Journal of Mathematics
Volume 2019 (2019), Article ID 8059135, 11 pages
https://doi.org/10.1155/2019/8059135

Research Article

A Halpern-Type Iteration Method for Bregman Nonspreading Mapping and Monotone Operators in Reflexive Banach Spaces

F. U. Ogbuisi,1,2 L. O. Jolaoso,¹ and F. O. Isiogugu1,2,3

¹School of Mathematics, Statistics and Computer Science, University of Kwazulu-Natal, Durban, South Africa
²Department of Mathematics, University of Nigeria, Nsukka, Nigeria
³DST-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Johannesburg, South Africa

Correspondence should be addressed to F. U. Ogbuisi; ferdinard.ogbuisi@unn.edu.ng

Received 14 May 2019; Accepted 22 July 2019; Published 2 September 2019

Guest Editor: Jianhua Chen

Copyright © 2019 F. U. Ogbuisi et al. 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.

Abstract.

In this paper, we introduce an iterative method for approximating a common solution of monotone inclusion problem and fixed point of Bregman nonspreading mappings in a reflexive Banach space. Using the Bregman distance function, we study the composition of the resolvent of a maximal monotone operator and the antiresolvent of a Bregman inverse strongly monotone operator and introduce a Halpern-type iteration for approximating a common zero of a maximal monotone operator and a Bregman inverse strongly monotone operator which is also a fixed point of a Bregman nonspreading mapping. We further state and prove a strong convergence result using the iterative algorithm introduced. This result extends many works on finding a common solution of the monotone inclusion problem and fixed-point problem for nonlinear mappings in a real Hilbert space to a reflexive Banach space.

1. Introduction

Let E be a real reflexive Banach space with a norm and be the dual space of E. We denote the value of at by A mapping A is called a monotone mapping if for any , we have

A monotone mapping is said to be maximal monotone if its graph, , is not properly contained in the graph of any other monotone operator. A basic problem that arises in several branches of applied mathematics [1–7] is to find such that

One of the methods for solving this problem is the well-known proximal point algorithm (PPA) introduced by Martinet [8]. Let H be a Hilbert space and let I denote the identity operator on The PPA generates for any starting point a sequence in H by

where A is a maximal monotone mapping and is a given sequence of positive real numbers. It has been observed that (3) is equivalent to

This algorithm was further developed by Rockafellar [5], who proved that the sequence generated by (3) converges weakly to an element of when is nonempty and Furthermore, Rockafellar [5] asked if the sequence generated by (3) converges strongly in general. This question was answered in the negative by Güler [9] who presented an example of a subdifferential for which the sequence generated by (3) converges weakly but not strongly. Also, the works of Bruck and Reich [10] and Bauschke et al. [11] are very important in this direction. For more recent results on PPA, see [12–14].

The problem of finding the zeros of the sum of two monotone mappings A and B, is to find a point such that

has recently received attention due to its significant importance in many physical problems. One classical method for solving problem (5) is the forward-backward splitting method [15], which is as follows: for ,

where This method combines the proximal point algorithm and the gradient projection algorithm. In [16], Lions and Mercier introduced the following splitting iterative methods in a real Hilbert space H:

where The first one is called Peaceman–Rachford algorithm and the second one is called Douglas–Rachford algorithm [15]. It was noted that both algorithms converge weakly in general [16, 17].

Many authors have studied the approximation of zero of the sum of two monotone operators (in Hilbert space) and accretive operators (in Banach spaces), but the approximation of the sum of two monotone operators in more general Banach spaces other the Hilbert spaces has not enjoyed such popularity.

Throughout this paper, is a proper lower semicontinuous and convex function, and the Fenchel conjugate of f is the function defined by

We denote by domf the domain of f, that is, the set For any and the right-hand derivative of f at x in the direction of t is defined by

The function f is said to be Gâteaux differentiable at x if the limit as in (9) exists for any In this case, coincides with the value of the gradient at The function f is said to be Gâteaux differentiable if it is Gâteaux differentiable for any . The function f is Fréchet differentiable at x if the limit is attained with and uniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for and

The function f is said to be Legendre if it satisfies the following two conditions:

 (L1) intdom and the subdifferential is single-valued in its domain (L2) tdom and is single-valued on its domain

The class of Legendre functions in infinite dimensional Banach spaces was first introduced and studied by Bauschke et al. in [18]. Their definition is equivalent to conditions (L1) and (L2) because the space E is assumed to be reflexive (see [18], Theorems 5.4 and 5.6, p. 634). It is well known that in reflexive Banach spaces, (see [19], p. 83). When this fact is combined with conditions (L1) and (L2), we obtain

It also follows that f is Legendre if and only if is Legendre (see [18], Corollary 5.5, p. 634) and that the functions f and are Gâteaux differentiable and strictly convex in the interior of their respective domains.

