Bregman Distance and Strong Convergence of Proximal-Type Algorithms

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 functional Φ(||·||) . 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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Li-Wei Kuo 1 and D. R. Sahu 2

Recommended by Dumitru Motreanu

1, Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan
2, Department of Mathematics, Banaras Hindu University, Varanasi 221005, India

Received 31 October 2012; Revised 7 March 2013; Accepted 7 March 2013

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 Y9;x,jYA; for the value of a functional j in ^X* at x in X . As usual, _xλ [arrow right]x and _xλ ...x stand for the norm and weak convergence of a net {_xλ } to x in X , respectively.

A continuous strictly increasing function [straight phi]:^...+ [arrow right]^...+ is said to be a gauge if [figure omitted; refer to PDF] The mapping _{J[straight phi]} :X[arrow right]^2X* defined by [figure omitted; refer to PDF] is called the duality mapping with gauge [straight phi] . In the special case where [straight phi](t)=t , the duality mapping _{J[straight phi]} =:J is the classical normalized duality mapping . In the case [straight phi](t)=^tp-1 , p>1 , the duality mapping _{J[straight phi]} =:_Jp is called the generalized duality mapping and it is given by [figure omitted; refer to PDF]

For a gauge [straight phi] , the function Φ:^...+ [arrow right]^...+ defined by [figure omitted; refer to PDF] is a continuous convex strictly increasing differentiable function on ^...+ with ^{Φ[variant prime]} (t)=[straight phi](t) and _{limt[arrow right]+∞} Φ(t)/t=+∞ . Therefore, Φ has a continuous inverse function ^Φ-1 .

We recall the Bregman Distance _{D[straight phi]} and function ^{D[straight phi]f} studied in [1]. Let X be a real smooth Banach space. The Bregman distance _{D[straight phi]} (x,y) between x and y in X is defined by [figure omitted; refer to PDF] One can see from Lemma 3 that _{D[straight phi]} (x,y)...5;0 . In the case [straight phi](t)=^tp-1 , p∈(1,∞) , the distance _{D[straight phi]} (x,y)=:_Dp (x,y) is called the p-Lyapunov functional studied in [2] and it is given by [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] is the Lyapunov functional . It is obvious that [figure omitted; refer to PDF] See Brègman [3], Butnariu and Iusem [4], and Censor and Lent [5].

Let C be a nonempty closed convex subset of a smooth Banach space X . The generalized projection _ΠC :X[arrow right]C is defined by [figure omitted; refer to PDF] The metric projection operator _PC :X[arrow right]C defined by [figure omitted; refer to PDF] 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., ||_PC x-_PC y||...4;||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[arrow right]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]).

Section 2 contains preliminaries. In Section 3, we study the fundamental properties of Bregman distance _{D[straight phi]} and ([straight phi],f) -generalized projection operators, where f:X[arrow right]^...+ is a proper, convex, lower semicontinuous function. In Section 4, we discuss [straight phi] -firmly nonexpansive mappings and [straight phi] -resolvent operators. In Section 5, we establish strong convergence of the proximal-projection methods for finding fixed points of [straight phi] -firmly nonexpansive mappings, zeros of (not necessarily maximal) monotone operators, and solutions of generalized mixed equilibrium problems in Banach spaces using ([straight phi],f) -generalized projection operators ^{ΠC[straight phi],f} . Here, we do not assume the maximality of monotone operators and the uniform smoothness of Banach spaces.

2. Preliminaries

Let A:X[arrow right]^2X* be a set-valued operator. The set ...9F;(A)={x∈X:Ax...0;∅} is called the effective domain of A . The range of A is defined by ...(A)=_{∪x∈...9F;(A)} Ax . The operator A is said to be monotone if for any x,y in ...9F;(A) , we have [figure omitted; refer to PDF] A monotone operator A is said to be maximal if the graph ...A2;(A)={(x,y):x∈...9F;(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-1 0 is closed and convex.

In the rest of this paper, by [straight phi] we always mean a gauge and by Φ the corresponding function defined in (4). We list some properties of the duality mapping _{J[straight phi]} :X[arrow right]^2X* below (for more details see [13, 14]).

Proposition 1.

Let X be a real Banach space.

(i) _{J[straight phi]} is norm-to-weak * upper semicontinuous;

(ii) for each x in X, the set _{J[straight phi]} (x) is convex and weakly closed in ^X* ;

(iii): _{J[straight phi]} (-x)=-_{J[straight phi]} (x) and _{J[straight phi]} (λx)=([straight phi](||λx||)/[straight phi](||x||))_{J[straight phi]} (x) for all nonzero x in X, λ>0;

(iv) there holds [figure omitted; refer to PDF]

(v) _{J[straight phi]} is maximal monotone;

(vi) if X is strictly convex, then _{J[straight phi]} is strictly monotone; that is, [figure omitted; refer to PDF]

(vii): if X is strictly convex and reflexive, then _{J [straight phi]} is single-valued monotone and demicontinuous.

The following result is well known. We include a proof for completeness.

Lemma 2.

