# A Strong Convergence Theorem for a Finite Family of Bregman Demimetric Mappings in a Banach Space under a New Shrinking Projection Method

Authors: Bijan Orouji; Ebrahim Soori; Donal O’Regan; Ravi P. Agarwal
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, using a new shrinking projection method and new generalizedk-demimetric mappings, we consider the strong convergence for finding a common point of the sets of zero points of maximal monotone mappings, common fixed points of a finite family of Bregman k-demimetric mappings, and common zero points of a finite family of Bregman inverse strongly monotone mappings in a reflexive Banach space. To the best of our knowledge, such a theorem for Bregman k-demimetric mapping is the first of its kind in a Banach space. This manuscript is online on arXiv by the link http://arxiv.org/abs/2107.13254.


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## Body

## 1. Introduction

LetH be a Hilbert space and let C be a nonempty, closed, and convex subset of H. Let T:C⟶H be a mapping. Then, we denote by FT the set of fixed points of T. For a real number t with 0≤t≤1, a mapping U:C⟶H is said to be a t-strict pseudocontraction [1] if(1)Ux−Uy2≤x−y2+tx−Ux−y−Uy2,for all x,y∈C. In particular, if t=0, then U is nonexpansive, i.e.,(2)Ux−Uy≤x−y,∀x,y∈C.IfU is a t-strict pseudocontraction and FU≠∅, then we get that, for x∈C and p∈FU,(3)Ux−p2≤x−p2+tx−Ux2.From this inequality, we get that(4)Ux−x2+x−p2+2Ux−x,x−p≤x−p2+tx−Ux2.Then, we get that(5)2x−Ux,x−p≥1−tx−Ux2.A mappingU:C⟶H is said to be a generalized hybrid [2] if there exist real numbers α,β such that(6)αUx−Uy2+1−αx−Uy2≤βUx−y2+1−βx−y2,for all x,y∈C. Such a mapping U is said to be a α,β-generalized hybrid. The class of generalized hybrid mappings covers several well-known mappings. A 1,0-generalized hybrid mapping is nonexpansive. For α=2 and β=1, it is nonspreading [3, 4], i.e.,(7)2Ux−Uy2≤Ux−y2+Uy−x2,∀x,y∈C.Forα=3/2 and β=1/2, it is also a hybrid [5], i.e.,(8)3Ux−Uy2≤x−y2+Ux−y2+Uy−x2,∀x,y∈C.In general, nonspreading mappings and hybrid mappings are not continuous (see [6]). If U is a generalized hybrid and FU≠∅, then we get that, for x∈C and p∈FU,(9)αp−Ux2+1−αp−Ux2≤βp−x2+1−βp−x2,and hence, Ux−p2≤x−p2. From this, we have that(10)2x−p,x−Ux≥x−Ux2.LetE be a smooth Banach space and let G be a maximal monotone mapping with G−10≠∅. Then, for the metric resolvent Jλ of G for a positive number λ>0, we obtain from [7, 8] that, for x∈E and p∈G−10=FJλ,(11)Jλx−p,Jx−Jλx≥0.Then, we get(12)Jλx−x+x−p,Jx−Jλx≥0,and hence,(13)x−p,Jx−Jλx≥x−Jλx2,where J is the duality mapping on E. Motivated by (5), (10), and (13), Takahashi [9] introduced a nonlinear mapping in a Banach space as follows: let C be a nonempty, closed, and convex subset of a smooth Banach space E and let η be a real number with η∈−∞,1. A mapping U:C⟶E with FU≠∅ is said to be η-demimetric if, for x∈C and p∈FU,(14)2x−p,Jx−Ux≥1−ηx−Ux2.According to this definition, we have that at-strict pseudocontraction U with FU≠∅ is t-demimetric, an α,β-generalized hybrid mapping U with FU≠∅ is 0-demimetric, and the metric resolvent Jλ with G−10≠∅ is −1-demimetric.On the other hand, in 1967, Bregman [10] discovered an effective technique using the so-called Bregman distance function Df in the process of designing and analyzing feasibility and optimization algorithms. This led to a growing area of research in which Bregman’s technique is applied in various ways in order to design and analyze iterative algorithms for solving feasibility problems, equilibrium problems, fixed point problems for nonlinear mappings, and so on (see, e.g., [11, 12] and the references therein).In 2010, Reich and Sabach [11] using the Bregman distance function Df introduced the concept of Bregman strongly nonexpansive mappings and studied the convergence of two iterative algorithms for finding common fixed points of finitely many Bregman strongly nonexpansive operators in reflexive Banach spaces.In this paper, motivated by Takahashi [13], we generalize k-demimetric mappings by the Bregman distance, and using a new shrinking projection method, we deal with the strong convergence for finding a common point of the sets of zero points of maximal monotone mappings, common fixed points of a finite family of Bregman k-demimetric mappings, and common zero points of a finite family of Bregman inverse strongly monotone mappings in a reflexive Banach space (see [14]).


