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Strong convergence theorem for total quasi-ϕ-asymptotically nonexpansive mappings in a Banach space
Fixed Point Theory and Applications volume 2013, Article number: 297 (2013)
Abstract
In this paper, we prove strong convergence theorems to a point which is a fixed point of multi-valued mappings, a zero of an α-inverse-strongly monotone operator and a solution of the equilibrium problem. Next, we obtain strong convergence theorems to a solution of the variational inequality problem, a fixed point of multi-valued mappings and a solution of the equilibrium problem. The results presented in this paper are improvement and generalization of the previously known results.
1 Introduction
Let E be a real Banach space with dual , and let C be a nonempty closed convex subset of E. Let be an operator. A is called monotone if
α-inverse-strongly monotone if there exists a constant such that
L-Lipschitz continuous if there exists a constant such that
If A is α-inverse strongly monotone, then it is -Lipschitz continuous, i.e.,
A monotone operator A is said to be maximal if its graph is not properly contained in the graph of any other monotone operator.
Let A be a monotone operator. We consider the problem of finding such that
a point is called a zero point of A. Denote by the set of all points such that . This problem is very important in optimization theory and related fields.
Let A be a monotone operator. The classical variational inequality problem for an operator A is to find such that
The set of solutions of (1.2) is denoted by . This problem is connected with the convex minimization problem, the complementary problem, the problem of finding a point satisfying .
The value of at will be denoted by or . For each , the generalized duality mapping is defined by
for all . In particular, is called the normalized duality mapping. 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. It is obvious from the definition of ϕ that
Alber [1] introduced 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 of the minimization problem
existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the mapping J.
Iiduka and Takahashi [2] introduced the following iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly monotone operator A in a 2-uniformly convex and uniformly smooth Banach space E: and
where is the generalized projection from E onto C, J is the duality mapping from E into and is a sequence of positive real numbers. They proved that the sequence generated by (1.6) converges weakly to some element of . In connection, Iiduka and Takahashi [3] studied the following iterative scheme for finding a zero point of a monotone operator A in a 2-uniformly convex and uniformly smooth Banach space E:
where is the generalized projection from E onto , J is the duality mapping from E into and is a sequence of positive real numbers. They proved that the sequence converges strongly to an element of . Moreover, under the additional suitable assumption they proved that the sequence converges strongly to some element of . Some solution methods have been proposed to solve the variational inequality problem; see, for instance, [4–6].
A mapping is said to be ϕ-nonexpansive [7, 8] if
T is said to be quasi-ϕ-nonexpansive [7, 8] if and
T is said to be total quasi-ϕ-asymptotically nonexpansive, if and there exist nonnegative real sequences , with , as and a strictly increasing continuous function with such that
Let be the family of all nonempty subsets of C, and let be a multi-valued mapping. For a point , define an iterative sequence as follows:
A point is said to be an asymptotic fixed point of S if there exists a sequence in C such that converges weakly to p and
The asymptotic fixed point set of S is denoted by .
A multi-valued mapping S is said to be total quasi-ϕ-asymptotically nonexpansive if and there exist nonnegative real sequences , with , as and a strictly increasing continuous function with such that for all , ,
S is said to be closed if for any sequence and in C with if and , then .
A multi-valued mapping S is said to be uniformly asymptotically regular on C if
Every quasi-ϕ-asymptotically nonexpansive multi-valued mapping implies a quasi-ϕ-asymptotically nonexpansive mapping but the converse is not true.
In 2012, Chang et al. [9] introduced the concept of total quasi-ϕ-asymptotically nonexpansive multi-valued mapping and then proved some strong convergence theorem by using the hybrid shrinking projection method.
Let be a bifunction, the equilibrium problem is to find such that
The set of solutions of (1.8) is denoted by . The equilibrium problem is very general in the sense that it includes, as special cases, optimization problems, variational inequality problems, min-max problems, saddle point problem, fixed point problem, Nash EP. In 2008, Takahashi and Zembayashi [10, 11] introduced iterative sequences for finding a common solution of an equilibrium problem and a fixed point problem. Some solution methods have been proposed to solve the equilibrium problem; see, for instance, [12–21].
