Bregman strongly nonexpansive operators in reflexive Banach spaces

Bregman Strongly Nonexpansive Operators in Reflexive Banach Spaces Victoria Mart´ın-M´arquez∗ , Simeon Reich†and Shoham Sabach‡ November 14, 2012 Abstract We present a detailed study of right and left Bregman strongly nonexpansive operators in reflexive Banach spaces. We analyze, in particular, compositions and convex combinations of such operators, and prove the convergence of the Picard iterative method for operators of these types. Finally, we use our results to approximate common zeroes of maximal monotone mappings and solutions to convex feasibility problems. 2010 Mathematics Subject Classification: 26B25, 46T99, 47H04, 47H05, 47H09, 47H10, 52A41, 54C15. Key words and phrases. Bregman distance, Bregman strongly nonexpansive operator, Legendre function, monotone mapping, nonexpansive operator, reflexive Banach space, resolvent, totally convex function. 1 Introduction The theory and applications of nonexpansive operators in Banach spaces have been intensively studied for almost fifty years now [7,23,24,25]. There are several important classes of nonexpansive operators which have remarkable properties not shared by all such operators. We refer, for example, to strongly nonexpansive operators which were introduced in [16]. This class of operators is of particular significance in fixed point, iteration and convex optimization theories mainly because it is closed under composition. It encompasses other noteworthy classes of nonexpansive operators. For example, in uniformly convex Banach spaces all firmly nonexpansive operators as well as all averaged operators are strongly nonexpansive [16]. A related class of operators comprises the quasi-nonexpansive operators. These operators still enjoy relevant fixed point properties although nonexpansivity is only required for each fixed point [22]. In this paper, we are concerned with certain analogous classes of operators which are, in some sense, strongly nonexpansive not with respect to the norm, but with respect to Bregman distances [3,14,18,21]. Since these distances are not symmetric in general, it seems natural to distinguish between left and right Bregman strongly nonexpansive operators. The left variant has already been ∗Departamento de An´alisis Matem´atico, Universidad de Sevilla, Apdo. 1160, 41080-Sevilla, Spain. Email: [email protected]. †Department of Mathematics, The Technion – Israel Institute of Technology, 32000 Haifa, Israel. Email: [email protected]. ‡Department of Mathematics, The Technion – Israel Institute of Technology, 32000 Haifa, Israel. Email: [email protected]. 1 studied and applied in [31,35]. We have recently introduced and studied several classes of right Bregman nonexpansive operators in reflexive Banach spaces [28,29]. The present paper is devoted to a detailed study of right and left Bregman strongly nonexpansive operators. Our paper is organized as follows. Section 2 contains certain essential preliminary results regarding properties of Bregman distances. In the next section we establish in detail two fundamental properties of left Bregman strongly nonexpansive operators. These properties concern the compositions of finitely many such operators. In Section 4 we establish analogous results for right Bregman strongly nonexpansive operators. The next section is devoted to convex combinations of a given finite number of right Bregman strongly nonexpansive operators. In Section 6 we bring out the connections between left and right Bregman strongly nonexpansive operators. Finally, in the last two sections we prove the convergence of the Picard iteration for right Bregman strongly nonexpansive operators and then use our results to find common zeroes of maximal monotone mappings and to solve convex feasibility problems. 2 Preliminaries In this section we collect several definitions and results, which are pertinent to our study. Let Xbe a reflexive Banach space and let f:X→(−∞,+∞] be a function with effective domain dom f:= {x∈X:f(x)<+∞}. The Fenchel conjugate function f∗:X∗→(−∞,+∞], defined by f∗(u) = sup {hu, xi − f(x) : x∈dom f}. We say that fis admissible if it is proper, convex, lower semicontinuous and Gˆateaux differentiable on int dom f. Under these assumptions we know that fis continuous in int dom f(see [4, Fact 2.3, page 619]). Recall that fis called cofinite if dom f∗=X∗. The subdifferential of fis the set-valued mapping ∂f :X→2X∗defined by ∂f (x) := {u∈X∗:f(y)≥f(x) + hu, y −xi ∀y∈X}, x ∈X. The boundedness of fis inherited by the subdifferential and vice versa as the following result shows. Proposition 2.1 (cf. [18, Proposition 1.1.11, page 16]).If f:X→Ris continuous and convex, then ∂f : 2X∗→Xis bounded on bounded subsets if and only if fitself is bounded on bounded subsets. The function fis said to be Legendre if it satisfies the following two conditions. (L1) int dom f6=∅and ∂f is single-valued on its domain (dom ∂f ={x∈X:∂f (x)6=∅}). (L2) int dom f∗6=∅and ∂f∗is single-valued on its domain. When