Research Article | Open | Published:
An Application of Hybrid Steepest Descent Methods for Equilibrium Problems and Strict Pseudocontractions in Hilbert Spaces
Journal of Inequalities and Applicationsvolume 2011, Article number: 173430 (2011)
We use the hybrid steepest descent methods for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudocontraction mapping in the setting of real Hilbert spaces. We proved strong convergence theorems of the sequence generated by our proposed schemes.
Let be a real Hilbert space and a closed convex subset of , and let be a bifunction of into , where is the set of real numbers. The equilibrium problem for is to find such that
denoted the set of solution by . Given a mapping , let for all , then if and only if , that is, is a solution of the variational inequality. Numerous problems in physics, optimizations, and economics reduce to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem, see, for instance, [1, 2].
A mapping of into itself is nonexpansive if , for all . The set of fixed points of is denoted by . In 2007, Plubtieng and Punpaeng , S. Takahashi and W. Takahashi , and Tada and W. Takahashi  considered iterative methods for finding an element of .
Recall that an operator is strongly positive if there exists a constant with the property
In 2006, Marino and Xu  introduced the general iterative method and proved that for a given , the sequence is generated by the algorithm
where is a self-nonexpansive mapping on , is a contraction of into itself with and satisfies certain conditions, and is a strongly positive bounded linear operator on and converges strongly to a fixed-point of which is the unique solution to the following variational inequality:
, for , and is also the optimality condition for some minimization problem. A mapping is said to be -strictly pseudocontractive if there exists a constant such that
Note that the class of -strict pseudo-contraction strictly includes the class of nonexpansive mapping, that is, is nonexpansive if and only if is 0-srictly pseudocontractive; it is also said to be pseudocontractive if . Clearly, the class of -strict pseudo-contractions falls into the one between classes of nonexpansive mappings and pseudo-contractions.
The set of fixed points of is denoted by . Very recently, by using the general approximation method, Qin et al.  obtained a strong convergence theorem for finding an element of . On the other hand, Ceng et al.  proposed an iterative scheme for finding an element of and then obtained some weak and strong convergence theorems. Based on the above work, Y. Liu  introduced two iteration schemes by the general iterative method for finding an element of .
In 2001, Yamada  introduced the following hybrid iterative method for solving the variational inequality:
where is -Lipschitzian and -strongly monotone operator with , , , then he proved that if satisfyies appropriate conditions, the generated by (1.5) converges strongly to the unique solution of variational inequality
Motivated and inspired by these facts, in this paper, we introduced two iteration methods by the hybrid iterative method for finding an element of , where is a -strictly pseudocontractive non-self mapping, and then obtained two strong convergence theorems.
Throughout this paper, we always assume that is a nonempty closed convex subset of a Hilbert space . We write to indicate that the sequence converges weakly to . implies that converges strongly to . For any , there exists a unique nearest point in , denoted by , such that
Such a is called the metric projection of onto . It is known that is nonexpansive. Furthermore, for and , , for all .
It is widely known that satisfies Opial's condition , that is, for any sequence with , the inequality
holds for every with . In order to solve the equilibrium problem for a bifunction , let us assume that satisfies the following conditions:
(A1), for all ,
(A2) is monotone, that is, , for all ,
(A3)For all .
(A4) For each fixed , the function is convex and lower semicontinuous. Let us recall the following lemmas which will be useful for our paper.
Lemma 2.1 (see ).
Let be a bifunction from into satisfying (A1), (A2),(A3) and (A4) then, for any and , there exists such that
Further, if , then the following hold:
(1) is single-valued,
(2) is firmly nonexpansive, that is,
(4) is nonempty, closed and convex.
Lemma 2.2 (see ).
If is a -strict pseudo-contraction, then the fixed-point set is closed convex, so that the projection is well difened.
Lemma 2.3 (see ).
Let be a -strict pseudo-contraction. Define by for each , then, as , T is nonexpansive mapping such that .
Lemma 2.4 (see ).
In a Hilbert space , there holds the inequality
Lemma 2.5 (see ).
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in (0,1) and is a sequence in , such that
(ii) or Then .
3. Main Results
Throughout the rest of this paper, we always assume that is a -lipschitzian continuous and -strongly monotone operator with and assume that . . Let be mappings defined as Lemma 2.1. Define a mapping by , for all , where , then, by Lemma 2.3, is nonexpansive. We consider the mapping on defined by
where . By Lemmas 2.1 and 2.3, we have
It is easy to see that is a contraction. Therefore, by the Banach contraction principle, has a unique fixed-point such that
For simplicity, we will write for provided no confusion occurs. Next, we prove that the sequence converges strongly to a which solves the variational inequality
Let be a nonempty closed convex subset of a real Hilbert space and a bifunction from into satisfying (A1), (A2), (A3), and (A4). Let be a -strictly pseudocontractive nonself mapping such that . Let be an -Lipschitzian continuous and - monotone operator on with and , . Let be asequence generated by
where , , and satisfy if and satisfy the following conditions:
(ii) and ,
then converges strongly to a point which solves the variational inequality (3.4).
First, take . Since and , from Lemma 2.1, for any , we have
Then, since , we obtain that
Further, we have
It follows that .
