 Research
 Open Access
 Published:
Convergence theorems for modified generalized fprojections and generalized nonexpansive mappings
Journal of Inequalities and Applications volume 2014, Article number: 305 (2014)
Abstract
The purpose of this paper is to study a sequence of modified generalized fprojections in a reflexive, smooth, and strictly convex Banach space and show that Mosco convergence of their ranges implies their pointwise convergence to the generalized fprojection onto the limit set. Furthermore, we prove a strong convergence theorem for a countable family of αnonexpansive mappings in a uniformly convex and smooth Banach space using the properties of a modified generalized fprojection operator. Our main results generalize the results of Ziming Wang, Yongfu Su, and Jinlong Kang and enrich the research contents of αnonexpansive mappings.
MSC:47H05, 47H09, 47H10.
1 Introduction
Let E be a real Banach space and C be a nonempty closed convex subset of E. A mapping T:C\to C is said to be nonexpansive if
Lots of iterative schemes for nonexpansive mappings have been introduced (see [1–3]); furthermore, many strong convergence theorems for nonexpansive mappings have been proved. On the other hand, there are many nonlinear mappings which are more general than the nonexpansive mapping. Compared to the existing problem of a fixed point of those mappings, the iterative methods for finding a fixed point are also very useful in studying the fixed point theory and the theory of equations in other fields.
In 2007, Gobel and Pineda [4] introduced and studied a new mapping, called αnonexpansive mapping. The mapping is more general than the nonexpansive mapping.
Definition 1.1 For a given multiindex, \alpha =({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n}) satisfies {\alpha}_{i}\ge 0, i=1,2,\dots ,n and {\sum}_{i=1}^{n}{\alpha}_{i}=1. A mapping T:C\to C is said to be αnonexpansive if
In order to show that the class of αnonexpansive mappings is more general than the one of nonexpansive mappings, we give an example [4].
Example 1.2 Let E={R}^{1}, and
Then T is not nonexpansive but αnonexpansive.
Proof Obviously, T is not nonexpansive. Taking x=\frac{1}{2}, y=0, by the definition of Tx, we have
On the other hand, for every x,y\in [0,+\mathrm{\infty}), we have
Therefore, we can affirm that
where \alpha =({\alpha}_{1},{\alpha}_{2})=(0,1). Then T is an αnonexpansive mapping but not a nonexpansive one. □
If T is a nonexpansive selfmapping, we can imply that T must be an αnonexpansive one, where \alpha =({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=(\frac{1}{n},\dots ,\frac{1}{n}).
For technical reasons, we always assume that the first coefficient {\alpha}_{1} is nonzero, that is, {\alpha}_{1}>0. In this case the mapping T satisfies the Lipschitz condition
For the αnonexpansive mapping T, \alpha =({\alpha}_{1},{\alpha}_{2},{\alpha}_{3},\dots {\alpha}_{n}), it is obvious that the mapping
is nonexpansive. However, the nonexpansiveness of {T}_{\alpha} is much weaker than (1.2), for instance, it does not entail the continuity of T (see [4]).
In 2010, Klineam and Suantai [5] introduced the relation of fixed point sets between an αnonexpansive operator and a {T}_{\alpha} operator. They gave the following theorem.
Theorem 1.3 (see Theorem 3.1 of Klineam and Suantai [5])
Let C be a closed convex subset of a Banach space E and for all n\in N, let \alpha =({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n}) such that {\alpha}_{i}\ge 0, i=1,2,\dots ,n, {\alpha}_{1}>0, and {\sum}_{i=1}^{n}{\alpha}_{i}=1. Let T be an αnonexpansive mapping from C into itself. If {\alpha}_{1}>\frac{1}{\sqrt[n1]{2}}, then F(T)=F({T}_{\alpha}), where F(T) is the fixed point set of T.
At the same time, they have succeeded in proving the demiclosedness principle for the αnonexpansive mappings.
Theorem 1.4 (see Theorem 3.4 of Klineam and Suantai [5])