If a Banach space E has a uniformly Gâteaux differentiable norm, then J:E[arrow right]^E* is uniformly norm-to-weak* continuous on nonempty bounded subsets of E to ^E* .

Proof.

Suppose not, and there exist norm one vectors _xn ,_yn ,z in E and a constant ...>0 such that _yn -_xn [arrow right]0 , and Y9;z,J(_yn )-J(_xn )YA;...5;... , for all n∈... . For a fixed t>0 , define [figure omitted; refer to PDF] Observe that [figure omitted; refer to PDF] Hence, [figure omitted; refer to PDF] Choose t=(2/...)||_xn -_yn || . By the uniform Gâteaux differentiability of the norm, if n is large enough, both _an and _bn are less than (1/2)... , and so _an +_bn <... . We arrive at a contradiction.

Together with Proposition 1, the conclusion in Lemma 2 also holds for _{J[straight phi]} .

Let X be a Banach space and ψ:X[arrow right](-∞,+∞] a function. The function ψ is proper if dom(ψ):={x∈X:ψ(x)<+∞}...0;∅ . An element j in ^X* is said to be a subgradient of a proper convex function ψ:X[arrow right](-∞,+∞] at a point x in dom(ψ) if [figure omitted; refer to PDF] The set (possibly empty) [figure omitted; refer to PDF] of subgradients of ψ at x in dom(ψ) , is called the subdifferential of [straight phi] at x .

Lemma 3.

Let X be a smooth Banach space and _{J[straight phi]} :X[arrow right]^X* the duality mapping with gauge [straight phi] . Then _{J[straight phi]} (y)=∂Φ(||y||) for y∈X\{0} ; that is, [figure omitted; refer to PDF]

The proof of the following result is straightforward.

Lemma 4.

Let X be a real Banach space and [straight phi] 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.

Let X be a real Banach space and [straight phi] 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||...4;r} , where r>0 is arbitrarily given. That is, there exists a strictly increasing convex function _gr :^...+ [arrow right]^...+ with _gr (0)=0 such that [figure omitted; refer to PDF]

: for all x,y in _Br and t in [0,1] .

Proof.

First we note that the general case can be reduced to the case r=1 . Suppose it holds [figure omitted; refer to PDF] We will show that (20) holds for any r>0 . Set _{[straight phi]r} (t)=[straight phi](rt) and _gr (t)=rg(t/r) for t...5;0 . Then _{[straight phi]r} is still a gauge function, and let _Φr be the function corresponding to _{[straight phi]r} as defined in (4), r_Φr (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 [figure omitted; refer to PDF] 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 assume r=1 and g=_g1 .

( b ) [implies] ( a ) . Given x,y in X such that ||x||=||y||=1 and ||x-y||=[straight epsilon] . Setting t=1/2 in (20), we have [figure omitted; refer to PDF] It turns out that [figure omitted; refer to PDF] This verifies that X is uniformly convex.

( a ) [implies] ( 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<[straight epsilon]...4;2 , [figure omitted; refer to PDF] where _BX is the closed unit ball of X .

Claim . Consider μ([straight epsilon])>0 for 0<[straight epsilon]...4;2 .

Suppose on the contrary that μ([straight epsilon])=0 for some 0<[straight epsilon]...4;2 . Then we can find sequences {_xn } , {_yn } in _BX such that ||_xn -_yn ||...5;[straight epsilon] for all n , and [figure omitted; refer to PDF] Without loss of generality, we may assume that [figure omitted; refer to PDF] It then follows from (26) that [figure omitted; refer to PDF] The strict convexity of Φ together with (28) implies that α+β<γ if α...0;β . Since, on the other hand, by definition, γ...4;α+β . We therefore must have α=β , which together with (28) implies that α=β=γ/2 . If we set ^{xn[variant prime]} =_xn /||_xn || , ^{yn[variant prime]} =_yn /||_yn || , then ||^{xn[variant prime]} ||=||^{yn[variant prime]} ||=1 for all n ; moreover, from (27), we get [figure omitted; refer to PDF] This contradicts the uniform convexity of X , and verifies that μ([straight epsilon])>0 for all 0<[straight epsilon]...4;2 .

It turns out from (25) that [figure omitted; refer to PDF] 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, [figure omitted; refer to PDF] 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.

Let X be a real uniformly convex Banach space. Then there exists a strictly increasing convex function g:^...+ [arrow right]^...+ with g(0)=0 such that [figure omitted; refer to PDF]

Proof.

Since _{J[straight phi]} is the subdifferential of the functional Φ(||·||) , we have for _jx in _{J[straight phi]} (x) , x in X that [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] Taking limit as λ[arrow right]0 , we get [figure omitted; refer to PDF]

Lemma 7 (see [17, Theorem 3.11, page 952]).

Let X be a reflexive Banach space and _{J[straight phi]} :X[arrow right]^2X* the duality map with gauge [straight phi] . Suppose that A:X[arrow right]^2X* is maximal monotone. Then the operator A+_{J[straight phi]} is surjective.

