doi:10.1016/j.cam.2010.10.036
Chang, S-s, Chan, CK, Joseph Lee, HW: Modified block iterative algorithm for quasi-ϕ-asymptotically nonexpansive mappings and equilibrium problem in Banach spaces. Appl. Math. Comput. 217, 7520-7530 (2011). doi:10.1016/j.amc.2011.02.060
Yao, YH, Liou, YC, Kang, SM: Strong convergence of an iterative algorithm on an infinite countable family of nonexpansive mappings. Appl. Math. Comput. 208, 211-218 (2009). doi:10.1016/j.amc.2008.11.038
Zegeye, H, Ofoedu, EU, Shahzad, N: Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings. Appl. Math. Comput. 216, 3439-3449 (2010). doi:10.1016/j.amc.2010.02.054
Nilsrakoo, W, Saejung, S: Strong convergence theorems by Halpern-Mann iterations for relatively non-expansive mappings in Banach spaces. Appl. Math. Comput. 217, 6577-6586 (2011). doi:10.1016/j.amc.2011.01.040
Chang, S-s, Joseph Lee, HW, Chan, CK, Liu, J: Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications. Appl. Math. Comput. 218, 3187-3198 (2011). doi:10.1016/j.amc.2011.08.055
Zhang, S-s: The generalized mixed equilibrium problem in Banach space. Appl. Math. Mech. 30, 1105-1112 (2009). doi:10.1007/s10483-009-0904-6
Chang, S-s, Kim, JK, Wang, XR: Modified block iterative algorithm for solving convex feasibility problems in Banach spaces. J. Inequal. Appl. 2010, Article ID 869684 (2010). doi:10.1155/2010/869684
Chang, S-s, Joseph Lee, HW, Chan, CK, Yang, L: Approximation theorems for total quasi-ϕ-asymptotically nonexpansive mappings with applications. Appl. Math. Comput. 218, 2921-2931 (2011). doi:10.1016/j.amc.2011.08.036
Luchuan, C, Jenchih, Y: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 214, 186-201 (2008). doi:10.1016/j.cam.2007.02.022
Homaeipour, S, Razani, A: Weak and strong convergence theorems for relatively nonexpansive multi-valued mappings in Banach spaces. Fixed Point Theory Appl. 2011, Article ID 73 (2011). doi:10.1186/1687-1812-2011-73
Alber, YI: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartosator, AG (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15-50. Dekker, New York (1996)
Cioranescu, I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht (1990)
Deng, WQ, Bai, P: An implicit iteration process for common fixed points of two infinite families of asymptotically nonexpansive mappings in Banach spaces. J. Appl. Math. 2013, Article ID 602582 (2013)
Acknowledgements
The author wishes to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions. This study is supported by the Major Science Foundation of Yunnan province education department (08Z0081).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Qian, S. Strong convergence theorem for totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings under relaxed conditions. Fixed Point Theory Appl 2015, 213 (2015). https://doi.org/10.1186/s13663-015-0452-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13663-015-0452-9