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.

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, 𝑥_𝜆→𝑥 and 𝑥_𝜆⇀𝑥 stand for the norm and weak convergence of a net {𝑥_𝜆} to 𝑥 in 𝑋, respectively.

A continuous strictly increasing function 𝜑∶ℝ⁺→ℝ⁺ is said to be a gauge if 𝜑(0)=0,lim_𝑡→+∞𝜑(𝑡)=+∞. (1) The mapping 𝐽_𝜑∶𝑋→2^𝑋∗ defined by 𝐽_𝜑(𝑥)=𝑗∈𝑋^∗∶⟨𝑥,𝑗⟩=‖𝑥‖𝑗_∗,𝑗_∗=𝜑(‖𝑥‖),𝑥∈𝑋, (2) is called the duality mapping with gauge 𝜑. In the special case where 𝜑(𝑡)=𝑡, the duality mapping 𝐽_𝜑=∶𝐽 is the classical normalized duality mapping. In the case 𝜑(𝑡)=𝑡^𝑝−1, 𝑝>1, the duality mapping 𝐽_𝜑=∶𝐽_𝑝 is called the generalized duality mapping and it is given by 𝐽_𝑝(𝑥)∶=𝑗∈𝑋^∗∶⟨𝑥,𝑗⟩=‖𝑥‖𝑗_∗,𝑗_∗=‖𝑥‖^𝑝−1,𝑥∈𝑋. (3)

For a gauge 𝜑, the function Φ∶ℝ⁺→ℝ⁺ defined by Φ(𝑡)=^𝑡₀𝜑(𝑠)𝑑𝑠 (4) is a continuous convex strictly increasing differentiable function on ℝ⁺ with Φ^′(𝑡)=𝜑(𝑡) and lim_𝑡→+∞Φ(𝑡)/𝑡=+∞. Therefore, Φ has a continuous inverse function Φ⁻¹.

We recall the Bregman Distance 𝐷_𝜑 and function 𝐷^𝑓_𝜑 studied in [1]. Let 𝑋 be a real smooth Banach space. The Bregman distance 𝐷_𝜑(𝑥,𝑦) between 𝑥 and 𝑦 in 𝑋 is defined by 𝐷_𝜑𝑥,𝑦∶=Φ(‖𝑥‖)−Φ𝑦−𝑥−𝑦,𝐽_𝜑𝑦. (5) One can see from Lemma 3 that 𝐷_𝜑(𝑥,𝑦)≥0. In the case 𝜑(𝑡)=𝑡^𝑝−1,  𝑝∈(1,∞), the distance 𝐷_𝜑(𝑥,𝑦)=∶𝐷_𝑝(𝑥,𝑦) is called the 𝑝-Lyapunov functional studied in [2] and it is given by 𝐷_𝑝=‖𝑥‖𝑥,𝑦^𝑝𝑝+𝑦^𝑝𝑞−𝑥,𝐽_𝑝𝑦. (6) Note that 𝜙𝑥,𝑦∶=2𝐷₂𝑥,𝑦=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+𝑦² (7) is the Lyapunov functional. It is obvious that ‖𝑥‖−𝑦²≤𝜙𝑥,𝑦,∀𝑥,𝑦∈𝑋. (8) 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 Π_𝐶∶𝑋→𝐶 is defined by Π_𝐶(𝑥)=𝑥₀∈𝐶∶𝜙𝑥₀,𝑥=inf_𝑦∈𝐶𝜙𝑦,𝑥. (9) The metric projection operator 𝑃_𝐶∶𝑋→𝐶 defined by 𝑃_𝐶𝑥∶=argmin_𝑦∈𝐶12𝑥−𝑦²,𝑥∈𝑋, (10) 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 𝑃_𝐶 are nonexpansive (i.e., ‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦 in 𝐻). However, this is no longer true in the framework of Banach spaces. Instead, the generalized projections Π_𝐶 are needed. In [6], Alber generalized the metric projection operator 𝑃_𝐶 to generalized projection operators Π_𝐶∶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]).

Section 2 contains preliminaries. In Section 3, we study the fundamental properties of Bregman distance 𝐷_𝜑 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 Π_𝐶^𝜑,𝑓. Here, we do not assume the maximality of monotone operators and the uniform smoothness of Banach spaces.

2. Preliminaries

Let 𝐴∶𝑋→2^𝑋∗ 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 𝑗_𝑥∈𝐴𝑥,𝑗_𝑦∈𝐴𝑦⟹𝑥−𝑦,𝑗_𝑥−𝑗_𝑦≥0. (11) 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, by 𝜑 we always mean a gauge and by Φ the corresponding function defined in (4). We list some properties of the duality mapping 𝐽_𝜑∶𝑋→2^𝑋∗ below (for more details see [13, 14]).

Proposition 1. Let 𝑋 be a real Banach space.(i)𝐽_𝜑 is norm-to-weak* upper semicontinuous;(ii)for each 𝑥 in 𝑋, the set 𝐽_𝜑(𝑥) is convex and weakly closed in 𝑋^∗;(iii)𝐽_𝜑(−𝑥)=−𝐽_𝜑(𝑥) and 𝐽_𝜑(𝜆𝑥)=(𝜑(‖𝜆𝑥‖)/𝜑(‖𝑥‖))𝐽_𝜑(𝑥) for all nonzero 𝑥 in 𝑋,𝜆>0;(iv)there holds𝐽_𝜑(𝑥)=𝜑(‖𝑥‖)‖𝑥‖𝐽(𝑥),0≠𝑥∈𝑋; (12)(v)𝐽_𝜑 is maximal monotone;(vi)if 𝑋 is strictly convex, then 𝐽_𝜑 is strictly monotone; that is,𝑥−𝑦,𝑗_𝑥−𝑗_𝑦>0∀𝑥,𝑦∈𝑋,𝑗_𝑥∈𝐽_𝜑(𝑥),𝑗_𝑦∈𝐽_𝜑𝑦; (13)
(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 space 𝐸 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 vectors 𝑥_𝑛,𝑦_𝑛,𝑧 in 𝐸 and a constant 𝜖>0 such that 𝑦_𝑛−𝑥_𝑛→0, and ⟨𝑧,𝐽(𝑦_𝑛)−𝐽(𝑥_𝑛)⟩≥𝜖, for all 𝑛∈ℕ. For a fixed 𝑡>0, define 𝑎_𝑛𝑥∶=^𝑛+𝑡𝑧−𝑥_𝑛−𝑡𝑧,𝐽𝑥_𝑛𝑡,𝑏_𝑛𝑦∶=^𝑛+𝑡𝑧−𝑦_𝑛−𝑡𝑧,𝐽𝑦_𝑛𝑡. (14) Observe that 𝑎_𝑛≥⟨𝑥_𝑛+𝑡𝑧,𝐽𝑦_𝑛⟩−𝑥_𝑛+𝑡𝑧,𝐽𝑥_𝑛𝑡=𝑧,𝐽𝑦_𝑛−𝐽𝑥_𝑛+𝑥_𝑛,𝐽𝑦_𝑛−𝐽𝑥_𝑛𝑡,𝑏_𝑛≥⟨𝑦_𝑛+𝑡𝑧,𝐽𝑥_𝑛⟩−𝑦_𝑛+𝑡𝑧,𝐽𝑦_𝑛𝑡=𝑧,𝐽𝑦_𝑛−𝐽𝑥_𝑛+𝑦_𝑛,𝐽𝑦_𝑛−𝐽𝑥_𝑛𝑡. (15) Hence, 𝑎_𝑛+𝑏_𝑛≥2𝑧,𝐽𝑦_𝑛−𝐽𝑥_𝑛−𝑥_𝑛−𝑦_𝑛,𝐽𝑦_𝑛−𝐽𝑥_𝑛𝑡𝑥≥2𝜖−2^𝑛−𝑦_𝑛𝑡. (16) Choose 𝑡=(2/𝜖)‖𝑥_𝑛−𝑦_𝑛‖. By the uniform Gâteaux differentiability of the norm, if 𝑛 is large enough, both 𝑎_𝑛 and 𝑏_𝑛 are less than (1/2)𝜖, and so 𝑎_𝑛+𝑏_𝑛<𝜖. We arrive at a contradiction.

