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Convergence theorems for common solutions of various problems with nonlinear mapping
Journal of Inequalities and Applications volume 2014, Article number: 2 (2014)
Abstract
In this paper, motivated and inspired by Zegeye and Shahzad (Nonlinear Anal. 70:2707-2716, 2009), Qin et al. (J. Comput. Appl. Math. 225(1):20-30, 2009) and Kimura and Takahashi (J. Math. Anal. Appl. 357:356-363, 2009), we introduce a new hybrid projection iterative scheme that converges strongly to a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of common fixed points for a family of hemi-relatively nonexpansive mappings in a Banach space.
MSC:47H10, 47J20, 49J40.
1 Introduction
A real 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 . It is known that a uniformly convex Banach space is reflexive and strictly convex. 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 limit is attained uniformly for .
Let E be a real Banach space with the norm , and let denote the dual space of E. We denote by J the normalized duality mapping from E to defined by
where denotes the generalized duality pairing. It is well known that if is strictly convex, then J is single-valued, and if E is uniformly smooth, then J is uniformly norm-to-norm continuous on a bounded subset of E. Moreover, if E is a reflexive and strictly convex Banach space with a strictly convex dual, then is single-valued, one-to-one, surjective and it is the duality mapping from into E and thus and (see [1]). We note that J is the identity mapping in a Hilbert space.
A mapping is said to be monotone if for each ,
A mapping A is said to be γ-inverse strongly monotone if there exists a positive real number such that
If a mapping A is γ-inverse strongly monotone, then it is Lipschitz continuous with constant , i.e.,
A mapping A is said to be strongly monotone, if for each , there exists such that
A monotone mapping A is said to be maximal if its graph is not properly contained in the graph of any other monotone mapping. It is known that the monotone mapping A is maximal if and only if for ,
for every implies that .
Let C be a nonempty, closed convex subset of a Banach space E. For a bifunction , we assume that θ satisfies the following conditions:
(E1) for all ;
(E2) θ is monotone, i.e., for all ;
(E3) for each ,
(E4) for each , the function is convex and lower semi-continuous.
The generalized mixed equilibrium problem is to find such that
where ψ is a lower semicontinuous and convex function. The set of solutions of problem (1.1) is denoted by GMEP. Recently, Zhang [2] considered this problem. Some special cases of problem (1.1) are stated as follows.
If , then problem (1.1) reduces to the following mixed equilibrium problem of finding such that
which was considered by Ceng and Yao [3].
If , then problem (1.1) reduces to the following generalized equilibrium problem of finding such that
which was studied in [4].
If and , then problem (1.1) reduces to the following equilibrium problem of finding such that
The set of solutions of problem (1.2) is denoted by EP.
If and , then problem (1.1) reduces to the following classical variational inequality problem of finding such that
The set of solutions of problem (1.3) is denoted by .
Equilibrium problems, which were introduced in [5] in 1994, have had great impact and influence on the development of several branches of pure and applied sciences. They include numerous problems in economics, finance, physics, network, elasticity, optimization, variational inequalities, minimax problems, and semigroups; see, for instance, [2–4, 6–17] and the references therein.
As well known, if C is a nonempty, closed and 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 true to more general Banach spaces. In this connection, Alber [18] introduced a generalized projection operator in a Banach space E which is an analogue of the metric projection in Hilbert spaces. Consider the functional defined by
Observe that, in a Hilbert space H, (1.4) reduces to
The generalized projection is a mapping that assigns an arbitrary point to the minimum point of the functional , that is, , where is the solution to the minimization problem
The existence and uniqueness of the mapping follows from the properties of the functional and strict monotonicity of the mapping J (see, for example, [18] and [19]). In a Hilbert space, . It is obvious from the definition of the function ϕ that:
-
(1)
for all .
-
(2)
for all .
-
(3)
for all .
-
(4)
If E is a reflexive, strictly convex and smooth Banach space, then, for all ,
Remark 1.1 In (4), it is sufficient to show that if then . In fact, from (1) we have . This implies that . From the definition of J, we have . Therefore, we have . For more details, see [1].