Let C be a closed convex subset of a Banach space E and for all n\in N, let \alpha =({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n}) such that {\alpha}_{i}\ge 0, i=1,2,\dots ,n, {\alpha}_{1}>0, and {\sum}_{i=1}^{n}{\alpha}_{i}=1. Let T be an αnonexpansive mapping from C into itself. If {\alpha}_{1}>\frac{1}{\sqrt[n1]{2}}, if \{{x}_{n}\}\subset C converges weakly to x and \{{x}_{n}T{x}_{n}\} converges strongly to 0 as n\to \mathrm{\infty}, then x\in F(T).
Recently, Wang et al. [6] proposed the following hybrid algorithm for an αnonexpansive mapping in a Banach space:
As we know that if C is a nonempty closed convex subset of a Hilbert space H and recall that the (nearest point) projection {P}_{C} from H onto C assigns to each x\in H, and the unique point {P}_{C}x\in C satisfies the property \parallel x{P}_{C}x\parallel ={min}_{y\in C}\parallel xy\parallel, it is well known that {P}_{C} is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. We consider the functional defined by
where J is the normalized duality mapping and the Banach space is smooth. In this connection, Alber [7] introduced a generalized projection {\mathrm{\Pi}}_{C} from E to C as follows:
It is obvious from the definition of functional ϕ that
If E is a Hilbert space, then \varphi (y,x)={\parallel yx\parallel}^{2} and {\mathrm{\Pi}}_{C} becomes the metric projection of E onto C. 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 (y,x), 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 (y,x) and strict monotonicity of the normalized duality mapping J [8]. It is well known that the metric projection operator plays an important role in nonlinear functional analysis, optimization theory, fixed point theory, nonlinear programming, game theory, variational inequality, and complementarity problems, etc. [8, 9]. In 1994, Alber [7] introduced and studied the generalized projections from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. Moreover, Alber [8] presented some applications of the generalized projections to approximately solve variational inequalities and von Neumann intersection problem in Banach spaces. In 2005, Li [9] extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solve the variational inequality in Banach spaces. Later, Wu and Huang [10] introduced a new generalized fprojection operator in Banach spaces. They extended the definition of generalized projection operators introduced by Abler [7] and proved some properties of the generalized fprojection operator. In 2009, Fan et al. [11] presented some basic results for the generalized fprojection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces.
The purpose of this paper is to study a sequence of modified generalized fprojections in a reflexive, smooth, and strictly convex Banach space and show that Mosco convergence of their ranges implies their pointwise convergence to the generalized fprojection onto the limit set. Furthermore, we prove strong convergence theorem for a countable family of αnonexpansive mappings in a uniformly convex and smooth Banach space using the properties of a modified generalized fprojection operator. Our main results generalize the results of Wang et al. [6] and enrich the research contents of αnonexpansive mappings.
2 Preliminaries
A Banach space E is said to be strictly convex if \frac{\parallel x+y\parallel}{2}<1 for x,y\in E with \parallel x\parallel =\parallel y\parallel =1 and x\ne y. It is said to be uniformly convex if for each \u03f5>0 there is \delta >0 such that for x,y\in E with \parallel x\parallel ,\parallel y\parallel \le 1 and \parallel xy\parallel \ge \u03f5, \parallel x+y\parallel \le 2(1\delta ) holds. The space E is said to be smooth if the limit
exists for all x,y\in S(E)=\{x\in E:\parallel x\parallel =1\}. And E is said to be uniformly smooth if the limit (2.1) exists uniformly for all x,y\in S(E).
Remark 2.1 The following basic properties of a Banach space E can be found in Cioranescu [12]:

(i)
if E is uniformly convex, then E is reflexive and strictly convex;

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

(iii)
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.
Let E be a real Banach space with the dual {E}^{\ast}. We denote by J the normalized duality mapping from E to {2}^{{E}^{\ast}} defined by
Many properties of the normalized duality mapping J can be found in Takahashi [13] or Vainberg [14]. We list some properties below for easy reference:

(i)
J is a monotone and bounded operator in arbitrary Banach spaces;

(ii)
J is a strictly monotone operator in strictly convex Banach spaces;

(iii)
J is a continuous operator in smooth Banach spaces;

(iv)
J is a uniformly continuous operator on each bounded set in uniformly smooth Banach spaces;

(v)
J is a bijection in smooth, reflexive, and strictly convex Banach spaces;

(vi)
J is the identity operator in Hilbert spaces.
Next, we recall the concept of generalized fprojector operator, together with its properties. Let G:C\times {E}^{\ast}\to R\cup \{+\mathrm{\infty}\} be a functional defined as follows:
where \xi \in C, \phi \in {E}^{\ast}, ρ is a positive number and f:C\to R\cup \{+\mathrm{\infty}\} is proper, convex, and lower semicontinuous. From the definitions of G and f, it is easy to see the following properties:

(i)
G(\xi ,\phi ) is convex and continuous with respect to φ when ξ is fixed;

(ii)
G(\xi ,\phi ) is convex and lower semicontinuous with respect to ξ when φ is fixed.
Definition 2.2 ([10])
Let E be a real Banach space with its dual {E}^{\ast}. Let C be a nonempty, closed, and convex subset of E. We say that {\mathrm{\Pi}}_{C}^{f}:{E}^{\ast}\to {2}^{C} is a generalized fprojection operator if
For the generalized fprojection operator, Wu and Huang [10] proved the following basic properties.
Lemma 2.3 ([10])
Let E be a real reflexive Banach space with its dual {E}^{\ast}, and let C be a nonempty, closed, and convex subset of E. Then the following statements hold:

(i)
{\mathrm{\Pi}}_{C}^{f}\phi is a nonempty closed convex subset of C for all \phi \in {E}^{\ast}.

(ii)
If E is smooth, then for all \phi \in {E}^{\ast}, x\in {\mathrm{\Pi}}_{C}^{f}\phi if and only if
\u3008xy,\phi Jx\u3009+\rho f(y)\rho f(x)\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in C. 
(iii)
If E is strictly convex and f:C\to R\cup \{+\mathrm{\infty}\} is positive homogeneous (i.e., f(tx)=tf(x) for all t>0 such that tx\in C, where x\in C), then {\mathrm{\Pi}}_{C}^{f} is a singlevalued mapping.
Fan et al. [11] showed that the condition f is positive homogeneous, which appeared in Lemma 2.3, can be removed.
Lemma 2.4 ([11])
Let E be a real reflexive Banach space with its dual {E}^{\ast}, and let C be a nonempty, closed, and convex subset of E. Then if E is strictly convex, then {\mathrm{\Pi}}_{C}^{f} is a singlevalued mapping.
Recall that J is a singlevalued mapping when E is a smooth Banach space. There exists a unique element \phi \in {E}^{\ast} such that \phi =Jx for each x\in E. This substitution in (2.2) gives
Now, we consider the second generalized fprojection operator in a Banach space.
Definition 2.5 Let E be a real Banach space and C be a nonempty, closed, and convex subset of E. We say that {\mathrm{\Pi}}_{C}^{f}:E\to {2}^{C} is a generalized fprojection operator if
We know that the following lemmas hold for the operator {\mathrm{\Pi}}_{C}^{f}.
Lemma 2.6 ([15])
Let C be a nonempty, closed, and convex subset of a smooth and reflexive Banach space E. Then the following statements hold:

(i)
{\mathrm{\Pi}}_{C}^{f}x is a nonempty closed and convex subset of C for all x\in E.

(ii)
For all x\in E, \stackrel{\u02c6}{x}\in {\mathrm{\Pi}}_{C}^{f}x if and only if
\u3008\stackrel{\u02c6}{x}y,JxJ\stackrel{\u02c6}{x}\u3009+\rho f(y)\rho f(x)\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in C. 
(iii)
If E is strictly convex, then {\mathrm{\Pi}}_{C}^{f} is a singlevalued mapping.
Now, we introduce a modified generalized fprojection operator. Let G:C\times {E}^{\ast}\to R be a functional defined as follows:
where \xi \in C, \phi \in {E}^{\ast}, ρ is a positive number and f:C\to R is convex and weakly continuous. From the definitions of G and f, it is easy to see the following properties:

(i)
G(\xi ,\phi ) is convex and continuous with respect to φ when ξ is fixed;

