# Strong Convergence of a Monotone Projection Algorithm in a Banach Space

Authors: Songtao Lv
Journal: The Scientific World Journal
 (2013)

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## Abstract

In this paper, a common solution problem is investigated based on a Bregman projection. Strong convergence of the monotone projection algorithm for monotone operators and bifunctions is obtained in a reflexive Banach space.


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## Body

## 1. Introduction

Fixed point theory as an important branch of nonlinear analysis theory has been applied in the study of nonlinear phenomena. Lots of problems arising in economics, engineering, and physics can be studied based on fixed point techniques; see [1–21] and the references therein. Many well-known problems can be studied by using algorithms which are iterative in their nature. As an example, in computer tomography with limited data, each piece of information implies the existence of a convex set Cm in which the required solution lies. The problem of finding a point in the intersection ∩m=1NCm, where N≥1 is some positive integer, is then of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such point. The well-known convex feasibility problem which captures applications in various disciplines such as image restoration and radiation therapy treatment planning is to find a point in the intersection of common fixed point sets of a family of nonlinear mappings see, for example, [22–28].Interest in monotone operators stems mainly from their firm connection with equations of evolution which is an important class of nonlinear operators. It is known that many physically significant problems can be modelled by initial value problems of the form:(1)x′(t)+Ax(t)=0,x(0)=x0,
where A is an monotone operator in an appropriate Banach space. Typical examples, where such evolution equations occur, can be found in the heat, wave, or Schrödinger’s equations. If x(t) is dependent on t, then (1) is reduced to
(2)Au=0,
whose solutions correspond to the equilibrium points of the system (1). Consequently, considerable research efforts have been devoted, especially within the past 40 years or so, to methods for finding approximate solutions (when they exist) of (2). An early fundamental result in the theory of monotone operators, due to Browder [29], states that the initial value problem (1) is solvable if A is locally Lipschitz and accretive on E.The Krasnoselskii-Mann iterative algorithm is efficient for treating zero points of monotone operators. However, it is weak convergence only. In many disciplines, including economics, image recovery, and control theory, problems arise in infinite dimension spaces. In such problems, strong convergence (norm convergence) is often much more desirable than weak convergence for it translates the physically tangible property that the energy∥xn-x∥ of the error between the iterate xn and the solution x eventually becomes arbitrarily small. The importance of strong convergence is also underlined, where a convex function f is minimized via the proximal-point algorithm; it is shown that the rate of convergence of the value sequence {f(xn)} is better when {xn} converges strongly than when it converges weakly. Such properties have a direct impact when the process is executed directly in the underlying infinite dimensional space. Projection methods which were first introduced by Haugazeau [30] have been considered for treating zero points of monotone operators. The advantage of projection methods is that strong convergence of iterative sequences can be guaranteed without any compact assumptions.In this paper, we study common zero points of a family of maximal monotone operators and common solutions of a system of equilibrium problems based on the Bregman projection. Strong convergence of the monotone projection algorithm is obtained in a reflexive Banach space.


## 2. Preliminaries

LetE be a real Banach space, E* the dual space of E, and C a nonempty subset of E. Let f be a bifunction from C×C to ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem. Find x¯∈C such that
(3)f(x¯y)≥0,∀y∈C.
We use EP(f) to denote the solution set of the equilibrium problem (3). That is,
(4)EP(f)={p∈C:f(p,y)≥0,∀y∈C}.
Given a mapping B:C→E*, let
(5)f(x,y)=〈Bx,y-x〉,∀x,y∈C.
Then, x¯∈EP(f) if and only if x¯ is a solution of the following variational inequality. Find x¯ such that
(6)〈Bx¯y-x¯〉≥0,∀y∈C.In order to study the solution of the equilibrium problem (3), we assume that f satisfies the following conditions:  (A1)
f(x,x)=0, for  all  x∈C;(A2)
f is monotone; that is, f(x,y)+f(y,x)≤0, for  all  x,y∈C;(A3)
(7)limsupt↓0f(tz+(1-t)x,y)≤f(x,y),∀x,y,z∈C;(A4)
for eachx∈C, y↦f(x,y) is convex and weakly lower semicontinuous.For any convex functiong:E→(-∞,+∞], we denote the domain of g by Dom(g)={x∈E:g(x)<∞}. For any x∈intDom(g) and y∈E, we denote by g0(x,y) the right-hand derivative of g at x in the direction y; that is,
(8)g0(x,y)=limt→0+g(x+ty)-g(x)t.