## 2. Preliminaries

LetE be a reflexive real Banach space and C be a nonempty, closed, and convex subset of E. Throughout this paper, the dual space of E is denoted by E∗. The norm and duality pairing between E and E∗ are, respectively, denoted by . and .,.. Let xnn∈ℕ be a sequence in E, and we denote the strong convergence of xnn∈ℕ to x∈E as n⟶∞ by xn⟶x and the weak convergence by xn⇀x.Throughout this paper,f:E⟶−∞,+∞ is a proper, lower semicontinuous, and convex function. We denote by domf≔x∈E;fx<∞, the domain of f. The function f is said to be strongly coercive if limx⟶∞fx/x=+∞. Let x∈intdomf, and the subdifferential of f at x is the convex mapping set ∂f:E⟶2E∗ defined by(15)∂fx=ξ∈E∗:fx+y−x,ξ≤fy,∀y∈E,∀x∈E,and f∗:E∗⟶−∞,+∞ is the Fenchel conjugate of f defined by(16)f∗ξ=supξ,x−fx:x∈E.It is well known thatξ∈∂fx is equivalent to(17)fx+f∗ξ=x,ξ.For anyx∈intdomf and y∈E, we denote by f∘x,y the right-hand derivative of f at x in the direction y, that is,(18)f∘x,y≔limt⟶0+fx+ty−fxt.The functionf is called Ga^teaux differentiable at x, if the limit in (18) exists for any y∈E. In this case, the gradient of f at x is the linear function ∇f which is defined by y,∇fx≔f∘x,y for any y∈E. The function f is said to be Ga^teaux differentiable if it is Ga^teaux differentiable at each x∈intdomf. The function f is said to be Fréchet differentiable at x, if the limit in (18) is attained uniformly in y=1, for any y∈E. Finally, f is said to be uniformly Fréchet differentiable on a subset C of E, if the limit in (18) is attained uniformly for x∈C and y=1.Lemma 1 (see [11]).
Iff:E⟶ℝ is uniformly Fréchet differentiable and bounded on bounded subsets of E, then f is uniformly continuous on bounded subsets of E and ∇f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of E∗.Proposition 2 (see [15]).
Letf:E⟶ℝ be a convex function which is bounded on bounded subsets of E. Then, the following assertions are equivalent:(i)
f is strongly coercive and uniformly convex on bounded subsets of E(ii)
f∗ is Fréchet differentiable, and ∇f∗ is uniformly norm-to-norm continuous on bounded subsets of domf∗=E∗Definition 3.
The functionf is said to be “Legendre” if it satisfies the following two conditions:
(L1): intdomf≠∅, and ∂f is single-valued on its domain.
(L2): intdomf∗≠∅, and ∂f∗ is single-valued on its domain.Because here the spaceE is assumed to be reflexive, we always have ∂f−1=∂f∗ ([16], p. 83). This fact, when combined with the conditions (L1) and (L2), implies the following equalities:(19)∇f=∇f∗−1,ran∇f=dom∇f∗=intdomf∗,ran∇f∗=dom∇f=intdomf.In addition, the conditions (L1) and (L2), in conjunction with Theorem 5.4 of [17], imply that the functions f and f∗ are strictly convex on the interior of their respective domains and f is Legendre if and only if f∗ is Legendre.One important and interesting Legendre function is1/p.p,p∈1,2. When E is a uniformly convex and p-uniformly smooth Banach space with p∈1,2, the generalized duality mapping Jp:E⟶2E∗ is defined by(20)Jpx=jpx∈E∗:jpx,x=x.jpx,jpx=xp−1.In this case, the gradient∇f of f coincides with the generalized duality mapping Jp of E, ∇f=Jp,p∈1,2. Several interesting examples of Legendre functions are presented in [17–19].From now on, we always assume that the convex functionf:E⟶0,+∞ is Legendre.Definition 4 (see [20]).
