# Bregman Distance and Strong Convergence of  Proximal-Type Algorithms

Authors: Li-Wei Kuo; D. R. Sahu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The purpose of this paper is to discuss some fundamental properties of Bregman distance,
generalized projection operators, firmly nonexpansive mappings, and resolvent operators of
set-valued monotone operators corresponding to a functionalΦ(∥·∥). We further study some proximal point algorithms for finding zeros of monotone operators and solving generalized mixed
equilibrium problems in Banach spaces. Our results improve and extend some recent results concerning
generalized projection operators corresponding to Bregman distance.


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## Body

## 1. Introduction

In this paper,X denotes a real Banach space with norm ∥·∥, and X* denotes the Banach dual of X endowed with the dual norm  ∥·∥*. We write 〈x,j〉 for the value of a functional j in X* at x in X. As usual, xλ→x and xλ⇀x stand for the norm and weak convergence of a net {xλ} to x in X, respectively.A continuous strictly increasing functionφ:ℝ+→ℝ+ is said to be a gauge if
(1)φ(0)=0,limt→+∞φ(t)=+∞.
The mapping Jφ:X→2X* defined by
(2)Jφ(x)={j∈X*:〈x,j〉=∥x∥∥j∥*,∥j∥*=φ(∥x∥)},x∈X,
is called the  duality mapping with gauge φ. In the special case where φ(t)=t, the duality mapping Jφ=:J is the classical normalized duality mapping. In the case φ(t)=tp-1, p>1, the duality mapping Jφ=:Jp is called  the generalized duality mapping and it is given by
(3)Jp(x)∶={j∈X*:〈x,j〉=∥x∥∥j∥*,∥j∥*=∥x∥p-1},x∈X.For a gaugeφ, the function Φ:ℝ+→ℝ+ defined by
(4)Φ(t)=∫0t‍φ(s)ds
is a continuous convex strictly increasing differentiable function on ℝ+ with Φ′(t)=φ(t) and limt→+∞Φ(t)/t=+∞. Therefore, Φ has a continuous inverse function Φ-1.We recall the Bregman DistanceDφ and function Dφf studied in [1]. Let X be a real smooth Banach space. The Bregman distanceDφ(x,y) between x and y in X is defined by
(5)Dφ(x,y):=Φ(∥x∥)-Φ(∥y∥)-〈x-y,Jφ(y)〉.
One can see from Lemma 3 that Dφ(x,y)≥0. In the case φ(t)=tp-1,  p∈(1,∞), the distance Dφ(x,y)=:Dp(x,y) is called the p-Lyapunov functional studied in [2] and it is given by
(6)Dp(x,y)=∥x∥pp+∥y∥pq-〈x,Jp(y)〉.
Note that
(7)ϕ(x,y):=2D2(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2
is the Lyapunov functional. It is obvious that
(8)(∥x∥-∥y∥)2≤ϕ(x,y),∀x,y∈X.
See Brègman [3], Butnariu and Iusem [4], and Censor and Lent [5].LetC be a nonempty closed convex subset of a smooth Banach space X. The generalized projectionΠC:X→C is defined by
(9)ΠC(x)={x0∈C:ϕ(x0,x)=infy∈Cϕ(y,x)}.
The metric projection operator PC:X→C defined by
(10)PCx:=argminy∈C12∥x-y∥2,x∈X,
has been employed successfully in optimization, optimal control, approximation theory, and fixed point theory in the framework of Hilbert spaces. In such a framework, metric projections PC are nonexpansive (i.e., ∥PCx-PCy∥≤∥x-y∥ for all x,y in H). However, this is no longer true in the framework of Banach spaces. Instead, the generalized projections ΠC are needed. In [6], Alber generalized the metric projection operator PC to generalized projection operators ΠC:X→C from Hilbert spaces to uniformly smooth Banach spaces. Many applications of the generalized projections in Banach spaces are discussed in the recent literature (see [7–12]).Section2 contains preliminaries. In Section 3, we study the fundamental properties of Bregman distance Dφ and (φ,f)-generalized projection operators, where f:X→ℝ+ is a proper, convex, lower semicontinuous function. In Section 4, we discuss φ-firmly nonexpansive mappings and φ-resolvent operators. In Section 5, we establish strong convergence of the proximal-projection methods for finding fixed points of φ-firmly nonexpansive mappings, zeros of (not necessarily maximal) monotone operators, and solutions of generalized mixed equilibrium problems in Banach spaces using (φ,f)-generalized projection operators ΠCφ,f. Here, we do not assume the maximality of monotone operators and the uniform smoothness of Banach spaces.


## 2. Preliminaries

LetA:X→2X* be a set-valued operator. The set 𝒟(A)={x∈X:Ax≠∅} is called the effective domain of A. The range of A is defined by ℛ(A)=∪x∈𝒟(A)Ax. The operator A is said to be monotone if for any x,y in 𝒟(A), we have
(11)jx∈Ax,jy∈Ay⟹〈x-y,jx-jy〉≥0.
A monotone operator A is said to be maximal if the graph 𝒢(A)={(x,y):x∈𝒟(A),y∈Ax} of A is not a proper subset of the graph of another monotone operator. We know that if A is a maximal monotone operator, then the zero set A-10 is closed and convex.In the rest of this paper, byφ we always mean a gauge and by Φ the corresponding function defined in (4). We list some properties of the duality mapping Jφ:X→2X* below (for more details see [13, 14]).Proposition 1.
