Abstract
Generalized asymptotically quasi-ϕ-nonexpansive mappings and a Ky Fan inequality are investigated. A strong convergence theorem for common solutions to a fixed point problem of generalized asymptotically quasi-ϕ-nonexpansive mappings and a Ky Fan inequality is established in a Banach space.
MSC:47H05, 47J25, 90C33.
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1 Introduction
Iterative algorithms have been studied by many authors. The applications of iterative algorithms are found in a wide range of areas, including economics, image recovery and signal processing. Many well-known problems can be studied by using algorithms which are iterative in their nature; see [1–14] and the references therein. As an example, in computer tomography with limited data, each piece of information implies the existence of a convex set, in which the required solution lies. The problem of finding a point in the intersection of these convex subsets is then of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such a point.
Mann iteration, introduced by Mann [15], is an efficient tool to study fixed point problems of asymptotical nonexpansive mappings. However, Mann iteration is only weak convergence in infinite-dimensional spaces; see [10] and the references therein. The importance of strong convergence is underlined in [16], where a convex function f is minimized via the proximal-point algorithm: it is shown that the rate of convergence of the value sequence is better when converges strongly than when it converges weakly. Such properties have a direct impact when the process is executed directly in the underlying infinite-dimensional space. To obtain strong convergence of Mann iteration, projection methods, which were first introduced by Haugazeau [17], have been considered for modifying Mann iteration to obtain strong convergence. The advantage of projection methods is that strong convergence of iterative sequences can be guaranteed without any compact assumptions.
The organization of this paper is as follows. In Section 2, we provide some necessary concepts and lemmas. In Section 3, fixed point problems of generalized asymptotically quasi-ϕ-nonexpansive mappings and solutions of a Ky Fan inequality are investigated. A strong convergence theorem is established in a Banach space.
2 Preliminaries
Recall that the normalized duality mapping J from E to is defined by
where denotes the generalized duality pairing. Let be the unit sphere of E. Then the Banach space E is said to be smooth iff
exists for each . It is also said to be uniformly smooth iff 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 E is said to be strictly convex iff for all with and . It is said to be uniformly convex iff for any two sequences and in E such that and . Recall that E enjoys the Kadec-Klee property if for any sequence , and with , and , then as . For more details on the Kadec-Klee property, readers can refer to [18] and the references therein. It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property.
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 , . 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 [19] recently introduced a generalized projection operator in a Banach space E which is an analogue of the metric projection in Hilbert spaces. Recall that 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 follow from the properties of the functional and strict monotonicity of the mapping J; see, for example, [18]. In Hilbert spaces, . It is obvious from the definition of a function ϕ that
and
Remark 2.1 If E is a reflexive, strictly convex and smooth Banach space, then if and only if ; for more details, see [18] and the reference therein.
Let C be a nonempty subset of E and 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 , then . In this paper, we use → and ⇀ to denote strong convergence and weak convergence, respectively. Recall that a point p in C is said to be an asymptotic fixed point of T iff C contains a sequence which converges weakly to p so that . The set of asymptotic fixed points of T will be denoted by .
Recall that T is said to be relatively nonexpansive iff
Recall that T is said to be relatively asymptotically nonexpansive iff
where is a sequence such that as .
Recall that a mapping T is said to be quasi-ϕ-nonexpansive iff
Recall that a mapping T is said to be asymptotically quasi-ϕ-nonexpansive iff there exists a sequence with as such that
Remark 2.2 The class of asymptotically quasi-ϕ-nonexpansive mappings was considered in Zhou et al. [20] and Qin et al. [21]; see also [22] and [23].
Remark 2.3 The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings [24]. Quasi-ϕ-nonexpansive mappings and asymptotically quasi-ϕ-nonexpansive mappings do not require the restriction .
Remark 2.4 The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are generalizations of the class of quasi-nonexpansive mappings and the class of asymptotically quasi-nonexpansive mappings in Banach spaces.
Recall that T is said to be generalized asymptotically quasi-ϕ-nonexpansive if , and there exists a sequence with as and a sequence with as such that for all , and .
Remark 2.5 The class of generalized asymptotically quasi-ϕ-nonexpansive mappings was considered in Qin et al. [25]; see also [26].
