A New Iterative Algorithm for Pseudomonotone Equilibrium Problem and a Finite Family of Demicontractive Mappings

Ogbuisi, F. U.; Isiogugu, F. O.

doi:https://doi.org/10.1155/2020/3183529

Abstract and Applied Analysis

On this page

Abstract Introduction Preliminaries Numerical Example Data Availability Disclosure Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 3183529 | https://doi.org/10.1155/2020/3183529

A New Iterative Algorithm for Pseudomonotone Equilibrium Problem and a Finite Family of Demicontractive Mappings

F. U. Ogbuisi^1,2and F. O. Isiogugu^1,2,3

Academic Editor: Patricia J. Y. Wong

Received07 Apr 2019

Accepted22 Aug 2019

Published20 Mar 2020

Abstract

In this paper, we introduce a new iterative method in a real Hilbert space for approximating a point in the solution set of a pseudomonotone equilibrium problem which is a common fixed point of a finite family of demicontractive mappings. Our result does not require that we impose the condition that the sum of the control sequences used in the finite convex combination is equal to 1. Furthermore, we state and prove a strong convergence result and give some numerical experiments to demonstrate the efficiency and applicability of our iterative method.

1. Introduction

In this paper, we will always take C to be a nonempty closed and convex subset of a real Hilbert space H endowed with inner product and induced norm , and denotes the set of fixed points of a mapping , that is, .

Definition 1. A mapping is said to be(1)Nonspreading [1] if or equivalently(2)κ-Strictly pseudononspreading [1] if there exists such that(3)β-Strictly pseudocontractive [2] if there exists such that(4)ρ-Demicontractive [3] if there exists such thatObviously, the class of demicontractive mappings is more general than the class of quasi-nonexpansive mappings. Moreover, If T is κ-strictly pseudononspreading (or κ-strictly pseudocontractive) and , then T is κ-demicontractive.

Definition 2. A bifunction is(i)Strongly monotone on C with modulus (β-strongly monotone on C) if and only if(ii)monotone on C if and only if(iii)pseudomonotone on C if and ifLet be a bifunction such that , for all . We consider the equilibrium problem (EP) in the sense of Blum and Oettli [4], which is to findLet denote the set of solutions of EP (9). If , where , then EP (9) reduces to the variational inequality problem:EPs form a very important area of research and have recently been considered in many research papers. EP (9) is applied in solving problems from optimization, variational inequality, Kakutani fixed point, Nash equilibria in noncooperative game theory, and minimax problems [4, 5].
A popular method that has been applied to solve EP (9) is the subgradient projection method which is developed from the steepest descent projection method in smooth optimization. If bifunction f is convex, subdifferentiable with respect to the second argument, Lipschitz, and strongly monotone on C, then regularization parameters can be chosen such that the subgradient projection method is linearly convergent [6]. However, when f is only monotone, the subgradient projection method may not be convergent. In order to get a method that guarantees convergence for pseudomonotone equilibrium problems (that is, equilibrium problems for pseudomonotone bifunctions) the extragradient (or double projection) method developed by Korpelevich [7] was extended to equilibrium problems. However, the extragradient algorithms involve two projections on the admissible set C, which may be costly to compute if the nature of the admissible set C is complicated. In the light of the need to obtain a more efficient algorithm, the inexact subgradient algorithms using only one projection [8, 9] has been proposed for solving equilibrium problems with paramonotone equilibrium bifunctions. Some other methods that have been utilized to solve equilibrium problems include the auxiliary problem principle method [10], gap function method [11], and the Tikhonov and proximal point regularization methods [12–15].
Recently, the problem of finding a common point in and the set of fixed points of mappings has become an attractive and interesting subject [16–22]. This interest is because of the possible application of these problems to mathematical models whose constraints can be present as fixed points of mappings and/or (EP). Such a problem occurs, in particular, in the practical problems as signal processing, network resource allocation, image recovery (see [23, 24]).
In 2007, Tada and Takahashi [22] proposed the following iterative algorithm for approximating a common element of the set of solutions of equilibrium problem for monotone bifunctions and the set of fixed points of a nonexpansive mapping T.

Algorithm 1. where is the regularization parameter at iteration and is the metric projection onto . They assume that f is a monotone bifunction and obtained a strong convergence result.
Recently, Anh and Muu [25] proposed a new type of algorithm which uses only one projection and does not require any Lipschitz condition for the bifunctions for finding a common point in the solution set of the class of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mappings. More precisely, they gave an iteration scheme generated as follows.