(ii)
G(\xi ,\phi ) is convex and weakly lower semicontinuous with respect to ξ when φ is fixed.
Obviously, the other definitions and lemmas hold respectively.
Next, we give the following example [16] which shows that metric projection, generalized projection and generalized fprojection are different.
Example 2.7 Let X={R}^{3} be provided with the norm
This is a smooth strictly convex Banach space and C=\{x\in {R}^{3}{x}_{2}=0,{x}_{3}=0\} is a closed and convex subset of X. It is a simple computation; we get {P}_{C}(1,1,1)=(1,0,0), {\mathrm{\Pi}}_{C}(1,1,1)=(2,0,0).
We set \rho =1 is a positive number and define f:C\to R by
Then f is convex and weakly continuous. Simple computations show that
Let E be a Banach space, and let {C}_{1},{C}_{2},{C}_{3},\dots be a sequence of weakly closed subsets of E. We denote by sL{i}_{n}{C}_{n} the set of limit points of \{{C}_{n}\}, that is, x\in sL{i}_{n}{C}_{n} if and only if there exists \{{x}_{n}\}\subset E such that \{{x}_{n}\} converges strongly to x and that {x}_{n}\in {C}_{n} for all n\in N. Similarly, we denote by wL{s}_{n}{C}_{n} the set of cluster points of \{{C}_{n}\}, y\in wL{s}_{n}{C}_{n} if and only if there exists \{{y}_{{n}_{i}}\} such that \{{y}_{{n}_{i}}\} converges weakly to y and that \{{y}_{{n}_{i}}\}\in {C}_{{n}_{i}} for all i\in N. Using these definitions, we define the Mosco convergence [2] of {C}_{{n}_{i}}. If {C}_{0} satisfies
we say that {C}_{n} is a Mosco convergent sequence to {C}_{0} and write
Notice that the inclusion sL{i}_{n}{C}_{n}\subset wL{s}_{n}{C}_{n} is always true. Therefore, in order to show the existence of M{lim}_{n\to \mathrm{\infty}}{C}_{n}, it is sufficient to prove wL{s}_{n}{C}_{n}\subset sL{i}_{n}{C}_{n}. For more details, see [17].
3 Main results
3.1 Generalized Mosco convergence theorems
Theorem 3.1 Let E be a smooth, reflexive, and strictly convex Banach space and C be a nonempty closed convex subset of E. Let {C}_{1},{C}_{2},{C}_{3},\dots be nonempty closed convex subsets of C, f:E\to R be a convex and weakly continuous mapping with C\subset int(D(f)). If {C}_{0}=M{lim}_{n\to \mathrm{\infty}}{C}_{n} exists and is nonempty, then {C}_{0} is a closed convex subset of C and, for each x\in C, \{{\mathrm{\Pi}}_{{C}_{n}}^{f}x\} converges weakly to {\mathrm{\Pi}}_{{C}_{0}}^{f}x.
Proof It is easy to prove that {C}_{0} is closed and convex if {C}_{n} is a closed convex subset of C for each n\in N. Fix x\in C. For the sake of simplicity, we write {x}_{n} instead of {\mathrm{\Pi}}_{{C}_{n}}^{f}x for n\in N. Since {C}_{0}=M{lim}_{n\to \mathrm{\infty}}{C}_{n}, we have that for each y\in {C}_{0}, there exists \{{y}_{n}\}\subset E such that {y}_{n}\to y as n\to \mathrm{\infty} and that {y}_{n}\in {C}_{n} for each n\in N. From Lemma 2.6, we have
Hence, we obtain
thus,
Suppose that \{{x}_{n}\} is not bounded. Then there exists a subsequence \{{x}_{{n}_{i}}\} of \{{x}_{n}\} such that \parallel {x}_{{n}_{i}}\parallel \to \mathrm{\infty}. It follows that
for a sufficiently large number i\in N. As i\to \mathrm{\infty}, we obtain +\mathrm{\infty}\le \parallel x{y}_{{n}_{i}}\parallel <+\mathrm{\infty}. This is a contradiction. Hence we have that \{{x}_{n}\} is bounded.
Since \{{x}_{n}\} is bounded, there exists a subsequence, again denoted by \{{x}_{n}\}, such that it converges weakly to {x}_{0}\in C. From the definition of {C}_{0}, we get {x}_{0}\in {C}_{0}.
Now, we prove that {\mathrm{\Pi}}_{{C}_{0}}^{f}x={x}_{0}. From weak lower semicontinuity of the norm and weak continuity of f, we have
On the other hand, we get
So,
that is,