For a mapping , let for all . Then if and only if for all ; i.e., x is a solution of the variational inequality.
Motivated and inspired by the work mentioned above, in this paper, we introduce and prove strong convergence of a new hybrid projection algorithm for a fixed point of total quasi-ϕ-asymptotically nonexpansive multi-valued mappings, the solution of the equilibrium problem, a zero point of monotone operators. Moreover, we prove strong convergence to the solution of the variation inequality in a uniformly smooth and 2-uniformly convex Banach space.
2 Preliminaries
A Banach space E with the norm is called strictly convex if for all with and . Let be 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 . The modulus of convexity of E is the function defined by
A Banach space E is uniformly convex if and only if for all . Let p be a fixed real number with . A Banach space E is said to be p-uniformly convex if there exists a constant such that for all . Observe that every p-uniform convex is uniformly convex. Every uniformly convex Banach space E has the Kadec-Klee property, that is, for any sequence , if and , then .
Let E be a real Banach space with dual , E is uniformly smooth if and only if is a uniformly convex Banach space. If E is a uniformly smooth Banach space, then E is a smooth and reflexive Banach space.
Remark 2.1
-
If E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.
-
If E is reflexive smooth and strictly convex, then the normalized duality mapping J is single-valued, one-to-one and onto.
-
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 .
See [22] for more details.
Remark 2.2 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 (1.3) we have . This implies that . From the definition of J, one has . Therefore, we have (see [22, 23] for more details).
Lemma 2.3 (Beauzamy [24] and Xu [25])
If E is a 2-uniformly convex Banach space, then, for all , we have
where J is the normalized duality mapping of E and .
The best constant in the lemma is called the p-uniformly convex constant of E.
Lemma 2.4 (Beauzamy [24] and Zalinescu [26])
If E is a p-uniformly convex Banach space, and let p be a given real number with , then, for all , and ,
where is the generalized duality mapping of E and is the p-uniformly convex constant of E.
Lemma 2.5 (Kamimura and Takahashi [27])
Let E be a uniformly convex and smooth Banach space, and let , be two sequences of E. If and either or is bounded, then .
Lemma 2.6 (Alber [1])
Let C be a nonempty closed convex subset of a smooth Banach space E, and let . Then if and only if
Lemma 2.7 (Alber [1])
Let E be a reflexive strictly convex and smooth Banach space, C be a nonempty closed convex subset of E, and let . Then
Lemma 2.8 (Chang et al. [9])
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 total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequence and with , as and a strictly increasing continuous function with . If , then the fixed point set is a closed convex subset of C.
For solving the equilibrium problem for a bifunction , let us 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 lower semi-continuous.
Lemma 2.9 (Blum and Oettli [28])
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), and let and . Then there exists such that
Lemma 2.10 (Takahashi and Zembayashi [11])
Let C be a closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space E, and let f be a bifunction from to ℝ satisfying 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 [29], that is, for all ,
-
(3)
;
-
(4)
is closed and convex.
Lemma 2.11 (Takahashi and Zembayashi [11])
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), and let . Then, for and ,
Let A be an inverse-strongly monotone mapping of C into which is said to be hemicontinuous if for all , the mapping h of into , defined by , is continuous with respect to the weak∗ topology of . We define by the normal cone for C at a point , that is,
Theorem 2.12 (Rockafellar [30])
Let C be a nonempty, closed convex subset of a Banach space E, and let A be a monotone, hemicontinuous operator of C into . Let be an operator defined as follows:
Then B is maximal monotone and .
Theorem 2.13 (Takahashi [31])
Let C be a nonempty subset of a Banach space E, and let A be a monotone, hemicontinuous operator of C into with . Then
It is obvious that the set is a closed and convex subset of C and the set is a closed and convex subset of E.
Theorem 2.14 (Takahashi [31])
Let C be a nonempty compact convex subset of a Banach space E, and let A be a monotone, hemicontinuous operator of C into with . Then is nonempty.
We make use of the following mapping V studied in Alber [1]:
that is, .