Let C be a nonempty closed and convex subset of E, and let T be a mapping from C into itself. We denote by be the set of fixed points of T. A point p in C is said to be a weak asymptotic fixed point of T [20] if 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 from C into itself is called relatively nonexpansive [21–23] if and for all and . The asymptotic behavior of relatively nonexpansive mappings was studied in [21, 22] and [12].
A point p in C is said to be a strong asymptotic fixed point of T if C contains a sequence which converges strongly to p such that . The set of strong asymptotic fixed points of T is denoted by . A mapping T from C into itself is called relatively weak nonexpansive if and for all and . A mapping T is called hemi-relatively nonexpansive if and for all and .
Remark 1.2 (1) It is obvious that a relatively nonexpansive mapping is a relatively weak nonexpansive mapping (see [14]). In fact, for any mapping , we have . Therefore, if T is a relatively nonexpansive mapping, then .
-
(2)
The class of hemi-relatively nonexpansive mappings is more general than the class of relatively weak nonexpansive mappings.
The converse of Remark 1.2 is not true. In order to explain this better, we give the following example.
Example 1.1 ([24])
Let E be any smooth Banach space, and let be any element of E. We define a mapping as follows:
for . Then T is a hemi-relatively nonexpansive mapping but not a relatively nonexpansive mapping.
Remark 1.3 There are other examples of hemi-relatively nonexpansive mappings such as the generalized projections (or projections) from a smooth, strictly convex and reflexive Banach space, and others; see [25].
A mapping is said to be closed, if for any sequence with and , then .
In 2009, Kimura and Takahashi [26] proposed the following hybrid iteration method with a generalized projection for a family of relatively nonexpansive mappings in a Banach space E:
They proved that converges strongly to , where is the set of common fixed points of , and is the generalized projection of E onto a nonempty closed convex subset K of E.
Recently, Zegeye and Shahzad [14] introduced the following iterative scheme for finding a common element of the solution set of a variational inequality problem and a fixed point of a relatively weak nonexpansive mapping with γ-inverse strongly monotone mapping satisfying for all and (see, e.g., [27]):
On the other hand, Qin et al. [25] proposed the following hybrid iterative scheme:
where is a closed hemi-relatively nonexpansive mapping. Under suitable conditions, they proved that the sequence converges strongly to , where is the solution of an equilibrium problem for a bifunction .
In this paper, we introduce a new hybrid projection iterative scheme that converges strongly to a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of common fixed points for a family of hemi-relatively nonexpansive mappings in a Banach space.
2 Preliminaries
Let E be a normed linear space with . The modulus of smoothness of E is the function defined by
The space E is said to be smooth if , and E is called uniformly smooth if and only if . The modulus of convexity of E is the function defined by
E is called uniformly convex if and only if for every . Let . Then E is said to be p-uniformly convex if there exists a constant such that for all . Observe that every p-uniformly convex space is uniformly convex.
It is well known (see, for example, [28]) that
In the following, we shall need the following results.
Lemma 2.1 ([28])
Let E be a 2-uniformly convex and smooth Banach space. Then, for all , we have
where J is the normalized duality mapping of E, and () is the 2-uniformly convex constant of E.
Let E be a real smooth, strictly convex and reflexive Banach space, and let C be a nonempty closed convex subset of E. Then the following conclusions hold:
-
(i)
, , .
-
(ii)
Suppose and . Then
Lemma 2.3 ([30])
Let E be a strictly convex and smooth Banach space, C be a nonempty closed and convex subset of E and be a hemi-relatively nonexpansive mapping. Then is a closed convex subset of C.
Lemma 2.4 ([29])
Let E be a real smooth and uniformly convex Banach space, and let , be two sequences of E. If and either or is bounded, then .
Lemma 2.5 ([31])
Let E be a real smooth Banach space and be a maximal monotone mapping. Then is a closed and convex subset of E.