Hence we get {\mathrm{\Pi}}_{{C}_{0}}^{f}x={x}_{0}.
According to our consideration above, each sequence \{{x}_{n}\} has, in turn, a subsequence which converges weakly to the unique point {\mathrm{\Pi}}_{{C}_{0}}^{f}x. Therefore, the sequence \{{x}_{n}\} converges weakly to {\mathrm{\Pi}}_{{C}_{0}}^{f}x. This completes the proof. □
A Banach space E is said to have the KadecKlee property if a sequence \{{x}_{n}\} of E satisfying that {x}_{n}\rightharpoonup {x}_{0} and \parallel {x}_{n}\parallel \to \parallel {x}_{0}\parallel converges strongly to {x}_{0}. It is known that {E}^{\ast} has a Fréchet differentiable norm if and only if E is reflexive, strictly convex, and has the KadecKlee property; see, for example, [10].
Theorem 3.2 Let E be a smooth Banach space such that {E}^{\ast} has a Fréchet differentiable norm. Let C be a nonempty closed convex subset of E. Let {C}_{1},{C}_{2},{C}_{3},\dots be nonempty closed convex subsets of C, f:E\to R be a convex and weakly continuous mapping with C\subset int(D(f)). If {C}_{0}=M{lim}_{n\to \mathrm{\infty}}{C}_{n} exists and is nonempty, then {C}_{0} is a closed convex subset of C and, for each x\in C, \{{\mathrm{\Pi}}_{{C}_{n}}^{f}x\} converges strongly to {\mathrm{\Pi}}_{{C}_{0}}^{f}x.
Proof Fix x\in C arbitrarily. We write {x}_{n}={\mathrm{\Pi}}_{{C}_{n}}^{f}x and {x}_{0}={\mathrm{\Pi}}_{{C}_{0}}^{f}x. By Theorem 3.1, we obtain {x}_{n}\rightharpoonup {x}_{0}. Since {E}^{\ast} has a Fréchet differentiable norm, E has the KadecKlee property. Therefore, it is sufficient to prove that \parallel {x}_{n}\parallel \to \parallel {x}_{0}\parallel as n\to \mathrm{\infty}. Since {x}_{0}\in {C}_{0}, there exists a sequence \{{y}_{n}\}\subset C such that {y}_{n}\to {x}_{0} as n\to \mathrm{\infty} and {y}_{n}\in {C}_{n} for each n\in N. It follows that
Hence we obtain G({x}_{0},Jx)={lim}_{n\to \mathrm{\infty}}G({x}_{n},Jx). Since \u3008{x}_{n},J(x)\u3009 converges to \u3008{x}_{0},J(x)\u3009 and f is weakly continuous, we get
Using the KadecKlee property of E, we obtain that \{{x}_{n}\} converges strongly to {x}_{0}. This completes the proof. □
Definition 3.3 ([18])
Let C be a closed convex subset of a Banach space E, let {\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}} be a countable family of mappings of C into itself with the nonempty common fixed point set F. The {\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}} is said to be uniformly closed if {x}_{n}\to x and \parallel {x}_{n}{T}_{n}{x}_{n}\parallel \to 0 as n\to \mathrm{\infty} implies x\in F.
3.2 Strong convergence theorems
Lemma 3.4 (see Lemma 3.3 of Klineam and Suantai [5])
Let C be a closed convex subset of a Banach space E and for all n\in N, let \alpha =({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n}) such that {\alpha}_{i}\ge 0, i=1,2,\dots ,n, {\alpha}_{1}>0, and {\sum}_{i=1}^{n}{\alpha}_{i}=1. Let T be an αnonexpansive mapping from C into itself. If {\alpha}_{1}>\frac{1}{\sqrt[n1]{2}}, let \{{x}_{m}\} be a bounded sequence in C, then \parallel {x}_{m}T{x}_{m}\parallel \to 0 if and only if \parallel {x}_{m}{T}_{\alpha}{x}_{m}\parallel \to 0 as m\to \mathrm{\infty}.
Lemma 3.5 ([6])
Let C be a closed convex subset of a Banach space E, and for all n\in N, let \alpha =({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n}) such that {\alpha}_{i}\ge 0, i=1,2,\dots ,n, {\alpha}_{1}>0, and {\sum}_{i=1}^{n}{\alpha}_{i}=1. Let T be an αnonexpansive mapping from C into itself. If {\alpha}_{1}>\frac{1}{\sqrt[n1]{2}}, let \{{x}_{m}\}\subset C converge strongly to x and \parallel {x}_{m}T{x}_{m}\parallel \to 0 converge strongly to 0 as m\to \mathrm{\infty}, then x\in F(T).