The function g is said to be Gâteaux differentiable at x if limt→0+((g(x+ty)-g(x))/t) exists for any y. In this case, g0(x,y) coincides with ∇g(x), the value of the gradient ∇g of g at x. The function g is said to be Gâteaux differentiable if it is Gâteaux differentiable everywhere. The function g is said to be Fréchet differentiable at x if the limit is attained uniformly in ∥y∥=1. The function g is said to be Fréchet differentiable if it is Fréchet differentiable everywhere. It is well known that if a continuous convex function g:E→ℝ is Gâteaux differentiable, then ∇g is norm-to-weak* continuous. If g:E→ℝ is Fréchet differentiable, then ∇g is norm-to-norm continuous; for more details, see [31] and the references therein; for more details, see [32] and the references therein. The function g is said to be strongly coercive [32] if lim∥x∥→∞(f(x)/∥x∥)=∞. It is said to be bounded on bounded subsets of E if g(B) is bounded for each bounded subset B of E. Let E be a reflexive Banach space. For any proper, lower semicontinuous, and convex function: g:E→(-∞,+∞], the conjugate function g* of g is defined by
(9)g*(x*)=supx∈E{〈x,x*〉-g(x)},∀x*∈E*.
It is well known that g(x)+g*(x*)≥〈x,x*〉 for all (x,x*)∈E×E*. It is also known that (x,x*)∈∂g, where ∂g is the subdifferential of g, is equivalent to
(10)g(x)+g*(x*)=〈x,x*〉.
If g is a proper, lower semicontinuous, and convex function, then g* is a proper, weak* lower semicontinuous, and convex function.Next, we recall some facts about Bregman distance. Letg:E→ℝ be a convex and Gâteaux differentiable function. Then, Bregman distance corresponding to g is the function Dg:E×E→ℝ defined by
(11)Dg(x,y)=g(x)-g(y)-〈x-y,∇g(y)〉,∀x,y∈E.
It is clear that Dg(x,y)≥0 for all x,y∈E. If E is smooth and g(x)=∥x∥2 for all x∈E, we obtain that ∇g(x)=2Jx, where J is the generalized duality mapping. If C is a nonempty, closed and convex subset of a reflexive Banach space E and g is a strongly coercive Bregman function, then for each x∈E, there exists a unique x0∈C such that Dg(x0,x)=miny∈CDg(y,x). Bregman projection ProjCg from E onto C is defined by ProjCgx=x0 for all x∈E. It is also well known [32] that ProjCg has the following property:
(12)Dg(y,ProjCgx)+Dg(ProjCgx,x)≤Dg(y,x),Dg(y,ProjCgx)+Dg(hhhhhh)∀y∈C,x∈E.
Let Br:={z∈E:∥z∥<r}. A function g:E→ℝ is said to be uniformly convex on bounded subsets of E if fr(t)>0 for all r,t>0, where fr:[0,∞]→∞[0,∞] is defined by
(13)fr(t)=infx,y∈Br,∥x-y∥=t,α∈(0,1)((α(1-α))-1(αg(x)+(1-α)g(y)infx,y∈Br,∥x-y∥=t,α∈(0,1)=((h-g(αx+(1-α)y))infx,y∈Br,∥x-y∥=t,α∈(0,1)=hh×(α(1-α))-1),∀t≥0.
