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Iterative schemes for approximating solution of nonlinear operators in Banach spaces
Fixed Point Theory and Applications volume 2013, Article number: 199 (2013)
Abstract
The purpose of this paper is to present a new modified Halpern-Mann type iterative scheme by using the generalized f-projection operator for finding a common element in the set of zeroes of a system of maximal monotone operators, the set of fixed points of a totally quasi-ϕ-asymptotically nonexpansive mapping and the set of solutions of a system of generalized Ky Fan’s inequalities in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Furthermore, we show that our proposed iterative scheme converges strongly to a common element of the sets mentioned above.
MSC:47H05, 47H09, 47H10.
1 Introduction
In 1972, Ky Fan’s inequalities were first introduced by Fan [1]. The study concerning Ky Fan’s inequalities, fixed points of nonlinear mappings and their approximation algorithms constitutes a topic of intensive research efforts. Many well-known problems arising in various branches of science can be studied by using algorithms which are iterative in their nature. 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.
Many authors have considered a family of nonexpansive mappings to show the existence of fixed points and related topics. Especially, the well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings and the problem of finding an optimal point that minimizes a given cost function over the set of common fixed points of a family of nonexpansive mappings.
Solving the convex feasibility problem for a system of generalized Ky Fan’s inequalities is very general in the sense that it includes, as special cases, optimization problems, equilibrium problems, variational inequality problems, minimax problems. Moreover, the generalized Ky Fan’s inequality was shown in [2] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, vector equilibrium problems, Nash equilibria in noncooperative games. In other words, the generalized Ky Fan’s inequality and equilibrium problem are a unified model for several problems arising in physics, engineering, science, optimization, economics and related topics.
One of the most interesting and important problems in the theory of maximal monotone operators is to find a zero point of maximal monotone operators. This problem contains the convex minimization problem and the variational inequality problem. A popular method for approximating this problem is called the proximal point algorithm introduced by Martinet [3] in a Hilbert space. In 1976, Rockafellar [4] extended the knowledge of Martinet [3] and proved weak convergence of the proximal point algorithm. The proximal point algorithm of Rockafellar [4] is a successful algorithm for finding a zero point of maximal monotone operators. Thereafter, many papers have shown convergence theorems of the proximal point algorithm in various spaces (see [5–14]).
A point is a fixed point of S provided . We denote by the fixed point set of S, that is, . A point p in C is called an asymptotic fixed point of S [15] if C contains a sequence which converges weakly to p such that . The set of asymptotic fixed points of S is denoted by . Recently, Halpern and Mann iterative algorithms have been considered for approximations of common fixed points by many authors. For example, in 2011, Saewan and Kumam [16] introduced a modified Mann iterative scheme by using the generalized f-projection method for approximating a common fixed point of a countable family of relatively quasi-nonexpansive mappings. Chang et al. [17] considered a modified Halpern iterative scheme for approximating a common fixed point for a totally quasi-ϕ-asymptotically nonexpansive mapping. Recently, Li et al. [18] introduced a hybrid iterative scheme for approximation of a fixed point of relatively nonexpansive mappings by using the properties of generalized f-projection operators in a uniformly smooth real Banach space, which is also uniformly convex, and proved some strong convergence theorems for the hybrid iterative scheme.
On the other hand, Ofoedu and Shehu [19] extended the algorithm of Li et al. [18] to prove strong convergence theorems for a common solution of the set of solutions of a system of Ky Fan’s inequalities and the set of common fixed points of a pair of relatively quasi-nonexpansive mappings in a Banach space by using the generalized f-projection operator. Chang et al. [20] extended and improved the results of Qin and Su [21] to obtain strong convergence theorems for finding a common element of the set of solutions for a generalized Ky Fan’s inequality, the set of solutions for a variational inequality problem and the set of common fixed points for a pair of relatively nonexpansive mappings in a Banach space.
Motivated and inspired by the work mentioned above, in this paper, we introduce a new hybrid iterative scheme of the generalized f-projection operator based on the Halpern-Mann type iterative scheme for finding a common element of the set of zeroes of a system of maximal monotone operators, the set of fixed points of a totally quasi-ϕ-asymptotically nonexpansive mapping and the set of solutions of a system of generalized Ky Fan’s inequalities in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.