Let f be a bifunction from to ℝ, where ℝ denotes the set of real numbers, and let be a mapping. Consider the following Ky Fan inequality which is known as a generalized equilibrium problem. Find such that
We use to denote the solution set of inequality (2.3). That is,
If , then problem (2.3) is reduced to the following Ky Fan inequality which is known as an equilibrium problem. Find such that
We use to denote the solution set of inequality (2.4). That is,
If , then problem (2.3) is reduced to the classical variational inequality. Find such that
We use to denote the solution set of inequality (2.5). That is,
Recall that a mapping is said to be α-inverse-strongly monotone if there exists such that
For solving problem (2.3), let us assume that the nonlinear mapping is α-inverse-strongly monotone and the bifunction satisfies the following conditions:
-
(A1)
, ;
-
(A2)
F is monotone, i.e., , ;
-
(A3)
-
(A4)
for each , is convex and weakly lower semicontinuous.
Recently, many authors investigated the solutions of problems (2.3), (2.4) and (2.5) based on iterative methods; see [27–37]. In this paper, we investigate generalized asymptotically quasi-ϕ-nonexpansive mappings and problem (2.3). A strong convergence theorem for common solutions to a fixed point problem of generalized asymptotically quasi-ϕ-nonexpansive mappings and problem (2.3) is established in a Banach space.
In order to state our main results, we need the following lemmas, which play an import role in the paper.
Lemma 2.6 [28]
Let E be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of E. Let be an α-inverse-strongly monotone mapping and f be a bifunction satisfying conditions (A1)-(A4). Let be any given number and be any given point. Then there exists such that
Lemma 2.7 [28]
Let E be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of E. Let be an α-inverse-strongly monotone mapping and f be a bifunction satisfying conditions (A1)-(A4). Let be any given number and define a mapping as follows: for any ,
Then the following conclusions hold:
-
(1)
is single-valued;
-
(2)
is a firmly nonexpansive-type mapping, i.e., for all ,
-
(3)
;
-
(4)
is quasi-ϕ-nonexpansive;
-
(5)
-
(6)
is closed and convex.
Lemma 2.8 [19]
Let C be a nonempty closed convex subset of a smooth Banach space E and . Then if and only if
Lemma 2.9 [19]
Let E be a reflexive, strictly convex and smooth Banach space, C be a nonempty closed convex subset of E and . Then
Lemma 2.10 [25]
Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property and C be a nonempty closed convex subset of E. Let be a closed generalized asymptotically quasi-ϕ-nonexpansive mapping. Then is closed and convex.
Lemma 2.11 [38]
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 .
3 Main results
Theorem 3.1 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property and C be a nonempty closed convex subset of E. Let be a generalized asymptotically quasi-ϕ-nonexpansive mapping. Let f be a bifunction from to ℝ satisfying (A1)-(A4) and be an α-inverse-strongly monotone mapping. Assume that T is closed and asymptotically regular on C, and is nonempty and bounded. Let be a sequence generated in the following manner:
where , is a real number sequence in such that and is a real number sequence such that . Then the sequence converges strongly to , where is the generalized projection from E onto .
Proof First, we prove is closed and convex so that the projection is well defined. We see that is closed and convex. Assume that is closed and convex for some positive integer m. For , we find that
which is equivalent to
It is easy to see that is closed and convex. This proves that is closed and convex so that is well defined. Set . It follows from Lemma 2.7 that is quasi-ϕ-nonexpansive.
