Abstract
In this paper, we introduce three iterative methods for finding a common element of the set of fixed points for a continuous pseudocontractive mapping and the solution set of a variational inequality problem governed by continuous monotone mappings. Strong convergence theorems for the proposed iterative methods are proved. Our results improve and extend some recent results in the literature.
MSC: 47H05, 47H09, 47J25, 65J15.
Keywords:
pseudocontractive mapping; monotone mapping; strong convergence theorem1 Introduction
The theory of variational inequalities represents, in fact, a very natural generalization of the theory of boundary value problems and allows us to consider new problems arising from many fields of applied mathematics, such as mechanics, physics, engineering, the theory of convex programming, and the theory of control. While the variational theory of boundary value problems has its starting point in the method of orthogonal projection, the theory of variational inequalities has its starting point in the projection on a convex set.
Let C be a nonempty closed and convex subset of a real Hilbert space H. The classical variational inequality problem is to find such that for all , where A is a nonlinear mapping. The set of solutions of the variational inequality is denoted by . The variational inequality problem has been extensively studied in the literature; see [19] and the references therein. In the context of the variational inequality problem, this implies that , , where is a metric projection of H into C.
Let A be a mapping from C to H, then A is called monotone if and only if for each ,
An operator A is said to be strongly positive on H if there exists a constant such that
A mapping A of C into itself is called LLipschitz continuous if there exits a positive number L such that
A mapping A of C into H is called αinversestrongly monotone if there exists a positive real number α such that
for all ; see [914]. If A is an αinverse strongly monotone mapping of C into H, then it is obvious that A is Lipschitz continuous, that is, for all . Clearly, the class of monotone mappings includes the class of αinverse strongly monotone mappings.
A mapping A of C into H is called strongly monotone if there exists a positive real number such that
for all ; see [15]. Clearly, the class of strongly monotone mappings includes the class of strongly positive mappings.
Recall that a mapping T of C into H is called pseudocontractive if for each , we have
T is said to be a kstrict pseudocontractive mapping if there exists a constant such that
A mapping T of C into itself is called nonexpansive if for all . We denote by the set of fixed points of T. Clearly, the class of pseudocontractive mappings includes the class of nonexpansive and strict pseudocontractive mappings.
For a sequence of real numbers in and arbitrary , let the sequence in C be iteratively defined by and
where T is a nonexpansive mapping of C into itself. Halpern [16] was first to study the convergence of algorithm (1.3) in the framework of Hilbert spaces. Lions [17] and Wittmann [18] improved the result of Halpern by proving strong convergence of to a fixed point of T if the real sequence satisfies certain conditions. Reich [19], Shioji and Takahashi [20], and Zegeye and Shahzad [21] extended the result of Wittmann [18] to the case of a Banach space.
In 2000, Moudafi [22] introduced a viscosity approximation method and proved that if H is a real Hilbert space, for given , the sequence generated by the algorithm
where is a contraction mapping and satisfies certain conditions, converges strongly to the unique solution in C of the variational inequality
Moudafi [22] generalized Halpern’s theorems in the direction of viscosity approximations. In [23], Zegeye et al. extended Moudafi’s result to the case of Lipschitz pseudocontractive mappings in Banach spaces more general that Hilbert spaces.
In 2006, Marino and Xu [24] introduced the following general iterative method:
They proved that if the sequence of parameters satisfies appropriate conditions, then the sequence generated by (1.6) converges strongly to the unique solution of the variational inequality
which is the optimality condition for the minimization problem
where h is a potential function for γf (i.e., for ).
Recently, Zegeye and Shahzad [25] introduced an iterative method and proved that if C is a nonempty subset of a real Hilbert space H, is a pseudocontractive mapping and is a continuous monotone mapping such that . For defined by the following: for and , define
Then the sequence generated by and
where is a contraction mapping and and satisfy certain conditions, converges strongly to , where .
