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A monotone projection algorithm for fixed points of nonlinear operators
Fixed Point Theory and Applications volume 2013, Article number: 318 (2013)
Abstract
In this paper, a monotone projection algorithm is investigated for equilibrium and fixed point problems. Strong convergence theorems for common solutions of the two problems are established in the framework of reflexive Banach spaces.
MSC:47H09, 47J25, 90C33.
1 Introduction and preliminaries
Let E be a real Banach space with the dual . Recall that the normalized duality mapping J from E to is defined by
where denotes the generalized duality pairing. Let be the unit ball of E. Recall that 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 . 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 . It is well known that E is uniformly smooth if and only if is uniformly convex. In what follows, we use ⇀ and → to stand for weak and strong convergence, respectively. Recall that E enjoys the Kadec-Klee property iff for any sequence , and with , and , then as . It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property. Let E be a smooth Banach space. Let us consider the functional defined by
Recently, Alber [1] 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 follows from the properties of the functional and strict monotonicity of the mapping J. If E is a reflexive, strictly convex and smooth Banach space, then if and only if . In Hilbert spaces, . It is obvious from the definition of function ϕ that , .
Let ℝ be the set of real numbers. Let F be a bifunction from to ℝ, where ℝ denotes the set of real numbers. Let be a real-valued function and be a mapping. The so-called generalized mixed equilibrium problem is to find such that
We use to denote the solution set of the equilibrium problem. That is,
Next, we give some special cases:
If , then problem (1.1) is equivalent to finding such that
which is called the mixed equilibrium problem.
If , then problem (1.1) is equivalent to finding such that
which is called the mixed variational inequality of Browder type.
If , then problem (1.1) is equivalent to finding such that
which is called the generalized equilibrium problem.
If and , then problem (1.1) is equivalent to finding such that
which is called the equilibrium problem.
For solving the above problem, let us assume that the bifunction satisfies the following conditions:
-
(A1)
, ;
-
(A2)
F is monotone, i.e., , ;
-
(A3)
-
(A4)
for each , is convex and weakly lower semi-continuous.
Iterative algorithms have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity and optimization; see [2–27] and the references therein. The computation of solutions of nonlinear operator equations (inequalities) is important in the study of many real world problems. Recently, the study of the convergence of various iterative algorithms for solving various nonlinear mathematical models forms the major part of numerical mathematics.
Let C be a nonempty subset of E, and let be a mapping. In this paper, we use to stand for the fixed point set of T. Recall that T is said to be asymptotically regular on C iff for any bounded subset K of C, . Recall that T is said to be closed iff for any sequence such that and , then . 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 such that . The set of asymptotic fixed points of T will be denoted by . T is said to be relatively nonexpansive iff and
T is said to be relatively asymptotically nonexpansive iff and
where is a sequence such that as .
Recall that T is said to be quasi-ϕ-nonexpansive iff and
Recall that T is said to be asymptotically quasi-ϕ-nonexpansive iff there exists a sequence with as such that
Remark 1.1 The class of relatively asymptotically nonexpansive mappings, which is an extension of the class of relatively nonexpansive mappings, was first introduced in [28].
Remark 1.2 The class of asymptotically quasi-ϕ-nonexpansive mappings, which is an extension of the class of quasi-ϕ-nonexpansive mappings, was considered in [29–31]. 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 mappings do not require the restriction .
Recall that T is said to be generalized asymptotically quasi-ϕ-nonexpansive iff , and there exist two nonnegative sequences with and with as such that
Remark 1.3 The class of generalized asymptotically quasi-ϕ-nonexpansive mappings [32] is a generalization of the class of generalized asymptotically quasi-nonexpansive mappings in the framework of Banach spaces which was studied by Agarwal et al. [33].
In this paper, we consider a projection algorithm for a common solution of a family of generalized asymptotically quasi-ϕ-nonexpansive mappings and generalized mixed equilibrium problems. A strong convergence theorem is established in a Banach space. In order to prove our main results, we need the following lemmas.
Lemma 1.4 [21]
Let E be a uniformly convex Banach space, and let . Then there exists a strictly increasing, continuous and convex function such that and
for all and such that .
Lemma 1.5 [1]
Let E be a reflexive, strictly convex and smooth Banach space, let C be a nonempty closed convex subset of E and . Then
Lemma 1.6 [1]
Let C be a nonempty closed convex subset of a smooth Banach space E and . Then if and only if
Lemma 1.7 [32]
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 be a generalized asymptotically quasi-ϕ-nonexpansive mapping. Then is closed and convex.
Lemma 1.8 [34]
Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E. Let be a continuous and monotone mapping, let be convex and lower semi-continuous, and 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 firmly nonexpansive-type mapping, i.e., for all , ;
-
(2)
is closed and convex;
-
(3)
is quasi-ϕ-nonexpansive;
-
(4)
, .
