Abstract
We introduce new implicit and explicit iterative schemes for finding a common element of the set of fixed points of k-strictly pseudocontractive mapping and the set of zeros of the sum of two monotone operators in a Hilbert space. Then we establish strong convergence of the sequences generated by the proposed schemes to a common point of two sets, which is a solution of a certain variational inequality. Further, we find the unique solution of the quadratic minimization problem, where the constraint is the common set of two sets mentioned above. As applications, we consider iterative schemes for the Hartmann-Stampacchia variational inequality problem and the equilibrium problem coupled with fixed point problem.
MSC:47H05, 47H09, 47H10, 47J05, 47J07, 47J25, 47J20, 49M05.
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1 Introduction
Let H be a real Hilbert space with the inner product and the induced norm . Let C be a nonempty closed convex subset of H, and let be a self-mapping on C. We denote by the set of fixed points of T, that is, .
Let be a single-valued nonlinear mapping, and let be a multivalued mapping. Then we consider the monotone inclusion problem (MIP) of finding such that
The set of solutions of the MIP (1.1) is denoted by . That is, is the set of zeros of . The MIP (1.1) provides a convenient framework for studying a number of problems arising in structural analysis, mechanics, economics and others; see, for instance [1, 2]. Also, various types of inclusion problems have been extended and generalized, and there are many algorithms for solving variational inclusions. For more details, see [3–5] and the references therein.
The class of pseudocontractive mappings is one of the most important classes of mappings among nonlinear mappings. We recall that a mapping is said to be k-strictly pseudocontractive if there exists a constant such that
Note that the class of k-strictly pseudocontractive mappings includes the class of nonexpansive mappings as a subclass. That is, T is nonexpansive (i.e., , ) if and only if T is 0-strictly pseudocontractive. The mapping T is also said to be pseudocontractive if , and T is said to be strongly pseudocontractive if there exists a constant such that is pseudocontractive. Clearly, the class of k-strictly pseudocontractive mappings falls into the one between classes of nonexpansive mappings and pseudocontractive mappings. Also, we remark that the class of strongly pseudocontractive mappings is independent of the class of k-strictly pseudocontractive mappings (see [6]). Recently, many authors have been devoting the studies on the problems of finding fixed points for pseudocontractive mappings (see, for example, [7–10] and the references therein).
Recently, in order to study the MIP (1.1) coupled with the fixed point problem, many authors have introduced some iterative schemes for finding a common element of the set of solutions of the MIP (1.1) and the set of fixed points of a countable family of nonexpansive mappings (see [4, 5, 11] and the references therein).
Inspired and motivated by the above-mentioned recent works, in this paper, we introduce new implicit and explicit iterative schemes for finding a common element of the set of the solutions of the MIP (1.1) with a set-valued maximal monotone operator B and an inverse-strongly monotone mapping A and the set of fixed points of a k-strictly pseudocontractive mapping T. Then we establish results of the strong convergence of the sequences generated by the proposed schemes to a common point of two sets, which is a solution of a certain variational inequality. As a direct consequence, we find the unique solution of the quadratic minimization problem:
Moreover, as applications, we consider iterative algorithms for the Hartmann-Stampacchia variational inequality problem and the equilibrium problem coupled with fixed point problem of nonexpansive mappings.
2 Preliminaries and lemmas
Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. In the following, we write to indicate that the sequence converges weakly to x. implies that converges strongly to x.
Recall that a mapping is said to be contractive if there exists such that
A mapping A of C into H is called inverse-strongly monotone if there exists a positive real number α such that
for all . For such a case, A is called α-inverse-strongly monotone. If A is an α-inverse-strongly monotone mapping of C into H, then it is obvious that A is -Lipschitzian and continuous. Let B be a mapping of H into . The effective domain of B is denoted by , that is, . A multi-valued mapping B is said to be a monotone operator on H if for all , , and . A monotone operator B on H is said to be maximal if its graph is not properly contained in the graph of any other monotone operator on H. For a maximal monotone operator B on H and , we may define a single-valued operator , which is called the resolvent of B for r. Let B be a maximal monotone operator on H, and let . It is well known that for all and the resolvent is firmly nonexpansive, i.e.,
and that the resolvent identity
holds for all and . It is worth mentioning that the resolvent operator is nonexpansive and 1-inverse-strongly monotone, and that a solution of the MIP (1.1) is a fixed point of the operator for all (see [11]).