Hence, is bounded, and we also obtain that and are bounded. Notice that
By Lemma 2.1, we have
It follows that
Thus, from Lemma 2.4, (3.7), and (3.11), we obtain that
It follows that
Since , therefore
From (3.9), we derive that
Define by , then is nonexpansive with by Lemma 2.3. We note that
So by (3.15) and , we obtain that
Since is bounded, so there exists a subsequence which converges weakly to . Next, we show that . Since is closed and convex, is weakly closed. So we have . Let us show that . Assume that , Since and , it follows from the Opial's condition that
This is a contradiction. So, we get and .
Next, we show that . Since , for any , we obtain
From (A2), we have
Replacing by , we have
Since and , it follows from (A4) that , for all . Let for all and , then we have and hence . Thus, from (A1) and (A4), we have
and hence . From (A3), we have for all and hence . Therefore, . On the other hand, we note that
Hence, we obtain
It follows that
This implies that
Since , it follows from (3.27) that as . Next, we show that solves the variational inequality (3.4).
As a matter of fact, we have
and we have
Hence, for ,
Since is monotone (i.e., , for all . This is due to the nonexpansivity of ).
Now replacing in (3.30) with and letting , we obtain
That is, is a solution of (3.4). To show that the sequence converges strongly to , we assume that . Similiary to the proof above, we derive . Moreover, it follows from the inequality (3.31) that
Interchange and to obtain
Adding up (3.32) and (3.33) yields
Hence, , and therefore as ,
This is equivalent to the fixed-point equation
Let be a nonempty closed convex subset of a real Hilbert space and a bifunction from into satisfying (A1), (A2), (A3) and (A4). Let be a -strictly pseudocontractive nonself mapping such that . Let be an -Lipschitzian continuous and -strongly monotone operator on with . Suppose that , . Let and be sequences generated by and
where , if ,, and satisfy the following conditions:
(i), , , ,
(ii) and , ,
(iii), and ,
then and converge strongly to a point which solves the variational inequality(3.4).
We first show that is bounded. Indeed, pick any to derive that
By induction, we have
and hence is bounded. From (3.6) and (3.7), we also derive that and are bounded. Next, we show that . We have
On the other hand, we have
From and , we note that
Putting in (3.43) and in (3.44), we have
So, from (A2), we have
Since , without loss of generality, let us assume that there exists a real number a such that for all . Thus, we have
where . Next, we estimate . Notice that
From (3.48), (3.49), and (3.42), we obtain that
where is an appropriate constant such that
From (3.41) and (3.50), we obtain
where . Hence, few by Lemma 2.5, we have
From (3.48) and (3.50), and , we have
it follows that
From and (3.53), we have
For , we have
This implies that
Then, from (3.7) and (3.59), we derive that
Since , , we have
From (3.57) and (3.61), we obtain that
Define by , then is nonexpansive with by Lemma 2.3. Notice that
By (3.62) and , we obtain that
Next, we show that , where is a unique solution of the variational inequality (3.4). Indeed, take a subsequence of such that
Since is bounded, there exists a subsequence of which converges weakly to .
Without loss of generality, we can assume that . From (3.61) and (3.64), we obtain and . By the same argument as in the proof of Theorem 3.1, we have . Since , it follows that
From , we have
This implies that
where , , and .
It is easy to see that , , and by (3.66). Hence by Lemma 2.5, the sequence converges strongly to .
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.
Moudafi A, Théra M: Proximal and dynamical approaches to equilibrium problems. In Ill-Posed Variational Problems and Regularization Techniques (Trier, 1998), Lecture Notes in Econom. and Math. Systems. Volume 477. Springer, Berlin, Germany; 1999:187–201.
Plubtieng S, Punpaeng R: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,336(1):455–469. 10.1016/j.jmaa.2007.02.044
Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,331(1):506–515. 10.1016/j.jmaa.2006.08.036
Tada A, Takahashi W: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. Journal of Optimization Theory and Applications 2007,133(3):359–370. 10.1007/s10957-007-9187-z
Marino G, Xu H-K: A general iterative method for nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2006,318(1):43–52. 10.1016/j.jmaa.2005.05.028
Qin X, Shang M, Kang SM: Strong convergence theorems of modified Mann iterative process for strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,70(3):1257–1264. 10.1016/j.na.2008.02.009
Ceng L-C, Al-Homidan S, Ansari QH, Yao J-C: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. Journal of Computational and Applied Mathematics 2009,223(2):967–974. 10.1016/j.cam.2008.03.032
Liu Y: A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,71(10):4852–4861. 10.1016/j.na.2009.03.060
Yamada I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), Stud. Comput. Math.. Volume 8. Edited by: Butnariu D, Censor Y, Reich S. North-Holland, Amsterdam, The Netherlands; 2001:473–504.
Iiduka H, Takahashi W: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Analysis: Theory, Methods & Applications 2005,61(3):341–350. 10.1016/j.na.2003.07.023
Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis. An International Journal 2005,6(1):117–136.
Zhou H: Convergence theorems of fixed points for -strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2008,69(2):456–462. 10.1016/j.na.2007.05.032
Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6
Chang S-S: Some problems and results in the study of nonlinear analysis. Nonlinear Analysis: Theory, Methods & Applications 1997,30(7):4197–4208. 10.1016/S0362-546X(97)00388-X
Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059
M. Tian was supported in part by the Science Research Foundation of Civil Aviation University of China (no. 2010kys02). He was also Supported in part by The Fundamental Research Funds for the Central Universities (Grant no. ZXH2009D021).