Letf:E⟶−∞,+∞ be a convex and Ga^teaux differentiable function. The bifunction Df:domf×intdomf⟶0,+∞ defined by(21)Dfy,x≔fy−fx−∇fx,y−x,is called the Bregman distance with respect to f.It should be noted thatDf is not a distance in the usual sense of the term. Clearly, Dfx,x=0, but Dfy,x=0 may not imply x=y. In our case, when f is Legendre, this indeed holds ([17], Theorem 7.3 (vi), p. 642). In general, Df satisfies the three-point identity(22)Dfx,y+Dfy,z−Dfx,z=x−y,∇fz−∇fy,and the four-point identity(23)Dfx,y+Dfω,z−Dfx,z−Dfω,y=x−ω,∇fz−∇fy,for any x,ω∈domf and y,z∈intdomf. Over the last 30 years, Bregman distances have been studied by many researchers (see [17, 21–23]).Letf:E⟶−∞,+∞ be a convex function on E which is Ga^teaux differentiable on intdomf. The function f is said to be totally convex at a point x∈intdomf if its modulus of total convexity at x,vfx,.:0,+∞⟶0,+∞, defined by(24)vfx,t=infDfy,x:y∈domf,y−x=t,is positive whenever t>0. The function f is said to be totally convex when it is totally convex at every point of intdomf. The function f is said to be totally convex on bounded sets, if for any nonempty bounded set B⊆E, the modulus of total convexity of f on B,vfB,t is positive for any t>0, where vfB,.:0,+∞⟶0,+∞ is defined by(25)vfB,t=infvfx,t:x∈B∩intdomf.We remark in passing thatf is totally convex on bounded sets if and only if f is uniformly convex on bounded sets (see [24, 25]).Proposition 5 (see [24]).
Letf:E⟶−∞,+∞ be a convex function that its domain contains at least two points. If f is lower semicontinuous, then f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets.Lemma 6 (see [11]).
Ifx∈intdomf, then the following statements are equivalent:(i)
The functionf is totally convex at x(ii)
For any sequenceyn⊂domf,(26)limn⟶+∞Dfyn,x=0⇒limn⟶+∞yn−x=0.Recall that the functionf is called sequentially consistent [25], if for any two sequences xnn∈ℕ and ynn∈ℕ in E such that xnn∈ℕ is bounded,(27)limn⟶+∞Dfyn,xn=0⇒limn⟶+∞yn−xn=0.Lemma 7 (see [14]).
Ifdomf 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 8 (see [26]).
Letf:E⟶ℝ be a Ga^teaux differentiable and totally convex function. If x1∈E and the sequence Dfxn,x1 is bounded, then the sequence xn is also bounded.Lemma 9 (see [12]).
Letf:E⟶ℝ be a Legendre function such that ∇f∗ is bounded on bounded subsets of intdomf∗. Let x1∈E, and if Dfx1,xn is bounded, then the sequence xn is bounded too.Recall that the Bregman projection [10] with respect to f of x∈intdomf onto a nonempty, closed, and convex set C⊆intdomf is the unique vector ProjCfx∈C satisfying(28)DfProjCfx,x=infDfy,x:y∈C.Similar to the metric projection in Hilbert spaces, the Bregman projection with respect to totally convex andGa^teaux differentiable functions has a variational characterization ([25], corollary 4.4, p. 23).Lemma 10 (see [25]).
Suppose thatf is Ga^teaux differentiable and totally convex on intdomf. Let x∈intdomf and C⊆intdomf be a nonempty, closed, and convex set. Then, the following Bregman projection conditions are equivalent:(i)
z0=ProjCfx(ii)
z=z0 is the unique solution of the following variational inequality:(29)z−y,∇fx−∇fz≥0,∀y∈C.(iii)
z=z0 is the unique solution of the following variational inequality:(30)Dfy,z+Dfz,x≤Dfy,x,∀y∈C.LetE be a real Banach space and C be a nonempty subset of E. An element p∈C is called a fixed point of a single-valued mapping T:C⟶C, if p=Tp. The set of fixed points of T is denoted by FT.A pointx∈C is called an asymptotic fixed point of T if C contains a sequence xn which converges weakly to x and limn⟶+∞xn−Txn=0. We denote the asymptotic fixed points of T by F~T.LetC be a nonempty, closed, and convex subset of intdomf and T:C⟶C be a mapping. Now, T is said to be Bregman quasi-nonexpansive, if FT≠∅ and(31)Dfp,Tx≤Dfp,x,∀x∈C,p∈FT.LetC be a nonempty, closed, and convex subset of intdomf. An operator T:C⟶intdomf is said to be Bregman strongly nonexpansive with respect to a nonempty F~T, if(32)Dfy,Tx≤Dfy,x,∀x∈C,y∈F~T,and for any bounded sequence xn⊆C with(33)limn⟶∞Dfy,xn−Dfy,Txn=0,it follows that(34)limn⟶∞DfTxn,xn=0.A mappingB:E⟶2E∗ is called Bregman inverse strongly monotone on the set C, if C∩intdomf≠∅, and for any x,y∈C∩intdomf, ξ∈Bx, and η∈By, we have that(35)ξ−η,∇f∗∇fx−ξ−∇f∗∇fy−η≥0.LetB:E⟶2E∗ be a mapping. Then, the mapping defined by(36)Bλf≔∇f∗∘∇f−λB:E⟶E,is called an antiresolvent associated with B and λ for any λ>0.Suppose thatA is a mapping of E into 2E∗ for the real reflexive Banach space E. The effective domain of A is denoted by domA, that is, domA=x∈E:Ax≠∅. A multivalued mapping A on E is said to be monotone if x−y,u∗−v∗≥0 for all x,y∈domA, u∗∈Ax, and v∗∈Ay. A monotone operator A on E is said to be maximal if graph A, the graph of A, is not a proper subset of the graph of any monotone operator on E.LetE be a real reflexive Banach space, f:E⟶−∞,+∞ uniformly Fréchet differentiable and bounded on bounded subsets of E. Then for any λ>0, the resolvent of A defined by(37)ResAfx=∇f+λA−1∘∇fx,is a single-valued Bregman quasi-nonexpansive mapping from E onto domA and FResAf=A−10. We denote by Aλ=1/λ∇f−∇fResAf the Yosida approximation of A for any λ>0. We get from [26] (prop. 2.7, p. 10) that(38)Aλx∈AResAfx,∀x∈E,λ>0.See [11], too.Lemma 11 (see [27]).
