- Research
- Open access
- Published:
A strong convergence theorem for solutions of equilibrium problems and asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense
Fixed Point Theory and Applications volume 2013, Article number: 305 (2013)
Abstract
In this paper, common solutions to an equilibrium problem and a nonlinear operator equation involving a finite family of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense are discussed. Strong convergence theorems of common solutions are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property.
MSC:47H09, 47J25.
1 Introduction
Equilibrium problems have been revealed as a very powerful and important tool in the study of nonlinear phenomena. They have turned out to be very useful in studying optimization problems, differential equations, and minimax theorems and in certain applications to mechanics and economic theory; see [1–27] and the references therein. Important practical situations motivate the study of a system of equilibrium problems. For instance, the flow of fluid through a fissured porous medium and certain models of plasticity led to such problems; see, for instance, [28]. Because of their importance, they have been extensively analyzed. The aim of this paper is to present an iterative method for solving solutions of an equilibrium problem, which is known as the Ky Fan inequality, and a nonlinear operator equation involving a finite family of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense.
The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, an iterative algorithm is presented. A strong convergence theorem is established in a reflexive Banach space. Some results in Hilbert spaces are also discussed.
2 Preliminaries
Let E be a real Banach space. Recall that the normalized duality mapping J from E to is defined by
where denotes the generalized duality pairing. Recall that E is said to be strictly convex if for all with and . It is said to be uniformly convex if for any two sequences and in E such that and . Let be the unit sphere of E. Then the Banach space E is said to be smooth if exists for each . It is said to be uniformly smooth if the above limit is attained uniformly for . It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. It is also well known that if E is uniformly smooth if and only if is uniformly convex.
Recall that E enjoys the Kadec-Klee property if for any sequence , and with , and , then as . It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property.
Next, we assume that E is a smooth Banach space. Consider the functional defined by
Observe that in a Hilbert space H, the equality is reduced to , . As we all know, if C is a nonempty closed convex subset of a Hilbert space H and is the metric projection of H onto C, then is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [29] recently introduced a generalized projection operator in a Banach space E which is an analogue of the metric projection in Hilbert spaces. Recall 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 to the minimization problem
Existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the mapping J. In Hilbert spaces, . It is obvious from the definition of function ϕ that
and
Remark 2.1 If E is a reflexive, strictly convex and smooth Banach space, then if and only if ; for more details, see [29] and the reference therein.
Let f be a bifunction from to ℝ, where ℝ denotes the set of real numbers and let be a mapping. Consider the following equilibrium problem. Find such that
We use to denote the solution set of inequality (2.3). That is,
If , then problem (2.3) is reduced to the following Ky Fan inequality. Find such that
We use to denote the solution set of inequality (2.4). That is,
If , then problem (2.3) is reduced to the classical variational inequality. Find such that
We use to denote the solution set of inequality (2.5). That is,
Recall that a mapping is said to be α-inverse-strongly monotone if there exists such that
For solving problem (2.3), let us assume that the nonlinear mapping is α-inverse-strongly monotone and the bifunction satisfies the following conditions:
-
(A1)
, ;
-
(A2)
F is monotone, i.e., , ;
-
(A3)
-
(A4)
for each , is convex and weakly lower semi-continuous.
Let C be a nonempty subset of E and be a mapping. In this paper, we use to denote the fixed point set of T. T is said to be asymptotically regular on C if for any bounded subset K of C,
T is said to be closed if for any sequence such that and , then . In this paper, we use → and ⇀ to denote the strong convergence and weak convergence, respectively.
Recall that a point p in C is said to be an asymptotic fixed point of T iff C contains a sequence which converges weakly to p such that . The set of asymptotic fixed points of T will be denoted by .
A mapping T is said to be relatively nonexpansive iff
A mapping T is said to be relatively asymptotically nonexpansive iff
where is a sequence such that as .
Remark 2.2 The class of relatively asymptotically nonexpansive mappings was first considered in Agarwal et al. [30].
Recall that a mapping T is said to be quasi-ϕ-nonexpansive iff
Recall that a mapping T is said to be asymptotically quasi-ϕ-nonexpansive iff there exists a sequence with as such that
Remark 2.3 The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasi-ϕ-nonexpansive mappings and asymptotically quasi-ϕ-nonexpansive mappings do not require the restriction ; for more details, see [31–33] the reference therein.
Remark 2.4 The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are generalizations of the class of quasi-nonexpansive mappings and the class of asymptotically quasi-nonexpansive mappings in Banach spaces.