The function fr is called the gauge of the uniform convexity of g. If g:E→ℝ is a convex function which is uniformly convex on bounded subsets, then
(14)g(αx+(1-α)y)≤αg(x)+(1-α)g(y)-α(1-α)fr(∥x-y∥)
for all x,y∈Br and α∈(0,1), where fr is the gauge of the uniform convexity of g.LetT:C→C be a mapping. In this paper, we use F(T) to denote the fixed point set of T. T is said to be closed; if for any sequence {xn}⊂C such that limn→∞xn=x0 and limn→∞Txn=y0, then Tx0=y0.Letg:E→ℝ be a proper, lower semicontinuous and convex function. Recall that T:C→C is said to be Bregman quasi-nonexpansive, if and only if
(15)F(T)≠∅,Dg(p,Tx)≤Dg(p,x),hiihhhhhhhhh∀x∈C,p∈F(T).A pointp in C is said to be an asymptotic fixed point of T if and only if C contains a sequence {xn}, which converges weakly to p such that limn→∞∥xn-Txn∥=0. The set of asymptotic fixed points of T will be denoted by F~(T). T is said to be Bregman relatively nonexpansive if and only if F~(T)=F(T)≠∅ and Dg(p,Tx)≤Dg(p,x) for all x∈C and p∈F(T).LetA be a multivalued operator from E to E* with domain Dom(A)={z∈E:Az≠∅} and range Ran(A)=∪{Az:z∈Dom(A)}. An operator A is said to be monotone if and only if 〈x1-x2,y1-y2〉≥0 for each xi∈Dom(A) and yi∈Axi, i=1,2. A monotone operator A is said to be maximal if its graph Grap(A)={(x,y):y∈Ax} is not properly contained in the graph of any other monotone operator. We know that if A is a maximal monotone operator, then A-1(0) is closed and convex.LetE be a reflexive, strictly convex, and smooth Banach space, and let A be a maximal monotone operator from E to E*. From Rockafellar [33], we find that s>0 and x∈E; there exists a unique xs∈D(A) such that ∇gx∈∇gxs+sAxs. If Jsx=xs, then we can define a single-valued mapping Js:E→Dom(A) by Js=(∇g+sA)-1∇g and such a Js is called the resolvent of A. We know that Js is closed and A-1(0)=F(Js) for all s>0. From [34], we know that Js:E→Dom(A) is a Bregman quasi-nonexpansive.In this paper, we study a common solution problem based on Bregman projections. Strong convergence of the monotone projection algorithm for monotone operators and bifunctions is obtained in a reflexive Banach space.In order to introduce our main results, we also need the following lemmas.Lemma 1 (see [34]).
LetE be a Banach space and let g:E→ℝ be a Gâteaux differentiable function which is uniformly convex on bounded subsets of E. Let {xn} and {yn} be bounded sequences in E. Then,
(16)limn→∞Dg(xn,yn)=0⟺limn→∞∥xn-yn∥=0.Lemma 2 (see [35]).
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E. Let T:C→C be a Bregman relatively nonexpansive mapping. Then, F(T) is closed and convex.Lemma 3 (see [36]).
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E and let f be a bifunction from C×C to ℝ satisfying (A1)–(A4). Let r>0 and x∈E. Then, there exists z∈C such that f(z,y)+1/r〈y-z,∇gz-∇gx〉≥0, for all y∈C. Define a mapping Sr:E→C by Srx={z∈C:f(z,y)+(1/r)〈y-z,∇gz-∇x〉, for all y∈C}. Then, the following conclusions hold:  (1)
Sr is single-valued;(2)
Sr is a Bregman firmly nonexpansive-type mapping; that is, for all x,y∈E,
(17)〈Srx-Sry,∇gSrx-∇gSry〉≤〈Srx-Sry,∇x-∇gy〉;(3)
F(Sr)=EP(f);(4)
Sr is Bregman quasi-nonexpansive;(5)
(18)Dg(q,Srx)+Dg(Srx,x)≤ϕ(q,x),∀q∈F(Sr);(6)
EP(f) is closed and convex.


## 3. Main Results

Theorem 4.