3. Bregman Distance _{D[straight phi]} and Function ^{D[straight phi]f}

One can easily see that [figure omitted; refer to PDF] Noticing that for x in X , the scalar function _{D[straight phi]} (·,x) is coercive (see [18, Lemma 7.3(v)]).

Proposition 8.

Let X be a strictly convex smooth Banach space. Let x,y∈X . Then [figure omitted; refer to PDF]

Proof.

See [18, Lemma 7.3(vi)]

Proposition 9.

Let X be a smooth and uniformly convex Banach space. Then there exists a strictly increasing convex function g:^...+ [arrow right]^...+ with g(0)=0 such that [figure omitted; refer to PDF]

Proof.

Let u,v∈_BX . By Lemma 6 we have [figure omitted; refer to PDF]

As in Butnariu et al. [19], we can prove the following proposition.

Proposition 10.

Let X be a smooth and uniformly convex Banach space. Let {_xn } and {_yn } be two sequences in X such that _{D[straight phi]} (_xn ,_yn )[arrow right]0 . If {_yn } is bounded, then ||_xn -_yn ||[arrow right]0 .

Proof.

Assume {_yn } is bounded. From definition (5) we have [figure omitted; refer to PDF] It follows that [figure omitted; refer to PDF] Since {_yn } is bounded, _{D[straight phi]} (_xn ,_yn )[arrow right]0 and Φ(t)/t[arrow right]∞ as t[arrow right]∞ 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 {γ_xn } and {γ_yn } for a sufficiently small γ>0 ). By Proposition 9, there exists a strictly increasing convex function g:^...+ [arrow right]^...+ with g(0)=0 such that [figure omitted; refer to PDF] Since _{D[straight phi]} (_xn ,_yn )[arrow right]0 and g is strictly increasing, we immediately conclude that ||_xn -_yn ||[arrow right]0 .

Proposition 11 (see [20, Lemma 3.1]).

Let X be a smooth Banach space. Let u∈X and {_xn } be a sequence in X such that {_{D[straight phi]} (_xn ,u)} is bounded. Then {_xn } is bounded.

The statement in the following proposition is evident from the definition of _{D[straight phi]} (cf. [18, Lemma 7.3(ii)]).

Proposition 12.

Let X be a smooth Banach space. Then, for any fixed x in X , the scalar function _{D[straight phi]} (·,x) is continuous, weakly lower semicontinuous, and convex on X .

Let f:X[arrow right][0,+∞] be a proper, convex, lower semicontinuous function. Define [figure omitted; refer to PDF]

Some of the following basic properties of the Bregman distance _{D[straight phi]} and function ^{D[straight phi]f} are known in the literature (see [18-21]).

The following proposition can be deduced from Butnariu and Kassay [21, Lemma 2.1].

Proposition 13.

Let C be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space X . Let x∈X and let f:X[arrow right][0,+∞] be a proper, convex, lower semicontinuous function with C⊂dom(f) . Then there exists a unique element _x0 in C such that [figure omitted; refer to PDF]

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 [straight phi] . Using the idea of [1, 26], Proposition 13 allows us to extend generalized projections _ΠC as follows.

Definition 14.

In the setting of Proposition 13, we define the ([straight phi],f) -generalized projection from X onto C by [figure omitted; refer to PDF]

In case [straight phi](t)=t and f=0 , we notice that ^{ΠC[straight phi],f} coincides with _ΠC . In case that X is a Hilbert space and [straight phi](t)=t , denote ^{ΠC[straight phi],f} by ^PCf .

Applying the tools used in [1, 26, 27], we can establish the following results.

Proposition 15.

Let C be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space X and let f:X[arrow right][0,+∞] be a proper, convex, lower semicontinuous function with C⊂dom(f) .

(i) Let _x0 ∈C and x∈X. Then the following assertions are equivalent:

(a) _x0 =^{ΠC[straight phi],f} (x) ,

(b) Y9;z-_x0 ,_{J[straight phi]x0} -_{J[straight phi]} xYA;+f(z)-f(_x0 )...5;0 ,

: ∀z∈C .

(ii) Given x in X, one has [figure omitted; refer to PDF]

Proposition 16.

Let C be a nonempty closed convex subset of a smooth and uniformly convex Banach space X and f:X[arrow right][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[arrow right]+∞} f(_xn )=0 . If ^{D[straight phi]f} (_xn ,u)...4;^{D[straight phi]f} (^{ΠC[straight phi],f} (u),u) for all n in ... , then _{limn[arrow right]+∞xn} =^{ΠC[straight phi],f} (u) .

Proof.

Set ^{ΠC[straight phi],f} (u)=:u~ . Then, by assumption, we have ^{D[straight phi]f} (_xn ,u)...4;^{D[straight phi]f} (u~,u) for all n in ... . By the three-point identity (36), we have [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] that is, f(z)=0 . Using Proposition 15, we have [figure omitted; refer to PDF] which implies that _{limi[arrow right]+∞D[straight phi]} (_xni ,u~)=0 . It follows from Proposition 10 that _{limi[arrow right]∞xni} =u~ . This implies that _xn converges strongly to u~=^{ΠC[straight phi],f} (u) .