the subdifferential of fis single-valued, it coincides with the gradient ∂f =∇f[30, Definition 1.3, page 3] (this is the case when fis an admissible function, not necessarily Legendre). The class of Legendre functions in infinite dimensional Banach spaces was first introduced and studied by Bauschke, Borwein and Combettes in [4]. Their definition is equivalent to conditions (L1) and (L2) because the space Xis assumed to be reflexive (see [4, Theorems 5.4 and 5.6, page 634]). It is well known that in reflexive spaces ∇f= (∇f∗)−1(see [4, Theorem 5.10, page 636]). When this fact is combined with conditions (L1) and (L2), we obtain ran ∇f= dom ∇f∗= int dom f∗and ran ∇f∗= dom ∇f= int dom f. It also follows that fis Legendre if and only if f∗is Legendre (see [4, Corollary 5.5, page 634]) and that the functions fand f∗are Gˆateaux differentiable and strictly convex in the interior of 2 their respective domains. When the Banach space Xis smooth and strictly convex, in particular, a Hilbert space, the function (1/p)k·kpwith p∈(1,∞) is Legendre (cf. [4, Lemma 6.2, page 639]). For examples and more information regarding Legendre functions, see, for instance, [2,4]. The bifunction Df: dom f×int dom f→[0,+∞) given by Df(y, x) := f(y)−f(x)− h∇f(x), y −xi(1) is called the Bregman distance with respect to f(cf. [20]). With the function fwe associate the bifunction Wf: dom f∗×dom f→[0,+∞) defined by Wf(ξ, x) := f(x)− hξ, xi+f∗(ξ).(2) This function satisfies Wf(∇f(x), y) = Df(y, x) for all x∈int dom fand y∈dom f(cf. [27]). We now recall the definition of a totally convex function which was introduced in [17,18]. Definition 2.2 (Total convexity).The function fis called totally convex at a point x∈int dom f if its modulus of total convexity at x,νf(x, ·) : [0,+∞)→[0,+∞], defined by νf(x, t) := inf {Df(y, x) : y∈dom f, ky−xk=t}, is positive whenever t > 0. The function fis called totally convex when it is totally convex at every point of int dom f. Definition 2.3 (Total convexity on bounded subsets).The function fis called totally convex on bounded sets if, for any nonempty and bounded set E⊂X, the modulus of total convexity of fon E,νf(E, t), is positive for any t > 0, where νf(E, ·) : [0,+∞)→[0,+∞] is defined by νf(E, t) := inf {νf(x, t) : x∈E∩int dom f}. Relevant examples of functions fsatisfying the above properties can be found in [10,12,13,28]. The following result will be crucial for our analysis. The following result will play a crucial rule in our results (cf. [18, Lemma 2.1.2, page 67]). Proposition 2.4 (Property of total convexity on bounded subsets).The function f:X→ (−∞,+∞]is totally convex on bounded subsets of Xif and only if for any two sequences {xn}n∈N and {yn}n∈Nin int dom fand dom f, respectively, such that the first one is bounded, lim n→∞ Df(yn, xn) = 0 ⇒lim n→∞ kyn−xnk= 0. Next we recall two boundedness properties (cf. [26,34]). Proposition 2.5 (Boundedness property - left variable).Let the function f:X→(−∞,+∞]be admissible and totally convex at a point x∈int dom f. Let {xn}n∈N⊂dom f. If {Df(xn, x)}n∈N is bounded, then the sequence {xn}n∈Nis bounded too. Proof. Suppose that the sequence {xn}n∈Nis not bounded. Then it contains a subsequence {xnk}k∈Nsuch that lim k→∞ kxnkk= +∞. 3 Consequently, limk→∞ kxnk−xk= +∞and there exists some k0>0 such that kxnk−xk>1 for all k > k0. Since fis totally convex at x, the modulus of total convexity νf(x, ·) is strictly increasing and positive on (0,+∞) (see [18, Proposition 1.2.2, page 18]), in particular, νf(x, 1) >0 for all x∈X. So using [18, Proposition 1.2.2, page 18] again, we get that, for all k > k0, νf(x, kxnk−xk)≥ kxnk−xkνf(x, 1) → ∞. Thus {νf(x, kxn−xk)}n∈Nis not bounded. Since, by definition, νf(x, kxn−xk)≤Df(xn, x),(3) for all n∈N, this implies that the sequence {Df(xn, x)}n∈Ncannot be bounded. Proposition 2.6 (Boundedness property - right variable).Let f:X→Rbe an admissible function such that ∇f∗is bounded on bounded subsets of dom f∗=X∗. Let x∈Xand {xn}n∈N⊂X. If {Df(x, xn)}n∈Nis bounded, so is the sequence {xn}n∈N. Proof. Let βbe an upper bound of the sequence {Df(x, xn)}n∈N. Then from the definition of Wf (see (2)) we obtain f(x)− h∇f(xn), xi+f∗(∇f(xn)) = Wf(∇f(xn), x) = Df(x, xn)≤β. This implies that {∇f(xn)}n∈Nis contained in the sub-level set, levψ ≤(β−f(x)), of the function ψ=f∗− h·, xi. Since the function f∗is proper and lower semicontinuous, an application of the Moreau-Rockafellar Theorem (see [4, Fact 3.1, page 623]) shows that ψis coercive, that is, limkxk→∞ ψ(x) = +∞. Consequently, all sub-level sets of ψare bounded. Hence the sequence {∇f(xn)}n∈Nis bounded. By hypothesis, ∇f∗is bounded on bounded subsets of X∗. Therefore the sequence {xn}n∈N={∇f∗(∇f(xn))}n∈Nis bounded too, as claimed. 