Several interesting examples of the Legendre functions are presented in [18, 20, 21]. A very important example of Legendre function is the function with , where the Banach space E is smooth and strictly convex, and in particular, a Hilbert space. Throughout this article, we assume that the convex function is Legendre.

Definition 1. Let be a convex and Gâteaux differentiable function, the function dom intdom which is defined by

is called the Bregman distance [22–24].

The Bregman distance does not satisfy the well-known metric properties, but it does have the following important property, which is called the three-point identity: for any and

Let C be a nonempty subset of a Banach space E and be a mapping, then a point x is called fixed point of T if . The set of fixed point of T is denoted by . Also, a point is said to be an asymptotic fixed point of T if C contains a sequence which converges weakly to and [25]. The set of asymptotic fixed points of T is denoted by .

Definition 2 [26, 27]. Let C be a nonempty, closed, and convex subset of E. A mapping is called

(i)Bregman firmly nonexpansive (BFNE for short) if

(ii)Bregman strongly nonexpansive (BSNE) with respect to a nonempty if

 for all and and if whenever is bounded, and

 it follows that

(iii)Bregman quasi-nonexpansive if and

(iv)Bregman skew quasi-nonexpansive if and

(v)Bregman nonspreading if

It is easy to see that every Bregman nonspreading mapping T with is Bregman quasi-nonexpansive. Also Bregman nonspreading mappings include, in particular, the class of nonspreading functions studied by Takahashi et al. in [28, 29]. For more information on Bregman nonspreading mappings, see [30].
In a real Hilbert space H, the nonlinear mapping is said to be

(i)Nonexpansive if

(ii)Quasi-nonexpansive if and

(iii)Nonspreading if

Clearly, every nonspreading mapping T with is also quasi-nonexpansive mapping. The class of nonspreading mappings is very important due to its relation with maximal monotone operators (see, e.g., [28]).
Let be a maximal monotone operator. The resolvent of is defined by (see [26])

It is known that is a BFNE operator, single-valued, and (see [26]). If is a Legendre function which is bounded, uniformly Fréchet differentiable on bounded subsets of E, then is BSNE and (see [31]).
Assume that the Legendre function f satisfies the following range condition:

An operator is called Bregman inverse strongly monotone (BISM) if (dom) (dom), and for any and each and we have

The class of BISM mappings is a generalization of the class of firmly nonexpansive mappings in Hilbert spaces. Indeed, if then where I is the identity operator and (25) becomes

which means

Observe that

In other words, T is a (single-valued) firmly nonexpansive operator.
For any operator the antiresolvent operator of A is defined by

It is known that the operator A is BISM if and only if the antiresolvent is a single-valued BFNE (see [32], Lemma 3.2(c) and (d), p. 2109) and . For examples and further information on BISM, see [32].
Since the monotone inclusion problems have very close connections with both the fixed-point problems and the equilibrium problems, finding the common solutions of these problems has drawn many people’s attention and has become one of the hot topics in the related fields in the past few years [33, 34]. Furthermore, interest in finding the common solution of these problems has also grown because of the possible application of these problems to mathematical models whose constraints can be present as fixed points of mappings and/or monotone inclusion problems and/or equilibrium problems. Such a problem occurs, in particular, in the practical problems as signal processing, network resource allocation, and image recovery (see [35, 36]).
In this paper, we introduce an iterative method for approximating a common solution of monotone inclusion problem and fixed point of Bregman nonspreading mapping in a reflexive Banach space and prove a strong convergence of the sequence generated by our iterative algorithm. This result extends many works on finding common solution of monotone inclusion problem and fixed problem of nonlinear mapping in a real Hilbert space to a reflexive Banach space.