4. [straight phi] -Firmly Nonexpansive and [straight phi] -Resolvent Operators

Following [1], we study properties of [straight phi] -firmly nonexpansive mappings in Banach spaces.

Definition 17.

Let X be a smooth Banach space, _{J[straight phi]} :X[arrow right]^X* a duality mapping with gauge function [straight phi] , and C a nonempty subset of X . An operator T:C[arrow right]X is called [straight phi] -firmly nonexpansive if [figure omitted; refer to PDF]

In the case of [straight phi](t)=t , inequality (50) reduces to [figure omitted; refer to PDF] 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]): [figure omitted; refer to PDF]

We now give useful characterizations of [straight phi] -firmly nonexpansive mappings which can be deduced from the Bregman distance (5).

Proposition 18.

Let X be a smooth Banach space and C a nonempty closed convex subset of X . Let T:C[arrow right]C be a [straight phi] -firmly nonexpansive mapping. Then [figure omitted; refer to PDF]

The geometry of the fixed point set of [straight phi] -firmly nonexpansive mappings is established in Reich and Sabach [30, Lemma 15.5] as follows.

Proposition 19.

Let X be a strictly convex smooth Banach space and C a nonempty closed convex subset of X . Let T:C[arrow right]C be a [straight phi] -firmly nonexpansive mapping. Then the set F(T) of fixed points of T is closed and convex.

Let C be a nonempty closed convex subset of a smooth Banach space X and let T:C[arrow right]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 -T_xn ||[arrow right]0 ; see [31]. We denote the set of asymptotic fixed points of T by F^(T) . A mapping T:C[arrow right]C is relatively [straight phi]-nonexpansive if the following conditions are satisfied:

(i) F(T) is nonempty;

(ii) _{D[straight phi]} (p,Tx)...4;_{D[straight phi]} (p,x) for all x in C and p in F(T) ;

(iii): F^(T)=F(T) .

The class of relatively [straight phi] -nonexpansive mappings is larger than the class of relatively nonexpansive mappings (see [32]).

A mapping T:C[arrow right]C is [straight phi] -firmly quasinonexpansive if F(T)...0;∅ and [figure omitted; refer to PDF] 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.

Let X be a smooth Banach space, C a nonempty subset of X , and T:C[arrow right]C a [straight phi] -firmly quasinonexpansive mapping. Then [figure omitted; refer to PDF] for all x in C and p in F(T) .

The following supplements Reich and Sabach [30, Lemma 15.6].

Proposition 21.

Let X 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[arrow right]C be a [straight phi] -firmly nonexpansive mapping. Then F^(T)=F(T) .

Proof.

It is easy to see that F(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 -T_xn ||[arrow right]0 . We need to prove that u∈F(T) . Using (53), we get [figure omitted; refer to PDF] It is not hard to find that (56) is reduced to the relation [figure omitted; refer to PDF] We note that T_xn ...u . Indeed, for any norm one linear functional ξ of X , we have [figure omitted; refer to PDF] Both terms in the right-hand side approach zero. Consequently, [figure omitted; refer to PDF]

Claim . Consider Y9;T_xn -Tu,_{J[straight phi]} (_xn )-_{J[straight phi]} (T_xn )YA;[arrow right]0 .

Since _xn ...u and ||_xn -T_xn ||[arrow right]0 , there is a constant M>0 such that all ||_xn ||,||T_xn ||<M . It follows from the uniform norm to weak* continuity of _{J[straight phi]} on bounded subsets (Lemma 2); we have [figure omitted; refer to PDF] Observe that [figure omitted; refer to PDF] Moreover, |||_xn ||-||T_xn |||...4;||_xn -T_xn ||[arrow right]0 . By the uniform continuity of [straight phi] on [0,M] , we have [figure omitted; refer to PDF] The claim is thus verified.

We obtain from (57) and the claim that [figure omitted; refer to PDF] This is equivalent to [figure omitted; refer to PDF] The strict monotonicity of the duality mapping _{J[straight phi]} implies that equality must hold. Namely, Tu=u or u∈F(T) .

The resolvent of an operator A:X[arrow right]^2X* relative to a Gâteaux differentiable function f is introduced and studied in [1]. We define [straight phi] -resolvent operators following [1, 18].

Definition 22.

Let C be a nonempty closed convex subset of a smooth Banach space X and let _{J[straight phi]} :X[arrow right]^X* be the duality mapping with gauge [straight phi] . Suppose that A:X[arrow right]^2X* is an operator satisfying the range condition [figure omitted; refer to PDF] For each λ>0 , the [straight phi] -resolvent associated with operator A is the operator ^{Rλ[straight phi],A} :C[arrow right]^2X defined by [figure omitted; refer to PDF]

For x in C and λ in (0,∞) , we have [figure omitted; refer to PDF] If A is maximal monotone, then, by Lemma 7, we see that condition (65) holds for C=D(A)¯ .

Remark 23.