Lemma 2.15 (Alber [1])
Let E be a reflexive strictly convex smooth Banach space, and let V be as in (2.4). Then we have
Lemma 2.16 (Beauzamy [24] and Xu [25])
If E is a 2-uniformly convex Banach space, then, for all , we have
where J is the normalized duality mapping of E and .
Lemma 2.17 (Cho et al. [32])
Let E be a uniformly convex Banach space, and let be a closed ball of E. Then there exists a continuous strictly increasing convex function with such that
for all and with .
Lemma 2.18 (Pascali and Sburlan [33])
Let E be a real smooth Banach space, and let be a maximal monotone mapping. Then is a closed and convex subset of E and the graph of A is demiclosed in the following sense: if with and with , then and .
3 Main results
Theorem 3.1 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from to ℝ satisfying conditions (A1)-(A4), and let A be an α-inverse-strongly monotone mapping of E into . Let be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences , with , as and a strictly increasing continuous function with . Assume that S is uniformly asymptotically regular on C with and . For arbitrary , , generate a sequence by
where . Assume that the control sequences , , , and satisfy the following conditions:
-
1.
, and are sequences in such that , ,
-
2.
for some a, b with and is the 2-uniformly convex constant of E,
-
3.
for some ,
then converges strongly to .
Proof We will show that is closed and convex for all . Since is closed and convex. Suppose that is closed and convex for all . For any , we know that is equivalent to
That is, is closed and convex, hence is closed and convex for all .
We show by induction that for all . It is obvious that . Suppose that where . Let , we have
Since A is an α-inverse-strongly monotone mapping, we get
It follows from Lemma 2.17 that
Replacing (3.2) by (3.3) and (3.4), we get
From Lemma 2.11, we know that
Since S is a total quasi-ϕ-asymptotically nonexpansive multi-valued mapping and , it follows that
where .
This shows that , which implies that . Hence for all and the sequence is well defined.
From the definition of with and , it follows that
that is, is nondecreasing. By Lemma 2.7, we get
This implies that is bounded and so exists. In particular, by (1.4), the sequence is bounded. This implies is also bounded. So, we have , and are also bounded.
Since for all with , by Lemma 2.7, we have
taking , we have . This implies that is a Cauchy sequence. From Lemma 2.5, it follows that and is a Cauchy sequence. By the completeness of E and the closedness of C, we can assume that there exists such that
we also get that
Next, we show that .
(a) We show that . By the definition of , we have
Since exists, we get
It follows from Lemma 2.5 that
From the definition of and , we have as . By Lemma 2.5, it follows that
From , we also have
By using the triangle inequality, we get as . Since J is uniformly norm-to-norm continuous, we obtain as . On the other hand, we note that
In view of and as , we obtain that
From Lemma 2.17, we have
It follows from , (3.16), (3.11) and the property of g that
Since is uniformly norm-to-norm continuous, we obtain
From (3.10) it follows that
For , generate a sequence by
On the other hand, we have . Since S is uniformly asymptotically regular, it follows that
we have
that is, as . From the closedness of S, we have .
(b) We show that .
From the definition of and , we have as . By Lemma 2.5, it follows that . By the triangle inequality, we get as . From and from (3.10), it follows that
Since J is uniformly norm-to-norm continuous, we also have
Hence, from the definition of the sequence , it follows that
From (3.23) and the definition of the sequence , we have
that is,
Since A is Lipschitz continuous, it follows from (3.10) that
Again, since A is Lipschitz continuous and monotone so it is maximal monotone. It follows from Lemma 2.18 that .
(c) We show that .
From and as and applying (3.7) for any , we get , it follows that
Taking limit as on the both sides of the inequality, we have . From Lemma 2.5, it follows that
and
Since J is uniformly norm-to-norm continuous on bounded subsets of E, we obtain
Since for all , we have as and
From (A2), the fact that
taking the limit as in the above inequality and from the fact that as , it follows that for all . For any , define . Then , which implies that . Thus it follows from (A1) that
and so . From (A3) we have for all and so . Hence, by (a), (b) and (c), that is, .
Finally, we show that . From , we have for all . Since , we also have
Taking limit , we obtain
By Lemma 2.6, we can conclude that and as . The proof is completed. □
Next, we define and assume that for all and . We can prove the strong convergence theorem for finding the set of solutions of the variational inequality problem in a real uniformly smooth and 2-uniformly convex Banach space.