LetX be a real Banach space.(i)
Jφis norm-to-weak*upper semicontinuous;(ii)
for eachxinX, the setJφ(x)is convex and weakly closed inX*;(iii)
Jφ(-x)=-Jφ(x)andJφ(λx)=(φ(∥λx∥)/φ(∥x∥))Jφ(x)for all nonzeroxinX,λ>0;(iv)
there holds(12)Jφ(x)=φ(∥x∥)∥x∥J(x),0≠x∈X;(v)
Jφis maximal monotone;(vi)
ifXis strictly convex, thenJφis strictly monotone; that is,(13)〈x-y,jx-jy〉>0∀x,y∈X,jx∈Jφ(x),jy∈Jφ(y);
(vii)
if
X
is strictly convex and reflexive, then
J
φ
is single-valued monotone and demicontinuous.The following result is well known. We include a proof for completeness.Lemma 2.
If a Banach spaceE has a uniformly Gâteaux differentiable norm, then J:E→E* is uniformly norm-to-weak* continuous on nonempty bounded subsets of E to E*.Proof.
Suppose not, and there exist norm one vectorsxn,yn,z in E and a constant ϵ>0 such that yn-xn→0, and 〈z,J(yn)-J(xn)〉≥ϵ, for all n∈ℕ. For a fixed t>0, define
(14)an:=∥xn+tz∥-∥xn∥-t〈z,J(xn)〉t,bn:=∥yn+tz∥-∥yn∥-t〈z,J(yn)〉t.
Observe that
(15)an≥〈xn+tz,J(yn)〉-〈xn+tz,J(xn)〉t=〈z,J(yn)-J(xn)〉+〈xn,J(yn)-J(xn)〉t,bn≥〈yn+tz,J(xn)〉-〈yn+tz,J(yn)〉t=〈z,J(yn)-J(xn)〉+〈yn,J(yn)-J(xn)〉t.
Hence,
(16)an+bn≥2〈z,J(yn)-J(xn)〉-〈xn-yn,J(yn)-J(xn)〉t≥2ϵ-2∥xn-yn∥t.
Choose t=(2/ϵ)∥xn-yn∥. By the uniform Gâteaux differentiability of the norm, if n is large enough, both an and bn are less than (1/2)ϵ, and so an+bn<ϵ. We arrive at a contradiction.Together with Proposition1, the conclusion in Lemma 2 also holds for Jφ.LetX be a Banach space and ψ:X→(-∞,+∞] a function. The function ψ is proper if dom(ψ):={x∈X:ψ(x)<+∞}≠∅. An element j in X* is said to be a  subgradient  of a proper convex function ψ:X→(-∞,+∞] at a point x in dom(ψ) if
(17)ψ(x)-ψ(y)≤〈x-y,j〉,∀y∈X.
The set (possibly empty)
(18)∂ψ(x):={j∈X*:ψ(x)-ψ(y)≤〈x-y,j〉∀y∈X},
of subgradients of ψ at x in dom(ψ), is called the subdifferential of φ at x.Lemma 3.
LetX be a smooth Banach space and Jφ:X→X* the duality mapping with gauge φ. Then Jφ(y)=∂Φ(∥y∥) for y∈X∖{0}; that is,
(19)Φ(∥x∥)-Φ(∥y∥)-〈x-y,Jφ(y)〉≥0,∀x∈X.The proof of the following result is straightforward.Lemma 4.
LetX be a real Banach space and φ a gauge function.(a)
Φ(∥·∥) is a convex and continuous function on X.(b)
X is strictly convex if and only if Φ(∥·∥) is strictly convex.Lemma 5.
LetX be a real Banach space and φ a gauge function. Then the following assertions are equivalent:(a)
X is uniformly convex;(b)
Φ(∥·∥) is uniformly convex on the closed ball Br:={x∈X:∥x∥≤r}, where r>0 is arbitrarily given. That is, there exists a strictly increasing convex function gr:ℝ+→ℝ+ with gr(0)=0 such that
(20)Φ(∥tx+(1-t)y∥)≤tΦ(∥x∥)+(1-t)Φ(∥y∥)-t(1-t)gr(∥x-y∥)for allx,y in Br and t in [0,1].Proof.
First we note that the general case can be reduced to the caser=1. Suppose it holds
(21)Φ(∥tx+(1-t)y∥)≤tΦ(∥x∥)+(1-t)Φ(∥y∥)-t(1-t)g(∥x-y∥).
We will show that (20) holds for any r>0. Set φr(t)=φ(rt) and gr(t)=rg(t/r) for t≥0. Then φr is still a gauge function, and let Φr be the function corresponding to φr as defined in (4), rΦr(t)=Φ(rt). Let x,y∈Br and xr=x/r and yr=y/r. Then xr,yr∈BX. Applying (21) to Φr, we get
(22)Φ(∥tx+(1-t)y∥)=rΦr(∥txr+(1-t)yr∥)≤rtΦr(∥xr∥)+r(1-t)Φr(∥yr∥)-rt(1-t)g(∥xr-yr∥)=tΦ(∥x∥)+(1-t)Φ(∥y∥)-t(1-t)gr(∥x-y∥),
which is exactly (20). A similar argument also shows that the case r=1 can be deduced from any other case r>0.
Below, we assumer=1 and g=g1.
(
b
)
⇒
(
a
). Given x,y in X such that ∥x∥=∥y∥=1 and ∥x-y∥=ε. Setting t=1/2 in (20), we have
(23)Φ(∥x+y2∥)≤Φ(1)-14g(ε).
It turns out that
(24)1-∥x+y2∥≥1-Φ-1(Φ(1)-14g(ε))>0.
This verifies that X is uniformly convex.
