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On generalized quasi-ϕ-nonexpansive mappings and their projection algorithms
Fixed Point Theory and Applications volume 2013, Article number: 204 (2013)
Abstract
A fixed point problem of a generalized asymptotically quasi-ϕ-nonexpansive mapping and an equilibrium problem are investigated. A strong convergence theorem for solutions of the fixed point problem and the equilibrium problem is established in a Banach space.
1 Introduction and preliminaries
Let E be a real Banach space, and let be the dual space of E. We denote by J the normalized duality mapping from E to defined by
where denotes the generalized duality pairing. A Banach space E is said to be strictly convex if for all with and . It is said to be uniformly convex if for any two sequences and in E such that and . Let be the unit sphere of E. Then the Banach space E is said to be smooth provided
exists for each . It is also said to be uniformly smooth if the above limit is attained uniformly for . It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. It is also well known that E is uniformly smooth if and only if is uniformly convex.
Recall that a Banach space E enjoys the Kadec-Klee property if for any sequence , and with , and , then as . For more details on the Kadec-Klee property, the readers can refer to [1] and the references therein. It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property.
Let C be a nonempty subset of E. Let f be a bifunction from to ℝ, where ℝ denotes the set of real numbers. In this paper, we investigate the following equilibrium problem. Find such that
We use to denote the solution set of the equilibrium problem (1.1). That is,
Given a mapping , let
Then iff p is a solution of the following variational inequality. Find p such that
In order to study the solution problem of the equilibrium problem (1.1), we assume that f satisfies the following conditions:
-
(A1)
, ;
-
(A2)
f is monotone, i.e., , ;
-
(A3)
-
(A4)
for each , is convex and weakly lower semi-continuous.
As we all know, if C is a nonempty closed convex subset of a Hilbert space H and is the metric projection of H onto C, then is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [2] recently introduced a generalized projection operator in a Banach space E, which is an analogue of the metric projection in Hilbert spaces.
Next, we assume that E is a smooth Banach space. Consider the functional defined by
Observe that in a Hilbert space H, the equality is reduced to , . The generalized projection is a map that assigns to an arbitrary point the minimum point of the functional , that is, , where is the solution to the minimization problem
Existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the mapping J; see, for example, [1] and [2]. In Hilbert spaces, . It is obvious from the definition of function ϕ that
and
Remark 1.1 If E is a reflexive, strictly convex and smooth Banach space, then if and only if ; for more details, see [1] and [3].
Let be a mapping. In this paper, we use to denote the fixed point set of T. T is said to be asymptotically regular on C if, for any bounded subset K of C, . T is said to be closed if, for any sequence such that and , . In this paper, we use → and ⇀ to denote the strong convergence and weak convergence, respectively.
A point p in C is said to be an asymptotic fixed point of T [3] iff C contains a sequence which converges weakly to p such that . The set of asymptotic fixed points of T will be denoted by . T is said to be relatively nonexpansive [4, 5] iff and for all and . T is said to be relatively asymptotically nonexpansive [6, 7] iff and there exists a sequence with as such that for all , and . T is said to be quasi-ϕ-nonexpansive [8, 9] iff and for all and . T is said to be asymptotically quasi-ϕ-nonexpansive [10–12] iff and there exists a sequence with as such that for all , and .
Remark 1.2 The class of asymptotically quasi-ϕ-nonexpansive mappings is more general than the class of relatively asymptotically nonexpansive mappings which requires the restriction .
Remark 1.3 The classes of asymptotically quasi-ϕ-nonexpansive mappings and quasi-ϕ-nonexpansive mappings are the generalizations of asymptotically quasi-nonexpansive mappings and quasi-nonexpansive mappings in Hilbert spaces.
Recently, Qin et al. [13] introduced a class of generalized asymptotically quasi-ϕ-nonexpansive mappings. Recall that a mapping T is said to be generalized asymptotically quasi-ϕ-nonexpansive iff and there exist a sequence with as and a sequence with as such that for all , and .
Remark 1.4 The class of generalized asymptotically quasi-ϕ-nonexpansive mappings is a generalization of the class of generalized asymptotically quasi-nonexpansive mappings which was studied in [14].
Recently, fixed point and equilibrium problems (1.1) have been intensively investigated based on iterative methods; see [15–28]. The projection method which grants strong convergence of the iterative sequences is one of efficient methods for the problems. In this paper, we investigate the equilibrium problem (1.1) and a fixed point problem of the generalized quasi-ϕ-nonexpansive mapping based on a projection method. A strong convergence theorem for solutions of the equilibrium and the fixed point problem is established in a Banach space.
In order to state our main results, we need the following lemmas.
Lemma 1.5 [2]
Let E be a reflexive, strictly convex, and smooth Banach space, let C be a nonempty, closed, and convex subset of E, and . Then
Lemma 1.6 [2]
Let C be a nonempty, closed, and convex subset of a smooth Banach space E, and . Then if and only if
Lemma 1.7 [11]
Let E be a reflexive, strictly convex, and smooth Banach space such that both E and have the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let be a closed asymptotically quasi-ϕ-nonexpansive mapping. Then is a closed convex subset of C.
Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Let f be a bifunction from to ℝ satisfying (A1)-(A4). Let and . Then there exists such that , . Define a mapping by . Then the following conclusions hold:
-
(1)
is a single-valued and firmly nonexpansive-type mapping, i.e., for all ,
-
(2)
is closed and convex;
-
(3)
is quasi-ϕ-nonexpansive;
-
(4)
, .
