# Convergence Results for a Common Solution of a Finite Family of Equilibrium Problems and Quasi-Bregman Nonexpansive Mappings in Banach Space

Authors: G. C. Ugwunnadi; Bashir Ali
Journal: Journal of Operators
 (2016)

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## Abstract

We introduce an iterative process for finding common fixed point of finite family of quasi-Bregman nonexpansive mappings which is a unique solution of some equilibrium problem.


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## Body

## 1. Introduction

LetE be a real reflexive Banach space and C a nonempty subset of E. Let T:C→C be a map, a point x∈C is called a fixed point of T if Tx=x, and the set of all fixed points of T is denoted by F(T). The mapping T is called L-Lipschitzian or simply Lipschitz if there exists L>0, such that Tx-Ty≤Lx-y,∀x,y∈C, and if L=1, then the map T is called nonexpansive.Letg:C×C→R be a bifunction. The equilibrium problem with respect to g is to find (1)z∈Csuch  thatgz,y≥0,∀y∈C.The set of solutions of equilibrium problem is denoted by EP(g). Thus (2)EPg≔z∈C:gz,y≥0,∀y∈C.Numerous problems in physics, optimization, and economics reduce to finding a solution of the equilibrium problem. Some methods have been proposed to solve equilibrium problem in Hilbert spaces; see, for example, Blum and Oettli [1], Combettes and Hirstoaga [2]. Recently, Tada and Takahashi [3, 4] and S. Takahashi and W. Takahashi [5] obtain weak and strong convergence theorems for finding a common element of the set of solutions of an equilibrium problem and set of fixed points of nonexpansive mapping in Hilbert space. In particular, Takahashi and Zembayashi [4] establish a strong convergence theorem for finding a common element of the two sets by using the hybrid method introduced in Nakajo and Takahashi [6]. They also proved such a strong convergence theorem in a uniformly convex and uniformly smooth Banach space.In 1967, Bregman [7] discovered an elegant and effective technique for using so-called Bregman distance function Df; see (3) in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman’s technique has been applied in various ways in order to design and analyze iterative algorithms for solving feasibility and optimization problems.Letf:E→(-∞,+∞] be a convex and Gâteaux differentiable function. The function Df:dom⁡f×intdom⁡f→[0,+∞) defined as (3)Dfy,x≔fy-fx-∇fx,y-xis called the Bregman distance with respect to f (see [8]). It is obvious from the definition of Df that (4)Dfz,x=Dfz,y+Dfy,x+∇fy-∇fx,z-y.We observed from (4) that, for any y1,y2,…,yN∈E, the following holds: (5)Dfy1,yN=∑k=2NDfyk-1,yk+∑k=3N∇fyk-1-∇fyk,yk-1-y1.Recall that the Bregman projection [7] of x∈intdom⁡f onto the nonempty closed and convex set C⊂dom⁡f is the necessarily unique vector PCf(x)∈C satisfying (6)DfPCfx,x=inf⁡Dfy,x:y∈C.A mapping T is said to be Bregman firmly nonexpansive [9], if, for all x,y∈C,(7)∇fTx-∇fTy,Tx-Ty≤∇fx-∇fy,Tx-Ty,or, equivalently, (8)DfTx,Ty+DfTy,Tx+DfTx,x+DfTy,y≤DfTx,y+DfTy,x.A point p∈C is said to be asymptotic fixed point of a map T, if, for any sequence {xn} in C which converges weakly to p, limn→∞xn-Txn=0. We denote by F^(T) the set of asymptotic fixed points of T. Let f:E→R; a mapping T:C→C is said to be Bregman relatively nonexpansive [10] if F(T)≠∅, F^(T)=F(T), and Df(p,T(x))≤Df(p,x) for all x∈C and p∈F(T). T is said to be quasi-Bregman relatively nonexpansive if F(T)≠∅, and Df(p,T(x))≤Df(p,x) for all x∈C and p∈F(T).Recently, by using the Bregman projection, in 2011 Reich and Sabach [9] proposed algorithms for finding common fixed points of finitely many Bregman firmly nonexpansive operators in a reflexive Banach space: (9)x0∈EQ0i=E,i=1,2,…,Nun∈Csuch  thatyni=Tixn+eni,Qn+1i=z∈Qni:∇fxn+eni-∇fyni,z-yni≤0,Cn=⋂i=1NCni,xn+1=PCn+1fx0,n≥0.Under some suitable conditions, they proved that the sequence generated