3 Composition of left Bregman strongly nonexpansive operators This section is devoted to the properties of left Bregman strongly nonexpansive operators. These properties originate in [31]. We include here complete proofs for the convenience of the reader. Definition 3.1 (Left Bregman strongly nonexpansive operators).We say that an operator T:K⊂ int dom f→int dom fis left Bregman strongly nonexpansive (L-BSNE) with respect to S⊂dom f if Df(p, Tx)≤Df(p, x) (4) for all p∈Sand x∈K, and if whenever {xn}n∈N⊂Kis bounded, p∈S, and lim n→∞ (Df(p, xn)−Df(p, Txn)) = 0,(5) it follows that lim n→∞ Df(Txn, xn) = 0.(6) Let f:X→(−∞,+∞] be admissible and let Kbe a nonempty subset of X. The fixed point set of an operator T:K→Xis the set {x∈K:Tx =x}. It is denoted by Fix (T). Recall that a point u∈Kis said to be an asymptotic fixed point [31] of Tif there exists a sequence {xn}n∈Nin Ksuch that xn* u as n→ ∞ (that is, {xn}n∈Nis weakly convergent to u) and kxn−Txnk → 0 as n→ ∞. We denote the asymptotic fixed point set of Tby d Fix (T). Next we list three types of left Bregman strong nonexpansivity. 4 Remark 3.2 (Types of left Bregman strong nonexpansivity).We will use the following particular cases. (i) An operator which satisfies (4)-(6) with respect to S:= Fix (T) is called properly L-BSNE. (ii) An operator which satisfies (4)-(6) with respect to S:= d Fix (T) is called strictly L-BSNE (this class of operators was first defined in [31]). (iii) An operator which satisfies (4)-(6) with respect to S:= Fix (T) = d Fix (T) is called fully L-BSNE.♦ The next result shows that the composition of a finite family of strictly L-BSNE operators is also strictly L-BSNE. Proposition 3.3 (Composition of strictly L-BSNE operators).Let f:X→Rbe an admissible function which is bounded, uniformly Fr´echet differentiable and totally convex on bounded subsets of X. Assume that ∇f∗is bounded on bounded subsets of dom f∗=X∗and let Kbe a nonempty subset of X. Let {Ti:i= 1, . . . , N}be Nstrictly L-BSNE operators from Kinto itself and consider the composition T=TN◦TN−1◦ · · · ◦ T1. Assume that the set b F= N \ i=1 d Fix (Ti) is not empty. Then (i) d Fix (T)⊂b F; (ii) moreover if d Fix (T)6=∅,Tis also strictly L-BSNE. Proof. Let u∈b F. First we claim that if the sequence {xn}n∈N⊂Kis bounded and lim n→∞ (Df(u, xn)−Df(u, Txn)) = 0,(7) then, for all i= 1, . . . , N, we have lim n→∞ yi n−yi−1 n= 0, where yi n:= Ti◦Ti−1◦ · · · ◦ T1xn,i= 1, . . . , N, under the conventions that y0 n=xnand T0=I, the identity operator. Now we prove this claim. Since each Ti,i= 1, . . . , N, is strictly L-BSNE operator we get from (4) Df(u, Txn) = Dfu, yN n≤Dfu, yN−1 n≤ · · · ≤ Df(u, T1xn)≤Df(u, xn).(8) Hence, from (7), we get for all i= 2, . . . , N, that lim n→∞ Dfu, yi−1 n−Dfu, yi n≤lim n→∞ (Df(u, xn)−Df(u, Txn)) = 0. Thus lim n→∞ Dfu, yi−1 n−Dfu, yi n= 0.(9) Since {xn}n∈Nis bounded, and both fand ∇fare bounded on bounded subsets, the sequence {Df(u, xn)}n∈Nis bounded too. Therefore if follows from (8) that Dfu, yi nn∈Nis bounded for 5 each i= 1, . . . , N. Since ∇f∗is bounded on bounded subsets of X∗, it follows from Proposition 2.6 that yi nn∈Nis bounded. This together with (9) implies that lim n→∞ Dfyi n, yi−1 n= 0 because Ti,i= 1, . . . , N, is strictly L-BSNE. Since yi−1 nn∈Nis bounded, we have from Proposition 2.4 that lim n→∞ yi n−yi−1 n= 0, as claimed. Now we will prove our assertions. (i) Let u∈b F. Given x∈d Fix (T), then there exists a sequence {xn}n∈N⊂Kconverging weakly to xsuch that lim n→∞ kxn−Txnk= 0.(10) Since fis uniformly Fr´echet differentiable on bounded subsets, it is also uniformly continuous on bounded subsets of X(see [1, Theorem 1.8, page 13]) and therefore we obtain from (10) lim n→∞ (f(Txn)−f(xn)) = 0.(11) In addition, from [33, Proposition 2.1, page 474] we also obtain that ∇fis uniformly continuous on bounded subsets of Xand thus again from (10) we get lim n→∞ k∇f(Txn)− ∇f(xn)k∗= 0.(12) From the definition of the Bregman distance (see (1)) we obtain that Df(u, xn)−Df(u, Txn) = [f(u)−f(xn)− h∇f(xn), u −xni] −[f(u)−f(Txn)− h∇f(Txn), u −Txni] =f(Txn)−f(xn)− h∇f(xn), u −xni +h∇f(Txn), u −Txni =f(Txn)−f(xn)− h∇f(xn)− ∇f(Txn), u −xni +h∇f(Txn), xn−Txni. The function fis bounded on bounded subsets of Xand therefore ∇fis also bounded on bounded subsets of X(see [18, Propostion 1.1.11, page 16]). Thus both sequences {xn}n∈N and {∇f(Txn)}n∈Nare bounded. Hence from these facts along with (10), (11) and (12), we deduce that lim n→∞ (Df(u, xn)−Df(u, Txn)) = 0.(13) As we have already proved, from this property we obtain that lim n→∞ yi n−yi−1 n= 0 for all i= 1, . . . , N. Now from (10) we obtain that yN nn∈N(note that yN n=Txn) converges weakly to xand since lim n→∞ yN n−yN−1 n= 0 6 we get that yN−1 nn∈Nalso converges weakly to x. Repeating this argument for all i= 1,...N −2, yields that the all sequences yi nn∈Nconverge weakly to x. Now, since lim n→∞ yi−1 n−Tiyi−1 n= 0 we get that x∈d Fix (Ti) for each i= 1,2, . . . , N, so that x∈b F, as asserted. (ii) Let u∈d Fix (T) and x∈K. From assertion (i) we already know that u∈b F. Since each Ti, i= 1,2, . . . , N, satisfies (4) we obtain that Df(u, Tx) = Df(u, TN◦TN−1◦ · · · ◦ T1x) ≤Df(u, TN−1◦ · · · ◦ T1x) ≤ . . . ≤Df(u, T1x) ≤Df(u, x). Hence Talso satisfies (4). Given u∈d Fix (T) and a bounded sequence {xn}n∈Nsuch that lim n→∞ (Df(u, xn)−Df(u, Txn)) = 0,(14) we now prove that (6) is valid. We have already proved that (14) yields lim n→∞ yi n−yi−1 n= 0 for all i= 1, . . . , N. Since kxn−Txnk ≤ xn−y1 n+y1 n−y2 n+· · · +yN−1 n−yN n we get that lim n→∞ kxn−Txnk= 0. The function fis bounded on bounded subsets of Xand therefore ∇fis also bounded on bounded subsets of X(see [18, Propostion 1.1.11, page 16]). Thus both sequences {xn}n∈N and {∇f(Txn)}n∈Nare bounded. Since fis uniformly continuous on bounded subsets of X (see [1, Theorem 1.8, page 13]), we have lim n→∞ (f(Txn)−f(xn)) = 0. So from the definition of the Bregman distance (see (1)) we obtain that lim n→∞ Df(Txn, xn)=0. Hence Tis strictly L-BSNE, as asserted. Proposition 3.4 (Composition of fully L-BSNE operators).Let f:X→Rbe an admissible function which is bounded, uniformly Frchet differentiable and totally convex on bounded subsets of X. Assume that ∇f∗is bounded on bounded subsets of dom f∗=X∗and let Kbe a nonempty 7 subset of X. For each i= 1, . . . , N, let Ti:K⊂X→Kbe a fully L-BSNE operator, and let T=TN◦TN−1◦ · · · ◦ T1. If F:= N \ i=1 Fix (Ti) is nonempty, then Tis also fully L-BSNE and Fix (T) = N \ i=1 Fix (Ti). Proof. Indeed, from Proposition 3.3 it follows that Fix (T)⊂d Fix (T)⊂ N \ i=1 d Fix (Ti) = N \ i=1 Fix (Ti) = F⊂Fix (T), which implies that all inclusions are equalities, as claimed. 4 Composition of right Bregman strongly nonexpansive operators This section is devoted to a detailed study of the compositions of right Bregman strongly nonexpansive operators. Definition 4.1 (Right Bregman Strongly Nonexpansive operators).We say that an operator T: K⊂dom f→int dom fis right Bregman strongly nonexpansive (R-BSNE) with respect to S⊂ int dom fif Df(Tx, p)≤Df(x, p) (15) for all p∈Sand x∈K, and if whenever {xn}n∈N⊂Kis bounded, p∈S, and lim n→∞ (Df(xn, p)−Df(Txn, p)) = 0,(16) it follows that lim n→∞ Df(xn, Txn) = 0.(17) Remark 4.2 (Types of right Bregman strong nonexpansivity).We will use the following particular cases. (i) An operator which satisfies (15)-(17) with respect to S:= Fix (T) is called properly R-BSNE. (ii) An operator which satisfies (15)-(17) with respect to S:= d Fix (T) is called strictly R-BSNE. (iii) An operator which satisfies (15)-(17) with respect to S:= Fix (T) = d Fix (T) is called fully R-BSNE.♦ Remark 4.3. When condition (15) is satisfied with Sbeing the set Fix (T), d Fix (T) or Fix (T) = d Fix (T), the operator Tis said to be properly R-QBNE, strictly R-QBNE or fully R-QBNE, respectively. The acronym QBNE stands for quasi-Bregman nonexpansive. ♦ 8 Proposition 4.4 (Composition of strictly R-BSNE operators).Let f:X→Rbe an admissible function which is bounded, uniformly continuous and totally convex on bounded subsets of X. Let Kbe a nonempty subset of X. Let {Ti:i= 1, . . . , N}be Nstrictly R-BSNE operators from Kinto itself, and T=TN◦TN−1◦ · · · ◦ T1. Assume that the set b F= N \ i=1 d Fix (Ti) is not empty. Then (i) d Fix (T)⊂b F; (ii) moreover, if d Fix (T)6=∅,Tis strictly R-BSNE. Proof. Let u∈b F. First we claim that if the sequence {xn}n∈N⊂Kis bounded and lim n→∞ (Df(xn, u)−Df(Txn, u)) = 0,(18) then, for all i= 1, . . . , N, we have lim n→∞ yi n−yi−1 n= 0, where yi n:= Ti◦Ti−1◦ · · · ◦ T1xn,i= 1, . . . , N, under the conventions that y0 n=xnand T0=I, the identity operator. Now we prove this claim. Since each Ti,i= 1, . . . , N, is strictly R-BSNE operator we get from (15) Df(Txn, u) = DfyN n, u≤Dfyi n, u≤Dfyi−1 n, u≤ · · · ≤ Df(T1xn, u)≤Df(xn, u).(19) Hence, from (18), we get for all i= 2, . . . , N, that lim n→∞ Dfyi−1 n, u−Dfyi n, u≤lim n→∞ (Df(xn, u)−Df(Txn, u)) = 0. Thus lim n→∞ Dfyi−1 n, u−Dfyi n, u= 0.(20) Since {xn}n∈Nis bounded and fis bounded on bounded subsets, the sequence {Df(xn, u)}n∈Nis bounded too. Therefore if follows from (19) that Dfyi n, un∈Nis bounded for each i= 1, . . . , N. Now it follows from Proposition 2.5 that yi nn∈Nis bounded. This together with (20) implies that lim n→∞ Dfyi−1 n, yi n= 0 because Ti,i= 1, . . . , N, is strictly R-BSNE. Since yi nn∈Nis bounded, Proposition 2.4 now implies that lim n→∞ yi n−yi−1 n= 0, as claimed. Now we will prove our assertions. 9 6 Connections between L-BSNE and R-BSNE operators Let f:X→(−∞,+∞] be a Legendre function, T:K⊂int dom f→int dom fan operator and let S⊂int dom fbe a nonempty set. The conjugate operator associated with Tis defined by T∗:= ∇f◦T◦ ∇f∗:∇f(K)→int dom f∗.