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A Halpern-Type Iteration Method for Bregman Nonspreading Mapping and Monotone Operators in Reflexive Banach Spaces
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.
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 . 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 bywhere 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 , who proved that the sequence generated by (3) converges weakly to an element of when is nonempty and Furthermore, Rockafellar  asked if the sequence generated by (3) converges strongly in general. This question was answered in the negative by Güler  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  and Bauschke et al.  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 thathas 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 , which is as follows: for ,where This method combines the proximal point algorithm and the gradient projection algorithm. In , 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 . 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 . Their definition is equivalent to conditions (L1) and (L2) because the space E is assumed to be reflexive (see , Theorems 5.4 and 5.6, p. 634). It is well known that in reflexive Banach spaces, (see , 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 , 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.
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 . 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 ifIt 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 .
In a real Hilbert space H, the nonlinear mapping is said to be(i)Nonexpansive if(ii)Quasi-nonexpansive if and(iii)Nonspreading ifClearly, 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., ).
Let be a maximal monotone operator. The resolvent of is defined by (see )It is known that is a BFNE operator, single-valued, and (see ). If is a Legendre function which is bounded, uniformly Fréchet differentiable on bounded subsets of E, then is BSNE and (see ).
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 haveThe 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) becomeswhich meansObserve thatIn other words, T is a (single-valued) firmly nonexpansive operator.
For any operator the antiresolvent operator of A is defined byIt is known that the operator A is BISM if and only if the antiresolvent is a single-valued BFNE (see , Lemma 3.2(c) and (d), p. 2109) and . For examples and further information on BISM, see .
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.
The Bregman projection  of onto the nonempty, closed, and convex subset C int (dom is defined as the necessarily unique vector satisfying
It is known from  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 . 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 byis 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
Then is nonnegative and Furthermore, by the subdifferential inequality, we have (see )
In addition, if is a proper lower semicontinuous function, then is a proper lower semicontinuous and convex function (see ). Hence, is convex in the second variable. Thus, for all ,where and with
Lemma 2 (see ). Let be a constant and let be a continuous uniformly convex function on bounded subsets of E. Thenfor all , , , and with , where is the gauge of uniform convexity of f.
Recall that a function f is said to be sequentially consisted (see ) if for any two sequences and in E such that the first one is bounded,
The following lemma follows from .
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 ). 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. ).
Lemma 5 (see ). 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 ). Let be a Gâteaux differentiable and totally convex function. If and the sequence is bounded, then the sequence is also bounded.
Lemma 7 (see ). 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 Ifand are nonempty, then T is also BSNE with .
Lemma 8 (Demiclosedness principle ). 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 ). Assume is a sequence of nonnegative real numbers satisfyingwhere is a sequence in and is a sequence in such that(i)(ii)Then,
Lemma 10 . 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 haveAlsoHence 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 haveThus, from (45), we haveWe 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 haveSince , then we haveand hence, by condition (iii) and the property of , we haveSince is uniformly norm-to-norm continuous on bounded subset of , we haveAgainSince is uniformly norm-to-norm continuous on bounded subsets of , we have thatNow, let , thenTherefore, we haveMore soSince and , we haveThus,Therefore, since is BSNE, we have that , which implies thatSetting for each , we haveThus,Therefore, from (47), we haveMoreover, since , thenand therefore, we have thatwhich impliesHence,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 haveChoose a subsequence of such thatSince , it follows from Lemma 1(ii) thatSince , , then,Hence, by Lemma 9 and (67), we conclude that . Therefore, converges strongly to .
Case B. Suppose that there exists a subsequence of such thatfor all . Then, by Lemma 10, there exists a nondecreasing sequence with as such thatfor all . Following the same line of arguments as in Case I, we have thatFrom (67), we haveSince