2 Preliminaries
A Banach space E with the norm is called strictly convex if for all with , where is the unit sphere of E. A Banach space E is called smooth if the limit
exists for each . It is also called uniformly smooth if the limit exists uniformly for all . In this paper, we denote the strong convergence and weak convergence of a sequence by and , respectively.
Let E be a real Banach space with the dual space and let C be a nonempty closed and convex subset of E. A mapping is said to be:
-
(1)
nonexpansive if
for all ;
-
(2)
quasi-nonexpansive if and
for all and ;
-
(3)
asymptotically nonexpansive if there exists a sequence with as such that
for all ;
-
(4)
asymptotically quasi-nonexpansive if and there exists a sequence with as such that
for all and ;
-
(5)
totally asymptotically nonexpansive if there exist nonnegative real sequences , with , as and a strictly increasing continuous function with such that
for all and .
A mapping is said to be uniformly L-Lipschitz continuous if there exists a constant such that
for all . A mapping is said to be closed if, for any sequence such that and , we have .
The normalized duality mapping is defined by
for all . If E is a Hilbert space, then , where I is the identity mapping. Consider the functional defined by
where J is the normalized duality mapping and denotes the duality pairing of E and . If E is a Hilbert space, then . It is obvious from the definition of ϕ that
for all .
A mapping is said to be:
-
(1)
relatively nonexpansive [22, 23] if and
for all and ;
-
(2)
relatively asymptotically nonexpansive [24] if and there exists a sequence with as such that
for all , and ;
-
(3)
for all ;
-
(4)
quasi-ϕ-nonexpansive [25, 26] if and
for all and ;
-
(5)
asymptotically ϕ-nonexpansive [26] if there exists a sequence with as such that
for all and ;
-
(6)
quasi-ϕ-asymptotically nonexpansive [26] if and there exists a sequence with as such that
for all , and ;
-
(7)
totally quasi-ϕ-asymptotically nonexpansive if and there exist nonnegative real sequences , with , as and a strictly increasing continuous function with such that
for all , and .
Lemma 1 [27]
Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let be a closed and totally quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences and with and as , respectively, and a strictly increasing continuous function with . If , then the set of fixed points of S is a closed convex subset of C.
Alber [28] introduced that the generalized projection is a mapping that assigns to an arbitrary point the minimum point of the functional , that is, , where is the solution of the minimization problem
The existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the mapping J (see, for example, [28–32]).
If E is a Hilbert space, then and becomes the metric projection . If C is a nonempty closed and convex subset of a Hilbert space H, then is nonexpansive.
Remark 1 The basic properties of a Banach space E related to the normalized duality mapping J are as follows (see [30]):
-
(1)
If E is an arbitrary Banach space, then J is monotone and bounded;
-
(2)
If E is a strictly convex Banach space, then J is strictly monotone;
-
(3)
If E is a smooth Banach space, then J is single-valued and semicontinuous;
-
(4)
If E is a uniformly smooth Banach space, then J is uniformly norm-to-norm continuous on each bounded subset of E;
-
(5)
If E is a reflexive smooth and strictly convex Banach space, then the normalized duality mapping J is single-valued, one-to-one and onto;
-
(6)
If E is a reflexive strictly convex and smooth Banach space and J is the duality mapping from E into , then is also single-valued, bijective and is also the duality mapping from into E, and thus and ;
-
(7)
If E is a uniformly smooth Banach space, then E is smooth and reflexive;
-
(8)
E is a uniformly smooth Banach space if and only if is uniformly convex;
-
(9)
If E is a reflexive and strictly convex Banach space, then is norm-weak∗-continuous.
Remark 2 If E is a reflexive, strictly convex and smooth Banach space, then if and only if . It is sufficient to show that if , then . From (2) we have
That is, . This implies that . From the definition of J, one has . Therefore, we have (see [30, 32, 33] for more details).
In 2006, Wu and Huang [34] introduced a new generalized f-projection operator in a Banach space. They extended the definition of the generalized projection operators introduced by Abler [35] and proved some properties of the generalized f-projection operator. Consider the functional defined by
for all , where ρ is a positive number and is proper, convex and lower semicontinuous. From the definition of G, Wu and Huang [34] proved the following properties:
-
(1)
is convex and continuous with respect to ϖ when y is fixed;
-
(2)
is convex and lower semicontinuous with respect to y when ϖ is fixed.