(33) This operator was first studied in [28], where its basic properties are collected in Proposition 2.7 there. Fact 6.1. If ∇f∗is bounded on bounded subsets of int dom f∗and Tis R-BSNE with respect to S, then T∗is L-BSNE with respect to ∇f(S). To see this, we first recall that Df∗(∇f(y),∇f(x)) = Df(x, y)∀x, y ∈int dom f. Since Tis R-QBNE with respect to S, this implies that T∗is L-QBNE with respect to ∇f(S). Now let {ξn}n∈Nbe a bounded sequence in ∇f(K) such that lim n→∞ (Df∗(η, ξn)−Df∗(η, T∗ξn)) = 0 for a point η∈ ∇f(S). Then the sequence {xn}n∈N⊂K, defined by xn=∇f∗(ξn), is bounded and lim n→∞ (Df(xn, p)−Df(Txn, p)) = 0 for p=∇f∗(η)∈S. Since Tis R-BSNE, it follows that lim n→∞ Df∗(T∗ξn, ξn) = lim n→∞ Df(xn, Txn) = 0, so T∗is indeed L-BSNE with respect to ∇f(S), as claimed. Analogously, it is possible to prove that if ∇fis bounded on bounded subsets of int dom f, then the converse implication holds. Therefore we arrive at the following result. Fact 6.2. If ∇fand ∇f∗are bounded on bounded subsets of int dom fand int dom f∗, respectively, then Tis R-BSNE with respect to Sif and only if T∗is L-BSNE with respect to ∇f(S). Thus, if S= Fix (T), since ∇f(Fix (T)) = Fix (T∗) (see [28, Proposition 2.7 (iii)]), we have the following relationship between properly right and left BSNE operators. Fact 6.3. If ∇fand ∇f∗are bounded on bounded subsets of int dom fand int dom f∗, respectively, then Tis properly R-BSNE if and only if T∗is properly L-BSNE. Furthermore, under certain continuity assumptions on the gradient mappings we obtain relations between the asymptotic fixed point sets of Tand T∗. If ∇f∗is uniformly continuous on bounded subsets of int dom f∗, by the definition of asymptotic fixed points it is easy to check that d Fix (T∗)⊂ ∇fd Fix (T). In an analogous way, we can show that when ∇fis uniformly continuous on bounded subsets of int dom f, then the reverse inclusion holds, resulting in the following statement (see [28, Proposition 2.7 (vi)]). Fact 6.4. If ∇fand ∇f∗are uniformly continuous on bounded subsets of int dom fand int dom f∗, respectively, then d Fix (T∗) = ∇fd Fix (T). 16 So in this case, if S=d Fix (T), Facts 6.2 and 6.4, yield the following equivalence between strictly right and left BSNE operators. Fact 6.5. If ∇fand ∇f∗are bounded and uniformly continuous on bounded subsets of int dom f and int dom f∗, respectively, then Tis strictly R-BSNE if and only if T∗is strictly L-BSNE. It is natural to wonder whether results regarding R-BSNE operators can be obtained from known results concerning L-BSNE operators by conjugation (see (33)) and vice versa. In this direction, bearing in mind, for instance, the connection we have just presented, we are able to show that Propositions 4.4 could be deduced from Propositions 3.3 under different suitable conditions, most of them imposed on the conjugate function. Proposition 6.6. Let f:X→(−∞,+∞]be a cofinite Legendre function such that f∗is totally convex on bounded subsets of X∗. Assume that ∇fand ∇f∗are bounded and uniformly continuous on bounded subsets of int dom fand X∗, respectively. Let {Ti:i= 1, . . . , N}be Nstrictly R-BSNE operators from Kinto itself, where K⊂X, and let T=TN◦TN−1◦ · · · ◦ T1. If the set b F= N \ i=1 d Fix (Ti)6=∅, then d Fix (T)⊂b F. Furthermore, if d Fix (T)is not empty, then Tis strictly R-BSNE. Proof. We consider, for each i= 1, . . . , N, the conjugate operator T∗ i:= ∇f◦Ti◦ ∇f∗:∇f(K)→ ∇f(K). Since ∇f∗is bounded and uniformly continuous on bounded subsets of int dom f∗, we know from Fact 6.5 that, for any i= 1, . . . , N,T∗ iis strictly L-BSNE with respect to d Fix (T∗) = ∇fd Fix (T). Therefore, if b F:= N \ i=1 d Fix (Ti) is nonempty, we can show that so is c F∗:= N \ i=1 d Fix (T∗ i). Indeed, x∈b Fif and only if x∈d Fix (Ti) = ∇f∗d Fix (T∗ i)for all i= 1, . . . , N. In other words, x∈b Fif and only if ∇f(x)∈d Fix (T∗ i) for any i= 1, . . . , N, which is equivalent to ∇f(x)∈c F∗. This means that ∇fb F=c F∗.(34) Thus the family {T∗ i:i= 1, . . . , N}satisfies the hypotheses of Proposition 3.3, and consequently, if we denote T∗=T∗ n◦ · · · ◦ T∗ 1, we see that d Fix (T∗)⊂c F∗.(35) Note that T∗is the conjugate operator of T=Tn◦ · · · ◦ T1, that is, T∗= (∇f◦Tn◦ ∇f∗)◦ · · · ◦ (∇f◦T1◦ ∇f∗) = ∇f◦Tn◦ · · · ◦ T1◦ ∇f∗=∇f◦T◦ ∇f∗ 17 and then d Fix (T) = ∇f∗d Fix (T∗). Hence one deduces from (34) and (35) that d Fix (T) = ∇f∗d Fix (T∗)⊂ ∇f∗c F∗=b F. If we assume that d Fix (T) is nonempty, then so is d Fix (T∗) and thus Proposition 3.3 assures us that T∗is strictly L-BSNE. It follows that Tis strictly R-BSNE, as asserted. Taking into account that the hypotheses of Proposition 6.6 seem to be stronger than those of Propositions 4.4, we see that the conjugation technique does not seem to lead to the best possible results. Regarding the convex combination operator, the results proved in Section 5can also be recovered from analogous results associated with the so-called block operator defined and analyzed in [27]. We recall here its definition and main properties [27]. Definition 6.7 (Block operator).Let {Ti: 1 ≤i≤N}be Noperators from Xinto Xand let the weights {wi}N i=1 ⊂(0,1) satisfy PN i=1 wi= 1. Then the block operator corresponding to {Ti: 1 ≤i≤N}and {wi: 1 ≤i≤N}is defined by TB:= ∇f∗◦ N X i=1 wi∇f◦Ti!.