LetE be a real reflexive Banach space and f:E⟶−∞,+∞ be a Legendre function which is totally convex on bounded subsets of E. Also, let C be a nonempty, closed, and convex subset of intdomf and T:C⟶2C be a multivalued Bregman quasi-nonexpansive mapping. Then, the fixed point set FT of T is a closed and convex subset of C.Lemma 12 (see[28]).
Assume thatf:E⟶ℝ is a Legendre function which is uniformly Fréchet differentiable and bounded on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E. Also, let Ti:i=1,⋯,N be N Bregman strongly nonexpansive mappings which satisfy F~Ti=FTi for each 1≤i≤N and let T=TNTN−1⋯T1. If FT and ⋂i=1NFTi are nonempty, then T is also Bregman strongly nonexpansive with FT=F~T.Lemma 13 (see [29]).
LetG:E⟶2E∗ be a maximal monotone operator and B:E⟶E∗ be a Bregman inverse strongly monotone mapping such that G+B−10∗≠∅. Also, let f:E⟶ℝ be a Legendre function which is uniformly Fréchet differentiable and bounded on bounded subsets of E. Then,(i)
G+B−10∗=FResλGf∘Bλf(ii)
ResλGf∘Bλf is a Bregman strongly nonexpansive mapping such that(39)FResλGf∘Bλf=F~ResλGf∘Bλf.(iii)
Dfu,ResλGf∘Bλfx+DfResλGf∘Bλfx,x≤Dfu,x, ∀u∈G+B−10∗, x∈E, and λ>0Lemma 14 (see [30]).
Letf:E⟶−∞,+∞ be a proper convex and lower semicontinuous Legendre function. Then, for any z∈E, for any xn⊆E and tii=1N⊆0,1 with ∑i=1Nti=1, the following holds(40)Dfz,∇f∗∑i=1Nti∇fxi≤∑i=1NtiDfz,xi.Proposition 15 (see [26], prop. 2.8, p. 10).
Letf be Ga^teaux differentiable and A:E⟶2E∗ be a maximal monotone operator such that A−10≠∅. Then(41)Dfq,x≥Dfq,ResrAfx+DfResrAfx,x,for all r>0, q∈A−10, and x∈E.Next, we generalize thek-demimetric notation introduced in [15].Definition 16.
LetE be a reflexive Banach space, f:E⟶−∞,+∞ be a Legendre function which is Ga^teaux differentiable, C be a nonempty, closed and convex subset of intdomf and let k∈−∞,1. A mapping T:C⟶intdomf with FT≠∅ is said to be Bregman k-demimetric, if for x∈C and q∈FT,(42)x−q,∇fx−∇fTx≥1−kDfx,Tx.Example 1.
Every Bregman quasi-nonexpansive mapping with the required conditions in Definition16 is a Bregman 0-demimetric mapping. Let p∈FT≠∅ and x∈C, and we have(43)Dfp,Tx≤Dfp,x⇒0≤Dfp,x−Dfp,Tx=Dfp,x+Dfx,Tx−Dfp,Tx−Dfx,Tx=p−x,∇fTx−∇fx−Dfx,Tx.
Therefore,(44)x−p,∇fx−∇fTx≥Dfx,Tx.Example 2.
From [31] (Lemma 2.1), every Bregman quasi-strictly pseudocontractive mapping with the required conditions is a Bregman k-demimetric mapping for k∈0,1.Example 3.
LetE be a reflexive Banach space, f:E⟶−∞,+∞ be a Legendre function which is Ga^teaux differentiable, and A:E⟶2E∗ be a maximal monotone operator with A−10≠∅ and r>0. Then, the f-resolvent ResrAf is Bregman 0-demimetric. In fact, from (22) and Proposition 15, we have that(45)Dfq,x≥Dfq,ResrAfx+DfResrAfx,x=Dfq,x+q−ResrAfx,∇fx−∇fResrAfx,for any x∈E and q∈A−10. Then, we obtain(46)ResrAfx−q,∇fx−∇fResrAfx≥0.