Now, we are in a position to prove that . Indeed, is obvious. Assume that for some positive integer m. Then, for , we have
which proves that . This implies that . Notice that . We find from Lemma 2.8 that for any . Since , we therefore find that
It follows from Lemma 2.9 that
This implies that the sequence is bounded. This in turn implies that the sequence is bounded. Since E is a uniform space, we find that E is reflexive. We may assume, without loss of generality, that . Next, we prove that . Since is closed and convex, we find that . This implies from that . On the other hand, we see from the weakly lower semicontinuity of the norm that
which implies that . Hence, we have . Since E enjoys the Kadec-Klee property, we find that as . In the light of and , we find that . This shows that is nondecreasing. We obtain that exists. It follows that
This implies that . In view of , we find that
It follows that
In view of (2.2), we see that . This implies that . That is,
This implies that is bounded. Since both E and are uniform, we find that both E and are reflexive. We may assume, without loss of generality, that . In view of the reflexivity of E, we see that . This shows that there exists an element such that . It follows that
It follows that
That is, , which in turn implies that . It follows that . Since E is uniformly smooth, we know that is uniformly convex. Therefore, enjoys the Kadec-Klee property, we obtain that . Since is demicontinuous and E enjoys the Kadec-Klee property, we obtain that as . Note that
It follows that
Since E is uniformly smooth, we know that is uniformly convex. In the light of Lemma 2.11, we find that
It follows that
Notice that
We find from (3.4) that
In view of the restriction on the sequences, we find from (3.5) that . Notice that
It follows that
The demicontinuity of implies that . Note that
This implies that . Since E has the Kadec-Klee property, we obtain that . Notice that
It follows from the uniformly asymptotic regularity of T that
That is, . From the closedness of T, we find . This proves . Next, we show that . It follows from Lemma 2.9 and (3.1) that
This yields that . This implies from (2.2) that . It follows that
We, therefore, find that
This shows that is bounded. 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 . In view of the restriction , we see that
Since , we find that
where
It follows from (A2) that
In view of (A4), we find 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 completes the proof .
Finally, what we need to prove is .
Letting in (3.2), we obtain that
From Lemma 2.8, we immediately find that . This completes the whole proof. □
Remark 3.2 Since the class of generalized asymptotically quasi-ϕ-nonexpansive mappings is a generalization of the class of asymptotically quasi-ϕ-nonexpansive mappings, Theorem 3.1 includes Kim’s [36] results as a special case.
Remark 3.3 Notice that every uniformly smooth and uniformly convex space is a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property, and every uniformly convex Banach space enjoys the Kadec-Klee property. We find that Theorem 3.1 is still valid in the framework of every uniformly smooth and uniformly convex space.
Next, we consider the solution of problem (2.4).
If the mapping T is closed quasi-ϕ-nonexpansive, which is more general than relatively nonexpansive mappings, we have the following.
Corollary 3.4 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property and C be a nonempty closed convex subset of E. Let be a quasi-ϕ-nonexpansive mapping and f be a bifunction from to ℝ satisfying (A1)-(A4). Assume that T is closed and is nonempty. Let be a sequence generated in the following manner:
where is a real number sequence in such that and is a real number sequence such that . Then the sequence converges strongly to , where is the generalized projection from E onto .
In the framework of Hilbert spaces, we find from Theorem 3.1 the following.
Theorem 3.5 Let E be a Hilbert space and C be a nonempty closed convex subset of E. Let be a generalized asymptotically quasi-nonexpansive mapping. Let f be a bifunction from to ℝ satisfying (A1)-(A4), and let be an α-inverse-strongly monotone mapping. Assume that T is closed and asymptotically regular on C, and is nonempty and bounded. Let be a sequence generated in the following manner:
where , is a real number sequence in such that and is a real number sequence such that . Then the sequence converges strongly to , where is the metric projection from E onto .
Proof In the framework of Hilbert spaces, we see that and the mapping J is reduced to the identity mapping. The desired conclusion can be immediately drawn from Theorem 3.1. □
For problem (2.4), we have the following result.
Corollary 3.6 Let E be Hilbert space and C be a nonempty closed convex subset of E. Let be a generalized asymptotically quasi-nonexpansive mapping. Let f be a bifunction from to ℝ satisfying (A1)-(A4). Assume that T is closed and asymptotically regular on C, and is nonempty and bounded. Let be a sequence generated in the following manner:
where , is a real number sequence in such that and is a real number sequence such that . Then the sequence converges strongly to , where is the metric projection from E onto .
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JS design the algorithm and give the main convergence analysis. MC participated in the design of the study. Both authors read and approved the final manuscript.
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Song, J., Chen, M. On generalized asymptotically quasi-ϕ-nonexpansive mappings and a Ky Fan inequality. Fixed Point Theory Appl 2013, 237 (2013). https://doi.org/10.1186/1687-1812-2013-237
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DOI: https://doi.org/10.1186/1687-1812-2013-237