Together with Proposition 1, the conclusion in Lemma 2 also holds for 𝐽_𝜑.

Let 𝑋 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 𝜓(𝑥)−𝜓𝑦≤𝑥−𝑦,𝑗,∀𝑦∈𝑋. (17) The set (possibly empty) 𝜕𝜓(𝑥)∶=𝑗∈𝑋^∗∶𝜓(𝑥)−𝜓𝑦≤⟨𝑥−𝑦,𝑗⟩∀𝑦∈𝑋, (18) of subgradients of 𝜓 at 𝑥 in dom(𝜓), is called the subdifferential of 𝜑 at 𝑥.

Lemma 3. Let 𝑋 be a smooth Banach space and 𝐽_𝜑∶𝑋→𝑋^∗ the duality mapping with gauge 𝜑. Then 𝐽_𝜑(𝑦)=𝜕Φ(‖𝑦‖) for 𝑦∈𝑋⧵{0}; that is, Φ(‖𝑥‖)−Φ𝑦−𝑥−𝑦,𝐽_𝜑𝑦≥0,∀𝑥∈𝑋. (19)

The proof of the following result is straightforward.

Lemma 4. Let 𝑋 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. Let 𝑋 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 𝐵_𝑟∶={𝑥∈𝑋∶‖𝑥‖≤𝑟}, where 𝑟>0 is arbitrarily given. That is, there exists a strictly increasing convex function 𝑔_𝑟∶ℝ⁺→ℝ⁺ with 𝑔_𝑟(0)=0 such that Φ()𝑦𝑥+()Φ𝑦()𝑔𝑡𝑥+1−𝑡≤𝑡Φ(‖‖)1−𝑡−𝑡1−𝑡_𝑟𝑥−𝑦 (20)  for all 𝑥,𝑦 in 𝐵_𝑟 and 𝑡 in [0,1].

Proof. First we note that the general case can be reduced to the case 𝑟=1. Suppose it holds Φ()𝑦𝑥+()Φ𝑦()𝑔.𝑡𝑥+1−𝑡≤𝑡Φ(‖‖)1−𝑡−𝑡1−𝑡𝑥−𝑦 (21) We will show that (20) holds for any 𝑟>0. Set 𝜑_𝑟(𝑡)=𝜑(𝑟𝑡) and 𝑔_𝑟(𝑡)=𝑟𝑔(𝑡/𝑟) for 𝑡≥0. Then 𝜑_𝑟 is still a gauge function, and let Φ_𝑟 be the function corresponding to 𝜑_𝑟 as defined in (4), 𝑟Φ_𝑟(𝑡)=Φ(𝑟𝑡). Let 𝑥,𝑦∈𝐵_𝑟 and 𝑥_𝑟=𝑥/𝑟 and 𝑦_𝑟=𝑦/𝑟. Then 𝑥_𝑟,𝑦_𝑟∈𝐵_𝑋. Applying (21) to Φ_𝑟, we get Φ𝑡𝑥+(1−𝑡)𝑦=𝑟Φ_𝑟𝑡𝑥_𝑟+(1−𝑡)𝑦_𝑟≤𝑟𝑡Φ_𝑟𝑥_𝑟+𝑟(1−𝑡)Φ_𝑟𝑦_𝑟−𝑟𝑡(1−𝑡)𝑔𝑥_𝑟−𝑦_𝑟𝑥+()Φ𝑦()𝑔=𝑡Φ(‖‖)1−𝑡−𝑡1−𝑡_𝑟𝑥−𝑦, (22) 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)⇒(a). Given 𝑥,𝑦 in 𝑋 such that ‖𝑥‖=‖𝑦‖=1 and ‖𝑥−𝑦‖=𝜀. Setting 𝑡=1/2 in (20), we have Φ𝑥+𝑦2(1)−1≤Φ4𝑔(𝜀). (23) It turns out that 1−𝑥+𝑦2≥1−Φ⁻¹Φ(1)−14𝑔(𝜀)>0. (24) 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, 𝜇(𝜀)1∶=inf2Φ𝑥+1(‖‖)2Φ𝑦1−Φ2𝑥+𝑦∶𝑥,𝑦∈𝐵_𝑋,𝑥−𝑦≥𝜀, (25) where 𝐵_𝑋 is the closed unit ball of 𝑋.
Claim. Consider 𝜇(𝜀)>0 for 0<𝜀≤2.
Suppose on the contrary that 𝜇(𝜀)=0 for some 0<𝜀≤2. Then we can find sequences {𝑥_𝑛}, {𝑦_𝑛} in 𝐵_𝑋 such that ‖𝑥_𝑛−𝑦_𝑛‖≥𝜀 for all 𝑛, and 12Φ𝑥_𝑛+12Φ𝑦_𝑛𝑥−Φ^𝑛+𝑦_𝑛2⟶0. (26) Without loss of generality, we may assume that 𝑥_𝑛⟶𝛼,𝑦_𝑛𝑥⟶𝛽,_𝑛+𝑦_𝑛⟶𝛾,𝑥_𝑛−𝑦_𝑛⟶𝜃≥𝜀. (27) It then follows from (26) that 12Φ(𝛼)+12Φ𝛽𝛾=Φ2. (28) 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 𝑥^′_𝑛=𝑥_𝑛/‖𝑥_𝑛‖, 𝑦^′_𝑛=𝑦_𝑛/‖𝑦_𝑛‖, then ‖𝑥^′_𝑛‖=‖𝑦^′_𝑛‖=1 for all 𝑛; moreover, from (27), we get 𝑥^′_𝑛+𝑦^′_𝑛2⟶1,𝑥^′_𝑛−𝑦^′_𝑛⟶𝜃𝛼>0. (29) This contradicts the uniform convexity of 𝑋, and verifies that 𝜇(𝜀)>0 for all 0<𝜀≤2.
It turns out from (25) that Φ121𝑥+2𝑦≤12Φ𝑥+1(‖‖)2Φ𝑦−14𝑔𝑥−𝑦, (30) for all 𝑥,𝑦 in 𝐵_𝑋, 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, Φ()𝑦𝑥+()Φ𝑦()𝑔𝑡𝑥+1−𝑡≤𝑡Φ(‖‖)1−𝑡−𝑡1−𝑡𝑥−𝑦 (31) for 𝑥,𝑦 in 𝐵_𝑋 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∶ℝ⁺→ℝ⁺ with 𝑔(0)=0 such that Φ𝑥+𝑦≥Φ(‖𝑥‖)+⟨𝑦,𝑗⟩+𝑔𝑦,∀𝑥,𝑦∈𝐵_𝑋,∀𝑗∈𝐽_𝜑(𝑥). (32)

