# Proximal Point Algorithms for Finding a Zero of a Finite Sum of Monotone Mappings in Banach Spaces

Authors: H. Zegeye; N. Shahzad
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We introduce an iterative process which converges strongly to a zero of a finite sum of monotone mappings under certain conditions. Applications to a convex minimization problem are included. Our theorems improve and unify most of the results that have been proved in this direction for this important class of nonlinear mappings.


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## Body

## 1. Introduction

LetC be a nonempty subset of a real Banach space E with dual E*. A mapping A:C→E* is said to be monotoneif for each x,y∈C, the following inequality holds:
(1)〈x-y,Ax-Ay〉≥0.
A monotone mapping A⊂E×E* is said to be maximal monotoneif its graph is not properly contained in the graph of any other monotone mapping. We know that if A is maximal monotone mapping, then A-1(0) is closed and convex (see [1] for more details).Monotone mappings were introduced by Zarantonello [2], Minty [3], and Kačurovskiĭ [4]. The notion of monotone in the context of variational methods for nonlinear operator equations was also used by Vaĭnberg and Kačurovskiĭ [5]. The central problem is to iteratively find a zero of a finite sum of monotone mappings A1,A2,…,AN in a Banach space E, namely, a solution to the inclusion problem
(2)0∈(A1+A2+⋯+AN)x.
It is known that many physically significant problems can be formulated as problems of the type (2). For instance, a stationary solution to the initial value problem of the evolution equation
(3)∂x∂t+Fx∋0,x(0)=x0
can be formulated as (2) when the governing maximal monotone F is of the form F:=A1+A2+⋯+AN (see, e.g., [6]). In addition, optimization problems often need [7] to solve a minimization problem of the form
(4)minx∈E{f1(x)+f2(x)+⋯+fN(x)},
where fi, i=1,2,…,N are proper lower semicontinuous convex functions from E to the extended real line R-:=(-∞,∞]. If in (2), we assume that Ai:=∂fi, for i=1,2,…,N, where ∂fi is the subdifferential operator of fi in the sense of convex analysis, then (4) is equivalent to (2). Consequently, considerable research efforts have been devoted to methods of finding approximate solutions (when they exist) of equations of the form (2) for a sum of a finite number of monotone mappings (see, e.g., [6, 8–12]).A well-known method for solving the equation0∈Ax in a Hilbert space H is the proximal point algorithm: x1=x∈H and
(5)xn+1=Jrnxn,(n=1,2,…),
where rn⊂(0,∞) and Jr=(I+rA)-1 for all r>0. This algorithm was first introduced by Martinet [10]. In 1976, Rockafellar [11] proved that if liminfn→∞rn>0 and A-1(0)≠∅, then the sequence {xn} defined by (5) converges weakly to an element of A-1(0). Later, many researchers have studied the convergence of the sequence defined by (5) in Hilbert spaces; see, for instance, [8, 12–18] and the references therein.In 2000, Kamimura and Takahashi [9] proved that for a maximal monotone mapping A in a Hilbert spaces H and Jr=(I+rA)-1 for all r>0, the sequence {xn} defined by
(6)xn+1=αnx+(1-αn)Jrnxn,n≥0,
where {αn}⊂[0,1] and {rn}⊂(0,∞) satisfy certain conditions, called Halpern type, converges strongly to a point in A-1(0).In a reflexive Banach spaceE and for a maximal monotone mapping A:E→2E*, Reich and Sabach [19] proved that the sequence {xn} defined by
(7)0=ξn+λn-1(∇f(yn)-∇f(xn)),ξn∈Ayn,Hn={z∈E:〈ξn,z-yn〉≤0},Wn={z∈E:〈∇f(x0)-∇f(xn),z-xn〉≤0},xn+1=ProjHn∩Wnf(x0),n=1,2,…,
where λn>0 and projCf is the Bergman projection of E on to a closed and convex subset C⊂E induced by a well-chosen convex function f, converges strongly to a point in A-1(0).Furthermore, many authors (see, e.g., [12, 20–25]) have studied strong convergence of an iterative process of Halpern type or proximal type to a common zero of a finite family of maximal monotone mappings in Hilbert spaces (or in Banach spaces).Regarding iterative solution of a zero of sum of two maximal monotone mappings, Lions and Mercier [6] introduced the nonlinear Douglas-Rachford splitting iterative algorithm which generates a sequence {vn} by the recursion
(8)vn+1=JλA(2JλB-I)vn+(I-JλB)vn,
where JλT denotes the resolvent of a monotone mapping T; that is, JλT:=(I+λT)-1. They proved that the nonlinear Douglas-Rachford algorithm (8) converges weakly to a point v, a solution of the inclusion,
(9)0∈(A+B)x,
for A+B maximal monotone mappings in Hilbert spaces.A natural question arises whether we can obtain an iterative scheme which converges strongly to a zero of sum of a finite number of monotone mappings in Banach spaces or not?Motivated and inspired by the work mentioned above, it is our purpose in this paper to introduce an iterative scheme (see (21)) which converges strongly to a zero of a finite sum of monotone mappings under certain conditions. Applications to a convex minimization problem are included. Our theorems improve the results of Lions and Mercier [6] and most of the results that have been proved in this direction.