(
a
)
⇒
(
b
). Assume that X is uniformly convex, which implies that Φ(∥·∥) is strictly convex by Lemma 4(b). Define a function μ on [0,2] by setting μ(0)=0, and for 0<ε≤2,
(25)μ(ε):=inf{12Φ(∥x∥)+12Φ(∥y∥)-Φ(12∥x+y∥):x,y∈BX,∥x-y∥≥ε},
where BX is the closed unit ball of X.
Claim. Consider μ(ε)>0 for 0<ε≤2.
Suppose on the contrary thatμ(ε)=0 for some 0<ε≤2. Then we can find sequences {xn}, {yn} in BX such that ∥xn-yn∥≥ε for all n, and
(26)12Φ(∥xn∥)+12Φ(∥yn∥)-Φ(∥xn+yn∥2)⟶0.
Without loss of generality, we may assume that
(27)∥xn∥⟶α,∥yn∥⟶β,∥xn+yn∥⟶γ,∥xn-yn∥⟶θ≥ε.
It then follows from (26) that
(28)12Φ(α)+12Φ(β)=Φ(γ2).
The strict convexity of Φ together with (28) implies that α+β<γ if α≠β. Since, on the other hand, by definition, γ≤α+β. We therefore must have α=β, which together with (28) implies that α=β=γ/2. If we set xn′=xn/∥xn∥, yn′=yn/∥yn∥, then ∥xn′∥=∥yn′∥=1 for all n; moreover, from (27), we get
(29)∥xn′+yn′2∥⟶1,∥xn′-yn′∥⟶θα>0.
This contradicts the uniform convexity of X, and verifies that μ(ε)>0 for all 0<ε≤2.
It turns out from (25) that
(30)Φ(∥12x+12y∥)≤12Φ(∥x∥)+12Φ(∥y∥)-14g(∥x-y∥),
for all x,y in BX, with g=4μ.
By the dyadic rational argument used in the proof of [15, Theorem 2.2], we can extend the inequality (30) to the case of a general convex combination of x and y, namely,
(31)Φ(∥tx+(1-t)y∥)≤tΦ(∥x∥)+(1-t)Φ(∥y∥)-t(1-t)g(∥x-y∥)
for x,y in BX and t in [0,1]. Note that g is increasing and continuous. By [16], the function g can also be assumed to be convex (the convexity of g is not needed in our argument throughout the rest of this paper however).Lemma 6.
LetX be a real uniformly convex Banach space. Then there exists a strictly increasing convex function g:ℝ+→ℝ+ with g(0)=0 such that
(32)Φ(∥x+y∥)≥Φ(∥x∥)+〈y,j〉+g(∥y∥),∀x,y∈BX,∀j∈Jφ(x).Proof.
SinceJφ is the subdifferential of the functional Φ(∥·∥), we have for jx in Jφ(x), x in X that
(33)〈y-x,jx〉≤Φ(∥y∥)-Φ(∥x∥)∀y∈X.
Let g be the function that satisfies (20) with r=2 and assume that x,y∈BX. Replacing y with x+λy,  0<λ<1, we obtain
(34)〈y,jx〉≤[Φ(∥x+λy∥)-Φ(∥x∥)]λ=[Φ(∥(1-λ)x+λ(x+y)∥)-Φ(∥x∥)]λ≤[(1-λ)Φ(∥x∥)+λΦ(∥x+y∥)-λ(1-λ)g(∥y∥)-Φ(∥x∥)](λ)-1≤Φ(∥x+y∥)-Φ(∥x∥)-(1-λ)g(∥y∥).
Taking limit as λ→0, we get
(35)〈y,jx〉=Φ(∥x+y∥)-Φ(∥x∥)-g(∥y∥).Lemma 7 (see [17, Theorem 3.11, page 952]).
LetX be a reflexive Banach space and Jφ:X→2X* the duality map with gauge φ. Suppose that A:X→2X* is maximal monotone. Then the operator A+Jφ is surjective.


## 3. Bregman DistanceDφ and Function Dφf

One can easily see that(36)Dφ(x,y)+Dφ(y,z)-Dφ(x,z)=〈x-y,Jφ(z)-Jφ(y)〉,∀x,y,z∈X.
Noticing that for x in X, the scalar function Dφ(·,x) is coercive (see [18, Lemma 7.3(v)]).Proposition 8.
LetX be a strictly convex smooth Banach space. Let x,y∈X. Then
(37)Dφ(x,y)=0⟺x=y.Proof.
See [18, Lemma 7.3(vi)]Proposition 9.
LetX be a smooth and uniformly convex Banach space. Then there exists a strictly increasing convex function g:ℝ+→ℝ+ with g(0)=0 such that
(38)Dφ(u,v)≥g(∥u-v∥)∀u,v∈BX.Proof.
Letu,v∈BX. By Lemma 6 we have
(39)Dφ(u,v)=Φ(∥v+(u-v)∥)-Φ(∥v∥)-〈u-v,Jφ(v)〉≥Φ(∥v∥)+〈u-v,Jφ(v)〉+g(∥u-v∥)-Φ(∥v∥)-〈u-v,Jφ(v)〉=g(∥u-v∥).As in Butnariu et al. [19], we can prove the following proposition.Proposition 10.
LetX be a smooth and uniformly convex Banach space. Let {xn} and {yn} be two sequences in X such that Dφ(xn,yn)→0. If {yn} is bounded, then ∥xn-yn∥→0.Proof.
Assume{yn} is bounded. From definition (5) we have
(40)Dφ(xn,yn)=Φ(∥xn∥)-Φ(∥yn∥)-〈xn-yn,Jφ(yn)〉.