For smooth X and [straight phi](t)=^tp-1 with p∈(1,+∞) , we have _{J[straight phi]} =_Jp and ^Rp,A =(_Jp +A^)-1_Jp . For p=2 , ^RA :=^R2,A =(J+A^)-1 [composite function]J and this kind of resolvent operators is studied in the literature (see [28, 33]).

Let C be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space X . Let A:X[arrow right]^2X* be a monotone operator satisfying the condition D(A)⊂C⊂^{J[straight phi]-1} ...(_{J[straight phi]} +λA) , where λ>0 . Using the smoothness and strict convexity of X , we obtain that ^{Rλ[straight phi],A} is single-valued. The conditions D(A)⊂C⊂^{J[straight phi]-1} ...(_{J[straight phi]} +λA) ensure that ^{Rλ[straight phi],A} is the single-valued [straight phi] -resolvent operator form C into D(A) . In other words, [figure omitted; refer to PDF]

Following [18, 28], we have the following proposition.

Proposition 24.

Let C be a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach space X and let _{J[straight phi]} :X[arrow right]^X* be the duality mapping with gauge [straight phi] . Let A:X[arrow right]^2X* be a monotone operator satisfying the condition D(A)⊂C⊂^{J[straight phi]-1} ...(_{J[straight phi]} +λA) , where λ is a positive real number. Let ^{Rλ[straight phi],A} be a resolvent of A , where

(a) ^{Rλ[straight phi],A} is [straight phi]-firmly nonexpansive mapping from C into C,

(b) F(^{Rλ[straight phi],A} )=^A-1 0 .

5. Convergence Theorems

Let C be a nonempty subset of a Banach space X and ...AF;={T(t):t>0} a family of mappings from C into C with _...t>0 ...F(T(t))...0;∅ . Let ...A2;={_Gt :t>0} be a family of mappings from C into C such that _...t>0 ...F(T(t))⊆_...t>0 ...F(_Gt ) . We say the family ...AF;:={T(t):t>0} has property (...9C;)with respect to the family ...A2;={_Gt :t>0} if the following assertion holds: [figure omitted; refer to PDF]

If ...AF;=...A2; and the above condition holds, then we say ...AF;:={T(s):s>0} has property (...9C;) .

Remark 25.

If ...AF; is a singleton, that is, ...AF;={T} , or T(s)=T for all s>0 , then {T} always has property (...9C;) .

We now give some examples.

Example 26.

Let C be a nonempty closed convex subset of a Banach space X and T a nonexpansive mapping from C into C with F(T)...0;∅ . Assume _bt in ... with 0<a...4;_bt ...4;b<1 for all t>0 . Define _Gt :C[arrow right]C by _Gt x=(1-_bt )x+_bt Tx for all x in C . Then T has property (...9C; ) with respect to the family {_Gt :t>0} .

Proof.

Let {_xt}t>0 be a bounded net in C such that ||_xt -_Gt (_xt )||[arrow right]0 as t[arrow right]∞ . Note that [figure omitted; refer to PDF] and 0<a...4;_bt ...4;b<1 for all t>0 . Therefore, ||_xt -T_xt ||[arrow right]0 as t[arrow right]+∞ .

The following example shows that the family {^RtA :t>0} of resolvent operators of a maximal monotone operator A enjoys property (...9C;) .

Example 27.

Let C 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: [figure omitted; refer to PDF] Let {_zt}t>0 be a bounded net in C such that ||_zt -^RtA_zt ||[arrow right]0 as t[arrow right]+∞ . Then ||_zt -^RrA_zt ||[arrow right]0 as t[arrow right]+∞ for each r>0 .

Proof.

Let r , t>0 . By Takahashi [34], we have [figure omitted; refer to PDF] Using (72), we have [figure omitted; refer to PDF]

We now discuss the problem of finding common fixed points of a sequence of [straight phi] -firmly nonexpansive mappings. Our proximal-projection method is based on (a not necessarily Bregman distance) function ^{D[straight phi]f} . The proof is based on the technique in [20].

Theorem 28.

Let X be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Let [straight phi] be a gauge function, C a nonempty closed convex subset of X , and f:X[arrow right][0,+∞] a proper, convex, lower semicontinuous function with C⊂dom(f) . Let T:C[arrow right]C be a [straight phi] -firmly nonexpansive mapping and ...AF;:={_Tn } a sequence of [straight phi] -firmly nonexpansive self-mappings on C such that F(T)=_...n∈... ...F(_Tn )...0;∅ and ...AF; has property (...9C; ) with respect to T . For u in C and _C1 =C with _x1 =^{Π_C1 [straight phi],f} (u) , define a sequence {_xn } in C as follows: [figure omitted; refer to PDF] Then {_xn } converges strongly to ^{ΠF(T)[straight phi],f} (u) .

Proof.

We proceed the proof in the following steps:

Step 1 . {_xn } is well defined.

Note that all _Cn are closed and convex. For p in F(...AF;) and n in ... , we obtain from (54) that [figure omitted; refer to PDF] It follows that p∈_Cn and hence F(...AF;)⊂_Cn . Therefore, {_xn } is well defined.

Step 2 . {_xn } is bounded.