Proof. Since 𝐽_𝜑 is the subdifferential of the functional Φ(‖⋅‖), we have for 𝑗_𝑥 in 𝐽_𝜑(𝑥), 𝑥 in 𝑋 that 𝑦−𝑥,𝑗_𝑥≤Φ𝑦−Φ(‖𝑥‖)∀𝑦∈𝑋. (33) Let 𝑔 be the function that satisfies (20) with 𝑟=2 and assume that 𝑥,𝑦∈𝐵_𝑋. Replacing 𝑦 with 𝑥+𝜆𝑦,  0<𝜆<1, we obtain 𝑦,𝑗_𝑥≤Φ𝑥+𝜆𝑦−Φ(‖𝑥‖)𝜆=Φ(1−𝜆)𝑥+𝜆𝑥+𝑦−Φ(‖𝑥‖)𝜆≤)Φ𝑥()𝑔𝑦𝑥𝜆)(1−𝜆(‖‖)+𝜆Φ𝑥+𝑦−𝜆1−𝜆−Φ(‖‖)(⁻¹≤Φ𝑥+𝑦−Φ(‖𝑥‖)−(1−𝜆)𝑔𝑦. (34) Taking limit as 𝜆→0, we get ⟨𝑦,𝑗_𝑥⟩=Φ𝑥+𝑦−Φ(‖𝑥‖)−𝑔𝑦. (35)

Lemma 7 (see [17, Theorem 3.11, page 952]). Let 𝑋 be a reflexive Banach space and 𝐽_𝜑∶𝑋→2^𝑋∗ the duality map with gauge 𝜑. Suppose that 𝐴∶𝑋→2^𝑋∗ is maximal monotone. Then the operator 𝐴+𝐽_𝜑 is surjective.

3. Bregman Distance 𝐷_𝜑 and Function 𝐷^𝑓_𝜑

One can easily see that 𝐷_𝜑𝑥,𝑦+𝐷_𝜑𝑦,𝑧−𝐷_𝜑(𝑥,𝑧)=⟨𝑥−𝑦,𝐽_𝜑(𝑧)−𝐽_𝜑𝑦⟩,∀𝑥,𝑦,𝑧∈𝑋. (36) Noticing that for 𝑥 in 𝑋, the scalar function 𝐷_𝜑(⋅,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 𝐷_𝜑𝑥,𝑦=0⟺𝑥=𝑦. (37)

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∶ℝ⁺→ℝ⁺ with 𝑔(0)=0 such that 𝐷_𝜑(𝑢,𝑣)≥𝑔(‖𝑢−𝑣‖)∀𝑢,𝑣∈𝐵_𝑋. (38)

Proof. Let 𝑢,𝑣∈𝐵_𝑋. By Lemma 6 we have 𝐷_𝜑(𝑢,𝑣)=Φ(‖𝑣+(𝑢−𝑣)‖)−Φ(‖𝑣‖)−⟨𝑢−𝑣,𝐽_𝜑(𝑣)⟩≥Φ(‖𝑣‖)+⟨𝑢−𝑣,𝐽_𝜑(𝑣)𝑣⟩+𝑔(‖𝑢−𝑣‖)−Φ(‖‖)−⟨𝑢−𝑣,𝐽_𝜑(𝑣)⟩=𝑔(‖𝑢−𝑣‖). (39)

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 {𝑥_𝑛} and {𝑦_𝑛} be two sequences in 𝑋 such that 𝐷_𝜑(𝑥_𝑛,𝑦_𝑛)→0. If {𝑦_𝑛} is bounded, then ‖𝑥_𝑛−𝑦_𝑛‖→0.

Proof. Assume {𝑦_𝑛} is bounded. From definition (5) we have 𝐷_𝜑𝑥_𝑛,𝑦_𝑛=Φ𝑥_𝑛−Φ𝑦_𝑛−𝑥_𝑛−𝑦_𝑛,𝐽_𝜑𝑦_𝑛. (40) It follows that Φ𝑥_𝑛−𝑥_𝑛𝜑𝑦_𝑛≤Φ𝑥_𝑛−⟨𝑥_𝑛,𝐽_𝜑𝑦_𝑛⟩=Φ𝑦_𝑛−𝑥_𝑛𝜑𝑦_𝑛+𝐷_𝜑𝑥_𝑛,𝑦_𝑛≤Φ𝑦_𝑛+𝐷_𝜑𝑥_𝑛,𝑦_𝑛. (41) Since {𝑦_𝑛} is bounded, 𝐷_𝜑(𝑥_𝑛,𝑦_𝑛)→0 and Φ(𝑡)/𝑡→∞ as 𝑡→∞ it follows from (41) that {‖𝑥_𝑛‖} is bounded, too.
We may now assume that {𝑥_𝑛} and {𝑦_𝑛} both lie in the closed unit ball 𝐵_𝑋 (otherwise consider the rescaled sequences {𝛾𝑥_𝑛} and {𝛾𝑦_𝑛} 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 𝑔𝑥_𝑛−𝑦_𝑛≤𝐷_𝜑𝑥_𝑛,𝑦_𝑛,∀𝑛∈ℕ. (42) Since 𝐷_𝜑(𝑥_𝑛,𝑦_𝑛)→0 and 𝑔 is strictly increasing, we immediately conclude that ‖𝑥_𝑛−𝑦_𝑛‖→0.