by (9) converges strongly to ⋂i=1N‍F(Ti) and applied the result for the solution of convex feasibility and equilibrium problems.In 2011, Chen et al. [11] introduced the concept of weak Bregman relatively nonexpansive mappings in a reflexive Banach space and gave an example to illustrate the existence of a weak Bregman relatively nonexpansive mapping and the difference between a weak Bregman relatively nonexpansive mapping and a Bregman relatively nonexpansive mapping. They also proved strong convergence of the sequences generated by the constructed algorithms with errors for finding a fixed point of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings under some suitable conditions.Recently in 2014, Alghamdi et al. [12] proved a strong convergence theorem for the common fixed point of finite family of quasi-Bregman nonexpansive mappings. Pang et al. [13] proved weak convergence theorems for Bregman relatively nonexpansive mappings, while Zegeye and Shahzad in [14, 15] proved a strong convergence theorem for the common fixed point of finite family of right Bregman strongly nonexpansive mappings and Bregman weak relatively nonexpansive mappings in reflexive Banach space, respectively.In 2015 Kumam et al. [16] introduced the following algorithm: (10)x1=x∈Czn=Resgfxnyn=∇f∗βn∇f∗xn+1-βn∇f∗Tnznxn+1=∇f∗αn∇f∗xn+1-αn∇f∗Tnyn,where Tn,n∈N, is a Bregman strongly nonexpansive mapping. They proved that the sequence {xn} which is generated by algorithm (10) converges strongly to the point PΩfx, where Ω≔F(T)∩EP(g).Motivated and inspired by the above works, in this paper, we prove a new strong convergence theorem for finite family of quasi-Bregman nonexpansive mapping and system of equilibrium problem in a real Banach space.


## 2. Preliminaries

LetE be a real reflexive Banach space with the norm · and E∗ the dual space of E. Throughout this paper, we will assume f:E→(-∞,+∞] is a proper, lower semicontinuous, and convex function. We denote by domf≔{x∈E:f(x)<+∞} the domain of f.Letx∈intdom⁡f; the subdifferential of f at x is the convex set defined by (11)∂fx=x∗∈E∗:fx+x∗,y-x≤fy,∀y∈E,where the Fenchel conjugate of f is the function f∗:E∗→(-∞,+∞] defined by (12)f∗x∗=sup⁡x∗,x-fx:x∈E.We know that the Young-Fenchel inequality holds: (13)x∗,x≤fx+f∗x∗,∀x∈E,x∗∈E∗.A function f on E is coercive [17] if the sublevel set of f is bounded; equivalently, (14)limx→+∞fx=+∞.A function f on E is said be strongly coercive [18] if (15)limx→+∞fxx=+∞.For any x∈intdom⁡f and y∈E, the right-hand derivative of f at x in the direction y is defined by (16)f∘x,y≔limt→0+fx+ty-fxt.The function f is said to be Gâteaux differentiable at x if limt→0+f(x+ty)-f(x)/t exists for any y. In this case, f∘(x,y) coincides with ∇f(x), the value of the gradient ∇f of f at x. The function f is said to be Gâteaux differentiable if it is Gâteaux differentiable for any x∈intdom⁡f. The function f is said to be Fréchet differentiable at x if this limit is attained uniformly in y=1. Finally, f is said to be uniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for x∈C and y=1. It is known that if f is Gâteaux differentiable (resp., Fréchet differentiable) on intdomf, then f is continuous and its Gâteaux derivative ∇f is norm-to-weak∗ continuous (resp., continuous) on int domf (see also [19, 20]). We will need the following results.Lemma 1 (see [21]).
Iff:E→R is uniformly Fréchet differentiable and bounded on bounded subsets of E, then ∇f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of E∗.Definition 2 (see [22]).
The functionf is said to be (i)
essentially smooth, if∂f is both locally bounded and single-valued on its domain,(ii)
essentially strictly convex, if(∂f)-1 is locally bounded on its domain and f is strictly convex on every convex subset of dom⁡∂f,(iii)
Legendre, if it is both essentially smooth and essentially strictly convex.Remark 3.