Recall that T is said to be asymptotically quasi-ϕ-nonexpansive in the intermediate sense iff and the following inequality holds:
Put
It follows that as . Then (2.6) is reduced to the following:
Remark 2.5 The class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense was first considered by Qin and Wang [34]; see also [35].
Remark 2.6 The class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense is a generalization of the class of asymptotically quasi-nonexpansive mappings in the intermediate sense, which was considered by Kirk [36] in the framework of Banach spaces.
In order to state our main results, we also need the following lemmas.
Lemma 2.7 [29]
Let C be a nonempty closed convex subset of a smooth Banach space E and . Then if and only if
Lemma 2.8 [29]
Let E be a reflexive, strictly convex, and smooth Banach space, let C be a nonempty closed convex subset of E and . Then
Lemma 2.9 [37]
Let E be a smooth, strictly convex, and reflexive Banach space, and let C be a nonempty closed convex subset of E. Let be an α-inverse-strongly monotone mapping and f be a bifunction satisfying conditions (A1)-(A4). Let be any given number, and let define a mapping as follows: for any ,
Then the following conclusions hold:
-
(1)
is single-valued;
-
(2)
is a firmly nonexpansive-type mapping, i.e., for all ,
-
(3)
;
-
(4)
is quasi-ϕ-nonexpansive;
-
(5)
-
(6)
is closed and convex.
Lemma 2.10 [38]
Let E be a smooth and uniformly convex Banach space, and let . Then there exists a strictly increasing, continuous, and convex function such that and
for all and .
3 Main results
Theorem 3.1 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property, and let C be a nonempty closed and convex subset of E. Let f be a bifunction from to ℝ satisfying (A1)-(A4), and let N be some positive integer. Let be a -inverse-strongly monotone mapping. Let be an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense for every . Assume that is closed asymptotically regular on C and is nonempty and bounded. Let be a sequence generated in the following manner:
where , is a real number sequence in for every , is a real number sequence in , where k is some positive real number. Assume that and for every . Then the sequence converges strongly to , where is the generalized projection from E onto .
Proof Since is closed and convex for every , we obtain from Lemma 2.9 that the common element set is closed and convex. Next, we show that both and are closed and convex. From the definition of and , it is obvious that is closed and is closed and convex. We show that is convex since is equivalent to
It follows that is convex. This in turn shows that is well defined. Putting , from Lemma 2.9 we see that is quasi-ϕ-nonexpansive. Now, we are in a position to prove that . Let ,
We have . This implies that . On the other hand, we see that . Suppose that for some m. There exists an element such that . In view of Lemma 2.7, we find that
Since , we arrive at
for every . We therefore find that . It follows that . This shows that the sequence is well defined.
Next, we prove that the sequence is bounded. It follows from the definition of and Lemma 2.7 that . In view of Lemma 2.8, we find that
for each . This implies that is bounded. It follows from (2.1) that is also bounded. Since , we find from Lemma 2.7 that . Therefore, we obtain that is nondecreasing. So there exists the limit of . It follows from Lemma 2.8 that
This shows that . Since , we find that . Since the space is reflexive, we may assume, without loss of generality, that . Since is closed and convex, we find that . This implies from that . On the other hand, we see from the weakly lower semicontinuity of that
which implies that . Hence, we have . In view of the Kadec-Klee property of E, we find that as . In view of (2.1), we see that . This implies that . That is,
This implies that is bounded. Note that both E and are reflexive. We may assume, without loss of generality, that . In view of the reflexivity of E, we see that . This shows that there exists an element such that . It follows that
Taking on the both sides of the equality above yields that
That is, , which in turn implies that . It follows that . Since enjoys the Kadec-Klee property, we obtain from (3.3) that . Since is demi-continuous and E enjoys the Kadec-Klee property, we obtain that , as . Note that . It follows that
Since J is uniformly norm-to-norm continuous on any bounded sets, we have
On the other hand, we have
We, therefore, find that
Since E is uniformly smooth, we know that is uniformly convex. In view of Lemma 2.10, we find that
It follows that . In view of the restriction on the sequences, we find from (3.6) that . It follows that . In the same way, we obtain that , . Notice that . It follows that . The demicontinuity of implies that . Note that
This implies that . Since E has the Kadec-Klee property, we obtain that . On the other hand, we have
It follows from the asymptotic regularity of that . That is, . From the closedness of , we find for every . This proves . Now, we state that . In view of Lemma 2.9, we find that
It follows from (3.6) that . This implies that . It follows from (3.4) that . It follows that
This shows that is bounded. Since is reflexive, we may assume that . In view of , we see that there exists such that . It follows that . Taking on the both sides of the equality above yields that
That is, , which in turn implies that . It follows that . Since enjoys the Kadec-Klee property, we obtain that as . Note that is demi-continuous. It follows that . Since E enjoys the Kadec-Klee property, we obtain that as . Note that . This implies that . Since J is uniformly norm-to-norm continuous on any bounded sets, we have . In view of the assumption , we see that
Since , we find that
where for every . It follows from (A2) that
It follows from (3.7) that
For and , define . It follows that , which yields that . It follows from (A1) and (A4) that
That is,
Letting , we obtain from (A3) that , . That is, for every . This implies that . This completes the proof that . Since . In view of Lemma 2.7, we find that
Since , we arrive at
Letting in the above inequality, we obtain that
It follows from Lemma 2.7 that . This completes the proof. □
Remark 3.2 Theorem 3.1 includes the corresponding results in the literature as special cases. Since every uniformly convex Banach space enjoys the Kadec-Klee property, the framework of the space can be applicable to , .