In a real Hilbert space H, we have
for all and . For every point , there exists a unique nearest point in C, denoted by , such that
is called the metric projection of H onto C. It is well known that is nonexpansive, and is characterized by the property
It is also well known that H satisfies the Opial condition, that is, for any sequence with , the inequality
holds for every with . For these facts, see [12].
We need the following lemmas for the proof of our main results.
Lemma 2.1 In a real Hilbert space H, the following inequality holds:
Lemma 2.2 [12]
For all and with , the following equality holds:
Lemma 2.3 [13]
Let H be a Hilbert space, let C be a closed convex subset of H. If T is a k-strictly pseudocontractive mapping on C, then the fixed point set is closed convex, so that the projection is well defined, and .
Lemma 2.4 [13]
Let H be a real Hilbert space, let C be a closed convex subset of H, and let be a k-strictly pseudocontractive mapping. Define a mapping by for all . Then, as , S is a nonexpansive mapping such that .
Lemma 2.5 [14]
Let C be a nonempty closed convex subset of a real Hilbert space H. Let the mapping be α-inverse strongly monotone, and let be a constant. Then we have
In particular, if , then is nonexpansive.
Lemma 2.6 [15]
Let be a maximal monotone operator, and let be a Lipschitz continuous mapping. Then the mapping is a maximal monotone operator.
Remark 2.1 Lemma 2.6 implies that is closed and convex if is a maximal monotone operator and is an inverse-strongly monotone mapping.
The following lemma is a variant of a Minty lemma (see [16]).
Lemma 2.7 Let C be a nonempty closed convex subset of a real Hilbert space H. Assume that the mapping is monotone and weakly continuous along segments, that is, weakly as . Then the variational inequality
is equivalent to the dual variational inequality
Lemma 2.8 [17]
Let and be bounded sequences in a real Banach space E, and let be a sequence in , which satisfies the following condition:
Suppose that for all , and
Then .
Lemma 2.9 [18]
Let be a sequence of non-negative real numbers satisfying
where and satisfy the following conditions:
-
(i)
and ;
-
(ii)
or .
Then .
3 Iterative schemes
Throughout the rest of this paper, we always assume as follows: Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let be an α-inverse strongly monotone mapping, and let B be a maximal monotone operator on H such that the domain of B is included in C. Let be the resolvent of B for . Let be a contractive mapping with constant , and let be a k-strictly pseudocontractive mapping. Define a mapping by , , where . Then, by Lemma 2.4, S is nonexpansive.
In this section, we introduce the following iterative scheme that generates a net in an implicit way:
where . We prove strong convergence of , as , to a point in , which is a solution of the following variational inequality:
Equivalently, .
If we take in (3.1), then we have
We also propose the following iterative scheme which generates a sequence in an explicit way:
where , and is an arbitrary initial guess, and establish the strong convergence of this sequence to a fixed point of T, which is also a solution of the variational inequality (3.2). If we take in (3.4), then we have
3.1 Strong convergence of the implicit algorithm
For , consider the following mapping on C defined by
By Lemma 2.5, we have
Since , is a contractive mapping. Therefore, by the Banach contraction principle, has a unique fixed point , which uniquely solves the fixed point equation
Now, we prove strong convergence of the sequence , and show the existence of , which solves the variational inequality (3.2).
Theorem 3.1 Suppose that . Then the net defined by the implicit method (3.1) converges strongly, as , to a point , which is the unique solution of the variational inequality (3.2).