Therefore,(47)ResrAfx−x+x−q,∇fx−∇fResrAfx≥0,and hence, from (22), we have that(48)x−q,∇fx−∇fResrAfx≥x−ResrAfx,∇fx−∇fResrAfx=Dfx,ResrAfx+DfResrAfx,x.
Thus,(49)x−q,∇fx−∇fResrAfx≥Dfx,ResrAfx,and then, we get that ResrAf is Bregman 0-demimetric.


## 3. Main Results

The following lemma is important and crucial in the proof of Theorem18.Lemma 17.
LetE be a reflexive Banach space andf:E⟶−∞,+∞ be a Legendre function which is Ga^teaux differentiable, and letC be a nonempty, closed, and convex subset of intdomf and let k be a real number with k∈−∞,1 and let T be a Bregman k-demimetric mapping of C into intdomf. Then, FT is closed and convex.Proof.
First, we show thatFT is closed. Consider a sequence qn such that qn⟶q and qn∈FT. We conclude from the definition of T that(50)q−qn,∇fq−∇fTq≥1−kDfq,Tq.
Sinceqn⟶q, we have 0≥1−kDfq,Tq. Then, from 1−k>0, we have that Dfq,Tq=0, and hence, q=Tq, and therefore, q∈FT. This implies that FT is closed.
Next, we show thatFT is convex. Suppose p,q∈FT and set z=αp+1−αq, where α∈0,1. Then, we have that(51)z−p,∇fz−∇fTz≥1−kDfz,Tz,z−q,∇fz−∇fTz≥1−kDfz,Tz.
Thus, fromα≥0 and 1−α≥0, we have that(52)αz−αp,∇fz−∇fTz≥α1−kDfz,Tz,1−αz−1−αq,∇fz−∇fTz≥1−α1−kDfz,Tz.
From these inequalities, we get that(53)0=z−z,∇fz−∇fTz≥1−kDfz,Tz.
Thus, we get thatDfz,Tz=0; hence, z=Tz; therefore, z∈FT. We conclude that FT is convex.Theorem 18.
LetE be a real reflexive Banach space. Suppose that f:E⟶ℝ is a proper, convex, lower semicontinuous, strongly coercive, Legendre function which is bounded on bounded subsets of E, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of intdomf. Let k1,k2,⋯,kM⊆−∞,1 and Tjj=1M be a finite family of Bregman kj-demimetric and Bregman quasi-nonexpansive and demiclosed mappings of C into itself. Suppose that Bii=1N is a finite family of Bregman inverse strongly monotone mappings of C into E and Bi,ηnfi=1N is the family of antiresolvent mappings of Bii=1N. Let A:E⟶2E∗ and G:E⟶2E∗ be maximal monotone mappings on E and let Qη=ResηGf=∇f+ηG−1∇f and Jr=ResrAf=∇f+rA−1∇f be the resolvents of G and A for η>0 and r>0, respectively. Assume that(54)Ω=A−10∩∩j=1MFTj∩∩i=1NBi+G−10∗≠∅.
Forx1∈C and C1=Q1=C, let xn be a sequence defined by(55)yn=∇f∗∑j=1Mξj1−λn∇f+λn∇fTjxn,zn=∇f∗∑i=1Nσi∇fQηnBi,ηnfyn,un=Jrnzn,Cn+1=z∈Cn:Dfz,yn≤Dfz,xn,Dfz,zn≤Dfz,ynandzn−z,∇fzn−∇fun≥Dfzn,un,xn+1=ProjCn+1∩Qnfx1,∀n∈ℕ,Qn+1=z∈Qn:xn+1−z,∇fx1−∇fxn+1≥0,where λn⊆0,1, ηn, rn⊆0,+∞, ξ1,ξ2,⋯,ξM, σ1,σ2,⋯,σN⊆0,1, and a,b,c∈ℝ satisfy the following:(1)
0<a≤λn, ∀n∈ℕ(2)
0<c≤rn, ∀n∈ℕ(3)
∑j=1Mξj=1 and ∑i=1Nσi=1
Then,xn converges strongly to a point ω0∈Ω where ω0=ProjΩfx1.Proof.
We divide the proof into several steps:(Step 1)
First, we prove thatΩ is a closed and convex subset of C.
SinceTjj=1M is a finite family of kj-Bregman demimetric mappings, by Lemma 17 and the condition Ω≠∅, FTj is nonempty, closed, and convex for 1≤j≤M. Also, it follows from Lemma 13 (i)–(ii) that Bi+G−10∗=FQηnBi,ηnf and QηnBi,ηnf is a Bregman strongly nonexpansive mapping, and therefore, from Lemma 12, we have that(56)FQηnBi,ηnf=F~QηnBi,ηnf.
