Abstract
In this paper, we investigate a common fixed point problem of a finite family of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense and an equilibrium problem. Strong convergence theorems of common solutions are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property.
MSC:47H09, 47J25, 90C33.
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1 Introduction-preliminaries
Let E be a real Banach space. Recall that 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 if
exists for each . It is said to be uniformly smooth if the above limit is attained uniformly for .
Recall that E has Kadec-Klee property if for any sequence , and with , and , then as . For more details of 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.
Recall that the normalized duality mapping J from E to is defined by
where denotes the generalized duality pairing. 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 if E is uniformly smooth if and only if is uniformly convex.
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 [2] recently introduced a generalized projection operator in a Banach space E which is an analog 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
The existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping J; see, for example, [1, 2]. In Hilbert spaces, . It is obvious from the definition of the 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, 2] and the references therein.
Let C be a nonempty subset of E and 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 , then . In this paper, we use → and ⇀ to denote the strong convergence and weak convergence, respectively.
Recall that 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 .
A mapping T is said to be relatively nonexpansive iff
A mapping T is said to be relatively asymptotically nonexpansive iff
where is a sequence such that as .
Remark 1.2 The class of relatively asymptotically nonexpansive mappings were first considered in [4]; see also, [5] and the references therein.
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 1.3 The class of quasi-ϕ-nonexpansive mappings was considered in [6]. The class of asymptotically quasi-ϕ-nonexpansive mappings which was investigated in [7] and [8] includes the class of quasi-ϕ-nonexpansive mappings as a special case.
Remark 1.4 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. Quasi-ϕ-nonexpansive mappings and asymptotically quasi-ϕ-nonexpansive do not require the restriction .
Remark 1.5 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 asymptotically quasi-ϕ-nonexpansive in the intermediate sense iff and the following inequality holds:
Putting
it follows that as . Then (1.3) is reduced to the following:
Remark 1.6 The class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense was first considered by Qin and Wang [9].
Remark 1.7 The class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense is a generalization of the class of asymptotically quasi-nonexpansive mappings in the intermediate sense, which was considered by Kirk [10], in the framework of Banach spaces.
Let f be a bifunction from to ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem. Find such that
We use to denote the solution set of the equilibrium problem (1.5). That is,
We remark here that the equilibrium problem was first introduced by Fan [11]. Given a mapping , let
Then if and only if p is a solution of the following variational inequality. Find p such that
To study the equilibrium problems (1.5), we may assume that F satisfies the following conditions:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A3) for each ,
(A4) for each , is convex and weakly lower semi-continuous.
Numerous problems in physics, optimization, and economics reduce to find a solution of (1.5). Recently, many authors have investigated common solutions of fixed point and equilibrium problems in Banach spaces; see, for example, [12–33] and the references therein.
In this paper, we consider a projection algorithm for treating the equilibrium problem and fixed point problems of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense.
In order to prove our main results, we need the following lemmas.
Lemma 1.8 [2]
Let E be a reflexive, strictly convex and smooth Banach space. Let C be a nonempty closed convex subset of E and let . Then
Lemma 1.9 [2]
Let C be a nonempty closed convex subset of a smooth Banach space E and let . Then if and only if
Lemma 1.10 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
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 1.11 [35]
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 and let C be a nonempty closed and convex subset of E. Let f be a bifunction from to ℝ satisfying (A1)-(A4) and let N be some positive integer. Let an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense for every . Assume that is closed 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 for every , is a real number sequence in , where k is some positive real number. Assume that and for every . Then the sequence converges strongly to , where is the generalized projection from E onto .
Proof First, we show that is closed and convex. From [9], we find that is closed and convex, which combines with Lemma 1.10 shows that is closed and convex. Next, we show that is closed and convex. It is obvious that is closed and convex. Suppose that is closed and convex for some positive integer h. For , we see that is equivalent to
It is to see that is closed and convex. This proves that is closed and convex. This in turn shows that is well defined. Putting , we from Lemma 1.10 see that is quasi-ϕ-nonexpansive. Now, we are in a position to prove that . Indeed, is obvious. Assume that for some positive integer h. Then, for , we have
which shows that . This implies that .