Lemma 3.6 ([6])
Let C be a closed convex subset of a uniformly convex and smooth Banach space E, and for all n\in N, let \alpha =({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n}) such that {\alpha}_{i}\ge 0, i=1,2,\dots ,n, {\alpha}_{1}>0, and {\sum}_{i=1}^{n}{\alpha}_{i}=1. Let T be an αnonexpansive mapping from C into itself. If {\alpha}_{1}>\frac{1}{\sqrt[n1]{2}}, then F(T) is closed and convex.
Theorem 3.7 Let C be a closed convex subset of a uniformly convex and smooth Banach space E, let {\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}} be a uniformly closed countable family of {\alpha}_{n}nonexpansive mappings of C into itself such that F:={\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})\ne \mathrm{\varnothing}, let {\alpha}_{n}=({\alpha}_{n1},{\alpha}_{n2},\dots ,{\alpha}_{n{N}_{0}}) such that {\alpha}_{ni}\ge 0, i=1,2,\dots ,{N}_{0}, {\alpha}_{n1}>0, and {\sum}_{i=1}^{{N}_{0}}{\alpha}_{ni}=1. Let f:E\to R be a convex and weakly continuous mapping with C\subset int(D(f)). For any given Gauss {x}_{0}\in E, {C}_{1}=C, and {x}_{1}={\mathrm{\Pi}}_{{C}_{1}}^{f}{x}_{0}, define a sequence \{{x}_{n}\} in C by the following algorithm:
where 0<a\le {\beta}_{n}\le 1 for all n\in N. If {\alpha}_{n1}>\frac{1}{\sqrt[{N}_{0}1]{2}}, then \{{x}_{n}\} converges strongly to {x}^{\ast}={\mathrm{\Pi}}_{F}^{f}{x}_{0}.
Proof Step 1. We show that {C}_{n} is closed and convex for each n\ge 0.
From the definitions of {C}_{n}, it is obvious that {C}_{n} is closed for each n\ge 0. Moreover, since \parallel {y}_{n}z\parallel \le \parallel {x}_{n}z\parallel is equivalent to
so {C}_{n} is convex for each n\ge 0.
Step 2. We show that F\subset {C}_{n} for all n\ge 0. For all p\in F, we have that
It implies that p\in {C}_{n} for all n\ge 0. So, we have F\subset {C}_{n} for all n\ge 0.
Step 3. We show that {lim}_{n\to \mathrm{\infty}}{x}_{n}={x}^{\ast}={\mathrm{\Pi}}_{\overline{C}}^{f}{x}_{0} and {x}^{\ast}\in F, where \overline{C}={\bigcap}_{n=1}^{\mathrm{\infty}}{C}_{n}. Indeed, since \{{C}_{n}\} is a decreasing sequence of closed convex subsets of E such that \overline{C}={\bigcap}_{n=1}^{\mathrm{\infty}}{C}_{n} is nonempty, it follows that
By Theorem 3.2, we get
Noticing that {x}_{n+1}={\mathrm{\Pi}}_{{C}_{n+1}}^{f}{x}_{0}\in {C}_{n+1}, we obtain that
In view of (3.2), we have that
and
From {y}_{n}=(1{\beta}_{n}){x}_{n}+{\beta}_{n}{T}_{n}{x}_{n}, we have
Because of the assumption that 0<a\le {\beta}_{n}\le 1, we have
Since \{{x}_{n}\} is uniformly closed, then {x}^{\ast}\in F.
Step 4. We show that {x}^{\ast}={\mathrm{\Pi}}_{F}^{f}{x}_{0}. Since {x}^{\ast}={\mathrm{\Pi}}_{\overline{C}}^{f}{x}_{0}\in F and F is a nonempty closed convex subset of \overline{C}={\bigcap}_{n=1}^{\mathrm{\infty}}{C}_{n}, we conclude that {x}^{\ast}={\mathrm{\Pi}}_{F}^{f}{x}_{0}. This completes the proof. □
Corollary 3.8 ([6])
Let C be a closed convex subset of a uniformly convex and smooth Banach space E, let T be an αnonexpansive mapping of C into itself such that F(T)\ne \mathrm{\varnothing}, let \alpha =({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{{N}_{0}}) such that {\alpha}_{i}\ge 0, i=1,2,\dots ,{N}_{0}, {\alpha}_{1}>0, and {\sum}_{i=1}^{{N}_{0}}{\alpha}_{i}=1. For any given Gauss {x}_{0}\in E, {C}_{1}=C, and {x}_{1}={\mathrm{\Pi}}_{{C}_{1}}{x}_{0}, define a sequence \{{x}_{n}\} in C by the following algorithm:
where 0<a\le {\beta}_{n}\le 1 for all n\in N. If {\alpha}_{1}>\frac{1}{\sqrt[{N}_{0}1]{2}}, then \{{x}_{n}\} converges strongly to {x}^{\ast}={\mathrm{\Pi}}_{F}{x}_{0}.