LetE be a reflexive Banach space and let g:E→R be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and  smooth on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E and let Λ be an index set. Let {si} be a positive real number sequence. Let fi be a bifunction from C×C to ℝ satisfying (A1)–(A4) and let Ai:E→E* be a maximal monotone operator such that Dom(Ai)⊂C for every i∈Λ. Assume that the common solution set CSS:=∩i∈ΛAi-1(0)⋂‍∩i∈ΛEF(fi) is nonempty. Let {xn} be a sequence generated in the following manner:
(19)x0∈Echosenarbitrarily,C(1,i)=C,C1=∩i∈ΛC(1,i),x1=ProjC1gx0,y(n,i)=∇g*(α(n,i)∇g(xn)+(1-α(n,i))∇gJsiAixn),u(n,i)∈Csuchthatfi(u(n,i),y)+1r(n,i)〈y-u(n,i),∇g(u(n,i))-∇g(y(n,i))〉≥0,fihhh+1r(n,i)〈y-u(n,i),∇g(u(n,i))-∇g(y(n,i))〉≥∀y∈C,C(n+1,i)={z∈C(n,i):Dg(z,u(n,i))≤Dg(z,xn)},Cn+1=∩i∈ΛC(n+1,i),xn+1=ProjCn+1gx1,
where ∇g is the right-hand derivative of g, JsiAi=(∇g+siAi)-1∇g, {α(n,i)} is a real sequence in [0,1] such that liminfn→∞α(n,i)(1-α(n,i))>0, and {r(n,i)} is a real sequence in [a,∞), where a is some positive real number, for every i∈Λ. Then, the sequence {xn} converges strongly to ProjCSSgx1, where ProjCSSg is the Bregman projection from E onto CSS.Proof.
In view of Lemma3, we find that the common solution set CSS is closed and convex. Next, we prove that Cn is closed and convex. It suffices to show that, for each fixed but arbitrary i∈Λ, C(n,i) is closed and convex. This can be proved by induction into n. It is obvious that C(1,i)=C is closed and convex. Assume that C(k,i) is closed and convex for some k≥1. For z1,z2∈C(k+1,i), we see that z1,z2∈C(k,i). It follows that z=tz1+(1-t)z2∈C(k,i), where t∈(0,1). Notice that
(20)Dg(z1,u(k,i))≤Dg(z1,xk),Dg(z2,u(k,i))≤Dg(z2,xk).
The above inequalities are equivalent to
(21)〈z1,∇g(x)-∇g(u(k,i))〉≤g(u(k,i))-g(xk)-〈u(k,i),∇g(u(k,i))〉+〈x,∇g(x)〉,〈z2,∇g(x)-∇g(u(k,i))〉≤g(u(k,i))-g(xk)-〈u(k,i),∇g(u(k,i))〉+〈x,∇g(x)〉.
Multiplying t and (1-t) on both sides of (21), respectively, yields that
(22)〈z,∇g(x)-∇g(u(k,i))〉≤g(u(k,i))-g(xk)-〈u(k,i),∇g(u(k,i))〉+〈x,∇g(x)〉.
That is,
(23)Dg(z,u(k,i))≤Dg(z,xk),
where z∈C(k,i). This implies that C(k+1,i) is closed and convex. We conclude that C(n,i) is closed and convex. This in turn implies that Cn=∩i∈ΛC(n,i) is closed and convex. This implies that ProjCn+1gx1 is well defined.
Next, we show thatCSS⊂Cn. CSS⊂C1=C is clear. Suppose that CSS⊂C(k,i) for some positive integer k. For any w∈CSS⊂C(k,i), we see that
(24)Dg(w,u(k,i))=Dg(w,Sr(k,i)y(k,i))≤Dg(w,y(k,i))=Dg(w,∇g*(α(k,i)∇g(xk)hhhhhhhhhhhhh+(1-α(k,i))∇gJsiAixk))=V(w,α(k,i)∇g(xk)+(1-α(k,i))∇g(JsiAixk))=g(w)-α(k,i)〈w,∇g(xk)〉-(1-α(k,i))×〈w,∇g(JsiAixk)〉+g*(α(k,i)∇g(xk)+(1-α(k,i))∇g(JsiAixk))≤α(k,i)V(w,∇g(xk))+(1-α(k,i))V(w,∇g(JsiAixk))≤α(k,i)Dg(w,xk)+(1-α(k,i))Dg(w,JsiAixk)≤Dg(w,xk),
which shows that w∈C(k+1,i). This implies that CSS⊂C(n,i). This yields that CSS⊂∩i∈ΛC(n,i). This completes the proof that CSS⊂Cn.