Thus,QηnBi,ηnf is a family of Bregman quasi-nonexpansive mappings. Using Ω≠∅ and Lemma 11, we see that Bi+G−10∗=FQηnBi,ηnf is a nonempty, closed, and convex set. We also know that A−10 is closed and convex. Then, Ω is nonempty, closed, and convex. Therefore, ProjΩf is well defined.(Step 2)
We prove thatCn and Qn are closed and convex subsets of C and Ω⊆Cn+1∩Qn,∀n∈ℕ.
In fact, it is clear thatC1=C is closed and convex. Suppose that Ck is closed and convex for some k≥1. Note that(57)Dfz,yk≤Dfz,xk⇔fz−fyk−∇fyk,z−yk≤fz−fxk−∇fxk,z−xk⇔∇fxk,z−xk−∇fyk,z−yk≤fyk−fxk.
Similarly, we have that(58)Dfz,zk≤Dfz,yk⇔∇fyk,z−yk−∇fzk,z−zk≤fzk−fyk.
Thus, from the fact thatDf.,x is continuous for each fixed x and using the above inequalities z∈Ck:Dfz,yk≤Dfz,xk and z∈Ck:Dfz,zk≤Dfz,ykwhich are closed and convex. We have also z∈Ck:zk−z,∇fzk−∇fuk≥Dfzk,uk which is closed and convex. Therefore, Ck+1 is closed and convex. From mathematical induction, we have that Cn is a closed and convex subset in C with Cn+1⊆Cn for all n∈ℕ.
Also, it is clear thatQ1=C is closed and convex. Suppose that Qk is closed and convex for some k≥1. Hence, z∈Qk:xk+1−z,∇fx1−∇fxk+1≥0 is closed and convex; i.e., Qk+1 is a closed and convex subset of Qk. Therefore, by mathematical induction, Qn is a closed and convex subset of C with Qn+1⊆Qn for all n∈ℕ. Next, we show that Ω⊆Cn for all n≥1. Clearly, Ω⊆C1=C. Assume that Ω⊆Ck for some k∈ℕ. Note from Lemma 14 that(59)Dfz,yk=Dfz,∇f∗∑j=1Mξj1−λk∇f+λk∇fTjxk=Dfz,∇f∗∑j=1Mξj∇f∇f∗1−λk∇f+λk∇fTjxk≤∑j=1MξjDfz,∇f∗1−λk∇f+λk∇fTjxk≤∑j=1Mξj1−λkDfz,xk+λkDfz,Tjxk≤∑j=1Mξj1−λkDfz,xk+λkDfz,xk=Dfz,xk,for all z∈Ω. Furthermore, since Bi is a Bregman inverse strongly monotone mapping for all 1≤i≤N and hence from Lemmas 13 and 14, we have that(60)Dfz,zk=Dfz,∇f∗∑i=1Nσi∇fQηkBi,ηkfyk≤∑i=1NσiDfz,QηkBi,ηkfyk≤∑i=1NσiDfz,yk=Dfz,yk,for all z∈Ω. Also, since Jrk is the resolvent of A and uk=Jrkzk, we have from (22) and Proposition 15 that(61)zk−z,∇fzk−∇fuk=Dfz,zk+Dfzk,uk−Dfz,uk=Dfz,zk+Dfzk,uk−Dfz,Jrkzk≥Dfz,zk+Dfzk,uk−Dfz,zk=Dfzk,uk,for all z∈Ω. From (59)–(61), we have that Ω⊆Ck+1. Therefore, we have by mathematical induction that Ω⊆Cn for all n∈ℕ.
Now, we shall show thatΩ⊆Qn for all n∈ℕ. Note that Ω⊆Q1=C. Assume that Ω⊆Qk for some k∈ℕ. Thus, Ω⊆Ck+1∩Qk for some k∈ℕ. From xk+1=ProjCk+1∩Qkfx1 and Lemma 10, we have that(62)xk+1−z,∇fx1−∇fxk+1≥0,∀z∈Ck+1∩Qk.
SinceΩ⊆Ck+1∩Qk, we have that(63)xk+1−z,∇fx1−∇fxk+1≥0,∀z∈Ω.
Then, we getΩ⊆Qk+1. By mathematical induction, we have that Ω⊆Qn for all n∈ℕ. This implies that xn is well defined.(Step 3)
We show thatlimn⟶∞Dfxn,x1 exists.
SinceΩ is nonempty, closed, and convex, there exists a ω0∈Ω such that ω0=ProjΩfx1. From xn+1=ProjCn+1∩Qnfx1, we get that(64)Dfxn+1,x1≤Dfz,x1,for all z∈Cn+1∩Qn. From ω0∈Ω⊆Cn+1∩Qn, we obtain that(65)Dfxn+1,x1≤Dfω0,x1.
