Convergence theorems for modified generalized f-projections and generalized nonexpansive mappings
© Cheng et al.; licensee Springer. 2014
Received: 18 March 2014
Accepted: 14 July 2014
Published: 19 August 2014
The purpose of this paper is to study a sequence of modified generalized f-projections in a reflexive, smooth, and strictly convex Banach space and show that Mosco convergence of their ranges implies their pointwise convergence to the generalized f-projection onto the limit set. Furthermore, we prove a strong convergence theorem for a countable family of α-nonexpansive mappings in a uniformly convex and smooth Banach space using the properties of a modified generalized f-projection operator. Our main results generalize the results of Ziming Wang, Yongfu Su, and Jinlong Kang and enrich the research contents of α-nonexpansive mappings.
MSC:47H05, 47H09, 47H10.
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  introduced and studied a new mapping, called α-nonexpansive mapping. The mapping is more general than the nonexpansive mapping.
In order to show that the class of α-nonexpansive mappings is more general than the one of nonexpansive mappings, we give an example .
Then T is not nonexpansive but α-nonexpansive.
where . Then T is an α-nonexpansive mapping but not a nonexpansive one. □
If T is a nonexpansive self-mapping, we can imply that T must be an α-nonexpansive one, where .
is nonexpansive. However, the nonexpansiveness of is much weaker than (1.2), for instance, it does not entail the continuity of T (see ).
In 2010, Klin-eam and Suantai  introduced the relation of fixed point sets between an α-nonexpansive operator and a operator. They gave the following theorem.
Theorem 1.3 (see Theorem 3.1 of Klin-eam and Suantai )
Let C be a closed convex subset of a Banach space E and for all , let such that , , , and . Let T be an α-nonexpansive mapping from C into itself. If , then , where is the fixed point set of T.
At the same time, they have succeeded in proving the demiclosedness principle for the α-nonexpansive mappings.
Theorem 1.4 (see Theorem 3.4 of Klin-eam and Suantai )
Let C be a closed convex subset of a Banach space E and for all , let such that , , , and . Let T be an α-nonexpansive mapping from C into itself. If , if converges weakly to x and converges strongly to 0 as , then .
The existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the normalized duality mapping J . 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  introduced and studied the generalized projections from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. Moreover, Alber  presented some applications of the generalized projections to approximately solve variational inequalities and von Neumann intersection problem in Banach spaces. In 2005, Li  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  introduced a new generalized f-projection operator in Banach spaces. They extended the definition of generalized projection operators introduced by Abler  and proved some properties of the generalized f-projection operator. In 2009, Fan et al.  presented some basic results for the generalized f-projection 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 f-projections in a reflexive, smooth, and strictly convex Banach space and show that Mosco convergence of their ranges implies their pointwise convergence to the generalized f-projection onto the limit set. Furthermore, we prove strong convergence theorem for a countable family of α-nonexpansive mappings in a uniformly convex and smooth Banach space using the properties of a modified generalized f-projection operator. Our main results generalize the results of Wang et al.  and enrich the research contents of α-nonexpansive mappings.
exists for all . And E is said to be uniformly smooth if the limit (2.1) exists uniformly for all .
if E is uniformly convex, then E is reflexive and strictly convex;
a Banach space E is uniformly smooth if and only if is uniformly convex;
each uniformly convex Banach space E has the Kadec-Klee property, i.e., for any sequence , if and , then .
J is a monotone and bounded operator in arbitrary Banach spaces;
J is a strictly monotone operator in strictly convex Banach spaces;
J is a continuous operator in smooth Banach spaces;
J is a uniformly continuous operator on each bounded set in uniformly smooth Banach spaces;
J is a bijection in smooth, reflexive, and strictly convex Banach spaces;
J is the identity operator in Hilbert spaces.
is convex and continuous with respect to φ when ξ is fixed;
is convex and lower semi-continuous with respect to ξ when φ is fixed.
Definition 2.2 ()
For the generalized f-projection operator, Wu and Huang  proved the following basic properties.
Lemma 2.3 ()
is a nonempty closed convex subset of C for all .