## 2. Preliminaries

LetE be a Banach space and let S(E)={x∈E:∥x∥=1}. Then, a Banach space E is said to be smooth provided that the limit
(10)limt→0∥x+ty∥-∥x∥t
exists for each x,y∈S(E). The norm of E is said to be uniformly smooth if the limit (10) is attained uniformly for (x,y) in S(E)×S(E) (see [1]).Themodulus of convexity of E is the function δE:(0,2]→[0,1] defined by
(11)δE(ϵ):=inf{1-∥x+y2∥:∥x∥=∥y∥=1;ϵ=∥x-y∥}.E is called uniformly convex if and only if δE(ϵ)>0, for every ϵ∈(0,2] (see [26]).Lemma 1 (see [27]).
LetE be a smooth, strictly convex, and reflexive Banach space. Let C be a nonempty closed convex subset of E, and let A:C⊂E→E* be a monotone mapping. Then, A is maximal if and only if R(J+rA)=E*, for all r>0, where J is the normalized duality mapping from E into 2E* defined, for each x∈E, by
(12)Jx:={f*∈E*:〈x,f*〉=∥x∥2=∥f*∥2},
where 〈·,·〉 denotes the generalized duality pairing between members of E and E*. We recall that E is smooth if and only if J is single valued (see [1]). If E=H, a Hilbert space, then the duality mapping becomes the identity map on H.Lemma 2 (see [27]).
LetE be a reflexive with E* as its dual. Let A:D(A)⊆E→E*, and let B:D(B)⊆E→E* be maximal monotone mappings. Suppose that D(A)∩intD(B)≠∅. Then, A+B is a maximal monotone mapping.Lemma 3 (see [28]).
LetE be a reflexive with E* as its dual. Let A:D(A)⊆E→E* be maximal monotone mapping, and let B:D(B)⊆E→E* be monotone mappings such that D(B)=E, B is hemicontinuous (i.e., continuous from the segments in E to the weak star topology in E*) and carries bounded sets into bounded sets. Then, A+B is maximal monotone mapping.LetE be a smooth Banach space with dual E*. Let the Lyapunov function ϕ:E×E→ℝ, introduced by Alber [29], be defined by
(13)ϕ(y,x)=∥y∥2-2〈y,Jx〉+∥x∥2,forx,y∈E,
where J is the normalized duality mapping from E into 2E*. If E=H, a Hilbert space, then (13) reduces to ϕ(x,y)=∥x-y∥2, for x,y∈H.LetE be a reflexive, strictly convex, and smooth Banach space, and let C be a nonempty closed and convex subset of E. The generalized projection mapping, introduced by Alber [29], is a mapping ΠC:E→C that assigns an arbitrary point x∈E to the minimizer, x-, of ϕ(·,x) over C; that is, ΠCx=x-, where x- is the solution to the minimization problem
(14)ϕ(x-,x)=min{ϕ(y,x),y∈C}.
We know the following lemmas.Lemma 4 (see [23]).
LetE be a real smooth and uniformly convex Banach space, and let {xn} and {yn} be two sequences of E. If either {xn} or {yn} is bounded and ϕ(xn,yn)→0, as n→∞, then xn-yn→0, as n→∞.Lemma 5 (see [29]).
LetC be a convex subset of a real smooth Banach space E, and let x∈E. Then x0=ΠCx if and only if
(15)〈z-x0,Jx-Jx0〉≤0,∀z∈C.We make use of the functionV:E×E*→ℝ defined by
(16)V(x,x*)=∥x∥2-2〈x,x*〉+∥x∥2,∀x∈E,x*∈E,
studied by Alber [29]. That is, V(x,y)=ϕ(x,J-1x*), for all x∈E and x*∈E*.In the sequel, we will make use of the following lemmas.Lemma 6 (see [29]).
LetE be a reflexive strictly convex and smooth Banach space with E* as its dual. Then,
(17)V(x,x*)+2〈J-1x*-x,y*〉≤V(x,x*+y*),
for all x∈E and x*,y*∈E*.Lemma 7 (see [30]).
LetE be a smooth and strictly convex Banach space, C be a nonempty closed convex subset of E, and A⊂E×E* be a maximal monotone mapping. Let Qr be the resolvent of A defined by Qr=(J+rA)-1J, for r>0 and {rn} a sequence of (0,∞) such that limn→∞rn=∞. If  {xn} is a bounded sequence of C such that Qrnxn⇀z, then z∈A-1(0).Lemma 8 (see [31]).
LetE be a smooth and strictly convex Banach space, C be a nonempty closed convex subset of E, and A⊂E×E* be a maximal monotone mapping, and A-1(0) is nonempty. Let Qr be the resolvent of A defined by Qr=(J+rA)-1J, for r>0. Then, for each r>0(18)ϕ(p,Qrx)+ϕ(Qrx,x)≤ϕ(p,x),
for all p∈A-1(0) and x∈C.Lemma 9 (see [32]).
Let{an} be a sequence of nonnegative real numbers satisfying the following relation:
(19)an+1≤(1-αn)an+αnδn,n≥n0,
where {αn}⊂(0,1) and {δn}⊂R satisfying the following conditions: limn→∞αn=0, ∑n=1∞‍αn=∞, and limsupn→∞δn≤0. Then, limn→∞an=0.Lemma 10 (see [33]).
Let{an} be the sequences of real numbers such that there exists a subsequence {ni} of {n} such that ani<ani+1, for all i∈N. Then, there exists a nondecreasing sequence {mk}⊂N such that mk→∞, and the following properties are satisfied by all (sufficiently large) numbers k∈N:
(20)amk≤amk+1,ak≤amk+1.
In fact, mk=max{j≤k:aj<aj+1}.