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 Δ be an index set and N be an integer. Let be a continuous and monotone mapping and be a lower semi-continuous and convex function. Let be a bifunction from to ℝ satisfying (A1)-(A4) for every . Let be an identity mapping, and let be a generalized asymptotically quasi-ϕ-nonexpansive mapping 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 r is some positive real number, and . Assume that and for every . Then the sequence converges strongly to , where is the generalized projection from E onto Ψ.
Proof The proof is split into five steps.
Step 1. Show that the common solution set Ψ is convex and closed.
This step is clear in view of Lemma 1.7 and Lemma 1.8.
Step 2. Show that the set is convex and closed.
To show Step 2, it suffices to show, for any fixed but arbitrary , that is convex and closed. This can be proved by induction. It is clear that is convex and closed. Assume that is closed and convex for some . We next prove that is convex and closed. It is clear that is closed. We only prove they are convex. Indeed, , we find that , and
and
Notice that the above two inequalities are equivalent to the following inequalities, respectively:
and
These imply that
Since is convex, we see that . Notice that the above inequality is equivalent to
This proves that is convex. This proves that is closed and convex. This completes Step 2.
Step 3. Show that .
It suffices to claim that for every . Note that . Suppose that for some m and for every . Then, for , we have
which proves that . This completes Step 3.
Step 4. Show that , where .
In view of Lemma 1.5, we find that for . This shows that the sequence is bounded. It follows that is also bounded. Since the framework of the space is reflexive, we may, without loss of generality, assume that , where . Note that . It follows that
This gives that . Hence, we have . Since the space E enjoys the Kadec-Klee property, we find that as .
Now, we are in a position to show that . By the construction of , we have that and . It follows that
Letting , we obtain that . In view of , we see that
We, therefore, obtain that . It follows that . It follows that . This implies that is bounded. Note that E is reflexive and is also reflexive. We may assume that . In view of the reflexivity of E, we see that . This shows that there exists such that . It follows that . Taking on the 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 . Since is demicontinuous, it follows that . Since E enjoys the Kadec-Klee property, we obtain that as . Note that . This gives that
Since J is uniformly norm-to-norm continuous on any bounded sets, we have
Notice that
It follows from (2.2) and (2.3) that
From (2.1), we find that , where . In view of , we find from Lemma 1.8 that
From (2.4), we obtain that
This implies that as . Since as , we arrive at . It follows that . Since is also reflexive, we may assume that . In view of , we see that there exists such that . It follows that
Taking on the 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 . Since , we find that . Since J is uniformly norm-to-norm continuous on any bounded sets, we have . From the assumption , we see that . Notice that
where . From (A2), we find that
Taking the limit as , we find that , . For and , define . It follows that , which yields that . It follows from conditions (A1) and (A4) that . This yields that . Letting , we find from condition (A3) that , . This implies that for every .
Next, we state . Since E is uniformly smooth, we know that is uniformly convex. It follows from Lemma 1.4 that
This yields that
In view of , we see from (2.4) that It follows from the property of g that
Since as and is demicontinuous, we obtain that . Note that . This implies that as . Since enjoys the Kadec-Klee property, we see that
On the other hand, we have . Combining (2.5) with (2.6), one obtains that . Since is demicontinuous, one sees that . Notice that . This yields that . Since the space E enjoys the Kadec-Klee property, we obtain that . Note that . Since T is asymptotically regular, we find that . That is, as . It follows from the closedness of that for every . This completes Step 4.
Step 5. Show that .
Since , we see that
Since , we find that
Letting , we arrive at
From Lemma 1.6, we can immediately obtain that . This completes the proof. □
Remark 2.2 Theorem 2.1 mainly improves the corresponding results in Kim [20], Yang et al. [21], Hao [23], Qin et al. [31], Qin et al. [35].
Remark 2.3 The framework of the space in Theorem 2.1 can be applicable to , .
If and , 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, and let C be a nonempty closed and convex subset of E. Let F be a bifunction from to ℝ satisfying (A1)-(A4). Let be a generalized asymptotically quasi-ϕ-nonexpansive mapping for every . Assume that each is closed asymptotically regular on C and is nonempty and bounded. Let be a sequence generated in the following manner:
where , , and are real number sequences in , is a real number sequence in , where r is some positive real number, and . Assume that and . Then the sequence converges strongly to , where is the generalized projection from E onto .
Remark 2.5 Corollary 2.4 mainly improves the corresponding results in Qin et al. [31]. To be more clear, the mapping is extended from quasi-ϕ-nonexpansive mappings to generalized asymptotically quasi-ϕ-nonexpansive mappings and the framework of spaces is extended from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space.
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Wu, C., Sun, L. A monotone projection algorithm for fixed points of nonlinear operators. Fixed Point Theory Appl 2013, 318 (2013). https://doi.org/10.1186/1687-1812-2013-318
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DOI: https://doi.org/10.1186/1687-1812-2013-318