We denote the normal cone for C at a point by , that is,
Lemma 2.6 ([32])
Let C be a nonempty closed convex subset of a Banach space E, and let A be a monotone and hemicontinuous mapping of C into with . Let be a mapping defined as follows:
Then T is maximal monotone and .
Remark 2.1 It is well known that the monotone and hemicontinuous mapping A with is maximal (see, e.g., [1]). Note that Lemma 2.6 is for the monotone and hemicontinuous mapping.
Remark 2.2 Let C be a nonempty closed convex subset of a Banach space E, and let A be a monotone and hemicontinuous mapping from C into with . Then
It is obvious that the set is a closed convex subset of C and the set is a closed convex subset of E (see [27]).
We make use of the function defined by
for all and , which was studied by Alber [18]. That is,
for all and . We know the following lemma.
Lemma 2.7 ([18])
Let E be a reflexive, strictly convex and smooth Banach space with as its dual. Then
for all and .
Lemma 2.8 ([2])
Let C be a closed subset of a smooth, strictly convex and reflexive Banach space E. Let be a continuous and monotone mapping, be a lower semicontinuous and convex function, and θ be a bifunction from to ℝ satisfying (E1)-(E4). Then, for and , there exists such that
Define a mapping by
for all . Then the following conclusions hold:
-
(i)
is single-valued;
-
(ii)
is a firmly nonexpansive type mapping [33], i.e., for all ,
-
(iii)
;
-
(iv)
GMEP is a closed and convex subset of C;
-
(v)
, , .
Remark 2.3 ([2])
The mapping defined by (2.1) is a relatively nonexpansive mapping. Thus, it is a hemi-relatively nonexpansive mapping.
3 An iterative scheme for a family of hemi-relatively nonexpansive mappings
In this section, we introduce a new hybrid iterative scheme for a common element of the solution set of problem (1.1), the solution set of problem (1.3) for an inverse strongly monotone mapping and the set of common fixed points of a family of hemi-relatively nonexpansive mappings.
Theorem 3.1 Let E be a real uniformly smooth and 2-uniformly convex Banach space and C be a nonempty, closed and convex subset of E. Let be a γ-inverse strongly monotone mapping and be a continuous and monotone mapping. Let be a lower semicontinuous and convex function and θ be a bifunction from to ℝ satisfying (E1)-(E4). Let be a family of closed hemi-relatively nonexpansive mappings of C into itself having
where is the set of common fixed points of . Assume that for all and . Suppose that , where c is the constant in Lemma 2.1. Let for some . Let be the sequence generated by
where J is the normalized duality mapping, and is a sequence in satisfying . Then converges to , where is the generalized projection of E onto Ω.
Proof Step 1. We prove that and both are closed and convex subsets of C and with , . In fact, it is obvious that is closed and convex, and is closed for each , . Since
and
is convex for each , . Hence is closed and convex for all , .
Step 2. For any given , from (v) of Lemma 2.8 and that is hemi-relatively nonexpansive, we have
for each . From Lemma 2.1, Lemma 2.7, and the assumption of A, we obtain
From (3.2) and (3.3),
for each . Thus
Therefore, and . Suppose that . Then, the methods in (3.2) and (3.3) imply that
which implies that . Since , it follows from Lemma 2.2 that
It implies that . Hence . Therefore, . Then, by induction on n, for all , . From Lemma 2.5 and Lemma 2.6, we know that is closed and convex set. Therefore, Ω is closed and convex. The sequence generated by (3.1) is well defined.
Step 3. We prove that is a Cauchy sequence. Let . From the definition of , and Lemma 2.2, we have and
Thus is bounded. Moreover, since
we have
which implies that is nondecreasing. It follows that the limit of exists. By the construction , one has that
for any positive integer . From Lemma 2.2, we have
Letting in (3.5), we have
Thus, Lemma 2.4 implies that
This implies that is a Cauchy sequence.