Next, we show that the sequence{xn} is bounded. It follows from (12) that
(25)Dg(xn,x1)≤Dg(ProjCSSgx1,x1)-Dg(ProjCSSgx1,xn)≤Dg(ProjCSSgx1,x1).
This implies that the sequence {Dg(xn,x1)} is bounded. It follows that the sequence {xn} is also bounded. In view of the construction of the set Cn, we find that Cm⊂Cn and xm=projCmgx1∈Cm⊂Cn for any positive integer m≥n. It follows from (12) that
(26)Dg(xm,xn)=Dg(xm,ProjCngx1)≤Dg(xm,x1)-Dg(ProjCngx1,x1)=Dg(xm,x1)-Dg(xn,x1).
It follows that
(27)Dg(xn,x1)≤Dg(xn,x1)+Dg(xm,xn)≤Dg(xm,x1).
This shows that the sequence {Dg(xn,x1)} is nondecreasing. Hence, the limit limn→∞Dg(xn,x1) exists. In view of xn=ProjCngx1, we find that
(28)Dg(xn,x1)≤Dg(xn+1,x1).
This concludes that limn→∞Dg(xn,x1) exists. Let m,n→∞ in (26); we find that Dg(xm,xn)→0. In view of Lemma 1, we obtain that ∥xm-xn∥→0 as m,n→∞. This yields that {xn} is a Cauchy sequence. Since C is closed and convex, we see that there exists an x¯∈C; that is,
(29)limn→∞∥xn-x¯∥=0.
Next, we show that x¯∈CSS. It follows from (26) that
(30)limn→∞Dg(xn+1,xn)=0.
Therefore, we obtain that
(31)limn→∞∥xn+1-xn∥=0.
Since xn+1∈Cn+1, we find that
(32)Dg(xn+1,u(n,i))≤Dg(xn+1,xn).
In view of (30), we find that
(33)limn→∞Dg(xn+1,u(n,i))=0.
It follows that
(34)limn→∞∥xn+1-u(n,i)∥=0.
Notice that
(35)∥xn-u(n,i)∥≤∥xn-xn+1∥+∥xn+1-u(n,i)∥.
It follows from (29) and (34) that limn→∞∥xn-u(n,i)∥=0. In view of Lemma 1, we find that
(36)limn→∞Dg(u(n,i),xn)=0.
Since ∇g is uniformly norm-to-norm continuous on any bounded subset of E, we find that
(37)limn→∞∥∇g(u(n,i))-∇g(xn)∥=0.
Notice that
(38)|Dg(w,xn)-Dg(w,u(n,i))|=|Dg(u(n,i),xn)+w-u(n,i),〈∇g(u(n,i))-∇g(xn)〉|≤|Dg(u(n,i),xn)|+∥w-u(n,i)∥∥∇g(u(n,i))-∇g(xn)∥.
It follows from (36) and (37) that
(39)limn→∞|Dg(w,xn)-Dg(w,u(n,i))|=0.
In view of (14), we find that
(40)Dg(w,u(n,i))=Dg(w,Sr(n,i)y(n,i))≤Dg(w,y(n,i))=Dg(+(1-α(n,i))∇g(JsiAixn))w,∇g*((1-α(n,i))∇g(JsiAixn)α(n,i)∇g(xn)hhhhhhhhhhhh+(1-α(n,i))∇g(JsiAixn)))=V(w,α(n,i)∇g(xn)+(1-α(n,i))∇g(JsiAixn))=g(w)-α(n,i)〈w,∇g(xn)〉-(1-α(n,i))×〈w,∇g(JsiAixn)〉+g*(α(n,i)∇g(xn)+(1-α(n,i))∇g(JsiAixn))≤α(n,i)V(w,∇g(xn))+(1-α(n,i))V(w,∇g(JsiAixn))-ρ(∥∇g(JsiAixn)-∇g(xn)∥)≤α(n,i)Dg(w,xn)+(1-α(n,i))Dg(w,JsiAixn)-α(n,i)(1-α(n,i))×ρ(∥∇g(JsiAixn)-∇g(xn)∥)≤Dg(w,xn)-α(n,i)(1-α(n,i))×ρ(∥∇g(JsiAixn)-∇g(xn)∥),
where ρ is the gauge of the uniform convexity of g*. It follows that
(41)α(n,i)(1-α(n,i))ρ(∥∇g(JsiAixn)-∇g(xn)∥)≤Dg(w,xn)-Dg(w,u(n,i)).