Remark 3.2 (Qin et al. [7])
Let be the generalized projection from a smooth strictly convex and reflexive Banach space E onto a nonempty closed convex subset C of E. Then is a closed quasi-ϕ-nonexpansive mapping from E onto C with .
Corollary 3.3 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from to ℝ satisfying conditions (A1)-(A4), and let A be an α-inverse-strongly monotone mapping of C into satisfying for all and . Let be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences , with , as and a strictly increasing continuous function with . Assume that S is uniformly asymptotically regular on C with and . For arbitrary , , generate a sequence by
where . Assume that the control sequences , , , and satisfy the following conditions:
-
1.
, and are sequences in such that , ,
-
2.
for some a, b with and is the 2-uniformly convex constant of E,
-
3.
for some ,
then converges strongly to .
Proof For and is quasi-ϕ-nonexpansive mapping, we have
So, we can show that .
Define by Theorem 2.14, B is maximal monotone and . Let . Since , we get .
From , we have
On the other hand, since . Then, by Lemma 2.6, we have
and thus
It follows from (3.31) and (3.32) that
where . From as and (3.23), taking on the both sides of the equality above, we have . By the maximality of B, we have , that is, . From Theorem 3.1, we have . The proof is completed. □
Let A be a strongly monotone mapping with constant k, Lipschitz with constant , that is,
which implies that
It follows that
hence A is α-inverse-strongly monotone with . Therefore, we have the following corollaries.
Corollary 3.4 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from to ℝ satisfying conditions (A1)-(A4), and let be a strongly monotone mapping with constant k, Lipschitz with constant . Let be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences , with , as and a strictly increasing continuous function with . Assume that S is uniformly asymptotically regular on C with and . For arbitrary , , a sequence is generated by
where . Assume that the control sequences , , , and satisfy the following conditions:
-
1.
, and are sequences in such that , ,
-
2.
for some a, b with and is the 2-uniformly convex constant of E,
-
3.
for some ,
then converges strongly to .
Corollary 3.5 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from to ℝ satisfying conditions (A1)-(A4), and let be a strongly monotone mapping with constant k,Lipschitz with constant satisfying for all and . Let be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences , with , as and a strictly increasing continuous function with . Assume that S is uniformly asymptotically regular on C with and . For arbitrary , , generate a sequence by
where . Assume that the control sequences , , , and satisfy the following conditions:
-
1.
, and are sequences in such that , ,
-
2.
for some a, b with and is the 2-uniformly convex constant of E,
-
3.
for some ,
then converges strongly to .
Let F be a Fréchet differentiable functional in a Banach space E and ∇F be the gradient of F, denote . Baillon and Haddad [34] proved the following lemma.
Lemma 3.6 (Baillon and Haddad [34])
Let E be a Banach space. Let F be a continuously Fréchet differentiable convex functional on E and ∇F be the gradient of F. If ∇F is -Lipschitz continuous, then ∇F is an α-inverse strongly monotone mapping.
We replace A in Theorem 3.1 by ∇F, then we can obtain the following corollary.
Corollary 3.7 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from to ℝ satisfying conditions (A1)-(A4). Let F be a continuously Fréchet differentiable convex functional on E and ∇F be -Lipschitz continuous. Let be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences , with , as and a strictly increasing continuous function with . Assume that S is uniformly asymptotically regular on C with and . For an initial point , , define the sequence by
where , , , .
Assume that the control sequences , , , and satisfy the following conditions:
-
1.
, and are sequences in such that , and ,
-
2.
for some a, b with and the 2-uniformly convex constant of E,
-
3.
for some ,
then converges strongly to .
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This work was supported by Thaksin University Research Fund. Moreover, the author also would like to thank Faculty of Science, Thaksin University.
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Saewan, S. Strong convergence theorem for total quasi-ϕ-asymptotically nonexpansive mappings in a Banach space. Fixed Point Theory Appl 2013, 297 (2013). https://doi.org/10.1186/1687-1812-2013-297
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DOI: https://doi.org/10.1186/1687-1812-2013-297