Proposition 11 (see [20, Lemma 3.1]). Let 𝑋 be a smooth Banach space. Let 𝑢∈𝑋 and {𝑥_𝑛} be a sequence in 𝑋 such that {𝐷_𝜑(𝑥_𝑛,𝑢)} is bounded. Then {𝑥_𝑛} is bounded.

The statement in the following proposition is evident from the definition of 𝐷_𝜑 (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_𝜑(⋅,𝑥) is continuous, weakly lower semicontinuous, and convex on 𝑋.

Let 𝑓∶𝑋→[0,+∞] be a proper, convex, lower semicontinuous function. Define 𝐷^𝑓_𝜑𝑥,𝑦=𝐷_𝜑𝑥,𝑦+𝑓(𝑥),∀𝑥,𝑦∈𝑋. (43)

Some of the following basic properties of the Bregman distance 𝐷_𝜑 and function 𝐷^𝑓_𝜑 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→[0,+∞] be a proper, convex, lower semicontinuous function with 𝐶⊂dom(𝑓). Then there exists a unique element 𝑥₀ in 𝐶 such that 𝐷^𝑓_𝜑𝑥₀,𝑥=inf𝐷^𝑓_𝜑(𝑧,𝑥)∶𝑧∈𝐶. (44)

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 𝐷_𝑓-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 Π_𝐶 as follows.

Definition 14. In the setting of Proposition 13, we define the (𝜑,𝑓)-generalized projection from 𝑋 onto 𝐶 by Π_𝐶^𝜑,𝑓(𝑥)=𝑥₀,𝑥∈𝑋. (45)
In case 𝜑(𝑡)=𝑡 and 𝑓=0, we notice that Π_𝐶^𝜑,𝑓 coincides with Π_𝐶. In case that 𝑋 is a Hilbert space and 𝜑(𝑡)=𝑡, denote Π_𝐶^𝜑,𝑓 by 𝑃^𝑓_𝐶.

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→[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=Π_𝐶^𝜑,𝑓(𝑥),(b)⟨𝑧−𝑥₀,𝐽_𝜑𝑥₀−𝐽_𝜑𝑥⟩+𝑓(𝑧)−𝑓(𝑥₀)≥0,  ∀𝑧∈𝐶.(ii)Given 𝑥 in 𝑋, one has𝐷_𝜑𝑦,Π_𝐶^𝜑,𝑓(𝑥)+𝐷_𝜑Π_𝐶^𝜑,𝑓(𝑥),𝑥≤𝐷_𝜑𝑦,𝑥+𝑓𝑦−𝑓Π_𝐶^𝜑,𝑓(𝑥),∀𝑦∈𝐶. (46)

Proposition 16. Let 𝐶 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 {𝑥_𝑛} be a bounded sequence in 𝐶 such that lim_𝑛→+∞𝑓(𝑥_𝑛)=0. If 𝐷^𝑓_𝜑(𝑥_𝑛,𝑢)≤𝐷^𝑓_𝜑(Π_𝐶^𝜑,𝑓(𝑢),𝑢) for all 𝑛 in ℕ, then lim_𝑛→+∞𝑥_𝑛=Π_𝐶^𝜑,𝑓(𝑢).

Proof. Set Π_𝐶^𝜑,𝑓(𝑢)=∶𝑢. Then, by assumption, we have 𝐷^𝑓_𝜑(𝑥_𝑛,𝑢)≤𝐷^𝑓_𝜑(𝑢,𝑢) for all 𝑛 in ℕ. By the three-point identity (36), we have 𝐷^𝑓_𝜑𝑥_𝑛,𝑢=𝐷^𝑓_𝜑𝑥_𝑛,𝑢+𝐷^𝑓_𝜑(𝑢,𝑢)−⟨𝑥_𝑛−𝑢,𝐽_𝜑(𝑢)−𝐽_𝜑(𝑢)⟩−𝑓(𝑢)≤𝐷^𝑓_𝜑(𝑢,𝑢)+𝐷^𝑓_𝜑(𝑢,𝑢)−⟨𝑥_𝑛−𝑢,𝐽_𝜑(𝑢)−𝐽_𝜑(𝑢)⟩−𝑓(𝑢)≤⟨𝑢−𝑢,𝐽_𝜑(𝑢)−𝐽_𝜑(𝑢)⟩−⟨𝑥_𝑛−𝑢,𝐽_𝜑(𝑢)−𝐽_𝜑(𝑢)⟩+𝑓(𝑢)=⟨𝑢−𝑥_𝑛,𝐽_𝜑(𝑢)−𝐽_𝜑(𝑢)⟩+𝑓(𝑢). (47) Since {𝑥_𝑛} is a bounded sequence in 𝐶, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that {𝑥_𝑛𝑖} converges weakly to some element 𝑧 in 𝐶. Note that 0≤𝑓(𝑧)≤liminf_𝑖→+∞𝑓𝑥_𝑛𝑖=0; (48) that is, 𝑓(𝑧)=0. Using Proposition 15, we have limsup_𝑖→+∞𝐷^𝑓_𝜑𝑥_𝑛𝑖,𝑢≤limsup_𝑖→+∞⟨𝑢−𝑥_𝑛𝑖,𝐽_𝜑(𝑢)−𝐽_𝜑(𝑢)⟩+𝑓(𝑢)≤⟨𝑢−𝑧,𝐽_𝜑(𝑢)−𝐽_𝜑(𝑢)(𝑢)(𝑧),⟩+𝑓≤𝑓 (49) which implies that lim_𝑖→+∞𝐷_𝜑(𝑥_𝑛𝑖,𝑢)=0. It follows from Proposition 10 that lim_𝑖→∞𝑥_𝑛𝑖=𝑢. This implies that 𝑥_𝑛 converges strongly to 𝑢=Π_𝐶^𝜑,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. Let 𝑋 be a smooth Banach space, 𝐽_𝜑∶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 ⟨𝑇𝑥−𝑇𝑦,𝐽_𝜑(𝑇𝑥)−𝐽_𝜑𝑇𝑦⟩≤⟨𝑇𝑥−𝑇𝑦,𝐽_𝜑(𝑥)−𝐽_𝜑𝑦⟩,∀𝑥,𝑦∈𝐶. (50)
In the case of 𝜑(𝑡)=𝑡, inequality (50) reduces to 𝑇𝑥−𝑇𝑦,𝐽(𝑇𝑥)−𝐽𝑇𝑦≤𝑇𝑥−𝑇𝑦,𝐽(𝑥)−𝐽𝑦,∀𝑥,𝑦∈𝐶. (51) 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]): 𝑇𝑥−𝑇𝑦²≤𝑇𝑥−𝑇𝑦,𝑥−𝑦,∀𝑥,𝑦∈𝐶. (52)
We now give useful characterizations of 𝜑-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→𝐶 be a 𝜑-firmly nonexpansive mapping. Then 𝐷_𝜑𝑇𝑥,𝑇𝑦+𝐷_𝜑𝑇𝑦,𝑇𝑥+𝐷_𝜑(𝑇𝑥,𝑥)+𝐷_𝜑𝑇𝑦,𝑦≤𝐷_𝜑𝑇𝑥,𝑦+𝐷_𝜑𝑇𝑦,𝑥∀𝑥,𝑦∈𝐶. (53)