Definition 1 Let E be a real Banach space with its dual space and let C be a nonempty closed and convex subset of E. We say that is a generalized f-projection operator if
Recall that a Banach space E has the Kadec-Klee property [30, 32, 36] if for any sequence and with and , we have as . It is well known that if E is a uniformly convex Banach space, then E has the Kadec-Klee property.
Lemma 2 [34]
Let E be a real reflexive Banach space with its dual space and let C be a nonempty closed and convex subset of E. The following statements hold:
-
(1)
is a nonempty, closed and convex subset of C for all ;
-
(2)
If E is smooth, then for all , if and only if
for all ;
-
(3)
If E is strictly convex and is positive homogeneous (i.e., for all such that , where ), then is a single-valued mapping.
Recently, Fan et al. [37] showed that the condition, f is positive homogeneous, which appears in [[37], Lemma 2.1(iii)], can be removed.
Lemma 3 [37]
Let E be a real reflexive Banach space with its dual space and let C be a nonempty closed and convex subset of E. If E is strictly convex, then is single-valued.
Recall that J is a single-valued mapping when E is a smooth Banach space. There exists a unique element such that , where . This substitution in (5) gives the following:
Now, we consider the second generalized f projection operator in a Banach space (see [18]).
Definition 2 Let E be a real smooth Banach space and let C be a nonempty closed and convex subset of E. We say that is a generalized f-projection operator if
Lemma 4 [38]
Let E be a Banach space and let be a lower semicontinuous and convex function. Then there exist and such that
for all .
Lemma 5 [18]
Let E be a reflexive smooth Banach space and let C be a nonempty closed and convex subset of E. The following statements hold:
-
(1)
is a nonempty closed and convex subset of C for all ;
-
(2)
For all , if and only if
for all ;
-
(3)
If E is strictly convex, then is a single-valued mapping.
Lemma 6 [18]
Let E be a real reflexive smooth Banach space and let C be a nonempty closed and convex subset of E. Then, for any and ,
for all .
Lemma 7 [18]
Let E be a Banach space and let be a proper, convex and lower semicontinuous mapping with convex domain . If is a sequence in such that and , then .
Remark 3 Let E be a uniformly convex and uniformly smooth Banach space and for all . Then Lemma 6 reduces to the property of the generalized projection operator considered by Alber [28].
If for all and , then the definition of a totally quasi-ϕ-asymptotically nonexpansive S is equivalent to the following:
For and there exist nonnegative real sequences , with , as , respectively, and a strictly increasing continuous function with such that
for all , and .
Let θ be a bifunction from to R, where R denotes the set of real numbers. The equilibrium problem (for short, (EP)) is to find such that
for all . The set of solutions of (EP) (7) is denoted by .
For solving the equilibrium problem for a bifunction , let us assume that θ satisfies the following conditions:
-
(A1)
for all ;
-
(A2)
θ is monotone, i.e., for all ;
-
(A3)
for all ,
-
(A4)
for all , is convex and lower semicontinuous.
For example, let B be a continuous and monotone operator of C into and define
for all . Then θ satisfies (A1)-(A4).
Lemma 8 [2]
Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E and let θ be a bifunction from to R satisfying the conditions (A1)-(A4). Then, for any and , there exists such that
for all .
Lemma 9 [39]
Let C be a closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space E and let θ be a bifunction from to R satisfying the conditions (A1)-(A4). For all and , define a mapping as follows:
Then the following hold:
-
(1)
is single-valued;
-
(2)
is a firmly nonexpansive-type mapping [40], that is, for all ,
-
(3)
;
-
(4)
is closed and convex.
Lemma 10 [39]
Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E and let θ be a bifunction from to R satisfying the conditions (A1)-(A4). Then, for any , and ,
An operator is said to be monotone if
for all . A point is called a zero point of A if
We denote the set of zeroes of the operator A by , that is,
A monotone is said to be maximal if its graph is not property contained in the graph of any other monotone operator. If A is maximal monotone, then the solution set is closed and convex.
Let E be a smooth strictly convex and reflexive Banach space, let C be a nonempty closed convex subset of E and let be a monotone operator satisfying . Then the resolvent of A is defined by
is a single-valued mapping from E to . On the other hand, for all .