This shows thatDfxn,x1 is a bounded sequence. By Lemma 10 (iii), we have that(66)0≤Dfxn+1,xn=Dfxn+1,ProjCn∩Qn−1fx1≤Dfxn+1,x1−DfProjCn∩Qn−1fx1,x1=Dfxn+1,x1−Dfxn,x1,for all n>1. Therefore,(67)Dfxn,x1≤Dfxn+1,x1.
This implies thatDfxn,x1 is bounded and nondecreasing. Then, limn⟶∞Dfxn,x1 exists. In view of Lemma 8, we deduce that the sequence xn is bounded. Also, from (66), we have that(68)limn⟶∞Dfxn+1,xn=0.
Since the functionf is totally convex on bounded sets, by Lemma 7, we have that(69)limn⟶∞xn+1−xn=0.(Step 4)
We prove thatxn is a Cauchy sequence in C.
We haveCm⊆Cn and Qm⊆Qn for any m,n∈ℕ with m≥n. From xn=ProjCn∩Qn−1fx1∈Cn∩Qn−1 and Lemma 10, we have that(70)Dfxm,xn=Dfxm,ProjCn∩Qn−1fx1≤Dfxm,x1−DfProjCn∩Qn−1fx1,x1=Dfxm,x1−Dfxn,x1.
Lettingm,n⟶∞ in (70), we deduce that Dfxm,xn⟶0. In view of Lemma 7, since the function f is totally convex on bounded sets, we get that xm−xn⟶0 as m,n⟶∞. Thus, xn is a Cauchy sequence. Since E is a Banach space and C is closed and convex, we conclude that there exists x¯∈C such that(71)limn⟶∞xn−x¯=0.(Step 5)
We prove thatlimn⟶∞∇fxn−∇fyn=0.
Using (68) and from xn+1∈Cn+1, we get that(72)Dfxn+1,yn≤Dfxn+1,xn⟶0asn⟶∞.
Then,limn⟶∞Dfxn+1,yn=0. Since the function f is totally convex on bounded sets and yn is bounded, by Lemma 7, we have that(73)limn⟶∞xn+1−yn=0.
Using (69) and (73), we have that(74)xn−yn≤xn−xn+1+xn+1−yn⟶0asn⟶∞.
Then,(75)limn⟶∞xn−yn=0.
Since from Lemma1, ∇f is uniformly continuous, we have that(76)limn⟶∞∇fxn−∇fyn=0.(Step 6)
We prove thatlimn⟶∞∇fyn−∇fzn=0.
Using (72) and from xn+1∈Cn+1, we get that(77)Dfxn+1,zn≤Dfxn+1,yn⟶0asn⟶∞.
Then,limn⟶∞Dfxn+1,zn=0. Since function f is totally convex on bounded sets and zn is bounded, by Lemma 7, we have that(78)limn⟶∞xn+1−zn=0.
Using (69) and (78), we have that(79)limn⟶∞xn−zn=0.
Applying (75) and (79), we get that(80)limn⟶∞yn−zn=0.
Since∇f is uniformly continuous, we have that(81)limn⟶∞∇fyn−∇fzn=0.(Step 7)
We prove thatlimn⟶∞∇fzn−∇fun=0.
Usingxn+1∈Cn+1 and from (22), we get that(82)Dfzn,un≤zn−xn+1,∇fzn−∇fun=Dfxn+1,zn+Dfzn,un−Dfxn+1,un.
Then,(83)0≤Dfxn+1,zn−Dfxn+1,un.
Therefore, from (77), we have that(84)Dfxn+1,un≤Dfxn+1,zn⟶0asn⟶∞,and then, limn⟶∞Dfxn+1,un=0. Since the function f is totally convex on bounded sets and from Proposition 15 and Lemma 9, un is bounded; hence, by Lemma 7, we have that(85)limn⟶∞xn+1−un=0.
Using (69) and (85), we conclude that(86)limn⟶∞xn−un=0.
Now, by (79) and (86), we get that(87)limn⟶∞zn−un=0.
Since from Lemma1, ∇f is uniformly continuous, we have that(88)limn⟶∞∇fzn−∇fun=0.(Step 8)
We prove thatx¯∈Ω.
SinceTj is Bregman kj-demimetric for all 1≤j≤M, we get that(89)xn−z,∇fxn−∇fyn=xn−z,∇fxn−∇f∇f∗∑j=1Mξj1−λn∇f+λn∇fTjxn=∑j=1Mξjxn−z,∇fxn−1−λn∇f+λn∇fTjxn=∑j=1Mξjλnxn−z,∇fxn−∇fTjxn≥∑j=1Mξjλn1−kDfxn,Tjxn≥∑j=1Mξja1−kDfxn,Tjxn,for all z∈∩j=1MFTj. We have from (76) that(90)limn⟶∞Dfxn,Tjxn=0,for all 1≤j≤M. Since the function f is totally convex on bounded sets, by Lemma 7, we have that(91)limn⟶∞xn−Tjxn=0,∀j∈1,⋯,M.