LetE be a reflexive Banach space. Then we have the following:(i)
f is essentially smooth if and only if f∗ is essentially strictly convex (see [22], Theorem 5.4).(ii)
(∂f)-1=∂f∗ (see [20]).(iii)
f is Legendre if and only if f∗ is Legendre (see [22], Corollary 5.5).(iv)
Iff is Legendre, then ∇f is a bijection satisfying ∇f=(∇f∗)-1, ran∇f=dom∇f∗=intdomf∗, and ran∇f∗=domf=intdomf (see [22], Theorem 5.10).
The following result was proved in [23] (see also [24]).Lemma 4.
LetE be a Banach space, let r>0 be a constant, let ρr be the gauge of uniform convexity of g, and let g:E→R be a convex function which is uniformly convex on bounded subsets of E. Then, (i)
for anyx,y∈Br and α∈(0,1), (17)gαx+1-αy≤αgx+1-αgy-α1-αρrx-y,(ii)
for anyx,y∈Br, (18)ρrx-y≤Dgx,y,(iii)
if, in addition,g is bounded on bounded subsets and uniformly convex on bounded subsets of E then, for any x∈E, y∗,z∗∈Br, and α∈(0,1), (19)Vgx,αy∗+1-αz∗≤αVgx,y∗+1-αVgx,z∗-α1-αρr∗y∗-x∗.Lemma 5 (see [25]).
LetE be a Banach space, let r>0 be a constant, and let f:E→R be a continuous and convex function which is uniformly convex on bounded subsets of E. Then (20)f∑k=0∞αkxk≤∑k=0∞αkfxk-αiαjρrxi-xj,for all i,j∈N∪{0},xk∈Br,αk∈(0,1), and k∈N∪{0} with ∑k=0∞‍αk=1, where ρr is the gauge of uniform convexity of f.We know the following two results; see [18].Theorem 6.
LetE be a reflexive Banach space and let f:E→R be a convex function which is bounded on bounded subsets of E. Then the following assertions are equivalent: (1)
f is strongly coercive and uniformly convex on bounded subsets of E.(2)
dom⁡f∗=E∗, f∗ is bounded on bounded subsets and uniformly smooth on bounded subsets of E∗.(3)
dom⁡f∗=E∗, f∗ is Fréchet differentiable and ∇f is uniformly norm-to-norm continuous on bounded subsets of E∗.Theorem 7.
LetE be a reflexive Banach space and let f:E→R be a continuous convex function which is strongly coercive. Then the following assertions are equivalent: (1)
f is bounded on bounded subsets and uniformly smooth on bounded subsets of E.(2)
f∗ is Fréchet differentiable and f∗ is uniformly norm-to-norm continuous on bounded subsets of E∗.(3)
dom⁡f∗=E∗, f∗ is strongly coercive and uniformly convex on bounded subsets of E∗.The following result was first proved in [26] (see also [27]).Lemma 8.
LetE be a reflexive Banach space, let f:E→R be a strongly coercive Bregman function, and let V be the function defined by (21)Vx,x∗=fx-x,x∗+f∗x∗,x∈E,x∗∈E∗.Then the following assertions hold: (1)
Df(x,∇f(x∗))=V(x,x∗) for all x∈E and x∗∈E∗.(2)
V(x,x∗)+〈∇f∗(x∗)-x,y∗〉≤V(x,x∗+y∗) for all x∈E and x∗,y∗∈E∗.Examples of Legendre functions were given in [22, 28]. One important and interesting Legendre function is 1/p·p(1<p<∞) when E is a smooth and strictly convex Banach space. In this case the gradient ∇f of f is coincident with the generalized duality mapping of E; that is, ∇f=Jp(1<p<∞). In particular, ∇f=I, the identity mapping in Hilbert spaces. In the rest of this paper, we always assume that f:E→(-∞,+∞] is Legendre.Concerning the Bregman projection, the following are well known.Lemma 9 (see [26]).
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let f:E→R be a Gâteaux differentiable and totally convex function and let x∈E. Then (a)
z=PCf(x) if and only if 〈∇f(x)-∇f(z),y-z〉≤0,∀y∈C.(b)
Df(y,PCf(x))+Df(PCf(x),x)≤Df(y,x),∀x∈E,y∈C.Letf:E→(-∞,+∞] be a convex and Gâteaux differentiable function. The modulus of total convexity of f at x∈intdom⁡f is the function vf(x,·):[0,+∞)→[0,+∞] defined by (22)vfx,t≔inf⁡Dfy,x:y∈dom⁡f,y-x=t,The function f is called totally convex at x if vf(x,t)>0 whenever t>0. The function f is called totally convex if it is totally convex at any point x∈intdom⁡f and is said to be totally convex on bounded sets if vf(B,t)>0 for any nonempty bounded subset B of E and t>0, where the modulus of total convexity of the function f on the set B is the function vf:intdom⁡f×[0,+∞)→[0,+∞] defined by (23)vfB,t≔inf⁡vfx,t:x∈B∩dom⁡f.Lemma 10 (see [29]).