The geometry of the fixed point set of 𝜑-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→𝐶 be a 𝜑-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→𝐶 be a mapping. A point 𝑢 in 𝐶 is an asymptotic fixed point of 𝑇 if 𝐶 contains a sequence {𝑥_𝑛} such that 𝑥_𝑛⇀𝑢 and ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0; see [31]. We denote the set of asymptotic fixed points of 𝑇 by 𝐹(𝑇). A mapping 𝑇∶𝐶→𝐶 is relatively 𝜑-nonexpansive if the following conditions are satisfied:(i)𝐹(𝑇) is nonempty; (ii)𝐷_𝜑(𝑝,𝑇𝑥)≤𝐷_𝜑(𝑝,𝑥) for all 𝑥 in 𝐶 and 𝑝 in 𝐹(𝑇);(iii)𝐹(𝑇)=𝐹(𝑇).

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

A mapping 𝑇∶𝐶→𝐶 is 𝜑-firmly quasinonexpansive if 𝐹(𝑇)≠∅ and ⟨𝑇𝑥−𝑝,𝐽_𝜑(𝑥)−𝐽_𝜑(𝑇𝑥)⟩≥0 (54) 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→𝐶 a 𝜑-firmly quasinonexpansive mapping. Then 𝐷_𝜑𝑝,𝑇𝑥+𝐷_𝜑(𝑇𝑥,𝑥)≤𝐷_𝜑𝑝,𝑥 (55) 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→𝐶 be a 𝜑-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 {𝑥_𝑛} in 𝐶 such that 𝑥_𝑛⇀𝑢 and ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0. We need to prove that 𝑢∈𝐹(𝑇). Using (53), we get 𝐷_𝜑(𝑇𝑢,𝑢)≤𝐷_𝜑𝑇𝑥_𝑛,𝑢+𝐷_𝜑𝑇𝑢,𝑥_𝑛−𝐷_𝜑𝑇𝑥_𝑛,𝑇𝑢−𝐷_𝜑𝑇𝑢,𝑇𝑥_𝑛−𝐷_𝜑𝑇𝑥_𝑛,𝑥_𝑛. (56) It is not hard to find that (56) is reduced to the relation 𝐷_𝜑(𝑇𝑢,𝑢)≤Φ(‖𝑇𝑢‖)−Φ(‖𝑢‖)+𝑢−𝑇𝑥_𝑛,𝐽_𝜑(𝑢+)𝑇𝑥_𝑛−𝑇𝑢,𝐽_𝜑𝑥_𝑛−𝐽_𝜑𝑇𝑥_𝑛+𝑇𝑥_𝑛−𝑇𝑢,𝐽_𝜑(𝑇𝑢). (57) We note that 𝑇𝑥_𝑛⇀𝑢. Indeed, for any norm one linear functional 𝜉 of 𝑋, we have 𝑇𝑥_𝑛−𝑢,𝜉=𝑇𝑥_𝑛−𝑥_𝑛,𝜉+𝑥_𝑛−𝑢,𝜉. (58) Both terms in the right-hand side approach zero. Consequently, 𝑇𝑥_𝑛−𝑇𝑢,𝐽_𝜑(𝑇𝑢)⟶𝑢−𝑇𝑢,𝐽_𝜑(𝑇𝑢). (59)
Claim. Consider ⟨𝑇𝑥_𝑛−𝑇𝑢,𝐽_𝜑(𝑥_𝑛)−𝐽_𝜑(𝑇𝑥_𝑛)⟩→0.
Since 𝑥_𝑛⇀𝑢 and ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0, there is a constant 𝑀>0 such that all ‖𝑥_𝑛‖,‖𝑇𝑥_𝑛‖<𝑀. It follows from the uniform norm to weak* continuity of 𝐽_𝜑 on bounded subsets (Lemma 2); we have ⟨𝑇𝑢,𝐽_𝜑𝑥_𝑛−𝐽_𝜑𝑇𝑥_𝑛⟩⟶0. (60) Observe that 𝜖_𝑛∶=𝑇𝑥_𝑛−𝑥_𝑛,𝐽_𝜑𝑥_𝑛⟶0. (61) Moreover, |‖𝑥_𝑛‖−‖𝑇𝑥_𝑛‖|≤‖𝑥_𝑛−𝑇𝑥_𝑛‖→0. By the uniform continuity of 𝜑 on [0,𝑀], we have 𝑇𝑥_𝑛,𝐽_𝜑𝑥_𝑛−𝐽_𝜑𝑇𝑥_𝑛=𝑇𝑥_𝑛,𝐽_𝜑𝑥_𝑛−𝑇𝑥_𝑛,𝐽_𝜑𝑇𝑥_𝑛=𝑥_𝑛𝜑𝑥_𝑛+𝜖_𝑛−𝑇𝑥_𝑛𝜑𝑇𝑥_𝑛=𝑥_𝑛−𝑇𝑥_𝑛𝜑𝑥_𝑛+𝑇𝑥_𝑛𝜑𝑥_𝑛−𝜑𝑇𝑥_𝑛+𝜖_𝑛⟶0. (62) The claim is thus verified.
We obtain from (57) and the claim that 𝐷_𝜑(𝑇𝑢,𝑢)≤Φ(‖𝑇𝑢‖)−Φ(‖𝑢‖)+⟨𝑢−𝑇𝑢,𝐽_𝜑(𝑇𝑢)⟩. (63) This is equivalent to ⟨𝑇𝑢−𝑢,𝐽_𝜑(𝑇𝑢)−𝐽_𝜑(𝑢)⟩≤0. (64) The strict monotonicity of the duality mapping 𝐽_𝜑 implies that equality must hold. Namely, 𝑇𝑢=𝑢 or 𝑢∈𝐹(𝑇).

The resolvent of an operator 𝐴∶𝑋→2^𝑋∗ relative to a Gâteaux differentiable function 𝑓 is introduced and studied in [1]. We define 𝜑-resolvent operators following [1, 18].