In this paper, motivated and inspired by the method of Marino and Xu [24] and the work of Zegeye and Shahzad [25], we introduce a viscosity approximation method for finding a common fixed point of a set of fixed points of continuous pseudocontractive mappings more general than nonexpansive mappings and a solution set of the variational inequality problem for continuous monotone mappings more general than αinverse strongly monotone mappings in a real Hilbert space. Our result extend and unify most of the results that have been proved for important classes of nonlinear operators.
Let C be a nonempty closed and convex subset of a real Hilbert space H. Let be a continuous pseudocontractive mapping and a continuous monotone mapping, respectively. For and , let be defined by (1.8) and (1.9).
We consider the three iterative methods given as follows:
where A is a strongly monotone and LLipschitzian continuous operator and is a contraction mapping. We prove in Section 3 that if and of parameters satisfy appropriate conditions, then the sequences , and converge strongly to .
2 Preliminaries
Let C be a closed and convex subset of a real Hilbert space H. For every , there exists a unique nearest point in C, denoted by , such that
is called the metric projection of H onto C. We know that is a nonexpansive mapping of H onto C. In connection with metric projection, we have the following lemma.
Lemma 2.1Let H be a real Hilbert space. The following identity holds:
Lemma 2.2LetCbe a nonempty closed convex subset of a Hilbert spaceH. Letand. Thenif and only if
Lemma 2.3[26]
Letbe a sequence of nonnegative real numbers such that
where
Lemma 2.4[27]
LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Letbe a continuous monotone mapping. Then, forand, there existssuch that
Moreover, by a similar argument as in the proof of Lemmas 2.8 and 2.9 in[28], Zegeye[27]obtained the following lemmas.
Lemma 2.5[27]
LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Letbe a continuous monotone mapping. Forand, define a mappingas follows:
for all. Then the following hold:
(2) is a firmly nonexpansive type mapping, i.e., for all,
In the sequel, we shall make use of the following lemmas.
Lemma 2.6[27]
LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Letbe a continuous pseudocontractive mapping. Then, forand, there existssuch that
Lemma 2.7[27]
LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Letbe a continuous pseudocontractive mapping. Forand, define a mappingas follows:
for all. Then the following hold:
(2) is a firmly nonexpansive type mapping, i.e., for all,
Lemma 2.8[15]
Letand letfbe anαcontraction of a real Hilbert spaceHinto itself, and letAbe astrongly monotone andLLipschitzian continuous operator ofHinto itself withand. Takeμ, γto be real numbers as follows:
3 Main results
Now, we prove our main theorems.
Theorem 3.1LetHbe a real Hilbert space, be a continuous pseudocontractive mapping andbe a continuous monotone mapping such that. Letand letfbe anαcontraction ofHinto itself, and letAbe astrongly monotone andLLipschitzian continuous operator ofCintoHwithand. Takeμ, γto be real numbers as follows:
For, letbe a sequence generated by (1.11), whereandare such that, , , and. Then the sequenceconverges strongly to, where.
Proof Since as , we may assume, without loss of generality, for all n. For , it implies that is a contraction of H into itself. Since H is a real Hilbert space, there exists a unique element such that .
Let , and let , where . Then we have from Lemma (2.5) and (2.7) that
Moreover, from (1.11) and (3.1), we get that
It follows from induction that
Thus is bounded, and hence so are , and . Next, to show that , we have
where .
Moreover, since , , we get that
Putting in (3.5) and in (3.6), we get that
Adding (3.7) and (3.8), we have
which implies that
Now, using the fact that is monotone, we get that
and hence
Without loss of generality, let us assume that there exists a real number b such that for all . Then we have
and hence from (3.13) we obtain that
where .
Furthermore, from (3.4) and (3.14) we have that
Hence by Lemma 2.3, we have
Consequently, from (3.14) and (3.16), we have that
Moreover, since , , we get that
and
Putting in (3.18) and in (3.19), we get that
and
Adding (3.20) and (3.21), we have
which implies that
Now, using the fact that is pseudocontractive, we get that
and hence
Thus, using the method in (3.13) and (3.14), we have that
where .