Let p∈F(...AF;) . It follows from Proposition 15; we have [figure omitted; refer to PDF] It follows that {_{D[straight phi]} (_xn ,u)} is bounded and hence from Proposition 11, we obtain that {_xn } is bounded.

Step 3 . Consider ||_xn -T_xn ||[arrow right]0 .

Note that _xn+1 ∈_Cn+1 ⊂_Cn . It follows from Proposition 15 that [figure omitted; refer to PDF] This implies that [figure omitted; refer to PDF] Therefore, the sequence {^{D[straight phi]f} (_xn ,u)} (see (43)) is increasing. Note that {^{D[straight phi]f} (_xn ,u)} is bounded by (76). It follows that _{limn[arrow right]+∞}^{D[straight phi]f} (_xn ,u) exists. By (78), we obtain [figure omitted; refer to PDF] One can see that _{limn[arrow right]+∞} f(_xn )=0 . Using Proposition 10, we obtain that _{limn[arrow right]+∞} ||_xn -_xn+1 ||=0 . Since _xn+1 ∈_Cn+1 , we have [figure omitted; refer to PDF] Using (79), we obtain that [figure omitted; refer to PDF] and hence _{limn[arrow right]+∞D[straight phi]} (_yn ,_xn )=0 . Note that _{xn } is bounded. Then, one can see from Proposition 10 that {_yn } is bounded and [figure omitted; refer to PDF] This implies that [figure omitted; refer to PDF] Since the family ...AF;:={_Tn :n∈...} has property (...9C; ) with respect to T , it follows from (83) that _{limn[arrow right]+∞} ||_xn -T_xn ||=0 .

Step 4 . The sequence {_xn } converges strongly to ^{ΠF(T)[straight phi],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)[straight phi],f} is well defined. Note that _xn =^{Π_Cn [straight phi],f} (u) and F(T) is contained in _Cn ; we have [figure omitted; refer to PDF] Therefore, we conclude from Proposition 16 that {_xn } converges strongly to ^{ΠF(T)[straight phi],f} (u) .

Theorem 29.

Let X be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Let [straight phi] be a gauge function, C a nonempty closed convex subset of X , and f:X[arrow right][0,+∞] a proper, convex, lower semicontinuous function with C⊂dom(f) . Let A:X[arrow right]^2X* be a monotone operator with ^A-1 0...0;∅ satisfying the following condition: [figure omitted; refer to PDF] Let λ be a positive real number; for u in C and _C1 =C with _x1 =^{Π_C1 [straight phi],f} (u) , define a sequence {_xn } in C as follows: [figure omitted; refer to PDF] Then {_xn } converges strongly to ^{ΠA-1 0[straight phi],f} (u) .

Proof.

Set T:=(_{J[straight phi]} +λA^)-1_{J[straight phi]} . Note that T is [straight phi] -firmly nonexpansive mapping from C into C and F(T)=^A-1 0 . Further, every singleton family {T} enjoys property (...9C; ). 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.

Let C be a nonempty closed convex subset of real Hilbert space H , and let f:X[arrow right][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-1 0...0;∅ . Let {_λn } be a sequence in (0,∞) such that _{limn[arrow right]+∞λn} =+∞ , for u in C , let _C1 =C with _x1 =^{P_C1 f} (u) , and define a sequence {_xn } in C as follows: [figure omitted; refer to PDF] Then {_xn } converges strongly to ^{PA-1 0f} (u) .

Proof.

Set T:=^RλA =(I+λA^)-1 for λ>0 and _Tn :=(I+_λn A^)-1 for all n in ... . Note that T is a firmly nonexpansive mapping from C into C and F(T)=^A-1 0 . From (83), we have _{limn[arrow right]+∞} ||_xn -_Tnxn ||=0 . Example 27 implies that ...AF;:={_Tr :r>0} has property (...9C; ). It follows that _{limn[arrow right]+∞} ||_xn -T_xn ||=0 . Therefore, Corollary 31 follows from Theorem 28.

Let C be a nonempty, closed, and convex subset of a Banach space X . Let Θ:C×C[arrow right]... be a bi-function, A:C[arrow right]^X* a nonlinear operator, and ψ:C[arrow right]... 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)...4;0 for all x,y in C .

(A3) for all x,y,z in C , _{lim supt[arrow down]0} Θ(tz+(1-t)x,y)...4;Θ(x,y) .

(A4) for all x in C , Θ(x,·) is convex and lower semicontinuous.

Blum and Oettli [36] studied the following equilibrium problem (EP).

Find x^ in C such that [figure omitted; refer to PDF] The solution set of (88) is denoted by EP(Θ) .

Following [37], we have the following lemma.

Lemma 32.

Let C be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space X , and let [straight phi] be a gauge function. Let Θ:C×C[arrow right]... be a bi-function satisfying conditions (A1)-(A4), and let r>0 and x∈X . Then there exists z in C such that [figure omitted; refer to PDF]

We now consider the following generalized mixed equilibrium problem (GMEP): find z in C such that [figure omitted; refer to PDF] 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).