Proof First, we can show easily the uniqueness of a solution of the variational inequality (3.2). In fact, if and both are solutions to (3.2). Then we have
Adding up (3.6) and (3.7) yields
This implies that . So , and the uniqueness is proved. Below, we use to denote the unique solution of the variational inequality (3.2).
Now, we prove that is bounded. Set for all . Take . It is clear that and (by Lemma 2.4). Since is nonexpansive and A is α-inverse-strongly monotone, we have from Lemma 2.5 that
So, we have that
Moreover, from (3.1), it follows that
that is,
Hence, is bounded, and so are , , and .
From (3.8) and (3.10), we have
where is an appropriate constant. This means that
Since , we deduce that
From (2.1) and (2.3), we also obtain
So, we get
Since is a convex function, by (3.13), we have
We deduce from (3.14) that
where is an appropriate constant. Since and , we have
Observing that
by (3.16), we obtain
Let be a sequence such that as . Put , and . Since is bounded, there exists a subsequence of , which converges weakly to .
Next, we show that . Since C is closed and convex, C is weakly closed. So, we have . Let us show . Assume that (). Since and , it follows from the Opial condition and (3.17) that
which is a contradiction. So we get .
We shall show that . Since as , it follows that converges weakly to . We choose a subsequence of such that . Let . Noting that
we have that
and so,
Since B is monotone, we have for ,
Since and , we have . Then by (3.16) and (3.18), we obtain
Since B is maximal monotone, this implies that , that is, . Hence, we have . Thus, we conclude that .
On the one hand, we note that for ,
Then it follows that
Hence, we have
In particular,
Since , by (3.20), we obtain
Since , it follows from (3.21) that as .
Now, we return to (3.20) and take the limit as to get
In particular, solves the following variational inequality
or the equivalent dual variational inequality (see Lemma 2.7)
Finally, we show that the net converges strongly, as , to . To this end, let be another sequence such that as . Put , and . Let be a subsequence of , and assume that . By the same proof as the one above, we have . Moreover, it follows from (3.23) that
Interchanging and , we obtain
Adding up (3.24) and (3.25) yields
Hence,
that is, . Since , we have . Therefore, we conclude that as .
Note that is well defined by Lemma 2.3 and Remark 2.1. The variational inequality (3.2) can be rewritten as
By (2.4), this is equivalent to the fixed point equation
This completes the proof. □
From Theorem 3.1, we can deduce the following result.
Corollary 3.1 Suppose that . Then the net defined by the implicit method (3.3) converges strongly, as , to , which solves the following minimum norm problem: find such that
Proof From (3.22) with and , we have
Equivalently,
This obviously implies that
It turns out that for all . Therefore, is the minimum-norm point of . □
3.2 Strong convergence of the explicit algorithm
Now, using Theorem 3.1, we establish the strong convergence of an explicit iterative scheme for finding a solution of the variational inequality (3.2), where the constraint set is the common set of the fixed point set of the k-strictly pseudocontractive mapping T and the solution set of the MIP (1.1).
Theorem 3.2 Suppose that . Let and satisfy the following conditions:
-
(C1)
;
-
(C2)
;
-
(C3)
;
-
(C4)
and .
Let the sequence be generated iteratively by (3.4):
where is an arbitrary initial guess. Then the sequence converges strongly to a point in , which is the unique solution of the variational inequality (3.2).
Proof First, from condition (C1), without loss of generality, we assume that , and we note that . From now, we put .
We divide the proof several steps as follows.
Step 1. We show that for all and all (). Indeed, let . From , and Lemma 2.5, we get
Using (3.27), we have
Using an induction, we have
Hence, is bounded, and so are , , and .