## 3. Main Result

Theorem 11.
LetC and D be nonempty, closed and convex subsets of a smooth and uniformly convex real Banach space E with E* as its dual. Assume that C∩int(D)≠∅. Let A1:C→E* and A2,A3,…,AN:D→E* be maximal monotone mappings. Assume that F:=(A1+A2+⋯+AN)-1(0) is nonempty. Let {xn} be a sequence generated by
(21)x0=w∈C,chosenarbitrarily,xn+1=J-1(αnJw+(1-αn)J(J+rnA)-1Jxn),∀n≥0,
where A=A1+A2+⋯+AN, αn∈(0,1) and {rn} a sequence of (0,∞) satisfying: limn→∞αn=0, ∑n=1∞‍αn=∞, and limn→∞rn=∞. Then, {xn} converges strongly to p=ΠF(w).Proof.
Observe that by Lemma2, we have that A2+A3+⋯+AN is maximal monotone. In addition, since C∩int(D)≠∅, the same lemma implies that A=A1+A2+⋯+AN is maximal monotone. Now, let p=ΠF(w), and let wn:=Qrnxn:=(J+rnA)-1Jxn. Then, we have that xn+1=J-1(αnJw+(1-αn)Jwn), and since p∈A-1(0), from Lemma 8, we get that
(22)ϕ(p,wn)=ϕ(p,Qrnxn)≤ϕ(p,xn).
Now from (21), property of ϕ, and (22) we get that
(23)ϕ(p,xn+1)=ϕ(p,J-1(αnJw+(1-αn)Jwn))=∥p∥2-2〈p,αnJw+(1-αn)Jwn〉+∥αnJw+(1-αn)Jwn∥2≤∥p∥2-2αn〈p,Jw〉-2(1-αn)〈p,Jwn〉+αn∥Jw∥2+(1-αn)∥Jwn∥2=αnϕ(p,w)+(1-αn)ϕ(p,wn)≤αnϕ(p,w)+(1-αn)ϕ(p,xn).
Thus, by induction,
(24)ϕ(p,xn+1)≤max{ϕ(p,w),ϕ(p,x0)},∀n≥0,
which implies that {xn} is bounded. In addition, using Lemma 6 and property of ϕ, we obtain that
(25)ϕ(p,xn+1)=V(p,Jxn+1)≤V(p,Jxn+1-αn(Jw-Jp))-2〈xn+1-p,-αn(Jw-Jp)〉=ϕ(p,J-1(αnJp+(1-αn)Jwn))+2αn〈xn+1-p,Jw-Jp〉≤αnϕ(p,p)+(1-αn)ϕ(p,wn)+2αn〈xn+1-p,Jw-Jp〉=(1-αn)ϕ(p,wn)+2αn〈xn+1-p,Jw-Jp〉≤(1-αn)ϕ(p,xn)+2αn〈xn+1-p,Jw-Jp〉.
Furthermore, using property of ϕ and the fact that αn→0, as n→∞, imply that
(26)ϕ(wn,xn+1)=ϕ(wn,J-1(αnJw+(1-αn)Jwn))≤αnϕ(wn,w)+(1-αn)ϕ(wn,wn)≤αnϕ(wn,w)+(1-αn)ϕ(wn,wn)⟶0,≤ϕ(wn,w)(1-αn)(wn,wn)asn⟶∞,
which implies from Lemma 4 that
(27)wn-xn+1⟶0,asn⟶∞.
Now, following the method of proof of Lemma 3.2 of Maing’e [33], we consider two cases.
Case