Next, we state a result on Ky Fan inequality (2.4) and a single asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense.
Corollary 3.3 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property, and let C be a nonempty closed and convex subset of E. Let f be a bifunction from to ℝ satisfying (A1)-(A4). Let be an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. Assume that T is closed asymptotically regular on C and is nonempty and bounded. Let be a sequence generated in the following manner:
where , is a real number sequence in , is a real number sequence in , where k is some positive real number. Assume that . Then the sequence converges strongly to , where is the generalized projection from E onto .
If the mapping T is quasi-ϕ-nonexpansive, we find from Corollary 3.3 the following.
Corollary 3.4 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property, and let C be a nonempty closed and convex subset of E. Let f be a bifunction from to ℝ satisfying (A1)-(A4). Let be a quasi-ϕ-nonexpansive mapping. Assume that is nonempty. Let be a sequence generated in the following manner:
where is a real number sequence in , is a real number sequence in , where k is some positive real number. Assume that . Then the sequence converges strongly to , where is the generalized projection from E onto .
Finally, we give a result in the framework of Hilbert spaces based on Theorem 3.1.
Corollary 3.5 Let E be a Hilbert space, and let C be a nonempty closed and convex subset of E. Let f be a bifunction from to ℝ satisfying (A1)-(A4), and let N be some positive integer. Let be a -inverse-strongly monotone mapping. Let be an asymptotically quasi-nonexpansive mapping in the intermediate sense for every . Assume that is closed asymptotically regular on C and is nonempty and bounded. Let be a sequence generated in the following manner:
where , is a real number sequence in for every , is a real number sequence in , where k is some positive real number. Assume that and for every . Then the sequence converges strongly to , where is the metric projection from E onto .
Proof Since and in the framework of Hilbert spaces, we draw the desired conclusion immediately. □
References
Park S:A review of the KKM theory on -space or GFC-spaces. Adv. Fixed Point Theory 2013, 3: 355–382.
Cho SY, Li W, Kang SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013., 2013: Article ID 199
Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi- ϕ -nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 10
Wang ZM, Lou W: A new iterative algorithm of common solutions to quasi-variational inclusion and fixed point problems. J. Math. Comput. Sci. 2013, 3: 57–72.
Wang G, Sun S: Hybrid projection algorithms for fixed point and equilibrium problems in a Banach space. Adv. Fixed Point Theory 2013, 3: 578–594.
Zegeye H, Shahzad N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory 2012, 2: 374–397.
Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011, 24: 224–228. 10.1016/j.aml.2010.09.008
Qin X, Cho SY, Kang SM: Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications. J. Comput. Appl. Math. 2009, 233: 231–240. 10.1016/j.cam.2009.07.018
Lv S: Strong convergence of a general iterative algorithm in Hilbert spaces. J. Inequal. Appl. 2013., 2013: Article ID 19
Yuan Q: Some results on asymptotically quasi- ϕ -nonexpansive mappings in the intermediate sense. J. Fixed Point Theory 2012., 2012: Article ID 1
Cho SY, Qin X, Kang SM: Iterative processes for common fixed points of two different families of mappings with applications. J. Glob. Optim. 2013. 10.1007/s10898-012-0017-y
He RH: Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequalities in FC -spaces. Adv. Fixed Point Theory 2012, 2: 47–57.