Step 4. Now, we prove that for each and . Since , we obtain
for all and
By (3.5) and Lemma 2.4, we have
Hence,
for all . By the methods in (3.2) and (3.4), we have
for all , . Since J is uniformly continuous on the bounded sets, it follows from Lemma 2.8(v), (3.2) and (3.7) that for any given ,
for all . From Lemma 2.4,
Thus for each . Since J is uniformly continuous on bounded sets, we obtain
as . Since and is uniformly continuous on bounded sets, we obtain
for all . It follows from (3.6) and (3.9) that
for all . Since is a Cauchy sequence, there exists a point such that . It follows from (3.6) that . Since is closed, from (3.10) we get .
Step 5. Now, we show that . Let be a mapping as follows:
By Lemma 2.6, S is maximal monotone, and . Let (graph of S). Since , we have . Moreover, implies that
On the other hand, from and Lemma 2.2, we obtain that
Hence,
From (3.11) and (3.12), we obtain
Since J is uniformly continuous on bounded sets, by (3.6) we have
Thus, since S is maximal monotone, we have and . Next, we show that . Let
From (3.6) and (3.8), we obtain and for all . Since J is uniformly continuous, from (3.8) we have . Therefore, it follows from for some that . Since , we have
Combining the above inequality and (E2), we get
Taking the limit as in the above inequality and by (E4), we have for all , . For any and , define
Then . From (E1) and (E4), we have
i.e., , for all . Thus, from (E3) and let , we have for all , . This implies that . Therefore .
Step 6. Finally, we prove that . Since and by Lemma 2.2, we have
Taking the limit in (3.13) and from for all , , we obtain
Therefore, from Lemma 2.2, we have . □
Remark 3.1 An iterative scheme for finding a solution of the variational inequality problem for a mapping A that satisfies the following conditions in a 2-uniformly and uniformly smooth Banach space E:
-
(1)
A is inverse strongly monotone,
-
(2)
,
-
(3)
for all and .
If condition (3) holds, then we can prove a convergence theorem for variational inequality problems. To consider the general variational inequality problem for inverse strongly monotone mappings, we have to assume condition (3) (see [27]).
For a practical case, we may apply this theorem to a finite number of mappings as follows.
Corollary 3.1 Let E be a real uniformly smooth and 2-uniformly convex Banach space and C be a nonempty, closed and convex subset of E. Let be a γ-inverse strongly monotone mapping and be a continuous and monotone mapping. Let be a lower semicontinuous and convex function and θ be a bifunction from to ℝ satisfying (E1)-(E4). Let be a finite family of closed hemi-relatively nonexpansive mappings of C into itself having
where is the set of common fixed points. Assume that for all and . Suppose that , where c is the constant in Lemma 2.1. Let for some . Let be the sequence generated by
where J is the normalized duality mapping, and is a sequence in satisfying . Then converges to , where is the generalized projection of E onto Ω.
If is the Hilbert space, then is the identity mapping on H. Then Theorem 3.1 reduces to the following corollary.
Corollary 3.2 Let H be a real Hilbert space and C be a nonempty, closed and convex subset of H. Let be a continuous and monotone mapping. Let be a lower semicontinuous and convex function and θ be a bifunction from to ℝ satisfying (E1)-(E4). Let be a family of closed hemi-relatively nonexpansive mappings with
where is the set of common fixed points of . Assume that for all and . Suppose that , where c is the constant in Lemma 2.1. Let for some . Let be the sequence generated by
where is a sequence in satisfying . Then converges to , where is the metric projection of H onto Ω.
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Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions which improved the presentation of this paper. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2012R1A1A4A01010526).
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The main idea of this paper was proposed by KSK. KSK prepared the manuscript initially, JKK and WHL confirmed all the steps of proofs in this research. All authors read and approved the final manuscript.
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Kim, K.S., Kim, J.K. & Lim, W.H. Convergence theorems for common solutions of various problems with nonlinear mapping. J Inequal Appl 2014, 2 (2014). https://doi.org/10.1186/1029-242X-2014-2
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DOI: https://doi.org/10.1186/1029-242X-2014-2