2. Preliminaries

The Bregman projection [22] of onto the nonempty, closed, and convex subset C int (dom is defined as the necessarily unique vector satisfying

It is known from [37] that if and only if

We also have

Note that if E is a Hilbert space and , then the Bregman projection of x onto C, i.e., , is the metric projection .

Lemma 1 [37]. Let f be totally convex on int (domf). Let C be a nonempty, closed, and convex subset of int (domf) and ; if , then the following conditions are equivalent:

(i)(ii)(iii)

Let be a convex and Gâteaux differentiable function. The function f is said to be totally convex at if its modulus of totally convexity at x, that is, the function defined by

is positive for any The function f is said to be totally convex when it is totally convex at every point . In addition, the function f is said to be totally convex on bounded set if is positive for any nonempty bounded subset B, where the modulus of total convexity of the function f on the set B is the function defined by

For further details and examples on totally convex functions, see [37–39].

Let be a convex, Legendre, and Gâteaux differentiable function and let the function associated with f (see [23, 40]) be defined by

Then is nonnegative and Furthermore, by the subdifferential inequality, we have (see [41])

In addition, if is a proper lower semicontinuous function, then is a proper lower semicontinuous and convex function (see [42]). Hence, is convex in the second variable. Thus, for all ,

where and with

Lemma 2 (see [43]). Let be a constant and let be a continuous uniformly convex function on bounded subsets of E. Then

for all , , , and with , where is the gauge of uniform convexity of f.

Recall that a function f is said to be sequentially consisted (see [37]) if for any two sequences and in E such that the first one is bounded,

The following lemma follows from [44].

Lemma 3. If domf contains at least two points, then the function f is totally convex on bounded sets if and only if the function f is sequentially consistent.

Lemma 4 (see [45]). Let be a Legendre function and let be a BISM operator such that Then the following statements hold:

(i)(ii)For any and , we have

Remark 1. If the Legendre function f is uniformly Fréchet differentiable and bounded on bounded subsets of then the antiresolvent is a single-valued BSNE operator which satisfies (cf. [31]).

Lemma 5 (see [46]). If is uniformly Fréchet differentiable and bounded on bounded subsets of E, then is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of .

Lemma 6 (see [44]). Let be a Gâteaux differentiable and totally convex function. If and the sequence is bounded, then the sequence is also bounded.

Lemma 7 (see [45]). Assume that is a Legendre function which is uniformly Fréchet differentiable and bounded on bounded subset of E. Let C be a nonempty, closed, and convex subset of E. Let be BSNE operators which satisfy for each and let If

and are nonempty, then T is also BSNE with .

Lemma 8 (Demiclosedness principle [30]). Let C be a nonempty subset of a reflexive Banach space. Let be a strict convex, Gâteaux differentiable, and locally bounded function. Let be a Bregman nonspreading mapping. If in C and , then

Lemma 9 (see [47]). Assume is a sequence of nonnegative real numbers satisfying

where is a sequence in and is a sequence in such that

(i)(ii)

Then,

Lemma 10 [48]. Let be a sequence of real numbers such that there exists a nondecreasing subsequence of , that is, for all . Then there exists a nondecreasing sequence such that and the following properties are satisfied for all (sufficiently large number ): and , .

3. Main Results

Theorem 1. Let C be a nonempty, closed, and convex subset of a real reflexive Banach space E and a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of Let be a Bregman inverse strongly monotone operator, be a maximal monotone operator, and be a Bregman nonspreading mapping. Suppose Let and , and be sequences in such that . Given and arbitrarily, let and be sequences in E generated by

Suppose the following conditions are satisfied:

(i) and (ii)(iii)

Then converges strongly to where is the Bregman projection of E onto .

Proof. First we observe that and Thus, since and are BSNE operators and it then follows from Lemma 7 that is BSNE and
We next show that and are bounded.
Let then from (43), we have

Also

Hence is bounded. Therefore, by Lemma 6, is also bounded, and consequently, is also bounded.
We now show that converges strongly to To do this, we first show that if there exists a subsequence of such that then
Let and be the gauge of uniform convexity of the conjugate function . From Lemma 2 and (9), we have

Thus, from (45), we have

We consider the following two cases for the rest of the proof.

