Kyungpook Mathematical Journal 2021; 61(3): 523-558
Published online September 30, 2021
Copyright © Kyungpook Mathematical Journal.
Convergence Theorem for Finding Common Fixed Points of N-generalized Bregman Nonspreading Mapping and Solutions of Equilibrium Problems in Banach Spaces
Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo*
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
e-mail : email@example.com
School of Mathematics, Statistics and Computer Science University of Kwazulu-Natal Durban, South Africa
e-mail : firstname.lastname@example.org
Received: May 4, 2019; Revised: July 28, 2020; Accepted: August 4, 2020
In this paper, we study some fixed point properties of n-generalized Bregman nonspreading mappings in re exive Banach space. We introduce a hybrid iterative scheme for finding a common solution for a countable family of equilibrium problems and fixed point problems in re exive Banach space. Further, we give some applications and numerical example to show the importance and demonstrate the performance of our algorithm. The results in this paper extend and generalize many related results in the literature.
Keywords: nonspreading mapping, Bregman distance, equilibrium problem, fixed point prolem, reflexive Banach space.
The EP(g) was shown by Blum and Oettli  to cover several other optimization problems such as monotone inclusion problems, saddle point problems, minimization problems, variational inequality problems and Nash equilibria in non-cooperative games. In addition, there are many other important problems, for example, the complementarity problem and fixed point problems, which can be written in the form of EP(g) (1.1). Thus, the EP(g) is a unifying model for several problems arising in physics, engineering, science, optimization, economics etc.
In the last two decades, the existence of solutions of the EP(g) have been mentioned in many papers, see for instance [7, 11, 13, 26, 36, 39, 40 ], and several iterative methods have been proposed for solving EP(g) and related optimization problems, see for instance [1, 2, 4, 14, 15, 17, 18, 19, 28, 29, 32, 30, 31, 38, 41, 42] and reference therein. In solving the EP(g) (1.1) it is necessary to assume that the bifunction
g(x,x) =0for all x ∈ C;
gis monotone, that is g(x,y)+g(y,x) ≤ 0for all x,y ∈ C;
(A3) For all
x,y, z∈ C
(A4) For all
x ∈ C, g(x,⋅)is convex and lower semicontinuous.
is called a Bregman distance with respect to
From the definition, we know that the following properties are satisfied (see ):
(i) The three points identity, for any
(ii) Four point identity, for any
Definition 1.2. Let
(a) Bregman nonexpansive  if
Bregman nonspreading  if
Next, we introduce a
Definition 1.3. Let
Remark 1.4. From Definition 1.3,
which is called 2-generalized Bregman nonspreading in the sense of , where
(c) The class of generalized Bregman nonspreading mapping reduces to Bregman nonspreading  if
(d) The class of generalized Bregman nonspreading mapping reduces to Bregman nonexpansive  if
We now present an example of Bregman nonspreading mapping which is not nonspreading in the usual Hilbert space setting.
Example 1.5. Let
does not hold. Taking
By simple calculations, we obtain
We further give an example of 2-generalized Bregman nonspreading mapping which is not necessarily 1-generalized Bregman nonspreading.
Example 1.6. Let
It is easy to see that
Case I: Suppose
Case II: Suppose
Case III: Suppose
Choosing suitable choices of
In 2010, by making use of the Bregman projection, Reich and Sabach  studied some approximation methods for finding common zeros of maximal monotone operators in reflexive Banach spaces. They also studied some approximation techniques for finding common solutions of finitely many Bregman nonexpansive operators, see . In the same sense, Kassay et al.  studied the approximation of solutions of system of variational inequalities in reflexive Banach spaces. It is worth noting that extension of many theory from Hilbert space to general Banach space suffer some difficulties because many of the useful techniques employed in Hilbert space (for instance the inner product and the nonexpansiveness of resolvent operators) are no longer valid in Banach spaces setting.
Motivated by the works given in [21, 35, 46], we prove some properties of the
In this section, we recall some definitions and preliminary results which will be used in the sequel. We denote the strong convergence (resp. weak convergence) of a sequence
and the Frénchet conjugate of
If the limit in (2.1) exists as
The notion of Legendre function in infinite dimensional spaces was first introduced by Bauschke, Borwein and Combettes in . By their definition, the conditions (L1) and (L2) also yield that
One important and interesting example of Legendre function is
Definition 2.1. Let
It is known from  that z = Proj
We also have
Similar to the metric projection in Hilbert space, the Bregman projection also has a variational characterization which is given below.