(36) Proposition 6.8 (Block operator of strictly L-BSNE operators).Let f:X→Rbe a Legendre function which is bounded, uniformly Fr´echet differentiable and totally convex on bounded subsets of X. Assume that ∇f∗is bounded on bounded subsets of int dom f∗. If each Ti,i= 1, . . . , N, is a strictly L-BSNE operator from K⊂Xinto Xand the set b F:= \nd Fix (Ti):1≤i≤No6=∅, then d Fix (TB)⊂b F. Furthermore, if d Fix (TB)6=∅, then TBis also strictly L-BSNE. Proposition 6.9 (Block operator of properly L-BSNE operators).Let f:X→Rbe a Legendre function which is bounded, uniformly Fr´echet differentiable and totally convex on bounded subsets of X. Assume that ∇f∗and is bounded on bounded subsets of int dom f∗. If each Ti,i= 1, . . . , N, is a properly L-BSNE operator from K⊂Xinto Xand the set F:= \{Fix (Ti):1≤i≤N} 6=∅, then TBis also properly L-BSNE and F= Fix (TB). These propositions can be applied to deduce the theses in Propositions 5.3, and 5.6 under different conditions on the function fas we show in the following two propositions. Proposition 6.10. Let f:X→(−∞,+∞]be a cofinite Legendre function such that f∗is bounded, uniformly Fr´echet differentiable and totally convex on bounded subsets of X∗. Assume that ∇fand ∇f∗are bounded and uniformly continuous on bounded subsets of int dom fand X∗, respectively. Let {Ti:i= 1, . . . , N}be Nstrictly R-BSNE operators from K⊂Xinto int dom f. If the set b F= N \ i=1 d Fix (Ti)6=∅, then, given weights {wi}N i=1 ⊂(0,1) satisfying PN i=1 wi= 1, every asymptotic fixed point of the convex combination operator TCbelongs to b F, that is, d Fix (TC)⊂b F. Furthermore, if d Fix (TC)is not empty, then TCis strictly R-BSNE. 18 Proof. We consider the conjugate operators T∗ i=∇f◦Ti◦∇f∗from ∇f(K) into X∗,i= 1, . . . , N, which are strictly L-BSNE with respect to d Fix (T∗ i) = ∇fd Fix (Ti)(see Facts 6.4 and 6.5). In Proposition 6.6 we proved that ∇fb F=c F∗:= TN i=1 d Fix (T∗ i). So c F∗6=∅and Proposition 6.8 ensures that the asymptotic fixed points of the block operator T∗ B:= ∇f◦PN i=1 wi∇f∗◦T∗ i belong to c F∗, that is, d Fix (T∗ B)⊂c F∗. Note that T∗ Bis the conjugate operator of TC: T∗ B=∇f◦ N X i=1 wi∇f∗◦(∇f◦Ti◦ ∇f∗)! =∇f◦ N X i=1 wiTi◦ ∇f∗! =∇f◦ N X i=1 wiTi!◦ ∇f∗ =∇f◦TC◦ ∇f∗. Then, by Proposition 6.8, d Fix (TC) = ∇f∗d Fix (T∗ B)⊂ ∇f∗c F∗=b F. If d Fix (TC)6=∅, then so is d Fix (T∗ B) and thus Proposition 6.8 assures us that T∗ Bis strictly L-BSNE. So from Fact 6.5 it follows that TCis strictly R-BSNE. Following the same arguments used in the previous proposition, it is readily proved that Proposition 5.6 is a consequence of Proposition 6.9 under suitable conditions on fand f∗. Proposition 6.11. Let f:X→Rbe a cofinite Legendre function such that fand f∗are bounded, uniformly Fr´echet differentiable and totally convex on bounded subsets of Xand X∗, respectively. Assume that ∇fand ∇f∗are bounded on bounded subsets of int dom fand X∗, respectively. For each i= 1, . . . , N, let Ti:K⊂X→Xbe a properly R-BSNE operator. If the set F= N \ i=1 Fix (Ti)6=∅, then, given weights {wi}N i=1 ⊂(0,1) satisfying PN i=1 wi= 1, the convex combination operator TCis properly R-BSNE with respect to Fix (TC) = TN i=1 Fix (Ti). 7 Picard iteration for R-BSNE operators In this section we are concerned with the iterates of R-BSNE operators, their compositions and their convex combinations. Definition 7.1 (Weakly sequentially continuous mapping).A mapping B:X→X∗is called weakly sequentially continuous if for any sequence {xn}n∈N⊂X,xn* x implies that Bxn* Bx as n→ ∞. 19 Proposition 7.2 (Weak convergence).Let f:X→(−∞,+∞]be a Legendre function. Suppose that the sequence {xn}n∈N⊂int dom fis bounded and that lim n→∞ Df(xn, u) (37) exists for any weak subsequential limit uof {xn}n∈N. Then {xn}n∈Nconverges weakly. Proof. It suffices to prove the uniqueness of weak subsequential limits of {xn}n∈Nbecause, since {xn}n∈Nis bounded and Xis reflexive, we know that there is at least one. Assume that uand v are any two weak subsequential limits of {xn}n∈N. From (37) we know that lim n→∞ (Df(xn, u)−Df(xn, v)) exists. From the definition of the Bregman distance (see (1)) we get Df(xn, u)−Df(xn, v)=[f(xn)−f(u)− h∇f(u), xn−ui] −[f(xn)−f(v)− h∇f(v), xn−vi] =f(v)−f(u) + h∇f(v)− ∇f(u), xni +h∇f(u), ui − h∇f(v), vi and therefore lim n→∞ h∇f(v)− ∇f(u), xni exists. Since uand vare weak subsequential limit of {xn}n∈N, there are subsequences {xnk}k∈N and {xmk}k∈Nof {xn}n∈Nsuch that xnk* u and xmk* v as k→ ∞. Thus h∇f(v)− ∇f(u), ui= lim k→+∞h∇f(v)− ∇f(u), xnki = lim n→+∞h∇f(v)− ∇f(u), xni = lim k→+∞h∇f(v)− ∇f(u), xmki =h∇f(v)− ∇f(u), vi. Hence h∇f(v)− ∇f(u), v −ui= 0, which implies that u=vbecause fis strictly convex in int dom fwhich implies the strict monotonicity of ∇fin dom ∇f. Definition 7.3 (Asymptotic regularity).An operator T:K→Kis called asymptotically regular if, for any x∈K, we have lim n→∞ Tn+1x−Tnx= 0.(38) In the following result we prove that any R-BSNE operator is asymptotically regular. Proposition 7.4 (R-BSNE operators are asymptotically regular).Let f:X→(−∞,+∞]be an admissible function which is totally convex on bounded subsets of int dom f. Let Kbe a nonempty, closed and convex subset of int dom f. Let Tbe a strictly (properly)R-BSNE operator from Kinto itself such that d Fix (T)6=∅(Fix (T)6=∅). Then Tis asymptotically regular. Proof. Assume that Tis strictly R-BSNE (see Remark 4.2(ii)). Let u∈d Fix (T) and x∈K. From (15) we get that DfTn+1x, u≤Df(Tnx, u)≤ · · · ≤ Df(Tx, u). 