Case A. Suppose is monotonically nonincreasing. Then, converges and as . Thus, from (47), we have

Since , then we have

and hence, by condition (iii) and the property of , we have

Since is uniformly norm-to-norm continuous on bounded subset of , we have

Again

Since is uniformly norm-to-norm continuous on bounded subsets of , we have that

Now, let , then

Therefore, we have

More so

Since and , we have

Thus,

Therefore, since is BSNE, we have that , which implies that

Setting for each , we have

Thus,

Therefore, from (47), we have

Moreover, since , then

and therefore, we have that

which implies

Hence,

Since is bounded, there exists a subsequence of such that converges weakly to as Since it follows from Lemma 8 that Also, since , it implies that also converges weakly to Therefore, from (59), we have that , and hence,
Next, we show that converges strongly to .
Now from (43), we have

Choose a subsequence of such that

Since , it follows from Lemma 1(ii) that

Since , , then,

Hence, by Lemma 9 and (67), we conclude that . Therefore, converges strongly to .

Case B. Suppose that there exists a subsequence of such that

for all . Then, by Lemma 10, there exists a nondecreasing sequence with as such that

for all . Following the same line of arguments as in Case I, we have that

From (67), we have

Since , it follows from (74) that

Since , we obtain

Then from (73), it follows that as . Combining with (74), we obtain as . Since for all , we have as , which implies that as .
Therefore, from the above two cases, we conclude that converges strongly to .
This completes the proof.

Corollary 1. Let C be a nonempty, closed, and convex subset of a real reflexive Banach space E and a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of Let be a Bregman inverse strongly monotone operator, be a maximal monotone operator, and be a Bregman firmly nonexpansive mapping. Suppose Let and , and be sequences in such that . Given and arbitrarily, let and be sequences in E generated by

Suppose the following conditions are satisfied:

(i) and (ii)(iii)

Then, converges strongly to where is the Bregman projection of E onto .

Corollary 2. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let be a single-valued 1-inverse strongly monotone operator, be a maximal monotone operator, and be a firmly nonexpansive mapping. Suppose Let and , and be sequences in such that . Given and arbitrarily, let and be sequences in E generated by

Suppose the following conditions are satisfied:

(i) and (ii)(iii)

Then, converges strongly to where is the metric projection of H onto .

4. Application

In this section, we apply our result to obtain a common solution of variational inequality problem (VIP) and equilibrium problem (EP) in real reflexive Banach spaces.

Let C be a nonempty, closed, and convex subset of a real reflexive Banach space E. Suppose is a bifunction that satisfies the following conditions:

 A1  A2  A3  A4 is convex and lower semicontinuous, for each

The equilibrium problem with respect to is to find such that

We denote the set of solutions of (79) by EP(). The resolvent of a bifunction that satisfies (see [49]) is the operator defined by

Lemma 11 ([27], Lemma 1, 2). Let be a coercive Legendre function and let C be a nonempty, closed, and convex subset of Suppose the bifunction satisfies , then

(1)(2) is single valued(3) is Bregman firmly nonexpansive(4)(5) is a closed and convex subset of C(6), for all and for all

Let be a Bregman inverse strongly monotone mapping and let C be a nonempty, closed, and convex subset of The variational inequality problem corresponding to A is to find , such that

The set of solutions of (81) is denoted by

Lemma 12 (see [25, 46]). Let be a Bregman inverse strongly monotone mapping and be a Legendre and totally convex function that satisfies the range condition. If C is a nonempty, closed, and convex subset of , then

(1) is Bregman relatively nonexpansive mapping(2)

Now let be the indicator function of a closed convex subset C of E, defined by

The subdifferential of the indicator function , where C is a closed subset of a Banach space E and is the normal cone defined by

The normal cone is maximal monotone and the resolvent of the normal cone corresponds to the Bregman projection (see [50], Example 4.4) that is .

Therefore, if we let and , then the iterative algorithm (77) becomes

Thus, from Corollary 1, we obtain a strong convergence result for approximating a point

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work of the first author is based on the research supported wholly by the National Research Foundation (NRF) of South Africa (Grant no. 111992). The third author acknowledges the financial support from the 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 no. BA 2018/012). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF and CoE-MaSS.

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