Ifx∈dom⁡f, then the following statements are equivalent: (i)
The functionf is totally convex at x.(ii)
For any sequence{yn}⊂dom⁡f, (24)limn→+∞Dfyn,x=0⟹limn→+∞yn-x=0.Recall that the functionf is called sequentially consistent [26] if for any two sequences {xn} and {yn} in E such that the first one is bounded (25)limn→+∞Dfyn,xn=0⟹limn→+∞yn-xn=0.Lemma 11 (see [30]).
The functionf is totally convex on bounded sets if and only if the function f is sequentially consistent.Lemma 12 (see [31]).
Letf:E→R be a Gâteaux differentiable and totally convex function. If x0∈E and the sequence {Df(xn,x0)} is bounded, then the sequence {xn} is bounded too.Lemma 13 (see [31]).
Letf:E→R be a Gâteaux differentiable and totally convex function, x0∈E, and let C be a nonempty, closed, and convex subset of E. Suppose that the sequence {xn} is bounded and any weak subsequential limit of {xn} belongs to C. If Df(xn,x0)≤Df(PCf(x0),x0) for any n∈R, then {xn} converges strongly to PCf(x0).Lemma 14 (see [32]).
LetE be a real reflexive Banach space, let f:E→(-∞,+∞] be a proper lower semicontinuous function, and then f∗:E∗→(-∞,+∞] is a proper weak∗ lower semicontinuous and convex function. Thus, for all z∈E, one has (26)Dfz,∇f∗∑i=1Nti∇fxi≤∑i=1NtiDfz,xi.In order to solve the equilibrium problem, let us assume that a bifunctiong:C×C→R satisfies the following conditions [1]:(A1)
g(x,x)=0, ∀x∈C.(A2)
g is monotone; that is, g(x,y)+g(y,x)≤0,∀x,y∈C.(A3)
limsupt↓0g(x+t(z-x),y)≤g(x,y)∀x,z,y∈C.(A4)
The functiony↦g(x,y) is convex and lower semicontinuous.The resolvent of a bifunctiong [2] is the operator Resgf:E→2C defined by (27)Resgfx=z∈C:gz,y+∇fz-∇fx,y-z≥0,∀y∈C.From Lemma  1, in [33], if f:(-∞,+∞] is a strongly coercive and Gâteaux differentiable function and g satisfies conditions (A1)–(A4), then dom(Resgf)=E. The following lemma gives some characterization of the resolvent Resgf.Lemma 15 (see [33]).
LetE be a real reflexive Banach space and let C be a nonempty closed convex subset of E. Let f:E→(-∞,+∞] be a Legendre function. If the bifunction g:C×C→R satisfies the conditions (A1)–(A4), then, the following hold: (i)
Resgf is single-valued.(ii)
Resgf is a Bregman firmly nonexpansive operator.(iii)
F(Resgf)=EP(g).(iv)
EP(g) is closed and convex subset of C.(v)
For allx∈E and for all q∈F(Resgf), one has (28)Dfq,Resgfx+DfResgfx,x≤Dfq,x.Lemma 16 (see [34]).
Let{an} be a sequence of nonnegative real numbers satisfying the following relation: (29)an+1≤1-αnan+αnδn,n≥n0,where {αn}⊂(0,1) and {δn} is a real sequence satisfying the following conditions: (30)limn→∞αn=0,∑n=1∞=∞,aslimsupn→∞δn≤0.Then, limn→∞an=0.Lemma 17 (see [35]).
Let{an} be a sequence of real numbers such that there exists a subsequence {ni} of {n} such that ani<ani+1 for all i∈N. Then there exists a nondecreasing sequence {mk}⊂N such that mk→∞ and the following properties are satisfied by all (sufficiently large) numbers k∈N: (31)amk≤amk+1,ak≤amk+1.In fact, mk=max{j≤k:aj<aj+1}.


## 3. Main Results

We now prove the following theorem.Theorem 18.