Definition 22. Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝑋 and let 𝐽_𝜑∶𝑋→𝑋^∗ be the duality mapping with gauge 𝜑. Suppose that 𝐴∶𝑋→2^𝑋∗ is an operator satisfying the range condition 𝐷(𝐴)⊂𝐶⊂𝐽_𝜑⁻¹ℛ𝐽_𝜑+𝑟𝐴∀𝑟>0. (65) For each 𝜆>0, the 𝜑-resolvent associated with operator 𝐴 is the operator 𝑅_𝜆^𝜑,𝐴∶𝐶→2^𝑋 defined by 𝑅_𝜆^𝜑,𝐴𝑥=𝑧∈𝑋∶𝐽_𝜑(𝑥)∈𝐽_𝜑+𝜆𝐴𝑧,𝑥∈𝐶. (66)
For 𝑥 in 𝐶 and 𝜆 in (0,∞), we have 0∈𝐴𝑥⟺𝐽_𝜑(𝑥)∈𝐽_𝜑𝑥𝐽+𝜆𝐴⟺𝑥∈_𝜑+𝜆𝐴⁻¹𝐽_𝜑𝑥⟺𝑥∈𝐹𝑅_𝜆^𝜑,𝐴. (67) 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 𝜑(𝑡)=𝑡^𝑝−1 with 𝑝∈(1,+∞), we have 𝐽_𝜑=𝐽_𝑝 and 𝑅^𝑝,𝐴=(𝐽_𝑝+𝐴)⁻¹𝐽_𝑝. For 𝑝=2,  𝑅^𝐴∶=𝑅^2,𝐴=(𝐽+𝐴)⁻¹∘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→2^𝑋∗ be a monotone operator satisfying the condition 𝐷(𝐴)⊂𝐶⊂𝐽_𝜑⁻¹ℛ(𝐽_𝜑+𝜆𝐴), where 𝜆>0. Using the smoothness and strict convexity of 𝑋, we obtain that 𝑅_𝜆^𝜑,𝐴 is single-valued. The conditions 𝐷(𝐴)⊂𝐶⊂𝐽_𝜑⁻¹ℛ(𝐽_𝜑+𝜆𝐴) ensure that 𝑅_𝜆^𝜑,𝐴 is the single-valued 𝜑-resolvent operator form 𝐶 into 𝐷(𝐴). In other words, 𝑅_𝜆^𝜑,𝐴𝑥=𝐽_𝜑+𝜆𝐴⁻¹𝐽_𝜑𝑥,∀𝑥∈𝐶. (68)
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_𝜑∶𝑋→𝑋^∗ be the duality mapping with gauge 𝜑. Let 𝐴∶𝑋→2^𝑋∗ be a monotone operator satisfying the condition 𝐷(𝐴)⊂𝐶⊂𝐽_𝜑⁻¹ℛ(𝐽_𝜑+𝜆𝐴), where 𝜆 is a positive real number. Let 𝑅_𝜆^𝜑,𝐴 be a resolvent of 𝐴, where(a)𝑅_𝜆^𝜑,𝐴 is 𝜑-firmly nonexpansive mapping from 𝐶 into 𝐶,(b)𝐹(𝑅_𝜆^𝜑,𝐴)=𝐴⁻¹0.

5. Convergence Theorems

Let 𝐶 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={𝐺_𝑡∶𝑡>0} be a family of mappings from 𝐶 into 𝐶 such that ⋂_𝑡>0𝐹(𝑇(𝑡))⊆⋂_𝑡>0𝐹(𝐺_𝑡). We say the family 𝒯∶={𝑇(𝑡)∶𝑡>0} has property (𝒜)with respect to the family 𝒢={𝐺_𝑡∶𝑡>0} if the following assertion holds: foreachboundednet𝑥_𝑠𝑠>0in𝐶if𝑥_𝑠−𝐺_𝑠𝑥_𝑠⟶0as𝑠⟶+∞then𝑥_𝑠−𝑇(𝑡)𝑥_𝑠⟶0as𝑠⟶+∞,∀𝑡>0. (69)

If 𝒯=𝒢 and the above condition holds, then we say 𝒯∶={𝑇(𝑠)∶𝑠>0} has property (𝒜).

Remark 25. If 𝒯 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. 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)≠∅. Assume 𝑏_𝑡 in ℝ with 0<𝑎≤𝑏_𝑡≤𝑏<1 for all 𝑡>0. Define 𝐺_𝑡∶𝐶→𝐶 by 𝐺_𝑡𝑥=(1−𝑏_𝑡)𝑥+𝑏_𝑡𝑇𝑥 for all 𝑥 in 𝐶. Then 𝑇 has property (𝒜) with respect to the family {𝐺_𝑡∶𝑡>0}.

Proof. Let {𝑥_𝑡}_𝑡>0 be a bounded net in 𝐶 such that ‖𝑥_𝑡−𝐺_𝑡(𝑥_𝑡)‖→0 as 𝑡→∞. Note that 𝑥_𝑡−𝑇𝑥_𝑡=1𝑏_𝑡𝑥_𝑡−𝐺_𝑡𝑥_𝑡 (70) and 0<𝑎≤𝑏_𝑡≤𝑏<1 for all 𝑡>0. Therefore, ‖𝑥_𝑡−𝑇𝑥_𝑡‖→0 as 𝑡→+∞.

The following example shows that the family {𝑅^𝐴_𝑡∶𝑡>0} of resolvent operators of a maximal monotone operator 𝐴 enjoys property (𝒜).

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: 𝐷(𝐴)⊂𝐶⊂_𝑟>0ℛ(𝐼+𝑟𝐴). (71) Let {𝑧_𝑡}_𝑡>0 be a bounded net in 𝐶 such that ‖𝑧_𝑡−𝑅^𝐴_𝑡𝑧_𝑡‖→0 as 𝑡→+∞. Then ‖𝑧_𝑡−𝑅^𝐴_𝑟𝑧_𝑡‖→0 as 𝑡→+∞ for each 𝑟>0.