In view of the restrictions on the sequence {α(n,i)}, we find from (39) that limn→∞∥∇g(xn)-∇g(JsiAixn)∥=0. Since ∇g* is uniformly norm-to-norm continuous on bounded subsets of E*, we obtain that
(42)limn→∞∥xn-JsiAixn∥=0.
Since JsiAi is closed Bregman quasi-nonexpansive, we find that JsiAix¯=x¯. This proves that x¯∈∩i∈ΛAi-1(0).
Next, we show thatx¯∈∩i∈ΛEF(fi). Notice that
(43)Dg(u(n,i),y(n,i))≤Dg(w,y(n,i))-Dg(w,u(n,i))≤Dg(w,xn)-Dg(w,u(n,i)).
It follows from (39) that
(44)limn→∞Dg(u(n,i),y(n,i))=0.
Therefore, we find that limn→∞∥u(n,i)-y(n,i)∥=0. Since ∇g is uniformly norm-to-norm continuous on any bounded sets, we have limn→∞∥∇g(u(n,i))-∇(y(n,i))∥=0. From the assumption r(n,i)≥a, we see that
(45)limn→∞∥∇g(u(n,i))-∇g(y(n,i))∥r(n,i)=0.
In view of u(n,i)=Sr(n,i)y(n,i), we see that
(46)fi(u(n,i),y)+1r(n,i)〈y-u(n,i),∇g(u(n,i))-∇g(y(n,i))〉≥0,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀y∈C.
It follows from (A2) that
(47)∥y-u(n,i)∥∥∇g(u(n,i))-∇g(y(n,i))∥rn,i≥1r(n,i)〈y-u(n,i),∇g(u(n,i))-∇g(y(n,i))〉≥fi(y,u(n,i)),∀y∈C.
In view of (A4), we find from (45) that
(48)fi(y,x¯)≤0,∀y∈C.
For 0<ti<1 and y∈C, define y(t,i)=tiy+(1-ti)x¯. It follows that y(t,i)∈C, which yields that f(y(t,i),x¯)≤0. It follows from (A1) and (A4) that
(49)0=f(y(t,i),y(t,i))≤tif(y(t,i),y)+(1-ti)f(y(t,i),x¯)≤tif(y(t,i),y).
That is,
(50)f(y(t,i),y)≥0.
Letting ti↓0, we obtain from (A3) that fi(x¯,y)≥0, for all y∈C. This implies that x¯∈EP(fi) for every i∈Λ. This shows that x¯∈CSS.
Finally, we prove thatx¯=ProjCSSgx1. In view of xn=ProjCngx1, we conclude that
(51)〈z-xn,∇g(xn)-∇g(x1)〉,∀z∈Cn.
Since CSC⊂Cn, we arrive at
(52)〈w-xn,∇g(xn)-∇g(x1)〉,∀w∈CSC.
Letting n→∞ in the above inequality, we see that
(53)〈w-x¯,∇g(x¯)-∇g(x1)〉≥0,∀w∈CSS.
This yields that x¯=ProjCSSgx1. This completes the proof.If the index set is singleton, then we have the following result.Corollary 5.