LetC be a nonempty, closed, and convex subset of a real reflexive Banach space E and f:E→R a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of E. For each j=1,2,…,m, let gj be a bifunction from C×C to R satisfying (A1)–(A4) and let {Ti=1N} be a finite family of quasi-Bregman nonexpansive self-mapping of C such that F≔⋂i=1NF(Ti)≠∅, where F=F(TNTN-1TN-2⋯T2T1)=F(T1TNTN-1TN-2⋯T2)=⋯=F(TN-1TN-2⋯T2T1TN)≠∅ and Ω≔(⋂j=1mEP(gj))∩F≠∅. Let {xn}n=1∞ be a sequence generated by x1=x∈C,C1=C, and (32)x1∈Cuj,n=Resgjfxn,j=1,2,3,…,myn=PC∇f∗1-αn∇fuj,nxn+1=PC∇f∗βn∇fyn+1-βn∇fTnyn,where T[n]=Tn(modN) and {αn}n=1∞⊂(0,1) and {βn}n=1∞⊂[c,d]⊂(0,1) satisfying limn→∞αn=0 and ∑n=1∞‍αn=∞. Then {xn}n=1∞ converges strongly to PΩf(x), where PΩf is the Bregman projection of C onto Ω.Proof.
Letp=PΩf∈Ω from Lemma 15; we obtain (33)Dfp,uj,n=Dfp,Resgjfxn≤Dfp,xn.Now from (32), we obtain(34)Dfp,yn≤Dfp,∇f∗1-αn∇fuj,n=Dfp,∇f∗αn∇f0+1-αn∇fuj,n≤αnDfp,0+1-αnDfp,uj,n≤αnDfp,0+1-αnDfp,xn.Also from (32), (26), and (34), we have (35)Dfp,xn+1≤Dfp,∇f∗1-βn∇fyn+βn∇fTnyn≤1-βnDfp,yn+βnDfp,Tnyn≤1-βnDfp,yn+βnDfp,yn=Dfp,yn≤αnDfp,0+1-αnDfp,xn≤max⁡Dfp,0,Dfp,xn.Thus, by induction we obtain (36)Dfp,xn+1≤max⁡Dfp,0,Dfp,xn,∀n≥0,which implies that {xn} is bounded and hence {yn}, {T[n]yn}, {T[n]xn}, and {uj,n} are all bounded for each j=1,2,…,m. Now from (32) let zn≔∇f∗((1-αn)∇f(uj,n)). Furthermore since αn→0 as n→∞, we obtain (37)∇fzn-∇fuj,n=αn-∇fuj,n⟶0asn⟶∞.Since f is strongly coercive and uniformly convex on bounded subsets of E, f∗ is uniformly Fréchet differentiable on bounded sets. Moreover, f∗ is bounded on bounded sets; from (37), we obtain(38)limn→∞⁡zn-uj,n=0.On the other hand, in view of (3) in Theorem 6, we know that dom⁡f∗=E∗ and f∗ is strongly coercive and uniformly convex on bounded subsets. Let s=sup{∇f(yn),∇f(T[n]yn)} and ρs∗:E∗→R be the gauge of uniform convexity of the conjugate function f∗. Now from (32) and Lemmas 4 and 8, we obtain(39)Dfp,yn≤Dfp,zn=Vp,∇fzn≤Vp,∇fzn+αn∇fp+αn-∇fp,zn-p=Dfp,∇f∗1-αn∇fuj,n+αn∇fp+αn∇fp+αn-∇fp,zn-p≤αnDfp,p+1-αnDfp,uj,n+αn-∇fp,zn-p≤1-αnDfp,xn+αn-∇fp,zn-p,(40)Dfp,xn+1≤Dfp,∇f∗1-βn∇fyn+βn∇fTnyn=Vp,1-βn∇fyn+βn∇fTnyn=fp-p,1-βn∇fyn+βn∇fTnyn+f∗1-βn∇fyn+βn∇fTnyn≤1-βnfp+βnfp-1-βnp,∇fyn-βnp,∇fTnyn+1-βnf∗∇fTnyn+βnf∗∇fTnyn-βn1-βnρs∗∇fyn-∇fTnyn=1-βnVp,∇fyn+βnVp,∇fTnyn-βn1-βnρs∗∇fyn-∇fTnyn=1-βnDfp,yn+βnDfp,Tnyn-βn1-βnρs∗∇fyn-∇fTnyn≤1-βnDfp,yn+βnDfp,yn-βn1-βnρs∗∇fyn-∇fTnyn=Dfp,yn-βn1-βnρs∗∇fyn-∇fTnyn≤1-αnDfp,xn+αn-∇fp,zn-p-βn1-βnρs∗∇fyn-∇fTnyn(41)≤1-αnDfp,xn+αn-∇fp,zn-p.Now, we consider two cases.