It follows that
(41)Φ(∥xn∥)-∥xn∥φ(∥yn∥)≤Φ(∥xn∥)-〈xn,Jφ(yn)〉=Φ(∥yn∥)-∥xn∥φ(∥yn∥)+Dφ(xn,yn)≤Φ(∥yn∥)+Dφ(xn,yn).
Since {yn} is bounded, Dφ(xn,yn)→0 and Φ(t)/t→∞ as t→∞ it follows from (41) that {∥xn∥} is bounded, too.
We may now assume that{xn} and {yn} both lie in the closed unit ball BX (otherwise consider the rescaled sequences {γxn} and {γyn} for a sufficiently small γ>0). By Proposition 9, there exists a strictly increasing convex function g:ℝ+→ℝ+ with g(0)=0 such that
(42)g(∥xn∥-∥yn∥)≤Dφ(xn,yn),∀n∈ℕ.
Since Dφ(xn,yn)→0 and g is strictly increasing, we immediately conclude that ∥xn-yn∥→0.Proposition 11 (see [20, Lemma 3.1]).
LetX be a smooth Banach space. Let u∈X and {xn} be a sequence in X such that {Dφ(xn,u)} is bounded. Then {xn} is bounded.The statement in the following proposition is evident from the definition ofDφ (cf. [18, Lemma 7.3(ii)]).Proposition 12.
LetX be a smooth Banach space. Then, for any fixed x in X, the scalar function Dφ(·,x) is continuous, weakly lower semicontinuous, and convex on X.Letf:X→[0,+∞] be a proper, convex, lower semicontinuous function. Define
(43)Dφf(x,y)=Dφ(x,y)+f(x),∀x,y∈X.Some of the following basic properties of the Bregman distanceDφ and function Dφf are known in the literature (see [18–21]).The following proposition can be deduced from Butnariu and Kassay [21, Lemma 2.1].Proposition 13.
LetC be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space X. Let x∈X and let f:X→[0,+∞] be a proper, convex, lower semicontinuous function with C⊂dom(f). Then there exists a unique element x0 in C such that
(44)Dφf(x0,x)=inf{Dφf(z,x):z∈C}.Bregman projections are thoroughly studied and used for iteration schemes such as sequential subspace methods or split feasibility problems successfully (see, [22–24]). The notion of Df-proximal mappings was introduced and studied in [1]. Recently, the notion of Moreau proximal mapping [25] is generalized by Butnariu and Kassay [21] as the proximal mapping relative to f associated with a proper, convex, lower semicontinuous function φ. Using the idea of [1, 26], Proposition 13 allows us to extend generalized projections ΠC as follows.Definition 14.
In the setting of Proposition13, we define the (φ,f)-generalized projection from X onto C by
(45)ΠCφ,f(x)=x0,x∈X.
In caseφ(t)=t and f=0, we notice that ΠCφ,f coincides with ΠC. In case that X is a Hilbert space and φ(t)=t, denote ΠCφ,f by PCf.Applying the tools used in [1, 26, 27], we can establish the following results.Proposition 15.
LetC be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space X and let f:X→[0,+∞] be a proper, convex, lower semicontinuous function with C⊂dom(f).(i)
Letx0∈Candx∈X. Then the following assertions are equivalent:
(a)
x0=ΠCφ,f(x),(b)
〈z-x0,Jφx0-Jφx〉+f(z)-f(x0)≥0,∀z∈C.(ii)
GivenxinX, one has(46)Dφ(y,ΠCφ,f(x))+Dφ(ΠCφ,f(x),x)≤Dφ(y,x)+f(y)-f(ΠCφ,f(x)),∀y∈C.Proposition 16.
LetC be a nonempty closed convex subset of a smooth and uniformly convex Banach space X and f:X→[0,+∞] a proper, convex, lower semicontinuous function with C⊂dom(f). Let u∈X and let {xn} be a bounded sequence in C such that limn→+∞f(xn)=0. If Dφf(xn,u)≤Dφf(ΠCφ,f(u),u) for all n in ℕ, then limn→+∞xn=ΠCφ,f(u).Proof.
SetΠCφ,f(u)=:u~. Then, by assumption, we have Dφf(xn,u)≤Dφf(u~,u) for all n in ℕ. By the three-point identity (36), we have
(47)Dφf(xn,u~)=Dφf(xn,u)+Dφf(u,u~)-〈xn-u,Jφ(u~)-Jφ(u)〉-f(u)≤Dφf(u~,u)+Dφf(u,u~)-〈xn-u,Jφ(u~)-Jφ(u)〉-f(u)≤〈u~-u,Jφ(u~)-Jφ(u)〉-〈xn-u,Jφ(u~)-Jφ(u)〉+f(u~)=〈u~-xn,Jφ(u~)-Jφ(u)〉+f(u~).
Since {xn} is a bounded sequence in C, there exists a subsequence {xni} of {xn} such that {xni} converges weakly to some element z in C. Note that
(48)0≤f(z)≤liminfi→+∞f(xni)=0;
that is, f(z)=0. Using Proposition 15, we have
(49)limsupi→+∞Dφf(xni,u~)≤limsupi→+∞[〈u~-xni,Jφ(u~)-Jφ(u)〉+f(u~)]≤〈u~-z,Jφ(u~)-Jφ(u)〉+f(u~)≤f(z),
which implies that limi→+∞Dφ(xni,u~)=0. It follows from Proposition 10 that limi→∞xni=u~. This implies that xn converges strongly to u~=ΠCφ,f(u).


## 4.φ-Firmly Nonexpansive and φ-Resolvent Operators

Following [1], we study properties of φ-firmly nonexpansive mappings in Banach spaces.Definition 17.