Let r>0 . For a given point x in C , find z in C such that [figure omitted; refer to PDF] The existence of a solution of the auxiliary mixed equilibrium problem (91) is guaranteed by [37].

Let C be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space X and let [straight phi] be a gauge function. Let A:C[arrow right]^X* be continuous and monotone, Θ:C×C[arrow right]... a bi-function satisfying conditions (A1)-(A4), and ψ:C[arrow right]... a lower semicontinuous and convex function. For r>0 , define the mapping ^Tr(Θ,A,ψ) :X[arrow right]^2C as follows: [figure omitted; refer to PDF]

Lemma 33 (see [37]).

One has the following.

(1) ^Tr(Θ,A,ψ) is single-valued.

(2) ^Tr(Θ,A,ψ) is a [straight phi] -firmly nonexpansive mapping; that is, for all x,y in X , [figure omitted; refer to PDF]

(4) F(^Tr(Θ,A,ψ) )=GMEP(Θ,A,ψ) .

(5) GMEP(Θ,A,ψ) is closed and convex.

(6) For all x in F(^Tr(Θ,A,ψ) ) and y in X , one has [figure omitted; refer to PDF]

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.

Let C be a nonempty closed convex subset of a uniformly convex Banach space X with a uniformly Gâteaux differentiable norm. Let [straight phi] be a gauge function and let f:X[arrow right][0,∞] be a proper, convex, lower semicontinuous function with C⊂dom(f) . Let A:C[arrow right]^X* be continuous and monotone, Θ:C×C[arrow right]... a bi-function satisfying conditions (A1)-(A4), and ψ:C[arrow right]... a lower semicontinuous and convex function. Assume that GMEP(Θ,A,ψ,[straight phi])...0;∅ . For u in C , r>0 , and _C1 =C with _x1 =^{Π_C1 [straight phi],f} (u) , define a sequence {_xn } in C as follows: [figure omitted; refer to PDF] Then {_xn } strongly converges to ^{ΠF(Tr(Θ,A,ψ) )[straight phi],f} (u) .

Proof.

Note that ^Tr(Θ,A,ψ) is a [straight phi] -firmly nonexpansive mapping from C into C and F(^Tr(Θ,A,ψ) )=GMEP(Θ,A,ψ,[straight phi]) . Therefore, Theorem 34 follows from Theorem 28.

Acknowledgments

This paper was initiated while D. R. Sahu was visiting the National Sun Yat-Sen University, Kaohsiung, Taiwan as a visiting professor. He would like to thank the Department of Applied Mathematics there for the warm hospitality. Professor Ngai-Ching Wong gave the encouragement and useful comments which were very important for this research. Both authors are also very grateful to the referees for their careful reading and many helpful suggestions. This research is partially supported by Taiwan NSC Grant 99-2115-M-110-007-MY3 and the National Sun Yat-Sen University.

References

[1] H. H. Bauschke, J. M. Borwein, P. L. Combettes, "Bregman monotone optimization algorithms," SIAM Journal on Control and Optimization , vol. 42, no. 2, pp. 596-636, 2003.

[2] T. Bonesky, K. S. Kazimierski, P. Maass, F. Schöpfer, T. Schuster, "Minimization of Tikhonov functionals in Banach spaces," Abstract and Applied Analysis , vol. 2008, 2008.

[3] L. M. Brègman, "A relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming," USSR Computational Mathematics and Mathematical Physics , vol. 7, pp. 200-217, 1967.

[4] D. Butnariu, A. N. Iusem Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization , vol. 40, pp. xvi+202, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.

[5] Y. Censor, A. Lent, "An iterative row-action method for interval convex programming," Journal of Optimization Theory and Applications , vol. 34, no. 3, pp. 321-353, 1981.

[6] Y. I. Alber, "Generalized projection operators in Banach spaces: properties and applications," Proceedings of the Israel Seminar , vol. 1, of Functional Differential Equation, pp. 1-21, Ariel, Israel, 1993.

[7] Y. I. Alber, "Metric and generalized projection operators in Banach spaces: properties and applications," Theory and Applications of Nonlinear Operators of Accretive and Monotone Type , vol. 178, pp. 15-50, Dekker, New York, NY, USA, 1996.

[8] Y. Alber, S. Guerre-Delabriere, "On the projection methods for fixed point problems," Analysis , vol. 21, no. 1, pp. 17-39, 2001.

[9] Y. I. Alber, A. I. Notik, "On some estimates for projection operators in Banach spaces," Communications on Applied Nonlinear Analysis , vol. 2, no. 1, pp. 47-55, 1995.

[10] J. Li, "The generalized projection operator on reflexive Banach spaces and its applications," Journal of Mathematical Analysis and Applications , vol. 306, no. 1, pp. 55-71, 2005.

[11] W. Song, Z. Cao, "The generalized decomposition theorem in Banach spaces and its applications," Journal of Approximation Theory , vol. 129, no. 2, pp. 167-181, 2004.

[12] Y. Su, H.-K. Xu, X. Zhang, "Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications," Nonlinear Analysis. Theory, Methods & Applications , vol. 73, no. 12, pp. 3890-3906, 2010.