Case  1. Suppose that there exists n0∈N such that {Df(p,xn)} is nonincreasing. In this situation {Df(p,xn)} is convergent. Then from (40) we obtain(42)βn1-βnρs∗∇fyn-∇fTnyn⟶0asn⟶∞,which implies, by the property of ρs and since βn∈[c,d]⊂(0,1), (43)limn→∞∇fyn-∇fTnyn=0.Since f is strongly coercive and uniformly convex on bounded subsets of E, f∗ is uniformly Fréchet differentiable on bounded sets. Moreover, f∗ is bounded on bounded sets; from (43), we obtain (44)limn→∞yn-Tnyn=0.Now from (4), we obtain (45)Dfyn,Tnyn=Dfp,Tnyn-Dfp,yn+∇fTnyn-∇fyn,p-yn≤Dfp,yn-Dfp,yn+∇fTnyn-∇fyn,p-yn,and therefore (46)Dfyn,Tnyn≤∇fyn-∇fTnynp-yn⟶0⁡asn⟶∞.
Also, from (28) in Lemma 15, we have(47)Dfxn,uj,n=Dfxn,Resgjfxn≤Dfp,Resgjfxn-Dfp,xn≤Dfp,xn-Dfp,xn⟶0asn⟶∞.Then, we have from Lemma 10 that (48)limn→∞xn-uj,n=0.
Also, from (b) of Lemma9, we have (49)Dfyn,PCzn=Dfyn,zn=Dfyn,∇f∗∇f0+1-αn∇fuj,n≤αnDfyn,0+1-αnDfyn,uj,n≤αnDfyn,0+1-αnDfuj,n,uj,n⟶0asn⟶∞.Then, we have from Lemma 10 that (50)limn→∞yn-zn=0.
From (38) and (48), we obtain (51)limn→∞xn-zn=0.From (50) and (51), we obtain (52)limn→∞xn-yn=0.
Sincef is strongly coercive and uniformly convex on bounded subsets of E, f∗ is uniformly Fréchet differentiable on bounded sets. Moreover, f∗ is bounded on bounded sets; from (52), we obtain (53)limn→∞∇fxn-∇fzn=0.
Also from (44) and (52) (54)limn→∞xn-Tnyn=0.
Now from (4) and (34), we obtain (55)Dfxn,yn=Dfp,yn-Dfp,xn+∇fxn-∇fyn,p-xn≤αnDfp,0+1-αnDfp,xn-Dfp,xn+∇fxn-∇fyn,p-xn=αnDfp,0-Dfp,xn+∇fxn-∇fyn,p-xn;therefore, from (53), we obtain(56)Dfxn,yn≤αnDfp,0-Dfp,xn+∇fxn-∇fynp-xn⟶0asn⟶∞.Also (57)Dfxn,Tnyn=Dfp,Tnyn-Dfp,xn+∇fxn-∇fTnyn,p-xn≤Dfp,yn-Dfp,xn+∇fxn-∇fTnyn,p-xn≤αnDfp,0+1-αn-Dfp,xn+∇fxn-∇fTnyn,p-xn=αnDfp,0-Dfp,xn+∇fxn-∇fTnyn,p-xn;thus (58)Dfxn,Tnyn≤αnDfp,0-Dfp,xn+∇fTnyn-∇fxnp-xn⟶0asn⟶∞.Also, from (56) (59)DfTnxn,Tnyn≤Dfxn,yn⟶0asn⟶∞.Then, we have from Lemma 10 that (60)limn→∞Tnxn-Tnyn=0.Then from (32) and (44), we have (61)∇fxn+1-∇fyn=βn∇fTnyn-∇fyn⟶0⁡asn⟶∞.
This implies(62)xn+1-yn⟶0asn⟶∞,(63)xn-Tnxn≤xn-yn+yn-Tnyn+Tnyn-Tnxn;from (44), (52), and (60), we obtain (64)limn→∞xn-Tnxn=0.This implies that (65)limn→∞∇fxn-∇fTnxn=0.Also from (52) and (62), we obtain (66)xn+1-xn≤xn+1-yn+yn-xn⟶0asn⟶∞.But(67)xn+N-xn≤xn+N-xn+N-1+xn+N-1-xn+N-2+⋯+xn+1-xn⟶0,as n→∞. Hence (68)limn→∞xn+N-xn=0.