Proof. Let 𝑟, 𝑡>0. By Takahashi [34], we have 1𝑟𝑅^𝐴_𝑡𝑧_𝑡−𝑅^𝐴_𝑟𝑅^𝐴_𝑡𝑧_𝑡≤1𝑡𝑧_𝑡−𝑅^𝐴_𝑡𝑧_𝑡. (72) Using (72), we have 𝑧_𝑡−𝑅^𝐴_𝑟𝑧_𝑡≤𝑧_𝑡−𝑅^𝐴_𝑡𝑧_𝑡+𝑅^𝐴_𝑡𝑧_𝑡−𝑅^𝐴_𝑟𝑅^𝐴_𝑡𝑧_𝑡+𝑅^𝐴_𝑟𝑅^𝐴_𝑡𝑧_𝑡−𝑅^𝐴_𝑟𝑧_𝑡≤𝑟2+𝑡𝑧_𝑡−𝑅^𝐴_𝑡𝑧_𝑡⟶0as𝑡⟶+∞. (73)

We now discuss the problem of finding common fixed points of a sequence of 𝜑-firmly nonexpansive mappings. Our proximal-projection method is based on (a not necessarily Bregman distance) function 𝐷^𝑓_𝜑. 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 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∶={𝑇_𝑛} a sequence of 𝜑-firmly nonexpansive self-mappings on 𝐶 such that 𝐹(𝑇)=⋂_𝑛∈ℕ𝐹(𝑇_𝑛)≠∅ and 𝒯 has property (𝒜) with respect to 𝑇. For 𝑢 in 𝐶 and 𝐶₁=𝐶 with 𝑥₁=Π_𝐶^𝜑,𝑓₁(𝑢), define a sequence {𝑥_𝑛} in 𝐶 as follows: 𝑦_𝑛=𝑇_𝑛𝑥_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶⟨𝑧−𝑦_𝑛,𝐽_𝜑𝑥_𝑛−𝐽_𝜑𝑦_𝑛,𝑥⟩≤0_𝑛+1=Π_𝐶^𝜑,𝑓_𝑛+1(𝑢)∀𝑛∈ℕ. (74) Then {𝑥_𝑛} converges strongly to Π^𝜑,𝑓_𝐹(𝑇)(𝑢).

Proof. We proceed the proof in the following steps:
Step 1. {𝑥_𝑛} is well defined.
Note that all 𝐶_𝑛 are closed and convex. For 𝑝 in 𝐹(𝒯) and 𝑛 in ℕ, we obtain from (54) that ⟨𝑝−𝑦_𝑛,𝐽_𝜑𝑥_𝑛−𝐽_𝜑𝑦_𝑛⟩≤0. (75) It follows that 𝑝∈𝐶_𝑛 and hence 𝐹(𝒯)⊂𝐶_𝑛. Therefore, {𝑥_𝑛} is well defined.
Step 2. {𝑥_𝑛} is bounded.
Let 𝑝∈𝐹(𝒯). It follows from Proposition 15; we have 𝐷_𝜑𝑥_𝑛,𝑢=𝐷_𝜑Π_𝐶^𝜑,𝑓_𝑛(𝑢),𝑢≤𝐷_𝜑𝑝,𝑢−𝐷_𝜑𝑝,Π_𝐶^𝜑,𝑓_𝑛(𝑢)+𝑓𝑝−𝑓𝑥_𝑛≤𝐷_𝜑𝑝,𝑢+𝑓𝑝. (76) It follows that {𝐷_𝜑(𝑥_𝑛,𝑢)} is bounded and hence from Proposition 11, we obtain that {𝑥_𝑛} is bounded.
Step 3. Consider ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0.
Note that 𝑥_𝑛+1∈𝐶_𝑛+1⊂𝐶_𝑛. It follows from Proposition 15 that 𝐷_𝜑𝑥_𝑛+1,Π_𝐶^𝜑,𝑓_𝑛(𝑢)+𝐷_𝜑Π_𝐶^𝜑,𝑓_𝑛(𝑢),𝑢≤𝐷_𝜑𝑥_𝑛+1,𝑢+𝑓𝑥_𝑛+1−𝑓Π_𝐶^𝜑,𝑓_𝑛(𝑢). (77) This implies that 𝐷_𝜑𝑥_𝑛,𝑢≤𝐷_𝜑𝑥_𝑛+1,𝑢−𝐷_𝜑𝑥_𝑛+1,𝑥_𝑛+𝑓𝑥_𝑛+1−𝑓𝑥_𝑛. (78) Therefore, the sequence {𝐷^𝑓_𝜑(𝑥_𝑛,𝑢)} (see (43)) is increasing. Note that {𝐷^𝑓_𝜑(𝑥_𝑛,𝑢)} is bounded by (76). It follows that lim_𝑛→+∞𝐷^𝑓_𝜑(𝑥_𝑛,𝑢) exists. By (78), we obtain lim_𝑛→+∞𝐷^𝑓_𝜑𝑥_𝑛+1,𝑥_𝑛=0. (79) One can see that lim_𝑛→+∞𝑓(𝑥_𝑛)=0. Using Proposition 10, we obtain that lim_𝑛→+∞‖𝑥_𝑛−𝑥_𝑛+1‖=0. Since 𝑥_𝑛+1∈𝐶_𝑛+1, we have 0≤𝐷_𝜑𝑥_𝑛+1,𝑦_𝑛+𝐷_𝜑𝑦_𝑛,𝑥_𝑛≤𝐷_𝜑𝑥_𝑛+1,𝑦_𝑛+𝐷_𝜑𝑦_𝑛,𝑥_𝑛+⟨𝑦_𝑛−𝑥_𝑛+1,𝐽_𝜑𝑥_𝑛−𝐽_𝜑𝑦_𝑛⟩=𝐷_𝜑𝑥_𝑛+1,𝑥_𝑛. (80) Using (79), we obtain that lim_𝑛→+∞𝐷_𝜑𝑥_𝑛+1,𝑦_𝑛+𝐷_𝜑𝑦_𝑛,𝑥_𝑛=0, (81) and hence lim_𝑛→+∞𝐷_𝜑(𝑦_𝑛,𝑥_𝑛)=0. Note that {𝑥_𝑛} is bounded. Then, one can see from Proposition 10 that {𝑦_𝑛} is bounded and lim_𝑛→+∞𝑥_𝑛−𝑦_𝑛=0. (82) This implies that lim_𝑛→+∞𝑥_𝑛−𝑇_𝑛𝑥_𝑛=0. (83) Since the family 𝒯∶={𝑇_𝑛∶𝑛∈ℕ} has property (𝒜) with respect to 𝑇, it follows from (83) that lim_𝑛→+∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0.
Step 4. The sequence {𝑥_𝑛} converges strongly to Π^𝜑,𝑓_𝐹(𝑇)(𝑢).
Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀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 Π^𝜑,𝑓_𝐹(𝑇) is well defined. Note that 𝑥_𝑛=Π_𝐶^𝜑,𝑓_𝑛(𝑢) and 𝐹(𝑇) is contained in 𝐶_𝑛; we have 𝐷^𝑓_𝜑𝑥_𝑛,𝑢≤𝐷^𝑓_𝜑Π^𝜑,𝑓_𝐹(𝑇)(𝑢),𝑢. (84) Therefore, we conclude from Proposition 16 that {𝑥_𝑛} converges strongly to Π^𝜑,𝑓_𝐹(𝑇)(𝑢).