Proof Substituting T to {T}_{n} in the proof of Theorem 3.7 and putting f(x)\equiv 0, we can draw from Theorem 3.7 the desired conclusion immediately. □
Remark 3.9 Theorem 3.7 extends the main results of [6] from a single mapping to a countable family of mappings and from the generalized projection operator to the modified generalized fprojection operator by a new method.
References
Chidume C Lecture Notes in Mathematics 1965. In Geometric Properties of Banach Spaces and Nonlinear Iterations. Springer, London; 2009.
Wittmann R: Approximation of fixed points of nonexpansive mappings. Arch. Math. 1992, 58: 486–491. 10.1007/BF01190119
Berinde V Lecture Notes in Mathematics 1912. In Iterative Approximation of Fixed Points. 2nd edition. Springer, Berlin; 2007.
Goebel K, Pineda MA: On a type of generalized nonexpansiveness. 74. Proc. of the 8th International Conference on Fixed Point Theory and Its Application 2007, 660–665.
Klineam C, Suantai S: Fixed point theorems for α nonexpansive mappings. Appl. Math. Lett. 2010, 23: 728–731. 10.1016/j.aml.2010.02.018
Wang Z, Su Y, Kang J: Hybrid algorithm for an α nonexpansive mappings in a Banach space. J. Nonlinear Sci. Appl. 2012,5(1):56–63.
Alber Y: Generalized projection operators in Banach spaces: properties and applications. Israel Funct. Differ. Equ. 1. Proceedings of the Israel Seminar 1994, 1–21. Ariel
Alber Y: 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 A. Dekker, New York; 1996:15–50.
Li JL: The generalized projection operator on reflexive Banach spaces and its applications. J. Math. Anal. Appl. 2005, 306: 55–71. 10.1016/j.jmaa.2004.11.007
Wu KQ, Huang NJ: The generalized f projection operator with an application. Bull. Aust. Math. Soc. 2006, 73: 307–317. 10.1017/S0004972700038892
Fan JH, Liu X, Li JL: Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces. Nonlinear Anal. TMA 2009, 70: 3997–4007. 10.1016/j.na.2008.08.008
Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht; 1990.
Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama; 2000.
Vainberg MM: Variational Methods and Method of Monotone Operators. Wiley, New York; 1973.
Li X, Huang N, O’Regan D: Strong convergence theorems for relatively nonexpansive mappings in Banach spaces with applications. Comput. Math. Appl. 2010, 60: 1322–1331. 10.1016/j.camwa.2010.06.013
Alber Y, Butnariu D: Convergence of Bregman projection methods for solving consistent convex feasibility problems in reflexive Banach spaces. J. Optim. Theory Appl. 1997, 92: 33–61. 10.1023/A:1022631928592
Beer G Mathematics and Its Applications 268. In Topologies on Closed and Closed Convex Sets. Kluwer Academic, Dordrecht; 1993.
Zhang J, Su Y, Cheng Q: Uniformly closed replaced AKTT or ^{∗}AKTT condition to get strong convergence theorems for a countable family of relatively quasinonexpansive mappings and systems of equilibrium problems. Fixed Point Theory Appl. 2014., 2014: Article ID 103
Acknowledgements
This project is supported by the National Natural Science Foundation of China under grant (11071279).
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 article. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Cheng, Q., Su, Y. & Zhang, J. Convergence theorems for modified generalized fprojections and generalized nonexpansive mappings. J Inequal Appl 2014, 305 (2014). https://doi.org/10.1186/1029242X2014305
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X2014305
Keywords
 αnonexpansive mappings
 monotone hybrid algorithm
 modified generalized fprojection operator
 Mosco convergence
 common fixed point