Step 2. We show that . Put , and define
Then we have
Since is nonexpansive for (by Lemma 2.5), we have
By the resolvent identity (2.2) and (3.30), we get
where is an appropriate constant. Hence, from (3.29) and (3.31), we obtain
It follows from conditions (C1) and (C4) that
Thus, by Lemma 2.8, we have
Consequently, we obtain
Step 3. We show that for . From (3.4), (3.27) and Lemma 2.2, we have
where is an appropriate constant. From (3.34) and conditions (C3) and (C4), we deduce that
Since (by condition (C1)) and (by Step 2), we conclude that
Step 4. We show that . First, from (2.1) and (2.3), we get for ,
So, we have
Using (3.34) and (3.35), we obtain
where are appropriate constants. This implies that
Thus, from condition (C1), Step 2 and Step 3, we deduce that
Step 5. We show that . First, by (3.4), we have
and so,
By conditions (C1) and (C3) and Step 2, we obtain
This together with Step 4 yields that
Step 6. We show that
where with being defined by (3.1). We note that from Theorem 3.1, , and is the unique solution of the variational inequality (3.2). To show this, we can choose a subsequence of such that
Since is bounded, there exists a subsequence of , which converges weakly to w. Without loss of generality, we can assume that . By the same argument as in the proof of Theorem 3.1 together with Step 5, we have . Since is the unique solution of the variational inequality (3.2), we deduce that
Step 7. We show that , where with being defined by (3.1), and is the unique solution of the variational inequality (3.2). Indeed, from (3.4), we note that
Applying Lemma 2.1, we have
This implies that
where is an appropriate constant, and
From conditions (C1) and (C2) and Step 6, it is easy to see that , and . Hence, by Lemma 2.9, we conclude that as . This completes the proof. □
From Theorem 3.2, we deduce immediately the following result.
Corollary 3.2 Suppose that . Let and satisfy the following conditions:
-
(C1)
;
-
(C2)
;
-
(C3)
;
-
(C4)
.
Let the sequence be generated iteratively by (3.5):
where is an arbitrary initial guess. Then the sequence converges strongly to a point in , which is the unique solution of the minimum norm problem (3.26).
Proof The variational inequality (3.2) is reduced to the inequality
This is equivalent to for all . It turns out that for all and is the minimum-norm point of . □
Remark 3.1 It is worth pointing out that our iterative schemes (3.1) and (3.4) are new ones different from those in the literature. The iterative schemes (3.3) and (3.5) are also new ones different from those in the literature (see [5, 11] and the references therein).
4 Applications
Let H be a real Hilbert space, and let g be a proper lower semicontinuous convex function of H into . Then the subdifferential ∂g of g is defined as follows:
for all . From Rockafellar [19], we know that ∂g is maximal monotone. Let C be a closed convex subset of H, and let be the indicator function of C, i.e.,
Since is a proper lower semicontinuous convex function on H, the subdifferential of is a maximal monotone operator. It is well known that if , then the MIP (1.1) is equivalent to find such that
This problem is called Hartman-Stampacchia variational inequality (see [20]). The set of solutions of the variational inequality (4.2) is denoted by .
The following result is proved by Takahashi et al. [11].
Lemma 4.1 [11]
Let C be a nonempty closed convex subset of a real Hilbert space H, let be the metric projection from H onto C, let be the subdifferential of , and let be the resolvent of for , where is defined by (4.1) and . Then
Applying Theorem 3.2, we can obtain a strong convergence theorem for finding a common element of the set of solutions to the variational inequality (4.2) and the set of fixed points of a nonexpansive mapping.
Theorem 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone mapping of C into H, and let S be a nonexpansive mapping of C into itself such that . Let and satisfy the following conditions:
-
(C1)
;
-
(C2)
;
-
(C3)
;
-
(C4)
and .
Let the sequence be generated iteratively by
where is an arbitrary initial guess. Then the sequence converges strongly to a point in .
Proof Put . It is easy to show that . In fact,
From Lemma 4.1, we get for all . Hence, the desired result follows from Theorem 3.2. □
As in [11, 21], we consider the problem for finding a common element of the set of solutions of a mathematical model related to equilibrium problems and the set of fixed points of a nonexpansive mapping in a Hilbert space.
Let C be a nonempty closed convex subset of a Hilbert space H, and let us assume that a bifunction satisfies the following conditions:
-
(A1)
for all ;
-
(A2)
Θ is monotone, that is, for all ;
-
(A3)
for each ,
-
(A4)
for each , is convex and lower semicontinuous.