[13] R. P. Agarwal, D. O'Regan, D. R. Sahu Fixed Point Theory for Lipschitzian-Type Mappings with Applications , vol. 6, of Topological Fixed Point Theory and Its Applications, pp. x+368, Springer, New York, NY, USA, 2009.

[14] I. Cioranescu Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems , vol. 62, pp. xiv+260, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1990.

[15] T.-C. Lim, H. K. Xu, Z. B. Xu, "Some ^Lp inequalities and their applications to fixed point theory and approximation theory," Progress in Approximation Theory , pp. 609-624, Academic Press, New York, NY, USA, 1991.

[16] C. Zalinescu Convex Analysis in General Vector Spaces , pp. xx+367, World Scientific Publishing, River Edge, NJ, USA, 2002.

[17] J.-P. Penot, R. Ratsimahalo, "On the Yosida approximation of operators," Proceedings of the Royal Society of Edinburgh Section A , vol. 131, no. 4, pp. 945-966, 2001.

[18] H. H. Bauschke, J. M. Borwein, P. L. Combettes, "Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces," Communications in Contemporary Mathematics , vol. 3, no. 4, pp. 615-647, 2001.

[19] D. Butnariu, A. N. Iusem, E. Resmerita, "Total convexity for powers of the norm in uniformly convex Banach spaces," Journal of Convex Analysis , vol. 7, no. 2, pp. 319-334, 2000.

[20] S. Reich, S. Sabach, "Two strong convergence theorems for a proximal method in reflexive Banach spaces," Numerical Functional Analysis and Optimization , vol. 31, no. 1-3, pp. 22-44, 2010.

[21] D. Butnariu, G. Kassay, "A proximal-projection method for finding zeros of set-valued operators," SIAM Journal on Control and Optimization , vol. 47, no. 4, pp. 2096-2136, 2008.

[22] F. Schöpfer, T. Schuster, A. K. Louis, "Metric and Bregman projections onto affine subspaces and their computation via sequential subspace optimization methods," Journal of Inverse and Ill-Posed Problems , vol. 16, no. 5, pp. 479-506, 2008.

[23] F. Schöpfer, T. Schuster, A. K. Louis, "An iterative regularization method for the solution of the split feasibility problem in Banach spaces," Inverse Problems , vol. 24, no. 5, pp. 20, 2008.

[24] F. Schöpfer, T. Schuster, "Fast regularizing sequential subspace optimization in Banach spaces," Inverse Problems , vol. 25, no. 1, pp. 22, 2009.

[25] J.-J. Moreau, "Fonctions convexes duales et points proximaux dans un espace hilbertien," Comptes Rendus de l'Académie des Sciences , vol. 255, pp. 2897-2899, 1962.

[26] S. Kamimura, W. Takahashi, "Strong convergence of a proximal-type algorithm in a Banach space," SIAM Journal on Optimization , vol. 13, no. 3, pp. 938-945, 2002.

[27] D. Butnariu, E. Resmerita, "Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces," Abstract and Applied Analysis , vol. 2006, 2006.

[28] F. Kohsaka, W. Takahashi, "Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces," SIAM Journal on Optimization , vol. 19, no. 2, pp. 824-835, 2008.

[29] K. Goebel, W. A. Kirk Topics in Metric Fixed Point Theory , vol. 28, pp. viii+244, Cambridge University Press, Cambridge, UK, 1990.

[30] S. Reich, S. Sabach, "Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces," Fixed-Point Algorithms for Inverse Problems in Science and Engineering , vol. 49, pp. 301-316, Springer, New York, NY, USA, 2011.

[31] S. Reich, "A weak convergence theorem for the alternating method with Bregman distances," Theory and Applications of Nonlinear Operators of Accretive and Monotone Type , vol. 178, of Lecture Notes in Pure and Applied Mathematics, pp. 313-318, Dekker, New York, NY, USA, 1996.

[32] S.-Y. Matsushita, W. Takahashi, "A strong convergence theorem for relatively nonexpansive mappings in a Banach space," Journal of Approximation Theory , vol. 134, no. 2, pp. 257-266, 2005.

[33] G. Kassay, "The proximal points algorithm for reflexive Banach spaces," Universitatis Babes-Bolyai. Studia. Mathematica , vol. 30, pp. 9-7, 1985.

[34] W. Takahashi Nonlinear Functional Analysis, Fixed Point Theory and Its Applications , pp. iv+276, Yokohama Publishers, Yokohama, Japan, 2000.

[35] R. P. Agarwal, Y. J. Cho, X. Qin, "Generalized projection algorithms for nonlinear operators," Numerical Functional Analysis and Optimization , vol. 28, no. 11-12, pp. 1197-1215, 2007.

[36] E. Blum, W. Oettli, "From optimization and variational inequalities to equilibrium problems," The Mathematics Student , vol. 63, no. 1-4, pp. 123-145, 1994.

[37] S.-S. Zhang, "Generalized mixed equilibrium problem in Banach spaces," Applied Mathematics and Mechanics , vol. 30, no. 9, pp. 1105-1112, 2009.