Next, we prove that the sequence is bounded. Notice that . We find from Lemma 1.9 that , for any . Since , we find that
It follows from Lemma 1.8 that
This implies that the sequence is bounded. It follows from (1.1) that the sequence is also bounded. Since the space 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 that
which implies that . Hence, we have . In view of the Kadec-Klee property of E, we find that as . Since , and , we find that . This shows that is nondecreasing. We find from its boundedness that exists. It follows that
This implies that
In the light of , we find that
It follows from (2.3) that
In view of (1.1), we see that . This implies that . That is,
This implies that is bounded. Note 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
Taking on both sides of the equality aboven yields
That is, , which in turn implies that . It follows that . Since enjoys the Kadec-Klee property, we obtain from (2.5) that . Since is demicontinuous and E enjoys the Kadec-Klee property, we obtain , as . Note that
It follows that
Since J is uniformly norm-to-norm continuous on any bounded sets, we have
On the other hand, we have
We, therefore, find that
Since E is uniformly smooth, we know that is uniformly convex. In view of Lemma 1.11, we find that
It follows that
In view of the restriction on the sequences, we find from (2.8) that . It follows that
In the same way, we obtain
Notice that . It follows that
The demicontinuity of implies that . Note that
This implies from (2.9) that . Since E has the Kadec-Klee property, we obtain . On the other hand, we have
It follows from the uniformly asymptotic regularity of that
That is, . From the closedness of , we find for every . This proves .
Next, we show that . In view of Lemma 1.8, we find that
It follows from (2.8) that . This implies that . It follows from (2.6) that
It follows 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 the both sides of equality above yields that
That is, , which in turn implies that . It follows that . Since enjoys the Kadec-Klee property, we obtain as . Note that is demicontinuous. It follows that . Since E enjoys the Kadec-Klee property, we obtain 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
Since , we find that
It follows from (A2) that
In view of (A4), we find from (2.10) that
For and , define . It follows that , which yields . It follows from (A1) and (A4) that
That is,
Letting , we obtain from (A3) that , . This implies that .
Finally, we turn our attention to proving that .
Letting in (2.2), we obtain
In view of Lemma 1.9, we find that . This completes the proof. □
From the definition of quasi-ϕ-nonexpansive mappings, we see that every quasi-ϕ-nonexpansive mapping is asymptotically quasi-ϕ-nonexpansive in the intermediate sense. We also know that every uniformly smooth and uniformly convex space is a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property (note that every uniformly convex Banach space enjoys the Kadec-Klee property).
Remark 2.2 Theorem 2.1 can be viewed an extension of the corresponding results in Qin et al. [6], Kim [12], Qin et al. [22], Takahashi and Zembayashi [24], respectively. The space , where , satisfies the restriction in Theorem 2.1.
3 Applications
Theorem 3.1 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property and let C be a nonempty closed and convex subset of E. Let f be a bifunction from to ℝ satisfying (A1)-(A4). Let an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. Assume that T is closed 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 , where k is some positive real number, and are two real number sequence in . Assume that . Then the sequence converges strongly to , where is the generalized projection from E onto .
Proof Putting , we draw from Theorem 2.1 the desired conclusion immediately. □
Remark 3.2 If the mapping T in Theorem 3.1 is quasi-ϕ-nonexpansive, then the restrictions that T is closed asymptotically regular on C and is bounded will not be required anymore.
If , where I is the identity for every , then we find from Theorem 2.1 the following.
Theorem 3.3 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property and 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 number sequence in , where k is some positive real number. Then the sequence converges strongly to , where is the generalized projection from E onto .
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Huang, C., Ma, X. Some results on asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense and equilibrium problems. J Inequal Appl 2014, 202 (2014). https://doi.org/10.1186/1029-242X-2014-202
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DOI: https://doi.org/10.1186/1029-242X-2014-202