20 Thus limn→∞ Df(Tnx, u) exists and the sequence {Df(Tnx, u)}n∈Nis bounded. Now Proposition 2.5 implies that {Tnx}n∈Nis also bounded for any x∈K. Since the limit limn→∞ Df(Tnx, u) exists, we have lim n→∞ Df(Tnx, u)−DfTn+1x, u= 0. From (16) and (17) we get lim n→∞ DfTnx, Tn+1x= 0. Since {Tnx}n∈Nis bounded, we now obtain from Proposition 2.4 that lim n→∞ Tnx−Tn+1x= 0. In other words, Tis asymptotically regular. The proof in the case where Tis properly R-BSNE is identical when we take u∈Fix (T) (see Remark 4.2(i)). We now state and prove the main result of this section. Theorem 7.5 (Picard iteration).Let f:X→(−∞,+∞]be an admissible and totally convex function. Let Kbe a nonempty, closed and convex subset of int dom fand let T:K→Kbe a strictly R-QBNE operator. Then the following assertions hold. (i) If d Fix (T)is nonempty, then {Tnx}n∈Nis bounded for each x∈K. (ii) If, furthermore, Tis asymptotically regular, then, for each x∈K,{Tnx}n∈Nconverges weakly to an element of d Fix (T). Proof. (i) See [29, Proposition 3.2]. (ii) We know that {Tnx}n∈Nis bounded (by assertion (i)). Let a subsequence {Tnkx}k∈Nof {Tnx}n∈Nconverge weakly to some u. Define xn=Tnxfor any n∈N. Since Tis asymptotically regular, it follows from (38) that kxn−Txnk → 0 as n→ ∞. Therefore we have xnk* u and kxnk−Txnkk → 0 as k→ ∞, which means that u∈d Fix (T). Thus we have proved that any weak subsequential limit of {Tnx}n∈Nbelongs to d Fix (T). Since Tis strictly R-QBNE, it follows that the limit limn→∞ Df(Tnx, u) exists for any weak subsequential limit uof the sequence {Tnx}n∈N. The result now follows immediately from Proposition 7.2. Corollary 7.6 (Picard iteration for fully R-BSNE operators).Let the function f:X→(−∞,+∞] be an admissible function which is totally convex on bounded subsets of int dom f. Let Kbe a nonempty, closed and convex subset of int dom f. Let T:K→Kbe a fully R-BSNE operator with Fix (T)6=∅. Then {Tnx}n∈Nconverges weakly to an element in Fix (T)for each x∈K. Proof. This result follows immediately from Theorem 7.5 and Proposition 7.4. Remark 7.7. For the norm analog of this result, see [32, Corollary 2.4, page 286]. If Fix (T)6= d Fix (T), but d Fix (T)6=∅, then we only know that, for a strictly R-BSNE operator T, the sequence {Tnx}n∈Nconverges weakly to an element in d Fix (T) for each x∈K.♦ 21 Remark 7.8 (Common fixed point - composition case).Let f:X→Rbe an admissible function which is bounded, uniformly continuous and totally convex on bounded subsets of X. Let Kbe a nonempty, closed and convex subset of X. Let {Ti: 1 ≤i≤N}be Noperators from Kinto itself which are R-BSNE with respect to d Fix (Ti) = Fix (Ti)6=∅for each 1 ≤i≤Nand let T=TN◦TN−1◦ · · · ◦ T1. From Proposition 4.5 we obtain that if T{Fix (Ti):1≤i≤N} 6=∅, then Tis also R-BSNE with respect to d Fix (T) = Fix (T) = T{Fix (Ti) : 1 ≤i≤N}. From Theorem 7.5 we now get that {Tnx}n∈Nconverges weakly to a common fixed point of the given family of R-BSNE operators. Similarly, if we just assume that each Tiis strictly R-BSNE, 1≤i≤N, with d Fix (Ti)6=∅, then we get weak convergence of the sequence {Tnx}n∈Nto a common asymptotic fixed point. ♦ Remark 7.9 (Common fixed point - convex combination case).Let f:X→(−∞,+∞] be an admissible function which is bounded, uniformly continuous and totally convex on bounded subsets of int dom f. Let Kbe a nonempty, closed and convex subset of X. Let {Ti: 1 ≤i≤N}be Noperators from Kto int dom fwhich are R-BSNE with respect to d Fix (Ti) = Fix (Ti)6=∅for each 1 ≤i≤N. From Proposition 5.6 we know that if T{Fix (Ti):1≤i≤N} 6= ∅, then the convex combination operator TCis also R-BSNE with respect to d Fix (T) = Fix (T) = T{Fix (Ti):1≤i≤N}. Then Theorem 7.5 guarantees that {Tn Cx}n∈Nconverges weakly to a common fixed point of the given family of R-BSNE operators. Similarly, if we just assume that each Tiis strictly R-BSNE, 1 ≤i≤N, with d Fix (Ti)6=∅, then we get weak convergence of the sequence {Tn Cx}n∈Nto a common asymptotic fixed point. ♦ 8 Applications In this section we present two applications of the Picard iteration proposed in the previous section. The first application concerns common zeroes of maximal monotone mappings and the second is an algorithm for solving convex feasibility problems. 8.1 Zeroes of Maximal Monotone Operators Let A:X→2X∗be a maximal monotone set-valued mapping. The problem of finding an element x∈Xsuch that 0∗∈Ax is very important in Optimization Theory and Nonlinear Analysis. In this section we use the Picard iteration to find common zeroes of Nmaximal monotone set-valued mappings. Definition 8.1 (Conjugate ∇f-resolvent).Let A:X→2X∗be a set-valued mapping. The conjugate resolvent of Awith respect to f, or the conjugate ∇f-resolvent, is the operator CResf A: X∗→2X∗defined by CResf A:= (I+A◦ ∇f∗)−1.(39) In the following proposition we collect several properties of conjugate resolvents (cf. [28]). Proposition 8.2 (Properties of conjugate ∇f-resolvents).Let f:X→(−∞,+∞]be an admissible function and let A:X→2X∗be a set-valued mapping such that int dom f∩dom A6=∅. The following statements hold. (i) dom CResf A⊂int dom f∗. 