Theorem 29. Let 𝑋 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→2^𝑋∗ be a monotone operator with 𝐴⁻¹0≠∅ satisfying the following condition: 𝐷(𝐴)⊂𝐶⊂_𝑟>0𝐽_𝜑⁻¹ℛ𝐽_𝜑+𝑟𝐴. (85) Let 𝜆 be a positive real number; for 𝑢 in 𝐶 and 𝐶₁=𝐶 with 𝑥₁=Π_𝐶^𝜑,𝑓₁(𝑢), define a sequence {𝑥_𝑛} in 𝐶 as follows: 𝑦_𝑛=𝐽_𝜑+𝜆𝐴⁻¹𝐽_𝜑𝑥_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝑧−𝑦_𝑛,𝐽_𝜑𝑥_𝑛−𝐽_𝜑𝑦_𝑛,𝑥≤0_𝑛+1=Π_𝐶^𝜑,𝑓_𝑛+1(𝑢),∀𝑛∈ℕ. (86) Then {𝑥_𝑛} converges strongly to Π_𝐴^𝜑,𝑓₋₁₀(𝑢).

Proof. Set 𝑇∶=(𝐽_𝜑+𝜆𝐴)⁻¹𝐽_𝜑. 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. Let 𝐶 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 {𝜆_𝑛} be a sequence in (0,∞) such that lim_𝑛→+∞𝜆_𝑛=+∞, for 𝑢 in 𝐶, let 𝐶₁=𝐶 with 𝑥₁=𝑃^𝑓_𝐶1(𝑢), and define a sequence {𝑥_𝑛} in 𝐶 as follows: 𝑦_𝑛=𝐼+𝜆_𝑛𝐴⁻¹𝑥_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶⟨𝑧−𝑦_𝑛,𝑥_𝑛−𝑦_𝑛,𝑥⟩≤0_𝑛+1=𝑃^𝑓_𝐶𝑛+1(𝑢),∀𝑛∈ℕ. (87) Then {𝑥_𝑛} converges strongly to 𝑃^𝑓_𝐴−10(𝑢).

Proof. Set 𝑇∶=𝑅^𝐴_𝜆=(𝐼+𝜆𝐴)⁻¹ for 𝜆>0 and 𝑇_𝑛∶=(𝐼+𝜆_𝑛𝐴)⁻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 lim_𝑛→+∞‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖=0. Example 27 implies that 𝒯∶={𝑇_𝑟∶𝑟>0} has property (𝒜). It follows that lim_𝑛→+∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=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→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 all 𝑥,𝑦,𝑧 in 𝐶, limsup_𝑡↓0Θ(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤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 Θ𝑥,𝑦≥0,∀𝑦∈𝐶. (88) 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 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 Θ+1𝑧,𝑦𝑟⟨𝑦−𝑧,𝐽_𝜑𝑧−𝐽_𝜑𝑥⟩≥0,∀𝑦∈𝐶. (89)

We now consider the following generalized mixed equilibrium problem (GMEP): find 𝑧 in 𝐶 such that Θ𝑧,𝑦+⟨𝐴𝑧,𝑦−𝑧⟩+𝜓𝑦−𝜓(𝑧)≥0,∀𝑦∈𝐶. (90) 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 Θ𝑦(𝑧)+1𝑧,𝑦+⟨𝐴𝑧,𝑦−𝑧⟩+𝜓−𝜓𝑟⟨𝐽_𝜑𝑧−𝐽_𝜑𝑥,𝑦−𝑧⟩≥0,∀𝑦∈𝐶. (91) 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 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 𝑇_𝑟^{(Θ,𝐴,𝜓)}∶𝑋→2^𝐶 as follows: 𝑇_𝑟^{(Θ,𝐴,𝜓)}(𝑥)=𝑦(𝑧)+1𝑧∈𝐶∶Θ𝑧,𝑦+⟨𝐴𝑧,𝑦−𝑧⟩+𝜓−𝜓𝑟⟨𝐽_𝜑𝑧−𝐽_𝜑𝑥,𝑦−𝑧⟩≥0,∀𝑦∈𝐶. (92)

Lemma 33 (see [37]). One has the following.(1)𝑇_𝑟^{(Θ,𝐴,𝜓)} is single-valued. (2)𝑇_𝑟^{(Θ,𝐴,𝜓)} is a 𝜑-firmly nonexpansive mapping; that is, for all 𝑥,𝑦 in 𝑋, 𝑇_𝑟^{(Θ,𝐴,𝜓)}𝑥−𝑇_𝑟^{(Θ,𝐴,𝜓)}𝑦,𝐽_𝜑𝑇_𝑟^{(Θ,𝐴,𝜓)}𝑥−𝐽_𝜑𝑇_𝑟^{(Θ,𝐴,𝜓)}𝑦≤𝑇_𝑟^{(Θ,𝐴,𝜓)}𝑥−𝑇_𝑟^{(Θ,𝐴,𝜓)}𝑦,𝐽_𝜑𝑥−𝐽_𝜑𝑦. (93)(4)𝐹(𝑇_𝑟^{(Θ,𝐴,𝜓)})=𝐺𝑀𝐸𝑃(Θ,𝐴,𝜓).(5)𝐺𝑀𝐸𝑃(Θ,𝐴,𝜓) is closed and convex.(6) For all 𝑥 in 𝐹(𝑇_𝑟^{(Θ,𝐴,𝜓)}) and 𝑦 in 𝑋, one has 𝐷_𝜑𝑥,𝑇_𝑟^{(Θ,𝐴,𝜓)}𝑦+𝐷_𝜑𝑇_𝑟^{(Θ,𝐴,𝜓)}𝑦,𝑦≤𝐷_𝜑𝑥,𝑦. (94)

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 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=Π_𝐶^𝜑,𝑓₁(𝑢), define a sequence {𝑥_𝑛} in 𝐶 as follows: 𝑦_𝑛=𝑇_𝑟^{(Θ,𝐴,𝜓)}𝑥_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶⟨𝑧−𝑦_𝑛,𝐽_𝜑𝑥_𝑛−𝐽_𝜑𝑦_𝑛,𝑥⟩≤0_𝑛+1=Π_𝐶^𝜑,𝑓_𝑛+1(𝑢)∀𝑛∈ℕ. (95) Then {𝑥_𝑛} strongly converges to Π^𝜑,𝑓_{𝐹(𝑇𝑟(Θ,𝐴,𝜓))}(𝑢).

Proof. Note that 𝑇_𝑟^{(Θ,𝐴,𝜓)} is a 𝜑-firmly nonexpansive mapping from 𝐶 into 𝐶 and 𝐹(𝑇_𝑟^{(Θ,𝐴,𝜓)})=GMEP(Θ,𝐴,𝜓,𝜑). 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.