LetE be a reflexive Banach space and let g:E→R be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and smooth on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E and let s be a positive real number. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4) and let A:E→E* be a maximal monotone operator such that Dom(A)⊂C. Assume that the common solution set CSS:=A-1(0)⋂‍EF(f) is nonempty. Let {xn} be a sequence generated in the following manner:
(54)x0∈Echosenarbitrarily,C1=C,x1=ProjC1gx0,yn=∇g*(αn∇g(xn)+(1-αn)∇g(JsAxn)),un∈Csuchthatf(un,y)+1rn〈y-un,∇g(un)-∇g(yn)〉≥0,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀y∈C,Cn+1={z∈Cn:Dg(z,un)≤Dg(z,xn)},xn+1=ProjCn+1gx1,
where ∇g is the right-hand derivative of g. JsA=(∇g+sA)-1∇g, {αn} is a real sequence in [0,1] such that liminfn→∞αn(1-αn)>0, and {rn} is a real sequence in [a,∞), where a is some positive real number. Then, the sequence {xn} converges strongly to ProjCSSgx1, where ProjCSSg is the Bregman projection from E onto CSS.Remark 6.
Corollary5 improves Theorem 4 of Qin et al. [24] in the following aspects: (1) the framework of spaces is relaxed; (2) a more general notion of Bregman function is considered instead of the generalized duality mapping; (3) a more general notion of Bregman projection is considered instead of the generalized projection operator; (4) families of operators extend from a finite family to an infinite family.Finally, we give some results in the framework of Hilbert spaces.Theorem 7.
LetE be a Hilbert space and let C be a nonempty, closed, and convex subset of E. Let Λ be an index set. Let {si} be a positive real number sequence. Let fi be a bifunction from C×C to ℝ satisfying (A1)–(A4) and let Ai:E→E be a maximal monotone operator such that Dom(Ai)⊂C for every i∈Λ. Assume that the common solution set CSS:=∩i∈ΛAi-1(0)⋂‍∩i∈ΛEF(fi) is nonempty. Let {xn} be a sequence generated in the following manner:
(55)x0∈Echosenarbitrarily,C(1,i)=C,C1=∩i∈ΛC(1,i),x1=ProjC1x0,y(n,i)=α(n,i)xn+(1-α(n,i))JsiAixn,u(n,i)∈Csuchthatfi(u(n,i),y)+1r(n,i)〈y-u(n,i),u(n,i)-y(n,i)〉≥0,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀y∈C,C(n+1,i)={z∈C(n,i):∥z-u(n,i)∥2≤∥z-xn∥2},Cn+1=∩i∈ΛC(n+1,i),xn+1=ProjCn+1x1,
where JsiAi=(I+siAi)-1, {α(n,i)} is a real sequence in [0,1] such that liminfn→∞α(n,i)(1-α(n,i))>0, and {r(n,i)} is a real sequence in [a,∞), where a is some positive real number, for every i∈Λ. Then, the sequence {xn} converges strongly to ProjCSSx1, where ProjCSS is the metric projection from E onto CSS.Proof.
Puttingg(x)=(1/2)∥x∥2, we find that ∇g(x)=x for all x∈E. It follows that Dg(x,y)=∥x-y∥2 for all x,y∈E. From Theorem 4, we can immediately obtain the desired conclusion.If the index set is singleton, then we have the following result.Corollary 8.
LetE be a Hilbert space and let C be a nonempty, closed, and convex subset of E. Let s be a positive real number. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4) and let A:E→E be a maximal monotone operator such that Dom(A)⊂C. Assume that the common solution set CSS:=A-1(0)⋂‍EF(f) is nonempty. Let {xn} be a sequence generated in the following manner:
(56)x0∈Echosenarbitrarily,C1=C,x1=ProjC1x0,yn=αnxn+(1-αn)JsAxn,un∈Csuchthatf(un,y)+1rn〈y-un,un-yn〉≥0,∀y∈C,Cn+1={z∈Cn:∥z-un∥2≤∥z-xn∥2},xn+1=ProjCn+1x1,
where JsA=(I+sA)-1, {αn} is a real sequence in [0,1] such that liminfn→∞αn(1-αn)>0 and {rn} is a real sequence in [a,∞), where a is some positive real number. Then, the sequence {xn} converges strongly to ProjCSSx1, where ProjCSS is the metric projection from E onto CSS.



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*Source: 764909-2013-11-21.xml*