LetX be a smooth Banach space, Jφ:X→X* a duality mapping with gauge function φ, and C a nonempty subset of X. An operator T:C→X is called φ-firmly nonexpansive if
(50)〈Tx-Ty,Jφ(Tx)-Jφ(Ty)〉≤〈Tx-Ty,Jφ(x)-Jφ(y)〉,∀x,y∈C.
In the case ofφ(t)=t, inequality (50) reduces to
(51)〈Tx-Ty,J(Tx)-J(Ty)〉≤〈Tx-Ty,J(x)-J(y)〉,∀x,y∈C.
If T satisfies condition (51), we call T of firmly nonexpansive type. The class of firmly nonexpansive type operators is studied by Kohsaka and Takahashi [28]. When X is a Hilbert space, inequality (50) reduces to the following inequality about firmly nonexpansive operators in the classical sense (see Goebel and Kirk [29]):
(52)∥Tx-Ty∥2≤〈Tx-Ty,x-y〉,∀x,y∈C.
We now give useful characterizations ofφ-firmly nonexpansive mappings which can be deduced from the Bregman distance (5).Proposition 18.
LetX be a smooth Banach space and C a nonempty closed convex subset of X. Let T:C→C be a φ-firmly nonexpansive mapping. Then
(53)Dφ(Tx,Ty)+Dφ(Ty,Tx)+Dφ(Tx,x)+Dφ(Ty,y)≤Dφ(Tx,y)+Dφ(Ty,x)∀x,y∈C.The geometry of the fixed point set ofφ-firmly nonexpansive mappings is established in Reich and Sabach [30, Lemma 15.5] as follows.Proposition 19.
LetX be a strictly convex smooth Banach space and C a nonempty closed convex subset of X. Let T:C→C be a φ-firmly nonexpansive mapping. Then the set F(T) of fixed points of T is closed and convex.LetC be a nonempty closed convex subset of a smooth Banach space X and let T:C→C be a mapping. A point u in C is an asymptotic fixed point of T if C contains a sequence {xn} such that xn⇀u and ∥xn-Txn∥→0; see [31]. We denote the set of asymptotic fixed points of T by F^(T). A mapping T:C→C is relativelyφ-nonexpansive if the following conditions are satisfied:(i)
F(T) is nonempty;(ii)
Dφ(p,Tx)≤Dφ(p,x) for all x in C and p in F(T);(iii)
F^(T)=F(T).The class of relativelyφ-nonexpansive mappings is larger than the class of relatively nonexpansive mappings (see [32]).A mappingT:C→C is φ-firmly quasinonexpansive if F(T)≠∅ and
(54)〈Tx-p,Jφ(x)-Jφ(Tx)〉≥0
for all x in C and p in F(T). From the definition of the Bregman distance (5) and (54), the following proposition follows immediately.Proposition 20.
LetX be a smooth Banach space, C a nonempty subset of X, and T:C→C a φ-firmly quasinonexpansive mapping. Then
(55)Dφ(p,Tx)+Dφ(Tx,x)≤Dφ(p,x)
for all x in C and p in F(T).The following supplements Reich and Sabach [30, Lemma 15.6].Proposition 21.
LetX be a strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let C be a nonempty closed convex subset of X and let T:C→C be a φ-firmly nonexpansive mapping. Then F^(T)=F(T).Proof.
It is easy to see thatF(T)⊂F^(T). It remains to prove that F^(T)⊂F(T). For this, suppose that u∈F^(T). Then, there exists a sequence {xn} in C such that xn⇀u and ∥xn-Txn∥→0. We need to prove that u∈F(T). Using (53), we get
(56)Dφ(Tu,u)≤Dφ(Txn,u)+Dφ(Tu,xn)-Dφ(Txn,Tu)-Dφ(Tu,Txn)-Dφ(Txn,xn).
It is not hard to find that (56) is reduced to the relation
(57)Dφ(Tu,u)≤Φ(∥Tu∥)-Φ(∥u∥)+〈u-Txn,Jφ(u)〉+〈Txn-Tu,Jφ(xn)-Jφ(Txn)〉+〈Txn-Tu,Jφ(Tu)〉.
We note that Txn⇀u. Indeed, for any norm one linear functional ξ of X, we have
(58)〈Txn-u,ξ〉=〈Txn-xn,ξ〉+〈xn-u,ξ〉.
Both terms in the right-hand side approach zero. Consequently,
(59)〈Txn-Tu,Jφ(Tu)〉⟶〈u-Tu,Jφ(Tu)〉.
Claim. Consider 〈Txn-Tu,Jφ(xn)-Jφ(Txn)〉→0.
Sincexn⇀u and ∥xn-Txn∥→0, there is a constant M>0 such that all ∥xn∥,∥Txn∥<M. It follows from the uniform norm to weak* continuity of Jφ on bounded subsets (Lemma 2); we have
(60)〈Tu,Jφ(xn)-Jφ(Txn)〉⟶0.
Observe that
(61)ϵn∶=〈Txn-xn,Jφ(xn)〉⟶0.
Moreover, |∥xn∥-∥Txn∥|≤∥xn-Txn∥→0. By the uniform continuity of φ on [0,M], we have
(62)〈Txn,Jφ(xn)-Jφ(Txn)〉=〈Txn,Jφ(xn)〉-〈Txn,Jφ(Txn)〉=∥xn∥φ(∥xn∥)+ϵn-∥Txn∥φ(∥Txn∥)=(∥xn∥-∥Txn∥)φ(∥xn∥)+∥Txn∥(φ(∥xn∥)-φ(∥Txn∥))+ϵn⟶0.
The claim is thus verified.