Then the mathematical model related to the equilibrium problem (with respect to C) is find such that
for all . The set of such solutions is denoted by . The following lemma was given in [22, 23].
Let C be a nonempty closed convex subset of H, and let Θ be a bifunction of into ℝ satisfying (A1)-(A4). Then for any and , there exists such that
Moreover, if we define as follows:
for all , then the following hold:
-
(1)
is single-valued;
-
(2)
is firmly nonexpansive, that is, for any ,
-
(3)
;
-
(4)
is closed and convex.
We call such the resolvent of Θ for . The following lemma was given in Takahashi et al. [11].
Lemma 4.3 [11]
Let C be a nonempty closed convex subset of a real Hilbert space H, and let Θ be a bifunction of into ℝ satisfying (A1)-(A4). Let be a multivalued mapping of H into itself defined by
Then , and is a maximal monotone operator with . Moreover, for any and , the resolvent of Θ coincides with the resolvent of ; i.e.,
Applying Lemma 4.3 and Theorem 3.2, we can obtain the following results.
Theorem 4.2 Let C be a nonempty closed convex subset of a real Hilbert space H, and let Θ be a bifunction of into ℝ satisfying (A1)-(A4). Let be a maximal monotone operator with defined as in Lemma 4.3, and let be the resolvent of Θ for . Let A be an α-inverse strongly monotone mapping of C into H, and let S be a nonexpansive mapping from C into itself such that . Let and satisfy the following conditions:
-
(C1)
;
-
(C2)
;
-
(C3)
;
-
(C4)
and .
Let the sequence be generated iteratively by
where is an arbitrary initial guess. Then the sequence converges strongly to a point in .
Theorem 4.3 Let C be a nonempty closed convex subset of a real Hilbert space H, and let Θ be a bifunction of into ℝ satisfying (A1)-(A4). Let be a maximal monotone operator with defined as in Lemma 4.3, and let be the resolvent of Θ for , and let S be a nonexpansive mapping from C into itself such that . Let and satisfy the following conditions:
-
(C1)
;
-
(C2)
;
-
(C3)
;
-
(C4)
and .
Let the sequence be generated iteratively by
where is an arbitrary initial guess. Then the sequence converges strongly to a point in .
Proof Put in Theorem 4.2. From Lemma 4.3, we also know that for all . Hence, the desired result follows from Theorem 4.2. □
Remark 4.1 (1) As in Corollary 3.2, if we take in Theorems 4.1, 4.2 and 4.3, then we can obtain the minimum-norm point of , and , respectively.
(2) For several iterative schemes for zeros of monotone operators, variational inequality problems, generalized equilibrium problems, convex minimization problems, and fixed point problems, we can also refer to [24–29] and the references therein. By combining our methods in this paper and methods in [24–29], we will consider new iterative schemes for the above-mentioned problems coupled with the fixed point problems of nonlinear operators.
References
Chang SS: Existence and approximation of solutions for set-valued variational inclusions in Banach spaces. Nonlinear Anal. 2001, 47: 583–594. 10.1016/S0362-546X(01)00203-6
Demyanov VF, Stavroulakis GE, Polyakova LN, Panagiotopoulos PD 10. In Quasi-Differentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics. Kluwer Academic, Dordrecht; 1996.
Lin LJ, Wang SY, Chuang CS: Existence theorems of systems of variational inclusion problems with applications. J. Glob. Optim. 2008, 40: 751–764. 10.1007/s10898-007-9160-2
Peng JW, Wang Y, Shyu DS, Yao JC: Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems. J. Inequal. Appl. 2008., 2008: Article ID 720371 10.1155/2008/720371
Zhang SS, Lee JHW, Chan CK: Algorithms of common solutions to quasi variational inclusion and fixed point problems. Appl. Math. Mech. 2008, 29: 571–581. 10.1007/s10483-008-0502-y
Browder FE, Petryshn WV: Construction of fixed points of nonlinear mappings Hilbert space. J. Math. Anal. Appl. 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6
Acedo GL, Xu HK: Iterative methods for strictly pseudo-contractions in Hilbert space. Nonlinear Anal. 2007, 67: 2258–2271. 10.1016/j.na.2006.08.036
Jung JS: Strong convergence of iterative methods for k -strictly pseudo-contractive mappings in Hilbert spaces. Appl. Math. Comput. 2010, 215: 3736–3753.