22 (ii) ran CResf A⊂int dom f∗. (iii) ∇f∗Fix CResf A= int dom f∩A−1(0∗). (iv) Suppose, in addition, that Ais a monotone mapping. Then the following assertions also hold. (a) If f|int dom fis strictly convex, then the operator CResf Ais single-valued on its domain and properly R-BSNE (if Fix CResf A6=∅). (b) If f:X→Ris such that ran ∇f⊂ran (∇f+A), then dom CResf A= int dom f∗. Remark 8.3. In [28, Proposition 5.5] it was actually proved that, under the assumptions in (iv)(a), the operator CResf Ais right Bregman firmly nonexpansive, which is more restrictive than being properly R-BSNE. We know that, if fis Legendre, and bounded and uniformly continuous on bounded subsets of X, then for every right Bregman firmly nonexpansive operator T,d Fix (T) = Fix (T) (see [35]). So under these assumptions on f, the operator CResf Ais R-BSNE. The following proposition [28, Theorem 5.7] is essential for our convergence result. Proposition 8.4 (Surjectivity result).Let f:X→Rbe a strictly convex, cofinite and admissible function, and let A:X→2X∗be a set-valued monotone mapping. Then Ais maximal monotone if and only if dom CResf A=X∗. Now we present a variant of the Picard iterative method. Proposition 8.5 (Common zeroes).Let f:X→Rbe a cofinite Legendre function such that f∗ is uniformly continuous and totally convex on bounded subsets of X∗. Assume that ∇f∗is weakly sequentially continuous. Let Ai:X→2X∗,i= 1,2, . . . , N, be Nmaximal monotone set-valued mappings such that Z:= TN i=1 A−1 i(0∗)6=∅. For each ξ1∈X∗, consider the sequence {ξn}n∈N generated iteratively by ξn+1 = CResf AN◦CResf AN−1◦ · · · ◦ CResf A1(ξ1), n ≥1.(40) Then the sequence {∇f∗(ξn)}n∈Nconverges weakly to an element in Z. Proof. From Proposition 8.4 we know that each Ti= CResf Ai,i= 1,2, . . . , N, is an operator from X∗into itself. Since N \ i=1 Fix CResf Ai= N \ i=1 ∇fA−1 i(0∗)=∇f(Z)6=∅,(41) Proposition 8.2(iii), (iv)(a) and Remark 8.3 guarantee that each Ti,i= 1,2, . . . , N, is R-BSNE with respect to Fix (Ti) = d Fix (Ti). Now the result follows immediately from Remark 7.8 applied to X∗. Remark 8.6. Analogously to the previous case, using convex combinations of the conjugate resolvents instead of their composition, we can consider a different Picard iterative method defined by means of the scheme ξn+1 = N X i=1 wiCResf Ai(ξn), n ≥1. The sequence {∇f∗(ξn)}n∈Ngenerated by this scheme converges weakly to a common zero of the given maximal monotone set-valued mappings. ♦ 23 8.2 Convex feasibility problems Let Ki,i= 1, . . . , N, be Nnonempty, closed and convex subsets of X. The convex feasibility problem (CFP) is to find an element in the assumed nonempty intersection TN i=1 Ki. Given a set K⊂int dom f, the right Bregman projection [28] onto Kis the operator −−→ projf K: int dom f→Kdefined by −−→ projf K(x) := argminy∈K{Df(x, y)}={z∈K:Df(x, z)≤Df(x, y)∀y∈K}(42) for each x∈int dom f. It is clear that Fix −−→ projf Ki=Kifor any i= 1, . . . , N. We proved in [28] that the right Bregman projection −−→ projf Kiis strictly R-BSNE. If, in addition, f:X→Ris Legendre and uniformly continuous on bounded subsets of X, and ∇fis weakly sequentially continuous, then Fix −−→ projf Ki=d Fix −−→ projf Ki(cf. [28]). Therefore, if we take Ti=−−→ projf Kiin Remark 7.8, then we get an algorithm for solving convex feasibility problems. More precisely, we arrive at the following result (for relevant related results see [9, Theorem 4.1], [6, Theorem 4.1 and Section 5] and [8, Section 5]). Proposition 8.7 (Picard’s iteration for solving the CFP).Let f:X→Rbe a cofinite Legendre function which is bounded, uniformly continuous and totally convex on bounded subsets of X. Assume that ∇fis weakly sequentially continuous. Let Ki,i= 1, . . . , N, be Nnonempty, closed and convex subsets of Xsuch that K:= TN i=1 Ki6=∅. Then, for each x1∈X, the sequence {xn}n∈N generated iteratively by xn+1 =−−→ projf KN◦−−→ projf KN−1◦ · · · ◦ −−→ projf K1(xn), n ≥1, converges weakly to an element in TN i=1 Ki. Remark 8.8. Analogously, using convex combinations of the right Bregman projections instead of their composition, we can consider a different Picard iterative method defined by means of the scheme xn+1 = N X i=1 wi −−→ projf Ki(xn), n ≥1. The sequence generated by this scheme also converges weakly to a solution of the convex feasibility problem. ♦ Acknowledgements. The first author was partially supported by MEC (Grant MTM2009- 10696-C02-01) and by the Junta de Andaluc´ıa (Grant FQM-127); the second and third authors were partially supported by the Israel Science Foundation (Grant 647/07), the Graduate School of the Technion, the Fund for the Promotion of Research at the Technion and by the Technion VPR Fund. Part of the research pertaining to this paper was carried out when the authors were visiting each other’s institutions. They thank their colleagues for their kind hospitality. 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