Therefore, from (3.16) and the property of , we have that
Furthermore, since , we have that
Now, for , using Lemma 2.5, we get that
and hence
Therefore, we have
Hence
So, we have
Next, we show that
To show this equality, we choose a subsequence of such that
Since is bounded, there exists a subsequence of and such that . Without loss of generality, we may assume that . Since and H is closed and convex, we get that . Moreover, since as , we have that .
Now, we show that . Note that from the definition of , we have
Put for all and . Consequently, we get that . From (3.34) it follows that
From the fact that as , we obtain that as .
Since is monotone, we have that . Thus, it follows that
and hence
Letting and using the fact that is continuous, we obtain that
Furthermore, from the definition of we have that
Put for all and . Consequently, we get that . From (3.33) and pseudocontractivity of , it follows that
Then, since as , we obtain that as . Thus, as , it follows that
and hence
Letting and using the fact that is continuous, we obtain that
Now, let . Then we obtain that and hence .
Therefore, and since , by Lemma 2.2, implies that
Now, we show that as . From , we have that
This implies that
where , and . We put . It is easy to see that , and by (3.42). Hence, by Lemma 2.3, the sequence converges strongly to z. This completes the proof. □
Theorem 3.2LetHbe a real Hilbert space, be a continuous pseudocontractive mapping andbe a continuous monotone mapping such that. Letand letfbe anαcontraction ofHinto itself, and letAbe astrongly monotone andLLipschitzian continuous operator ofHinto itselfHwithand. Takeμ, γto be real numbers as follows:
For, letbe a sequence generated by (1.12), whereandare such that, , , and. The sequenceconverges strongly to, where.
Proof Let be the sequence given by and
From Theorem 3.1, . We claim that . Indeed, we estimate
It follows from , and Lemma 2.3 that .
Theorem 3.3LetHbe a real Hilbert space, be a continuous pseudocontractive mapping andbe a continuous monotone mapping such that. Letand letfbe anαcontraction ofHinto itself, and letAbe astrongly monotone andLLipschitzian continuous operator ofHinto itselfHwithand. Takeμ, γto be real numbers as follows:
For, letbe a sequence generated by (1.13), whereandare such that, , , and. The sequenceconverges strongly to, where.
Proof Define the sequences and by
It follows from induction that
Thus both and are bounded. We observe that
Thus Theorem 3.2 implies that converges to some point z. In this case, we also have
Hence the sequence converges to some point z. This completes the proof. □
Setting , , where I is the identity mapping in Theorem 3.1, we have the following result.
Corollary 3.4LetHbe a real Hilbert space, be a continuous pseudocontractive mapping andbe a continuous monotone mapping such that. Letfbe a contraction ofHinto itself and letbe a sequence generated byand
whereandare such that, , , and. The sequenceconverges strongly to, where.
In Theorem 3.1, , , is a constant mapping, then we get . In fact, we have the following corollary.
Corollary 3.5LetHbe a real Hilbert space, be a continuous pseudocontractive mapping andbe a continuous monotone mapping such that. Letbe a sequence generated byand
whereandare such that, , , and. The sequenceconverges strongly to, where.
In Theorem 3.1, and , where I is the identity mapping, then we have the following corollary.
Corollary 3.6LetHbe a real Hilbert space andbe a continuous monotone mapping such that. Letfbe a contraction ofHinto itself, and letbe a sequence generated byand
whereandare such that, , , and. The sequenceconverges strongly to, where.
Remark 3.7 Our results extend and unify most of the results that have been proved for these important classes of nonlinear operators. In particular, Theorem 3.1 extends Theorem 3.1 of Iiduka and Takahashi [12] and Zegeye et al.[25], Corollary 3.2 of Su et al.[29] in the sense that our convergence is for the more general class of continuous pseudocontractive and continuous monotone mappings. Corollary 3.4 also extends Theorem 4.2 of Iiduka and Takahashi [12] in the sense that our convergence is for the more general class of continuous pseudocontractive and continuous monotone mappings.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank Naresuan University and the university of Phayao. Moreover, the authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.
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