Jung JS: Some results on a general iterative method for k -strictly pseudo-contractive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 24 10.1186/1687-1812-2011-24
Morales CH, Jung JS: Convergence of paths for pseudo-contractive mappings in Banach spaces. Proc. Am. Math. Soc. 2000, 128: 3411–3419. 10.1090/S0002-9939-00-05573-8
Takahashi S, Takahashi W, Toyoda M: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 2010, 147: 27–41. 10.1007/s10957-010-9713-2
Takahashi W: Nonlinear Functional Analysis: Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2000.
Zhou H: Convergence theorems of fixed points for k -strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 2008, 69: 456–462. 10.1016/j.na.2007.05.032
Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2006, 128: 191–201. 10.1007/s10957-005-7564-z
Brézis H North-Holland Mathematics Studies 5. In Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam; 1973. Notas de Matemática (50)
Minty GJ: On the generalization of a direct method of the calculus of variations. Bull. Am. Math. Soc. 1967, 73: 315–321. 10.1090/S0002-9904-1967-11732-4
Suzuki T: Strong convergence of Krasnoselskii and Mann’s type sequences for one parameter nonexpansive semigroups without Bochner integral. J. Math. Anal. Appl. 2005, 305: 227–239. 10.1016/j.jmaa.2004.11.017
Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332
Rockafellar RT: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 1970, 33: 209–216. 10.2140/pjm.1970.33.209
Hartman P, Stampacchia G: On some non-linear elliptic differential-functional equations. Acta Math. 1966, 115: 271–310. 10.1007/BF02392210
Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in Hilbert spaces. Nonlinear Anal. 2008, 69: 1025–1033. 10.1016/j.na.2008.02.042
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.
Combettes PI, Hirstoaga SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.
Ceng LC, Ansari QH, Khan AR, Yao JC: Strong convergence on composite iterative schemes for zeros of m -accretive operators in Banach spaces. Nonlinear Anal. 2009, 70: 1830–1840. 10.1016/j.na.2008.02.083
Ceng LC, Ansari QH, Khan AR, Yao JC: Viscosity approximation methods for strongly positive and monotone operators. Fixed Point Theory 2009, 10: 35–71.
Ceng LC, Ansari QH, Yao JC: Some iterative methods for finding fixed point and for solving constrained convex minimization problems. Nonlinear Anal. 2011, 74(16):5286–5302. 10.1016/j.na.2011.05.005
Ceng LC, Ansari QH, Yao JC: Extragradient-projection method for solving constrained convex minimization problems. Numer. Algebra Control Optim. 2011, 1(3):341–359.
Ceng LC, Ansari QH, Wong MM, Yao JC: Mann type hybrid extragradient method for variational inequalities, variational inclusion and fixed point problems. Fixed Point Theory 2012, 13(2):403–422.
Zeng LC, Ansari QH, Shyu DS, Yao JC: Strong and weak convergence theorems for common solutions of generalized equilibrium problems and zeros of maximal monotone operators. Fixed Point Theory Appl. 2010., 2010: Article ID 590278
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The author would like to thank the anonymous referees for their careful reading and valuable suggestions, which improved the presentation of this manuscript, and the editor for his valuable comments along with providing some recent related papers. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2013021600).
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Jung, J.S. Iterative algorithms for monotone inclusion problems, fixed point problems and minimization problems. Fixed Point Theory Appl 2013, 272 (2013). https://doi.org/10.1186/1687-1812-2013-272
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DOI: https://doi.org/10.1186/1687-1812-2013-272