Algorithm 2. Pick . At each iteration , do the following:Inspired by Anh and Muu [25], Wangkeeree et al. [26] presented an iterative method for finding hierarchically an element in with respect to a nonexpansive mapping. Precisely, they considered the following problems:where T and S are nonexpansive mappings.
Other authors have also considered different algorithms which involve either projection mapping or projection mapping and the construction of sequences of sets and for approximating a common solution of pseudomonotone equilibrium problems and fixed point problems of nonexpansive mappings ( see, for example, [27–29]). Those methods are tasking and difficult to compute.
In 2018, Thong and Hieu [30] proposed the following iterative algorithm for the approximation of a common fixed point of a finite family of demicontractive operators. Let be a sequence in H defined byAmong other standard assumptions, they assumed that is a finite sequence of positive numbers such thatLet K be a nonempty closed and convex subset of a real Hilbert space H. Suppose that , is a countable finite family of mappings . In [31], the authors consider the horizontal iteration process generated from an arbitrary for the finite family of mappings , using a finite family of th control sequences as follows.
For N = 2,For N = 3,For an arbitrary but finite ,

1.1. Question

Is it possible to give an iterative algorithm and obtain a strong convergence result for finding a common element in the set of fixed points of a finite family of demicontractive mappings which also solves equilibrium problems for pseudomonotone bifunctions without imposing the type of condition in (15) on the control sequences?

In this paper, motivated by the works of Anh and Muu [25] and Wangkeeree et al. [26], we propose an iterative algorithm for finding a common element in the set of fixed points of a finite family of demicontractive mappings, which also solves equilibrium problems for pseudomonotone bifunctions and prove a strong convergence result which does not require such condition as in (15) on the control sequences. We further give a numerical experiment to demonstrate the performance of our iterative algorithm.

2. Preliminaries

In the sequel, we shall need the following definitions and lemmas. Let H be a real Hilbert space, and C a nonempty, closed, and convex subset of H. By , we denote the metric projection operator onto C, that is,

Lemma 1. Suppose that C is a nonempty, closed, and convex subset in H. Then, has the following properties:

Lemma 2 (see [32]). Let and be two nonnegative real sequences satisfying the following conditions:

Then, exists.

Lemma 3 (see [33]). Let H be a real Hilbert space, C a closed convex subset of H, and let be a continuous pseudocontractive mapping, then(i)F(T) is closed convex subset of C(ii) is demiclosed at zero, i.e., if is a sequence in C such that and , as , then

Lemma 4 (see [1]). Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and let be a ρ-strictly pseudononspreading mapping. If , then it is closed and convex.

Lemma 5 (see [1]). Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and let be a ρ-strictly pseudononspreading mapping. Then, is demiclosed at 0.

Definition 3. Let C be a nonempty closed and convex subset of a Hilbert space E. Let be a bifunction where is a convex function for each . Then, the ϵ-subdifferential (ϵ-diagonal subdifferential) of f at x denoted by is given by

Lemma 6. Let be a countable subset of the set of real numbers , where is an arbitrary integer. Then, the following holds:

Proof. This result has been proved in [31], but for the sake of completeness, we present the proof again here. For ,We assume it is true for N and prove for N + 1.

Remark 1. Lemma 6 holds if is replaced with .

Lemma 7 (see also [31]). Let t and u be arbitrary elements of a real Hilbert space H, and let be such that . Let and be a countable finite subset of H and , respectively. Define

Then,

Proof. Let , and . Observe that, for , . Using the well-known identitywhich holds for all and for all , we have

3. Main Results

We now give a strong convergence result for the approximation of a solution of a pseudomonotone equilibrium problem which is also a common fixed point of a finite family of demicontractive mappings.

Let be a bifunction that satisfies the following conditions: (B1) f is pseudomonotone on C, and is lower semicontinuous for all (B2) is sequentially weakly upper semicontinuous on C for each fixed point i.e., if is a sequence converging weakly to then ; (B3) is convex and subdifferentiable on E for every fixed ; (B4) If is bounded as then the sequence with is bounded; (B5) f is pseudomonotone on C with respect to every and satisfies the strict paramonotonicity property, i.e.,

It has been proved that under the conditions (B1)–(B3), the solution set of EP (9) is closed and convex [34].

Algorithm 3. Step 1. Let and be sequences of nonnegative real numbers and let , be -demicontractive mappings on . Suppose that the following conditions are satisfied:(1).(2).(3). Step 2.

Theorem 1. Let C be a closed and convex subset of a real Hilbert space H, and let be a bifunction that satisfies conditions (B1)–(B5). Let be a finite family of -demicontractive mappings such that is demiclosed at 0 for each i and . Then, the sequences and generated by Algorithm 3 converges strongly to a point , where .