Lemma 2.3.  (Characterization of Bregman Projection)) Let
(i)the vector ω is the Bregman projection of x onto C, with respect to
(ii) the vector ω is the unique solution of the variational inequality
(iii) the vector ω is the unique solution of the inequality
Definition 2.4. Let
(i) totally convex at
is positive for any
(ii) totally convex if it is totally convex at every point
(iii) totally convex on bounded subset B of E, if
(iv) cofinite if dom
(v) coercive if
(vi) sequentially consistent if for any two sequences
Lemma 2.5. () The function
Lemma 2.6. () Let
Lemma 2.7. () Let
Lemma 2.8. () If
Lemma 2.9. () Let
Lemma 2.10. () Let
Recall that a mapping
Lemma 2.11. () Let
Define the resolvent mapping
It is easy to see that the resolvent operator satisfies the following inequality: for all
3. Main Results
In this section, we present the existence and some properties of fixed points of
Proposition 3.1. Let E be a real reflexive Banach space and
Thus, we obtain
Therefore by Lemma 2.10,
The following results follow as direct consequences of Theorem 3.1.
Corollary 3.2. Let
Corollary 3.3. Let
Remark 3.4. Corollary 3.2 is a generalization of the corresponding result in Theorem 3.2 of , where the equivalence between the two assertions was shown for
We now show another important property of the fixed points of
Proposition 3.5. Let
which implies that
This means that
Next, we show that
Using Corollary 3.3 and Proposition 3.5, we prove the following common fixed point theorem for a commutative family of
Theorem 3.6. Let
Suppose that for some
The following result will be used in the sequel.
Proposition 3.7. Let E be a real reflexive Banach space and let C be a nonempty, closed and convex subset of E. Let
Hence, from the three points identity (1.2), we have
The following result is another important property which characterized the
Proposition 3.8. Let
We show that
Using the four points identity (1.3), we have
4. Convergence Analysis
In this section, we introduce a hybrid algorithm for finding common solutions of countable family of equilibrium problem and finite fixed points of n-generalized Bregman nonspreading mapping in reflexive Banach space.
Using the above definition, we have the following lemma.
Proposition 4.1. Let C be a nonempty, closed and convex subset of a real reflexive Banach space E and let
(iii) If in addition,
This implies that
Then by the property of
It follows that
Using three points identity (1.2), we obtain
Therefore from (4.3), we have
Taking limit as
By the norm-to-norm uniform continuity of
We next prove that
Taking limit as
Similarly, we have
Taking limit as
Following similar approach as above, we have
This together with the Bregman relative nonexpansiveness of each
We are now in position to introduce our iterative algorithm.
Theorem 4.2. Let C be a nonempty, closed and convex subset of a real reflexive Banach space E and
(ii) There exists
Then, the sequence
Step 1: We show that
It is clear that
Step 2: We prove that
In view of Lemma 2.6, we conclude that the sequence
Step 3: Next, we show that
Therefore the sequence
This yields that
By the uniform continuity of
Note that from (4.8), we have
Using the property of
By the uniform continuity of
Hence from (4.7), we get
Also from Lemma 2.11, we have for each
From the assumption (A2), we have
Taking the limit as
Taking the limit of the above inequality, we have
5. Application to Zeros of Maximal Monotone Operators
Sabach  showed that under some properties of the function
A monotone operator
The following result was proved for the mapping
Proposition 5.1. (Sabach ) Let
Based on the above result, we propose the following which can be obtain from Theorem 4.2 for finding common fixed point of finite family of n-generalized Bregman nonspreading mapping and zeros of maximal monotone operators in reflexive Banach space.
Theorem 5.2. Let C be a nonempty, closed and convex subset of a real reflexive Banach space E and
(ii) There exists
Then, the sequence
6. Numerical Example
We give a numerical example to demonstrate the performance of our algorithm (4.7).
Finally, we select the following values
Using Matlab 2016(b) and
Figure 1. Example 6.1, Top-Left: Case(i); Top-Right: Case(ii); Bottom: Case(iii).
The authors sincerely thank the reviewer for his careful reading, constructive comments and fruitful suggestions that substantially improved the manuscript. The first author acknowledges with thanks the bursary and financial support from Department of Science and Innovation and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF COE-MaSS) Doctoral Bursary. The second author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF.
- H. A. Abass, K. O. Aremu, L. O. Jolaoso and O. T. Mewomo, An inertial forward-backward splitting method for approximating solutions of certain optimization problems, J. Nonlinear Funct. Anal., 2020(2020), Art. ID 6, 20 pp.