SinceTj is demiclosed for all 1≤j≤M and from (71) wherexn⟶x¯ as n⟶∞, we have that x¯∈∩j=1MFTj. We now show that x¯∈∩i=1NBi+G−10∗. From (22) and Lemma 13 (iii), we get that(92)yn−z,∇fyn−∇fzn=yn−z,∇fyn−∇f∇f∗∑i=1Nσi∇fQηnBi,ηnfyn=∑i=1Nσiyn−z,∇fyn−∇fQηnBi,ηnfyn=∑i=1NσiDfz,yn+Dfyn,QηnBi,ηnfyn−Dfz,QηnBi,ηnfyn≥∑i=1NσiDfz,yn+Dfyn,QηnBi,ηnfyn−Dfz,yn=∑i=1NσiDfyn,QηnBi,ηnfyn,for all z∈∩i=1NBi+G−10∗ and i∈1,⋯,N. Using (81), from the above, we have that(93)limn⟶∞Dfyn,QηnBi,ηnfyn=0,for all 1≤i≤N. Since the function f is totally convex on bounded sets, by Lemma 7, we conclude that(94)limn⟶∞yn−QηnBi,ηnfyn=0.
On the other hand, it follows from (71) and (75) that(95)limn⟶∞yn−x¯=0.
Now, from (94) and (95), we have that x¯∈F~QηnBi,ηn for all i∈1,⋯,N. From Lemma 13, we conclude that(96)F~QηnBi,ηn=FQηnBi,ηn=Bi+G−10∗,∀i∈1,⋯,N,and therefore, x¯∈∩i=1NBi+G−10∗.
We now show thatx¯∈A−10. Using rn≥c, we have from (88) that(97)limn⟶∞1rn∇fzn−∇fun=0.
So applyingArn, the Yosida approximation of A, we have(98)limn⟶∞Arnzn=limn⟶∞1rn∇fzn−∇fun=0.
SinceArnzn∈Aun, for p,p∗∈A, we have from the monotonicity of A that p−un,p∗−Arnzn≥0 for all n∈ℕ. By (71) and (79), we have that(99)limn⟶∞zn−x¯=0.
From (87) and the above, we have un−x¯⟶0. Thus, we get p−x¯,p∗≥0. From the maximality of A, we have that x¯∈A−10. Therefore, x¯∈Ω.
Sinceω0=ProjΩfx1, x¯∈Ω, and xn−x¯⟶0, we have from (65) that(100)Dfω0,x1≤Dfx¯,x1=limn⟶∞Dfxn,x1≤limn⟶∞Dfω0,x1=Dfω0,x1,which implies that(101)limn⟶∞Dfxn,x1=Dfω0,x1.
Therefore,(102)Dfx¯,x1=Dfω0,x1.
From (55), (71), and ω0∈Ω⊆Qn+1 for all n∈ℕ and also since ∇f is uniformly continuous on bounded subsets and is bounded on bounded subsets, we have that(103)xn+1−ω0,∇fx1−∇fxn+1≥0⇒limn⟶∞xn+1−ω0,∇fx1−∇fxn+1≥0⇒x¯−ω0,∇fx1−∇fx¯≥0.
Now, from (22) and (102), we get that(104)Dfω0,x¯=Dfω0,x¯+Dfx¯,x1−Dfω0,x1=ω0−x¯,∇fx1−∇fx¯.
From (103) and the above, we conclude that Dfω0,x¯≤0, and therefore, Dfω0,x¯=0. Thus, ω0=x¯, and hence, xn⟶ω0. This completes the proof.Next, we prove a proposition to extend Theorem18.Proposition 19.
LetE be a real reflexive Banach space. Suppose that f:E⟶ℝ is a proper, convex, lower semicontinuous, Legendre function and is Ga^teaux differentiable of E. Let C be a nonempty, closed, and convex subset of intdomf. Suppose k∈−∞,0 and a mapping T:C⟶C with FT≠∅ is Bregman k-demimetric. Then, T is a Bregman quasi-nonexpansive mapping.Proof.
Letp∈FT and x∈C. Using (22) and Definition 16, we have(105)Dfp,Tx−Dfp,x=Dfx,Tx−Dfp,x−Dfx,Tx+Dfp,Tx=Dfx,Tx−p−x,∇fTx−∇fx≤Dfx,Tx−1−kDfx,Tx=kDfx,Tx≤0.
Then,T is a Bregman quasi-nonexpansive mapping.Remark 20.
Using Proposition19, for each 1≤j≤M, we may remove the condition that Tj is Bregman quasi-nonexpansive in Theorem 18 when kj≤0.Open Problem 1.
Can one generalize Proposition19 to Bregman k-demimetric mappings with k<1?



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