Lemma 1.9 [31]
Let E be a smooth and uniformly convex Banach space, and let . Then there exists a strictly increasing, continuous and convex function such that and
for all and .
2 Main results
Theorem 2.1 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let f be a bifunction from to ℝ satisfying (A1)-(A4), and let be a closed generalized asymptotically quasi-ϕ-nonexpansive mapping. Assume that T is asymptotically regular on C and that is nonempty and bounded. Let be a sequence generated in the following manner:
where , is a real sequence in such that , and is a real sequence in , where a is some positive real number. Then the sequence converges strongly to .
Proof In view of Lemma 1.7 and Lemma 1.8, we find that ℱ is closed and convex, so that is well defined for any . Next, we show that is closed and convex. It is obvious that is closed and convex. Suppose that is closed and convex for some . We now show that is also closed and convex. For , we see that . It follows that , where . Notice that
and
The above inequalities are equivalent to
and
Multiplying t and on both sides of (2.1) and (2.2), respectively, yields that
That is,
where . This gives that is closed and convex. Then is closed and convex. This shows that is well defined.
Next, we show that . is obvious. Suppose that for some . Fix . It follows that
which shows that . This implies that for each . In view of , from Lemma 1.6 we find that for any . It follows from that
It follows from Lemma 1.5 that
This implies that the sequence is bounded. It follows from (1.3) that the sequence is also bounded. Since the space is reflexive, we may assume that . Since is closed and convex, we find that . On the other hand, we see from the weak lower semicontinuity of the norm that
which implies that . Hence, we have . In view of the Kadec-Klee property of E, we find that as .
Now, we are in a position to prove that . Since and , we find that . This shows that is nondecreasing. It follows from the boundedness that exists. In view of the construction of , we arrive at
This implies that
In light of , we find that
Thanks to (2.6), we find that
In view of (1.3), we see that . It follows that . This is equivalent to
This implies that is bounded. Note that both E and are reflexive. We may assume that . In view of the reflexivity of E, we see that . This shows that there exists an element such that . It follows that
Taking on both sides of the equality above yields that
That is, , which in turn implies that . It follows that . Since enjoys the Kadec-Klee property, we obtain from (2.8) that
Since E enjoys the Kadec-Klee property, we obtain that as . Note that . It follows that
This gives that
Notice that
It follows from (2.9) and (2.10) that
Since E is uniformly smooth, we know that is uniformly convex. In view of Lemma 1.9, we see that
This implies that
In view of the restrictions on the sequence , we find from (2.11) that
Notice that
It follows from (2.12) that
The demicontinuity of implies that . Note that
This implies from (2.13) that . Since E has the Kadec-Klee property, we obtain that . Since
we find from the asymptotic regularity of T that , that is, as . It follows from the closedness of T that .
Next, we show that . In view of Lemma 1.8, we find from (2.4) that
It follows from (2.11) that . This implies that . In view of (2.9), we see that as . This implies that as . It follows that . Since is reflexive, we may assume that . In view of , we see that there exists such that . It follows that
Taking on both sides of the equality above yields that
That is, , which in turn implies that . It follows that . Since enjoys the Kadec-Klee property, we obtain that as . Note that is demicontinuous. It follows that . Since E enjoys the Kadec-Klee property, we obtain that as . Note that
This implies that
Since J is uniformly norm-to-norm continuous on any bounded sets, we have
From the assumption , we see that
In view of , we see that
It follows from (A2) that
By taking the limit as in the above inequality, from (A4) and (2.16) we obtain that
For and , define . It follows that , which yields that . It follows from (A1) and (A4) that
That is,
Letting , we obtain from (A3) that , . This implies that . This shows that .
Finally, we prove that . Letting in (2.5), we see that
In view of Lemma 1.6, we find that . This completes the proof. □
If T is asymptotically quasi-ϕ-nonexpansive, then Theorem 2.1 is reduced to the following.
Corollary 2.2 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let f be a bifunction from to ℝ satisfying (A1)-(A4), and let be a closed asymptotically quasi-ϕ-nonexpansive mapping. Assume that T is asymptotically regular on C and that is nonempty and bounded. Let be a sequence generated in the following manner:
where , is a real sequence in such that , and is a real sequence in , where a is some positive real number. Then the sequence converges strongly to .
If T is quasi-ϕ-nonexpansive, then Theorem 2.1 is reduced to the following.
Corollary 2.3 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let f be a bifunction from to ℝ satisfying (A1)-(A4), and let be a closed quasi-ϕ-nonexpansive mapping. Assume that is nonempty. Let be a sequence generated in the following manner:
where is a real sequence in such that , and is a real sequence in , where a is some positive real number. Then the sequence converges strongly to .
If T is the identity, then Theorem 2.1 is reduced to the following.
Corollary 2.4 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let f be a bifunction from to ℝ satisfying (A1)-(A4). Assume that is nonempty. Let be a sequence generated in the following manner:
where , is a real sequence in such that , and is a real sequence in , where a is some positive real number. Then the sequence converges strongly to .
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The study was supported by the Natural Science Foundation of Zhejiang Province (Y6110270).
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Hao, Y. On generalized quasi-ϕ-nonexpansive mappings and their projection algorithms. Fixed Point Theory Appl 2013, 204 (2013). https://doi.org/10.1186/1687-1812-2013-204
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DOI: https://doi.org/10.1186/1687-1812-2013-204