- T. O. Alakoya, L. O. Jolaoso and O. T. Mewomo, Modified inertia subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems, Optimization(2020), DOI:10.1080/02331934.2020.1723586.
- B. Ali, M. H. Harbau and L. H. Yusuf, Existence theorems for attractive points of semigroups of Bregman generalized nonspreading mappings in Banach spaces, Adv. Oper. Theory, 2(3)(2017), 257-268.
- K. O. Aremu, C. Izuchukwu and G. C. Ugwunnadi, O. T. Mewomo, On the proximal point algorithm and demimetric mappings in CAT(0) spaces, Demonstr. Math., 51(2018), 277-294.
- H. H. Bauschke and J. M. Borwein, Legendre functions and the method of random Bregman projection, J. Convex Anal., 4(1997), 27-67.
- H. H. Bauschke, J. M. Boorwein and P. L. Combettes, Essential smoothness, essential strict convexity and Legendre functions in Banach space, Comm. Contemp. Math, 3(2001), 615-647.
- E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math Stud., 63(1994), 123-145.
- J. M. Borwein, S. Reich and S. Sabach, A characterization of Bregman firmly nonexpansive operators using a new monotonicity concept, J. Nonlinear Convex Anal, 12(2011), 161-184.
- D. Butnariu and A. N. Iusem, Totally convex functions for fixed points computational and infinite dimensional optimization, Kluwer Academic Publishers, Dordrecht, The Netherland, (2000).
- D. Butnariu, E. Resmerita, Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal., (2006), 1-39, Article ID: 84919.
- P. Daniele, F. Giannessi, and A. Mougeri,(eds.), Equilibrium problems and variational models. Nonconvex optimization and its application, vol. 68. Kluwer Academic Publications, Norwell (2003).
- N. Hussain, E. Naraghirad and A. Alotaibi, Existence of common fixed points using Bregman nonexpansive retracts and Bregman functions in Banach spaces, Fixed Point Theory Appl., 2013, 2013:113.
- C. Izuchukwu, G. C. Ugwunnadi, O. T. Mewomo, A. R. Khan, and M. Abbas, Proximal-type algorithms for split minimization problem in p-uniformly convex metric space, Numer. Algorithms, 82(3)(2019), 909-935.
- l. O. Jolaoso, O. T. Alakoya, A. Taiwo and O. T. Mewomo, A parallel combination extragradient method with Armijo line searching for finding common solution of finite families of equilibrium and fixed point problems, Rend. Circ. Mat. Palermo II, (2019), DOI:10.1007/s12215-019-00431-2
- L. O. Jolaoso, T. O. Alakoya, A. Taiwo and O. T. Mewomo, Inertial extragradient method via viscosity approximation approach for solving Equilibrium problem in Hilbert space, Optimization, (2020), DOI:10.1080/02331934.2020.1716752.
- L. O. Jolaoso and O. T. Mewomo, On generalized Bregman nonspreading mappings and zero points of maximal monotone operator in a reflexive Banach space, Port. Math., 76(2019), 229-258. doi: 10.4171/PM/2034.
- L. O. Jolaoso, K. O. Oyewole, C. C. Okeke and O. T. Mewomo, A unified algorithm for solving split generalized mixed equilibrium problem and fixed point of nonspreading mapping in Hilbert space, Demonstr. Math., 51(2018), 211-232.
- L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem, Comput. Appl. Math., 39(1)(2020), Art. 38, 1-28.
- L. O. Jolaoso, A. Taiwo, T. O. Alakoya, and O. T. Mewomo, Strong convergence theorem for solving pseudo-monotone variational inequality problem using projection method in a reflexive Banach space, J. Optim. Theory Appl., (2020), DOI: 10.1007/s10957-020-01672-3.
- G. Kassay, S. Reich and S. Sabach, Iterative methods for solving system of variational inequalities in reflexive Banach spaces, SIAM J. Optim., 21(4)(2011), 1319-1344.
- K. R. Kazmi, and R. Ali, Common solution to an equilibrium problem and a fixed point problem for an asymptotically quasi-φ-nonexpansive mapping in intermediate sense, RACSAM, 111(2017), 877889.
- K. R. Kazmi, R. Ali and S. Yousuf, Generalized equilibrium and fixed point problems for Bregman relatively nonexpansive mappings in Banach spaces, J. Fixed Point Theory Appl., 20:151(2018), doi: 10.1007/s11784-018-0627-1.