We obtain from (57) and the claim that
(63)Dφ(Tu,u)≤Φ(∥Tu∥)-Φ(∥u∥)+〈u-Tu,Jφ(Tu)〉.
This is equivalent to
(64)〈Tu-u,Jφ(Tu)-Jφ(u)〉≤0.
The strict monotonicity of the duality mapping Jφ implies that equality must hold. Namely, Tu=u or u∈F(T).The resolvent of an operatorA:X→2X* relative to a Gâteaux differentiable function f is introduced and studied in [1]. We define φ-resolvent operators following [1, 18].Definition 22.
LetC be a nonempty closed convex subset of a smooth Banach space X and let Jφ:X→X* be the duality mapping with gauge φ. Suppose that A:X→2X* is an operator satisfying the range condition
(65)D(A)⊂C⊂Jφ-1ℛ(Jφ+rA)∀r>0.
For each λ>0, the φ-resolvent associated with operator A is the operator Rλφ,A:C→2X defined by
(66)Rλφ,Ax={z∈X:Jφ(x)∈(Jφ+λA)z},x∈C.
Forx in C and λ in (0,∞), we have
(67)0∈Ax⟺Jφ(x)∈(Jφ+λA)x⟺x∈(Jφ+λA)-1Jφx⟺x∈F(Rλφ,A).
If A is maximal monotone, then, by Lemma 7, we see that condition (65) holds for C=D(A)¯.Remark 23.
For smoothX and φ(t)=tp-1 with p∈(1,+∞), we have Jφ=Jp and Rp,A=(Jp+A)-1Jp. For p=2,  RA:=R2,A=(J+A)-1∘J and this kind of resolvent operators is studied in the literature (see [28, 33]).
LetC be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space X. Let A:X→2X* be a monotone operator satisfying the condition D(A)⊂C⊂Jφ-1ℛ(Jφ+λA), where λ>0. Using the smoothness and strict convexity of X, we obtain that Rλφ,A is single-valued. The conditions D(A)⊂C⊂Jφ-1ℛ(Jφ+λA) ensure that Rλφ,A is the single-valued φ-resolvent operator form C into D(A). In other words,
(68)Rλφ,Ax=(Jφ+λA)-1Jφx,∀x∈C.
Following [18, 28], we have the following proposition.Proposition 24.
LetC be a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach space X and let Jφ:X→X* be the duality mapping with gauge φ. Let A:X→2X* be a monotone operator satisfying the condition D(A)⊂C⊂Jφ-1ℛ(Jφ+λA), where λ is a positive real number. Let Rλφ,A be a resolvent of A, where(a)
Rλφ,Aisφ-firmly nonexpansive mapping fromCintoC,(b)
F(Rλφ,A)=A-10.


## 5. Convergence Theorems

LetC be a nonempty subset of a Banach space X and 𝒯={T(t):t>0} a family of mappings from C into C with ⋂t>0‍F(T(t))≠∅. Let 𝒢={Gt:t>0} be a family of mappings from C into C such that ⋂t>0‍F(T(t))⊆⋂t>0‍F(Gt). We say the family 𝒯:={T(t):t>0} has property(𝒜)with respect to the family𝒢={Gt:t>0} if the following assertion holds:
(69)foreachboundednet{xs}s>0inCifxs-Gsxs⟶0ass⟶+∞thenxs-T(t)xs⟶0ass⟶+∞,∀t>0.If𝒯=𝒢 and the above condition holds, then we say 𝒯:={T(s):s>0} has property(𝒜).Remark 25.
If𝒯 is a singleton, that is, 𝒯={T}, or T(s)=T for all s>0, then {T} always has property (𝒜).We now give some examples.Example 26.
LetC be a nonempty closed convex subset of a Banach space X and T a nonexpansive mapping from C into C with F(T)≠∅. Assume bt in ℝ with 0<a≤bt≤b<1 for all t>0. Define Gt:C→C by Gtx=(1-bt)x+btTx for all x in C. Then T has property (𝒜) with respect to the family {Gt:t>0}.Proof.
Let{xt}t>0 be a bounded net in C such that ∥xt-Gt(xt)∥→0 as t→∞. Note that
(70)∥xt-Txt∥=1bt∥xt-Gt(xt)∥
and 0<a≤bt≤b<1 for all t>0. Therefore, ∥xt-Txt∥→0 as t→+∞.The following example shows that the family{RtA:t>0} of resolvent operators of a maximal monotone operator A enjoys property (𝒜).Example 27.
LetC be a nonempty closed convex subset of a real Hilbert space H and let A⊂H×H be a monotone operator satisfying the following condition:
(71)D(A)⊂C⊂⋂r>0‍ℛ(I+rA).
Let {zt}t>0 be a bounded net in C such that ∥zt-RtAzt∥→0 as t→+∞. Then ∥zt-RrAzt∥→0 as t→+∞ for each r>0.Proof.
Letr, t>0. By Takahashi [34], we have
(72)1r∥RtAzt-RrARtAzt∥≤1t∥zt-RtAzt∥.
Using (72), we have
(73)∥zt-RrAzt∥≤∥zt-RtAzt∥+∥RtAzt-RrARtAzt∥+∥RrARtAzt-RrAzt∥≤(2+rt)∥zt-RtAzt∥⟶0ast⟶+∞.We now discuss the problem of finding common fixed points of a sequence ofφ-firmly nonexpansive mappings. Our proximal-projection method is based on (a not necessarily Bregman distance) function Dφf. The proof is based on the technique in [20].Theorem 28.