From the uniformly continuous∇f, we have from (66) that (69)limn→∞∇fxn+1-∇fxn=0.From (4), (35), and (69), we obtain (70)Dfxn,xn+1=Dfp,xn+1-Dfp,xn+∇fxn-∇fxn+,p-xn≤αnDfp,0+1-αnDfp,xn-Dfp,xn+∇fxn-∇fxn+1,p-xn,which implies (71)Dfxn,xn+1≤αnDfp,0-Dfp,xn+∇fxn+1-∇fxnp-xn⟶0⁡asn⟶∞.Also from quasi-Bregman nonexpansivity of T[n], we have (72)DfTnxn,Tnxn+1≤Dfxn,xn+1⟶0asn⟶∞,which implies (73)limn→∞Tnxn-Tnxn+1=0,and from the uniform continuous ∇f, we obtain (74)limn→∞∇fTnxn-∇fTnxn+1=0.
Also from (4) and (64), we obtain (75)Dfxn,Tnxn=Dfp,Tnxn-Dfp,xn+∇fxn-∇fTnxn,p-xn≤Dfp,xn-Dfp,xn+∇fxn-∇fTnxn,p-xn≤∇fTnxn-∇fxnp-xn⟶0asn⟶∞.From (64), (66), and (73), we obtain (76)xn-Tn+1xn≤xn-xn+1+xn+1-Tn+1xn+1+Tn+1xn+1-Tn+1xn⟶0asn⟶∞,which from uniform continuous ∇f implies (77)limn→∞∇fTnxn-∇fTn+1xn=0,and from (4) and (77), we obtain (78)Dfxn,Tn+1xn≤Dfp,Tn+1xn-Dfp,xn+∇fxn-∇fTn+1xn,p-xn≤Dfp,xn-Dfp,xn+∇fTn+1xn-∇fxnp-xn⟶0asn⟶∞.
From (4), (71), (77), and (78) (79)Dfxn+1,Tn+1xn=Dfxn+1,xn+Dfxn,Tn+1xn+∇fTn+1xn-∇fxn,xn-xn+1≤Dfxn+1,xn+Dfxn,Tn+1xn+∇fTn+1xn-∇fxnxn-xn+1⟶0asn⟶∞.Also from (4), (71), and (79) (80)Dfxn,Tn+1xn=Dfxn,xn+1+Dfxn+1,Tn+1xn+∇fxn+1-∇fTn+1xn+1,xn+1-xn=Dfxn,xn+1+Dfxn+1,Tn+1xn+∇fTn+1xn-∇fxn+1xn+1-xn⟶0⁡asn⟶∞.
Using the quasi-Bregman nonexpansivity ofT(i) for each i, we obtain the following finite table: (81)Dfxn+N,Tn+Nxn+N-1⟶0asn⟶∞DfTn+Nxn+N-1,Tn+NTn+N-1xn+N-2⟶0asn⟶∞⋮DfTn+N⋯Tn+2xn+1,Tn+N⋯Tn+1xn⟶0asn⟶∞.Then, applying Lemma 10 on each line above, we obtain (82)xn+N-Tn+Nxn+N-1⟶0asn⟶∞Tn+Nxn+N-1-Tn+NTn+N-1xn+N-2⟶0asn⟶∞⋮Tn+N⋯Tn+2xn+1-Tn+N⋯Tn+1xn⟶0asn⟶∞,and adding up this table, we obtain (83)xn+N-Tn+NTn+N-1⋯Tn+1xn⟶0asn⟶∞.Using this and (68), we obtain (84)limn→∞xn-Tn+NTn+N-1⋯Tn+1xn=0.Also from quasi-Bregman nonexpansivity of T(i), for each i, we have (85)DfTn+NTn+N-1⋯Tn+1xn,Tn+NTn+N-1⋯Tn+1yn≤Dfxn,yn⟶0,asn→∞. Then, we have from Lemma 10 that (86)Tn+NTn+N-1⋯Tn+1xn-Tn+NTn+N-1⋯Tn+1yn⟶0asn⟶∞.Since(87)yn-Tn+NTn+N-1⋯Tn+1yn≤yn-xn+xn-Tn+NTn+N-1⋯+Tn+1xn+Tn+NTn+N-1⋯Tn+1xn-Tn+NTn+N-1⋯Tn+1yn,then, from (52), (84), and (86), we obtain (88)limn→∞yn-Tn+NTn+N-1⋯Tn+1yn=0.Following the argument from (85), (86), and (88) by replacing yn with zn and using (51), we obtain(89)limn→∞zn-Tn+NTn+N-1⋯Tn+1zn=0.