- (ii)If E is smooth, then for all , if and only if
If E is strictly convex and is positive homogeneous (i.e., for all such that , where ), then is a single-valued mapping.
Fan et al.  showed that the condition f is positive homogeneous, which appeared in Lemma 2.3, can be removed.
Lemma 2.4 ()
Let E be a real reflexive Banach space with its dual , and let C be a nonempty, closed, and convex subset of E. Then if E is strictly convex, then is a single-valued mapping.
Now, we consider the second generalized f-projection operator in a Banach space.
We know that the following lemmas hold for the operator .
Lemma 2.6 ()
is a nonempty closed and convex subset of C for all .
- (ii)For all , if and only if
If E is strictly convex, then is a single-valued mapping.
is convex and continuous with respect to φ when ξ is fixed;
is convex and weakly lower semi-continuous with respect to ξ when φ is fixed.
Obviously, the other definitions and lemmas hold respectively.
Next, we give the following example  which shows that metric projection, generalized projection and generalized f-projection are different.
This is a smooth strictly convex Banach space and is a closed and convex subset of X. It is a simple computation; we get , .
Notice that the inclusion is always true. Therefore, in order to show the existence of , it is sufficient to prove . For more details, see .
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 be nonempty closed convex subsets of C, be a convex and weakly continuous mapping with . If exists and is nonempty, then is a closed convex subset of C and, for each , converges weakly to .
for a sufficiently large number . As , we obtain . This is a contradiction. Hence we have that is bounded.
Since is bounded, there exists a subsequence, again denoted by , such that it converges weakly to . From the definition of , we get .
Hence we get .
According to our consideration above, each sequence has, in turn, a subsequence which converges weakly to the unique point . Therefore, the sequence converges weakly to . This completes the proof. □
A Banach space E is said to have the Kadec-Klee property if a sequence of E satisfying that and converges strongly to . It is known that has a Fréchet differentiable norm if and only if E is reflexive, strictly convex, and has the Kadec-Klee property; see, for example, .
Theorem 3.2 Let E be a smooth Banach space such that has a Fréchet differentiable norm. Let C be a nonempty closed convex subset of E. Let be nonempty closed convex subsets of C, be a convex and weakly continuous mapping with . If exists and is nonempty, then is a closed convex subset of C and, for each , converges strongly to .
Using the Kadec-Klee property of E, we obtain that converges strongly to . This completes the proof. □
Definition 3.3 ()
Let C be a closed convex subset of a Banach space E, let be a countable family of mappings of C into itself with the nonempty common fixed point set F. The is said to be uniformly closed if and as implies .
3.2 Strong convergence theorems
Lemma 3.4 (see Lemma 3.3 of Klin-eam and Suantai )
Let C be a closed convex subset of a Banach space E and for all , let such that , , , and . Let T be an α-nonexpansive mapping from C into itself. If , let be a bounded sequence in C, then if and only if as .
Lemma 3.5 ()
Let C be a closed convex subset of a Banach space E, and for all , let such that , , , and . Let T be an α-nonexpansive mapping from C into itself. If , let converge strongly to x and converge strongly to 0 as , then .
Lemma 3.6 ()
Let C be a closed convex subset of a uniformly convex and smooth Banach space E, and for all , let such that , , , and . Let T be an α-nonexpansive mapping from C into itself. If , then is closed and convex.
where for all . If , then converges strongly to .
Proof Step 1. We show that is closed and convex for each .
so is convex for each .
It implies that for all . So, we have for all .
Since is uniformly closed, then .
Step 4. We show that . Since and F is a nonempty closed convex subset of , we conclude that . This completes the proof. □
Corollary 3.8 ()
where for all . If , then converges strongly to .
Proof Substituting T to in the proof of Theorem 3.7 and putting , we can draw from Theorem 3.7 the desired conclusion immediately. □
Remark 3.9 Theorem 3.7 extends the main results of  from a single mapping to a countable family of mappings and from the generalized projection operator to the modified generalized f-projection operator by a new method.
This project is supported by the National Natural Science Foundation of China under grant (11071279).
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