LetX be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Let φ be a gauge function, C a nonempty closed convex subset of X, and f:X→[0,+∞] a proper, convex, lower semicontinuous function with C⊂dom(f). Let T:C→C be a φ-firmly nonexpansive mapping and 𝒯:={Tn} a sequence of φ-firmly nonexpansive self-mappings on C such that F(T)=⋂n∈ℕ‍F(Tn)≠∅ and 𝒯 has property (𝒜) with respect to T. For u in C and C1=C with x1=ΠC1φ,f(u), define a sequence {xn} in C as follows:
(74)yn=Tnxn,Cn+1={z∈Cn:〈z-yn,Jφ(xn)-Jφ(yn)〉≤0},xn+1=ΠCn+1φ,f(u)∀n∈ℕ.
Then {xn} converges strongly to ΠF(T)φ,f(u).Proof.
We proceed the proof in the following steps:
Step  1. {xn} is well defined.
Note that allCn are closed and convex. For p in F(𝒯) and n in ℕ, we obtain from (54) that
(75)〈p-yn,Jφ(xn)-Jφ(yn)〉≤0.
It follows that p∈Cn and hence F(𝒯)⊂Cn. Therefore, {xn} is well defined.
Step  2. {xn} is bounded.
Letp∈F(𝒯). It follows from Proposition 15; we have
(76)Dφ(xn,u)=Dφ(ΠCnφ,f(u),u)≤Dφ(p,u)-Dφ(p,ΠCnφ,f(u))+f(p)-f(xn)≤Dφ(p,u)+f(p).
It follows that {Dφ(xn,u)} is bounded and hence from Proposition 11, we obtain that {xn} is bounded.
Step  3. Consider ∥xn-Txn∥→0.
Note thatxn+1∈Cn+1⊂Cn. It follows from Proposition 15 that
(77)Dφ(xn+1,ΠCnφ,f(u))+Dφ(ΠCnφ,f(u),u)≤Dφ(xn+1,u)+f(xn+1)-f(ΠCnφ,f(u)).
This implies that
(78)Dφ(xn,u)≤Dφ(xn+1,u)-Dφ(xn+1,xn)+f(xn+1)-f(xn).
Therefore, the sequence {Dφf(xn,u)} (see (43)) is increasing. Note that {Dφf(xn,u)} is bounded by (76). It follows that limn→+∞Dφf(xn,u) exists. By (78), we obtain
(79)limn→+∞Dφf(xn+1,xn)=0.
One can see that limn→+∞f(xn)=0. Using Proposition 10, we obtain that limn→+∞∥xn-xn+1∥=0. Since xn+1∈Cn+1, we have
(80)0≤Dφ(xn+1,yn)+Dφ(yn,xn)≤Dφ(xn+1,yn)+Dφ(yn,xn)+〈yn-xn+1,Jφ(xn)-Jφ(yn)〉=Dφ(xn+1,xn).
Using (79), we obtain that
(81)limn→+∞[Dφ(xn+1,yn)+Dφ(yn,xn)]=0,
and hence limn→+∞Dφ(yn,xn)=0.  Note that {xn} is bounded. Then, one can see from Proposition 10 that {yn} is bounded and
(82)limn→+∞∥xn-yn∥=0.
This implies that
(83)limn→+∞∥xn-Tnxn∥=0.
Since the family 𝒯:={Tn:n∈ℕ} has property (𝒜) with respect to T, it follows from (83) that limn→+∞∥xn-Txn∥=0.
Step  4. The sequence {xn} converges strongly to ΠF(T)φ,f(u).
Since{xn} is bounded, there exists a subsequence {xni} of {xn} such that xni⇀z∈C. Hence z∈F^(T)=F(T). From Proposition 19, F(T) is closed and convex. The nonemptiness of F(T) implies that the generalized projection ΠF(T)φ,f is well defined. Note that xn=ΠCnφ,f(u) and F(T) is contained in Cn; we have
(84)Dφf(xn,u)≤Dφf(ΠF(T)φ,f(u),u).
Therefore, we conclude from Proposition 16 that {xn} converges strongly to ΠF(T)φ,f(u).Theorem 29.
LetX be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Let φ be a gauge function, C a nonempty closed convex subset of  X, and f:X→[0,+∞] a proper, convex, lower semicontinuous function with C⊂dom(f). Let A:X→2X* be a monotone operator with A-10≠∅ satisfying the following condition:
(85)D(A)⊂C⊂⋂r>0‍Jφ-1ℛ(Jφ+rA).
Let λ be a positive real number; for u in C and C1=C with x1=ΠC1φ,f(u), define a sequence {xn} in C as follows:
(86)yn=(Jφ+λA)-1Jφxn,Cn+1={z∈Cn:〈z-yn,Jφ(xn)-Jφ(yn)〉≤0},xn+1=ΠCn+1φ,f(u),∀n∈ℕ.
Then {xn} converges strongly to ΠA-10φ,f(u).Proof.
SetT:=(Jφ+λA)-1Jφ. Note that T is φ-firmly nonexpansive mapping from C into C and F(T)=A-10. Further, every singleton family {T} enjoys property (𝒜). Therefore, Theorem 29 follows from Theorem 28.Remark 30.
Compared with other convergence theorems concerning proximal point algorithms in the literature (see, e.g., Agarwal et al. [35, Theorem 3.1]; Kamimura and Takahashi [26, Theorem 8]; Matsushita and Takahashi [32, Theorem 4.3]), Theorem 29 establishes a new proximal point algorithm for the problem of finding zeros of (not necessarily maximal) monotone operators in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm.We now derive an interesting new result.Corollary 31.