Proof. Let From and , we haveTherefore,Since we also haveFrom (33) and (34), we haveThat is,Therefore,Moreover, since and , we haveIt then follows from the definitions of and thatThus, from (33)–(39), we obtainButTherefore, from (40) and (41), we getNow, from Proposition 2.9, , and is -demicontractive, for , we havewhereFrom (42) and (43), we haveFurthermore, since then for all Therefore, by the pseudomonotonicity of f with respect to , we have , for all It then follows from that Hence, we have from (45) thatApplying Lemma 2 to (46), we obtain the existence ofSince f is pseudomonotone on C and , we have . Thus, from (45), we obtainSumming up the inequality (48), for every k, we haveSince the sequences and are bounded, then there exists a constant such that . Thus, we haveThus, from (49), we obtainThus, and .
Since and , we have thatFor any , suppose that is the subsequence of such thatand without loss of generality, we may assume that as for some .
Next, we show that is in . Observe that since is weakly upper semicontinuous, we haveBut since f is pseudomonotone with respect to and , we haveFrom (54) and (55), we conclude that Thus, by condition B6, we obtain that is in .
Since , we have thatFurthermore, with and , we havewhich implies that, and thereforeIt then follows from that . Since are demiclosed at , and . We have that .
Now, observe thatConsequently,We now show thatIt follows from (46) that, for all ,where , for all and .
Now, using Lemma 1, we haveThat is,For all , since is convex, we have . Therefore,Replacing with in (62), we have thatFrom (65) and (66), we haveHence,which implies that is a Cauchy sequence. Hence, strongly converges to some point . However, , we obtain in the limit thatThus, from (64), we conclude that , and consequently .

Corollary 1. Let C be a closed and convex subset of a real Hilbert space H, and let be a bifunction that satisfies conditions (B1)–(B5). Let be a finite family of -strictly pseudononspreading mappings, such that . Then, the sequences and generated by Algorithm 3 converge strongly to a point , where .

Corollary 2. Let C be a closed and convex subset of a real Hilbert space H and let be a bifunction that satisfies conditions (B1)–(B5). Let be a finite family of -strictly pseudocontractive mappings, such that . Then, the sequences and generated by Algorithm 3 converge strongly to a point where .

4. Numerical Example

We now give a numerical example to demonstrate the performance and efficiency of our proposed iterative algorithm. Let and . Set and define by

Observe that . Moreover, , and thus letting , we have that . Take and . Therefore, (32) is expressed as an iteration:

We make different choices of and use for stopping criterion. Figures 1–4 are the graphs of the numerical computations of Algorithm 4.2 (71) corresponding, respectively, to and

Data Availability

The data used to support the findings of this study are included within the article.

Disclosure

Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

The work of the first author was based on the research supported wholly by the National Research Foundation (NRF) of South Africa (grant number 111992). The second author acknowledges the financial support from Department of Science and Technology and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DST-NRF COE-MaSS) Postdoctoral Fellowship (grant numbers BA 2018/012).