For any , the Yosida approximation of A is defined by for all . We know that for all and . Since relatively quasi-nonexpansive mappings and quasi-ϕ-nonexpansive mappings are the same, we can see that is a quasi-ϕ-nonexpansive mapping (see [[41], Theorem 4.7]).
Lemma 11 [42]
Let E be a smooth strictly convex and reflexive Banach space, let C be a nonempty closed convex subset of E and let be a monotone operator satisfying . For any , let and be the resolvent and the Yosida approximation of A, respectively. Then the following hold:
-
(1)
for all and ;
-
(2)
for all ;
-
(3)
.
Lemma 12 [43]
Let E be a reflexive strictly convex and smooth Banach space. Then an operator is maximal monotone if and only if for all .
3 Main result
Now, we give the main results in this paper.
Theorem 1 Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. For each , let be a bifunction from to R satisfying the conditions (A1)-(A4). Let be a maximal monotone operator satisfying and for all and . Let be a closed and totally quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences , with , as , respectively, and a strictly increasing continuous function with . Let be a convex and lower semicontinuous function with and . Assume that S is uniformly L-Lipschitz continuous and . For any initial point , define and the sequence in C by
for each , where , and are the sequences in such that , and for each , for some . If, for each , , and , then the sequence converges strongly to a point .
Proof We split the proof into five steps.
Step 1. We first show that is closed and convex for all . From the definitions is closed and convex. Suppose that is closed and convex for all . For any , we know that is equivalent to the following:
Therefore, is closed and convex for all .
Step 2. We show that for all . Now, we show by induction that for all . It is obvious that . Suppose that for some . Define , when for all with and define when for all with . Let . Then we have
Since S is a totally quasi-ϕ-asymptotically nonexpansive mapping, from (10) we have
This shows that , which implies that and so for all and the sequence is well defined.
Step 3. We show that , , and as . Since is a convex and lower semi-continuous function, from Lemma 4, we known that there exist and such that
for all . Since , it follows that
For all and , we have
That is, is bounded and so are and . By using the fact that and , it follows from Lemma 6 and (3) that
This implies that is nondecreasing and so exists. Taking , we obtain
Since is bounded, E is reflexive and is closed and convex for all . We can assume that as . From the fact that , we get
for all . Since f is convex and lower semi-continuous, we have
By (15) and (16), we get
That is, , which implies that as . Since E has the Kadec-Klee property, we obtain
We also have
From (17), we get
From (17) and (18), we have . Since J is uniformly norm-to-norm continuous, it follows that
Moreover, since and (9), we have
is equivalent to the following:
Since , (14) and (19), we have
By (3), it follows that
as . Since J is uniformly norm-to-norm continuous, we obtain
as . This implies that is bounded in , Since is reflexive, we assume that as . In view of , there exists such that . It follows that
Taking on both sides of the equality above, since is weak lower semi-continuous, this yields that
From Remark 2, , which implies that . It follows that as . From (24) and the Kadec-Klee property of , we have as . Note that is norm-weak∗-continuous, that is, as . From (23) and the Kadec-Klee property of E, we have
From (11), we have
From (17), (19), (27) and the conditions , , it follows that for any , . Let for all . From Lemma 11(1), it follows that for any ,
Taking on both sides of the inequality above, we have
From (3), it follows that as . Since as , we have
as . Since J is uniformly norm-to-norm continuous on bounded subsets of E, it follows that
as . This implies that is bounded in . Since is reflexive, we can assume that as . In view of , there exists such that , and so
Taking on both sides of the equality above, from the weak lower semi-continuity of the norm , it follows that
From Remark 2, we have , which implies that and so as . From (29) and the Kadec-Klee property of , we have as . Since is norm-weak∗-continuous, that is, , from (28) and the Kadec-Klee property of E, it follows that
From (11), we have
From (17), (19), (27) and the conditions , , it follows that . From Lemma 9, it follows that for any and ,
Taking on both sides of the inequality above, we have
From (3), we have
as . Since , we have
as , and so
as . That is, is bounded in . Since is reflexive, we can assume that as . In view of , there exists such that . It follows that
Taking on both sides of the equality above, since is weak lower semi-continuous, it follows that
From Remark 2, , that is, . It follows that . From (36) and the Kadec-Klee property of , we have as . Since is norm-weak∗-continuous, that is, as . From (35) and the Kadec-Klee property of E, we have
Step 4. We show that . First, we show that . Let for each . Then, for any , it follows that for each ,
By Lemma 11, for each , we have
Since and as , we get as for all . From (3), it follows that
as for all . Since as , we also have
as for all . This implies that for each , is bounded. Since E is reflexive, without loss of generality, we can assume that as . Since is closed and convex for each , it is obvious that . Again, since
taking on both sides of the equality above, we have
That is, and it follows that for all ,
as . Thus, from (42), (44) and the Kadec-Klee property, it follows that
for all . We also have
for all , and so
for all . Since J is uniformly norm-to-norm continuous on bounded subsets of E and for each , we have
Let for each . Then we have
For any and for each , it follows from the monotonicity of that for all ,
for all . Letting in the inequality above, we get for all . Since is maximal monotone for all , we obtain .