Tanaka Y: A constructive version of Ky Fan’s coincidence theorem. J. Math. Comput. Sci. 2012, 2: 926–936.
Qin X, Cho SY, Kang SM: An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings. J. Glob. Optim. 2011, 49: 679–693. 10.1007/s10898-010-9556-2
Al-Bayati AY, Al-Kawaz RZ: A new hybrid WC-FR conjugate gradient-algorithm with modified secant condition for unconstrained optimization. J. Math. Comput. Sci. 2012, 2: 937–966.
Qin X, Cho SY, Kang SM: Iterative algorithms for variational inequality and equilibrium problems with applications. J. Glob. Optim. 2010, 48: 423–445. 10.1007/s10898-009-9498-8
Chen JH: Iterations for equilibrium and fixed point problems. J. Nonlinear Funct. Anal. 2013., 2013: Article ID 4
Yang S: A proximal point algorithm for zeros of monotone operators. Math. Finance Lett. 2013., 2013: Article ID 7
Song J, Chen M: On generalized asymptotically quasi- ϕ -nonexpansive mappings and a Ky Fan inequality. Fixed Point Theory Appl. 2013., 2013: Article ID 237
Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607–1618.
Takahashi W, Yao JC: Nonlinear operators of monotone type and convergence theorems with equilibrium problems in Banach spaces. Taiwan. J. Math. 2011, 15: 787–818.
Takahashi W, Wong NC, Yao JC: Iterative common solutions for monotone inclusion problems, fixed point problems and equilibrium problems. Fixed Point Theory Appl. 2012., 2012: Article ID 181
Noor MA, Noor KI, Waseem M: Decomposition method for solving system of linear equations. Eng. Math. Lett. 2013, 2: 34–41.
Qing Y, Kim JK: Weak convergence of algorithms for asymptotically strict pseudocontractions in the intermediate sense and equilibrium problems. Fixed Point Theory Appl. 2012., 2012: Article ID 132
Kim JK: A new iterative algorithm of pseudomonotone mappings for equilibrium problems in Hilbert spaces. J. Inequal. Appl. 2013., 2013: Article ID 128
Kang SM, Cho SY, Liu Z: Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings. J. Inequal. Appl. 2010., 2010: Article ID 827082
Wu C: Mann iteration for zero theorems of accretive operators. J. Fixed Point Theory 2013., 2013: Article ID 3
Showalter RE Math. Surveys Monogr. 49. In Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Am. Math. Soc., Providence; 1997.
Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos AG. Dekker, New York; 1996.
Agarwal RP, Cho YJ, Qin X: Generalized projection algorithms for nonlinear operators. Numer. Funct. Anal. Optim. 2007, 28: 1197–1215. 10.1080/01630560701766627
Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011
Zhou H, Gao G, Tan B: Convergence theorems of a modified hybrid algorithm for a family of quasi- ϕ -asymptotically nonexpansive mappings. J. Appl. Math. Comput. 2010, 32: 453–464. 10.1007/s12190-009-0263-4
Qin X, Cho SY, Kang SM: On hybrid projection methods for asymptotically quasi- ϕ -nonexpansive mappings. Appl. Math. Comput. 2010, 215: 3874–3883. 10.1016/j.amc.2009.11.031
Qin X, Wang L: On asymptotically quasi- ϕ -nonexpansive mappings in the intermediate sense. Abstr. Appl. Anal. 2012., 2012: Article ID 636217
Hecai Y, Aichao L: Projection algorithms for treating asymptotically quasi- ϕ -nonexpansive mappings in the intermediate sense. J. Inequal. Appl. 2013., 2013: Article ID 265
Kirk WA: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type. Isr. J. Math. 1974, 17: 339–346. 10.1007/BF02757136
Chang SS, Chan CK, Lee HWJ: A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces, with applications. Nonlinear Anal. 2010, 73: 2260–2270. 10.1016/j.na.2010.06.006
Zǎlinescu C: On uniformly convex functions. J. Math. Anal. Appl. 1983, 95: 344–374. 10.1016/0022-247X(83)90112-9
Acknowledgements
The authors are grateful to the two anonymous reviewers’ suggestions which improved the contents of the article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to this manuscript. Both authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Qing, Y., Lv, S. A strong convergence theorem for solutions of equilibrium problems and asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense. Fixed Point Theory Appl 2013, 305 (2013). https://doi.org/10.1186/1687-1812-2013-305
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-305