References

M. O. Osilike and F. O. Isiogugu, “Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 5, pp. 1814–1822, 2011.
View at: Publisher Site | Google Scholar
F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 20, no. 2, pp. 197–228, 1967.
View at: Publisher Site | Google Scholar
S. A. Naimpally and K. L. Singh, “Extensions of some fixed point theorems of Rhoades,” Journal of Mathematical Analysis and Applications, vol. 96, no. 2, pp. 437–446, 1983.
View at: Publisher Site | Google Scholar
E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” Mathematics student, vol. 63, no. 1–4, pp. 123–145, 1994.
View at: Google Scholar
L. Muu and W. Oettli, “Convergence of an adaptive penalty scheme for finding constrained equilibria,” Nonlinear Analysis: Theory, Methods & Applications, vol. 18, no. 12, pp. 1159–1166, 1992.
View at: Publisher Site | Google Scholar
L. D. Muu and T. D. Quoc, “Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot equilibrium model,” Journal of Optimization Theory and Applications, vol. 142, no. 1, pp. 185–204, 2009.
View at: Publisher Site | Google Scholar
G. M. Korpelevich, “An extragradient method for finding saddle points and for other problems,” Ekonomika i Matematicheskie Metody, vol. 12, no. 4, pp. 747–756, 1976.
View at: Google Scholar
J. Y. Bello Cruz, P. S. Santos, and S. Scheimberg, “A two-phase algorithm for a variational inequality formulation of equilibrium problems,” Journal of Optimization Theory and Applications, vol. 159, no. 3, pp. 562–575, 2012.
View at: Google Scholar
P. Santos and S. Scheimberg, “An inexact subgradient algorithm for equilibrium problems,” Journal of Computational and Applied Mathematics, vol. 30, no. 1, pp. 91–107, 2011.
View at: Google Scholar
M. A. Noor, “Auxiliary principle technique for equilibrium problems,” Journal of Optimization Theory and Applications, vol. 122, no. 2, pp. 371–386, 2004.
View at: Publisher Site | Google Scholar
G. Mastroeni, “Gap functions for equilibrium problems,” Journal of Global Optimization, vol. 27, no. 4, pp. 411–426, 2003.
View at: Google Scholar
P. G. Hung and L. D. Muu, “The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 17, pp. 6121–6129, 2011.
View at: Publisher Site | Google Scholar
A. N. Iusem and W. Sosa, “Iterative algorithms for equilibrium problems,” Optimization, vol. 52, no. 3, pp. 301–316, 2003.
View at: Publisher Site | Google Scholar
I. V. Konnov, “Regularization methods for nonmonotone equilibrium problems,” Journal of Nonlinear and Convex Analysis, vol. 10, no. 1, pp. 93–101, 2009.
View at: Google Scholar
A. Moudafi, “Proximal point algorithm extended to equilibrium problems,” Journal of Natural Geometry, vol. 15, no. 1-2, pp. 91–100, 2009.
View at: Google Scholar
P. N. Anh, “A hybrid extragradient method extended to fixed point problems and equilibrium problems,” Optimization, vol. 62, no. 2, pp. 271–283, 2013.
View at: Publisher Site | Google Scholar
P. N. Anh, “A logarithmic quadratic regularization method for solving pseudo-monotone equilibrium problem,” Acta Mathematica Vietnamica, vol. 34, no. 2, pp. 183–200, 2009.
View at: Google Scholar
P. N. Anh and D. X. Son, “A new iterative scheme for pseudomonotone equilibrium problems and a finite family of pseudocontractions,” Applied Mathematics & Information Sciences, vol. 29, pp. 1179–1191, 2011.
View at: Google Scholar
K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 8, pp. 2350–2360, 2007.
View at: Publisher Site | Google Scholar
J.-K. Kim, P. N. Anh, and Y.-M. Nam, “Strong convergence of an extragradient method for equilibrium problems and fixed point problems,” Journal of the Korean Mathematical Society, vol. 49, no. 1, pp. 187–200, 2012.
View at: Publisher Site | Google Scholar
A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem,” Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359–370, 2007.
View at: Publisher Site | Google Scholar
S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 506–515, 2007.
View at: Publisher Site | Google Scholar
H. Iiduka, “A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping,” Optimization, vol. 59, no. 5-6, pp. 873–885, 2010.
View at: Publisher Site | Google Scholar
P.-E. Maingé, “A hybrid extragradient-viscosity method for monotone operators and fixed point problems,” SIAM Journal on Control and Optimization, vol. 47, no. 3, pp. 1499–1515, 2008.
View at: Publisher Site | Google Scholar
P. N. Anh and L. D. Muu, “A hybrid subgradient algorithm for nonexpansive mappings and equilibrium problems,” Optimization Letters, vol. 8, no. 2, pp. 727–738, 2014.
View at: Publisher Site | Google Scholar
R. Wangkeeree, K. Rattanaseeha, and R. Wangkeeree, “A hybrid subgradient Algorithm for finding a common solution of pseudomonotone equilibrium problems and Hierarchical fixed point problems of nonexpansive mappings,” Thai Journal of Mathematics, vol. 16, no. 1, pp. 61–77, 2018.
View at: Google Scholar
P. N. Anh, N. D. Hien, N. X. Phuong, and V. T. Lingua, “A parallel subgradient method extended to variational inequalities involving nonexpansive mappings,” Applicable Analysis, pp. 1–17, 2019.
View at: Publisher Site | Google Scholar
P. N. Anh and N. X. Phuong, “A parallel extragradient-like projection method for unrelated variational inequalities and fixed point problems,” Journal of Fixed Point Theory and Applications, vol. 20, no. 2, 2018.
View at: Publisher Site | Google Scholar
P. N. Anh and D. D. Thanh, “Linesearch methods for equilibrium problems and an infinite family of nonexpansive mappings,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 38, no. 3, pp. 1157–1175, 2015.
View at: Publisher Site | Google Scholar
D. V. Thong and D. V. Hieu, “A new approximation method for finding common fixed points of families of demicontractive operators and applications,” Journal of Fixed Point Theory and Applications, vol. 20, no. 2, 2018.
View at: Publisher Site | Google Scholar
F. O. Isiogugu, C. Izuchukwu, and C. C. Okeke, New Iteration Scheme for Approximating a Common Fixed Point of a Finite Family of Mappings, Journal of Mathematics, In press.
K. K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301–308, 1993.
View at: Publisher Site | Google Scholar
Q. B. Zhang and C. Z. Cheng, “Strong convergence theorem for a family of Lipschitz pseudocontractive mappings in a Hilbert space,” Mathematical and Computer Modelling, vol. 48, no. 3-4, pp. 480–485, 2008.
View at: Publisher Site | Google Scholar
D. Q. Tran, M. L. Dung, and V. H. Nguyen, “Extragradient algorithms extended to equilibrium problems,” Optimization, vol. 57, no. 6, pp. 746–776, 2008.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2020 F. U. Ogbuisi and F. O. Isiogugu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

394

Downloads

1308

Citations