LetC be a nonempty closed convex subset of real Hilbert space H, and let f:X→[0,+∞] be a proper, convex, lower semicontinuous function with C⊂dom(f). Let A⊂H×H be a monotone operator satisfying condition (71) such that A-10≠∅. Let {λn} be a sequence in (0,∞) such that limn→+∞λn=+∞, for u in C, let C1=C with x1=PC1f(u), and define a sequence {xn} in C as follows:
(87)yn=(I+λnA)-1xn,Cn+1={z∈Cn:〈z-yn,xn-yn〉≤0},xn+1=PCn+1f(u),∀n∈ℕ.
Then {xn} converges strongly to PA-10f(u).Proof.
SetT:=RλA=(I+λA)-1 for λ>0 and Tn:=(I+λnA)-1 for all n in ℕ. Note that T is a firmly nonexpansive mapping from C into C and F(T)=A-10. From (83), we have limn→+∞∥xn-Tnxn∥=0. Example 27 implies that 𝒯:={Tr:r>0} has property (𝒜). It follows that limn→+∞∥xn-Txn∥=0. Therefore, Corollary 31 follows from Theorem 28.LetC be a nonempty, closed, and convex subset of a Banach space X. Let Θ:C×C→ℝ be a bi-function, A:C→X* a nonlinear operator, and ψ:C→ℝ a real-valued function. We assume the following conditions are all satisfied.(A1)
Θ(x,x)=0 for all x in C.(A2)
Θ is monotone; that is, Θ(x,y)+Θ(y,x)≤0 for all x,y in C.(A3)
for allx,y,z in C, limsupt↓0Θ(tz+(1-t)x,y)≤Θ(x,y).(A4)
for allx in C, Θ(x,·) is convex and lower semicontinuous.Blum and Oettli [36] studied the following equilibrium problem (EP).Findx^ in C such that
(88)Θ(x^,y)≥0,∀y∈C.
The solution set of (88) is denoted by EP(Θ).Following [37], we have the following lemma.Lemma 32.
LetC be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space X, and let φ be a gauge function. Let Θ:C×C→ℝ be a bi-function satisfying conditions (A1)–(A4), and let r>0 and x∈X. Then there exists z in C such that
(89)Θ(z,y)+1r〈y-z,Jφz-Jφx〉≥0,∀y∈C.We now consider the followinggeneralized mixed equilibrium problem (GMEP): find z in C such that
(90)Θ(z,y)+〈Az,y-z〉+ψ(y)-ψ(z)≥0,∀y∈C.
The solution set of (90) is denoted by GMEP(Θ,A,ψ). The following auxiliary generalized mixed equilibrium problem is an important tool for finding the solution of GMEP (90).Letr>0. For a given point x in C, find z in C such that
(91)Θ(z,y)+〈Az,y-z〉+ψ(y)-ψ(z)+1r〈Jφz-Jφx,y-z〉≥0,∀y∈C.
The existence of a solution of the auxiliary mixed equilibrium problem (91) is guaranteed by [37].LetC be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space X and let φ be a gauge function. Let A:C→X* be continuous and monotone, Θ:C×C→ℝ a bi-function satisfying conditions (A1)–(A4), and ψ:C→ℝ a lower semicontinuous and convex function. For r>0, define the mapping Tr(Θ,A,ψ):X→2C as follows:
(92)Tr(Θ,A,ψ)(x)={1rz∈C:Θ(z,y)+〈Az,y-z〉+ψ(y)-ψ(z)+1r〈Jφz-Jφx,y-z〉≥0,∀y∈C}.Lemma 33 (see [37]).
One has the following.(1)
Tr(Θ,A,ψ) is single-valued.(2)
Tr(Θ,A,ψ) is a φ-firmly nonexpansive mapping; that is, for all x,y in X,
(93)〈Tr(Θ,A,ψ)x-Tr(Θ,A,ψ)y,JφTr(Θ,A,ψ)x-JφTr(Θ,A,ψ)y〉≤〈Tr(Θ,A,ψ)x-Tr(Θ,A,ψ)y,Jφx-Jφy〉.(4)
F(Tr(Θ,A,ψ))=GMEP(Θ,A,ψ).(5)
GMEP(Θ,A,ψ) is closed and convex.(6)
For allx in F(Tr(Θ,A,ψ)) and y in X, one has
(94)Dφ(x,Tr(Θ,A,ψ)y)+Dφ(Tr(Θ,A,ψ)y,y)≤Dφ(x,y).The following theorem establishes the strong convergence of the proximal-projection method for solving generalized mixed equilibrium problems in the framework of uniformly convex Banach spaces.Theorem 34.
LetC be a nonempty closed convex subset of a uniformly convex Banach space X with a uniformly Gâteaux differentiable norm. Let φ be a gauge function and let f:X→[0,∞] be a proper, convex, lower semicontinuous function with C⊂dom(f). Let A:C→X* be continuous and monotone, Θ:C×C→ℝ a bi-function satisfying conditions (A1)–(A4), and ψ:C→ℝ a lower semicontinuous and convex function. Assume that GMEP(Θ,A,ψ,φ)≠∅. For u in C,  r>0, and C1=C with x1=ΠC1φ,f(u), define a sequence {xn} in C as follows:
(95)yn=Tr(Θ,A,ψ)xn,Cn+1={z∈Cn:〈z-yn,Jφ(xn)-Jφ(yn)〉≤0},xn+1=ΠCn+1φ,f(u)∀n∈ℕ.
Then {xn} strongly converges to ΠF(Tr(Θ,A,ψ))φ,f(u).Proof.
Note thatTr(Θ,A,ψ) is a φ-firmly nonexpansive mapping from C into C and F(Tr(Θ,A,ψ))=GMEP(Θ,A,ψ,φ). Therefore, Theorem 34 follows from Theorem 28.



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