- F. Kohsaka, Existence of fixed points of nonspreading mappings with Bregman distance, In: Nonlinear Mathematics for Uncertainty and its Applications, Advances in Intelligent and Soft Computing, Vol. 100, 49(2011), pp. 403-410.
- F. Kohsaka and W. Takahashi, Proximal point algorithms with Bregman funcBanach space, J. Nonlinear Convex Anal., 6(2005), 505523.
- L. -J. Lin, W. Takahashi and Z. -T. Yu, Attractive point theorems for generalized nonspreading mappings in Banach spaces, J. nonlinear and convex analysis, 14(1)(2013), 1-20.
- A. Moudafi, Second order differential proximal methods for equilibrium problems, J. Inequal. Pure Appl. Math., 4(1)(2003), 17.
- E. Naraghirad and J. -C. Yao, Bregman weak relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 2013(2013), Art. No. 141.
- F. U. Ogbuisi and O. T. Mewomo, On split generalized mixed equilibrium problems and fixed point problems with no prior knowledge of operator norm, J. Fixed Point Theory Appl., 19(3)(2016), 2109-2128.
- F. U. Ogbuisi and O. T. Mewomo, Convergence analysis of common solution of certain nonlinear problems, Fixed Point Theory, 19(1)(2018), 335-358.
- G. N. Ogwo, C. Izuchukwu, K. O. Aremu and O. T. Mewomo, A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space, Bull. Belg. Math. Soc. Simon Stevin, 27(2020), 127-152.
- O. K. Oyewole, H. A. Abass and O. T. Mewomo, Strong convergence algorithm for a fixed point constraint split null point problem, Rend. Circ. Mat. Palermo II, (2020), DOI:10.1007/s12215-020-00505-6.
- K. O. Oyewole, L. O. Jolaoso, C. Izuchuwu and O. T. Mewomo, On approximation of common solution of finite family of mixed equilibrium problems with µ-η relaxed monotone operator in a Banach space, Politehin. Uni. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 80(1)(2018), 175-190.
- S. Reich and S. Sabach, A strong convergence theorem for proximal type-algorithm in reflexive Banach spaces, J. Nonlinear Convex Anal., 10(2009), 471-485.
- S. Reich and S. Sabach, Two strong convergence theorem for a proximal method in reflexive Banach spaces, Numer. Funct. Anal. Optim., 31(13)(2010), 22-44.
- S. Reich and S. Sabach, Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces, Nonlinear Anal., 73(1)(2010), 122-135.
- S. Reich and S. Sabach, Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces, Contemp. Math., 568(2012), 225240.
- S. Sabach, Products of finite many resolvents of maximal monotone mappings in reflexive Banach space, SIAM J. Optim, 21(2011), 1289-1308.
- A. Taiwo, T. O. Alakoya and O. T. Mewomo, Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces, Numer. Algorithms, (2020), DOI: 10.1007/s11075-020-00937-2.
- A. Taiwo, L. O. Jolaoso and O. T. Mewomo, A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces, Comput. Appl. Math., 38(2)(2019), Article 77.
- A. Taiwo, L. O. Jolaoso and O. T. Mewomo, Parallel hybrid algorithm for solving pseudomonotone equilibrium and Split Common Fixed point problems, Bull. Malays. Math. Sci. Soc., 43(2020), 1893-1918.
- A. Taiwo, L. O. Jolaoso and O. T. Mewomo, Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert Spaces, J. Ind. Manag. Optim., (2020), DOI:10.3934/jimo.2020092.
- A. Taiwo, L. O. Jolaoso and O. T. Mewomo, General alternative regularization method for solving Split Equality Common Fixed Point Problem for quasi- pseudocontractive mappings in Hilbert spaces, Ricerche Mat., (2019), DOI: 10.1007/s11587-019-00460-0.
- W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and its Applications, Yokohama Publishers, Yokohama, 2000.
- W. Takahashi, N. -C. Wong and J. -C. Yao, Fixed point theorems for three new nonlinear mappings in Banach spaces, J. Nonlinear Convex Anal., 13(2012), 368-381.
- W. Takahashi, N. -C. Wong and J. -C. Yao, Fixed point theorems and convergence theorems for generalized nonspreading mappings in Banach spaces, J. Fixed Point Theory and Appl., 11(2012), 159-183.
- W. Takahashi and K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mmapings in Banach spaces, Nonlinear Anal., 70(2009) 45-57.