Next, we show that . For any and , we observe that
By Lemma 10, for , we have
Since and as , we get as for all . From (3), it follows that
as . Since as , we also have
as . Since is bounded and E is reflexive, without loss of generality, we assume that as . Since is closed and convex for each , it is obvious that . Again, since
taking on both sides of the equality above, we have
That is, and it follows that for all , it follows that
as . Thus, from (52), (54) and the Kadec-Klee property, it follows that
for all . We also have
for all , and so
for all . Since J is uniformly norm-to-norm continuous, we obtain
for all . From for all , we have
as and
for all . Thus, by (A2), we have
for all and as , and so for all . For any t with , define . Then , which implies that for all . Thus, from (A1), it follows that
and so for all . From (A3), we have for all and , that is, for all . This implies that .
Finally, we show that . Since is bounded, the mapping S is also bounded. From as and (9), we have
as . Since is norm-weak∗-continuous,
as .
On the other hand, in view of (59), it follows that
and so . Since E has the Kadee-Klee property, we get
for all . By using the triangle inequality, since S is uniformly L-Lipschitz continuous, we get
Since as , we get as , and so as . In view of the closedness of S, we have , which implies that . Hence .
Step 5. We show that . Since ℱ is a closed and convex set, it follows from Lemma 5 that is single-valued, which is denoted by . By the definitions of and , we also have
for all . By the definitions of G and f, we know that for any , is convex and lower semicontinuous with respect to ξ, and so
From the definition of , since , we conclude that and as . This completes the proof. □
Setting , and in Theorem 1, we have the following result.
Corollary 1 Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. For each , let be a bifunction from to R satisfying the conditions (A1)-(A4). Let be a maximal monotone operator satisfying and for all and . Let be a closed and quasi-ϕ-asymptotically nonexpansive mapping. Let be a convex and lower semicontinuous function with and . Assume that S is uniformly L-Lipschitz continuous and . For an initial point , define and the sequence in C by
for all , where , and are the sequences in with , and, for each , for some . If , and for all , then the sequence converges strongly to a point .
If for all in Theorem 1, then and and so we have the following corollary.
Corollary 2 Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. For each , let be a bifunction from to R satisfying the conditions (A1)-(A4). Let be a maximal monotone operator satisfying and for all and . Let be a closed and totally quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences , with , as , respectively, and a strictly increasing continuous function with . Assume that S is uniformly L-Lipschitz continuous and . For an initial point , define and the sequence in C by
for all , where , and are the sequences in with , and, for each , for some . If , and for each , then the sequence converges strongly to a point .
Setting , and in Theorem 1, we have the following corollary.
Corollary 3 Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. For each , let be a bifunction from to R satisfying the conditions (A1)-(A4). Let be a maximal monotone operator satisfying and for all and . Let be a closed and quasi-ϕ-asymptotically nonexpansive mapping. Assume that S uniformly L-Lipschitz continuous and . For an initial point , define and the sequence in C by
for all , where , and are the sequences in with , and, for each , for some . If , and for each , then the sequence converges strongly to a point .
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Acknowledgements
This work was supported by Thaksin University Research Fund and YJ Cho was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012-0008170).
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Saewan, S., Kanjanasamranwong, P., Kumam, P. et al. Iterative schemes for approximating solution of nonlinear operators in Banach spaces. Fixed Point Theory Appl 2013, 199 (2013). https://doi.org/10.1186/1687-1812-2013-199
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DOI: https://doi.org/10.1186/1687-1812-2013-199