Let{xni} be a subsequence of {xn}. Since {xn} is bounded and E is reflexive, without loss of generality, we may assume that xni⇀q for some q∈F and since xn-zn→0 as n→∞, then zni⇀q. Since the pool of mappings of T[n] is finite, passing to a further subsequence if necessary, we may further assume that, for some i∈{1,2,…,N}, from (89), we get (90)zni-Ti+N⋯Ti+1zni⟶0asi⟶∞,and also (91)limsupn→∞-∇fp,zn-p=limi→∞-∇fp,zni-p.
Noticing thatuj,n=Resgjf(xn) for each j=1,2,…,m, we obtain(92)gjuj,n,y+y-uj,n,∇fuj,n-∇fxn≥0,∀y∈C.Hence (93)gjuj,ni,y+y-uj,ni,∇fuj,ni-∇fxni≥0,∀y∈C.
From (A2), we note that, for eachj=1,2,…,m, (94)y-uj,n∇fuj,ni-∇fxnirni≥y-uj,n,∇fuj,n-∇fxn≥-gjuj,ni≥gjy,uj,ni,∀y∈C.Taking the limit as i→∞ in above inequality and from (A4) and uj,ni⇀q, we have gj(y,q)≤0 for each j=1,2,…,m. For 0<t<1 and y∈C, define yt=ty+(1-t)q. Noticing that y,q∈C, we obtain yt∈C, which yield that gj(yt,q)≤0. It follows from (A1) that (95)0=gjyt,yt≤tgjyt,y+1-tgjyt,q≤tgjyt,y.That is, for each j=1,2,…,m, we have gj(yt,y)≥0.
Lett↓0; from (A3), we obtain gj(q,y)≥0 for any y∈C, for each j=1,2,…,m. This implies that q∈⋂j=1mEP(gj). Hence q∈Ω. It follows from the definition of the Bregman projection that (96)limsupn→∞-∇fp,zn-p=limi→∞-∇fp,zni-p≤-∇fp,q-p≤0.It follows from Lemma 16 and (41) that Df(p,xn)→0 as n→∞. Consequently, from Lemma 10, we obtain xn→p as n→∞.
Case  2. Suppose Df(p,xn) is not monotone decreasing sequences; then set Φn≔Df(p,xn) and let τ:N→N be a mapping defined for all n≥N0 for some sufficiently large N0 by (97)τn≔max⁡k∈N:k≤n,Φk≤Φk+1.Then by Lemma 17τ(n) is a nondecreasing sequence such that τ(n)→∞ as n→∞ and Φτ(n)≤Φτ(n)+1, for n≥N0. Then from (40) and the fact that ατ(n)→0, we obtain that (98)ρs∗∇fyτn-∇fTτnyτn⟶0asτn⟶∞.Following the same argument as in Case  1, we obtain (99)yτn-Ti+N⋯Ti+1yτn⟶0asτn⟶∞,and also we obtain (100)limsupτn→∞-∇fp,yτn-p≤0.
Then from (41), we obtain that (101)0≤Dfp,xτn+1-Dfp,xτn≤ατn-∇fp,yτn-p-Dfp,xτn.
It follows from (101) and Φn≤Φτ(n)+1, ατ(n)>0 that (102)Dfp,xτn≤-∇fp,yτn-p⟶0,as τ(n)→∞. Thus (103)limτn→∞Φτn=limτn→∞Φτn+1=0.Furthermore, for n≥N0, if n≠τ(n) (i.e., τ(n)<n), because Φj>Φj+1 for τ(n)+1≤j≤n. it then follows that for all n≥N0 we have (104)0≤Φn≤max⁡Φτn,Φτn+1=Φτn+1.This implies that limn→∞Φn=0, and hence Df(p,xn)→0 as n→∞. Consequently, from Lemma 10, we obtain xn→p as n→∞. Therefore from the above two cases, we conclude that {xn} converges strongly to p∈Ω and this completes the proof.



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*Source: 2752158-2016-03-24.xml*