1. Suppose that there exists n0∈ℕ such that {ϕ(p,xn)} is nonincreasing for all n≥n0. In this situation, {ϕ(p,xn)} is convergent. Since {xn+1} is bounded and E is reflexive, we choose a subsequence {xni+1} of {xn+1} such that xni+1⇀z and limsupn→∞〈xn+1-p,Jw-Jp〉=limi→∞〈xni+1-p,Jw-Jp〉. Then, from (27), we get that
(28)wni⇀z,asi⟶∞.
Thus, by Lemma 7, we get that z∈A-1(0), and hence z∈F=(A1+A2+⋯+AN)-1(0). Therefore, by Lemma 5, we immediately obtain that limsupn→∞〈xn+1-p,Jw-Jp〉=limi→∞〈xni+1-p,Jw-Jp〉=〈z-p,Jw-Jp〉≤0. It follows from Lemma 9 and (25)  that ϕ(p,xn)→0, as n→∞. Consequently, xn→p.
Case
2. Suppose that there exists a subsequence {ni} of {n} such that
(29)ϕ(p,xni)<ϕ(p,xni+1),
for all i∈ℕ. Then, by Lemma 10, there exist a nondecreasing sequence {mk}⊂ℕ such that mk→∞, satisfying
(30)ϕ(p,xmk)≤ϕ(p,xmk+1),00ϕ(p,xk)≤ϕ(p,xmk+1),ϕ(p,xk)≤ϕ(p+)∀k∈ℕ.
Thus, following the method of proof of Case 1, we obtain that
(31)limsupk→∞〈xmk+1-p,Jw-Jp〉≤0.
Then, from (25),  we have that
(32)ϕ(p,xmk+1)≤(1-αmk)ϕ(p,xmk)+2αmk〈xmk+1-p,Jw-Jp〉.
Now, inequalities (30) and (32) imply that
(33)αmkϕ(p,xmk)≤ϕ(p,xmk)-ϕ(p,xmk+1)+2αmk〈xmk+1-p,Jw-Jp〉≤2αmk〈xmk+1-p,Jw-Jp〉.
In particular, since αmk>0, we get
(34)ϕ(p,xmk)≤2〈xmk+1-p,Jw-Jp〉.
Then, from (31), we obtain ϕ(p,xmk)→0, as k→∞. This together with (32) gives ϕ(p,xmk+1)→0, as k→∞. But ϕ(p,xk)≤ϕ(p,xmk+1), for all k∈N; thus, we obtain that xk→p. Therefore, from the above two cases, we can conclude that {xn} converges strongly to p, and the proof is complete.Theorem 12.
LetC be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space E with E* as its dual. Let A1:C→E* be maximal monotone mapping, and let A2,A3,…,AN:E→E* be bounded and hemicontinuous monotone mappings. Assume that F:=(A1+A2+⋯+AN)-1(0) is nonempty. Let {xn} be a sequence generated by
(35)x0=w∈C,chosenarbitrarily,xn+1=J-1(αnJw+(1-αn)J(J+rnA)-1Jxn),∀n≥0,
where A=A1+A2+⋯+AN, αn∈(0,1) and {rn} is a sequence of (0,∞) satisfying: limn→∞αn=0, ∑n=1∞‍αn=∞, and limn→∞rn=∞. Then, {xn} converges strongly to p=ΠF(w).Proof.
By Lemma3, we have that A=A1+A2+⋯+AN is maximal monotone, and hence following the method of proof of Theorem 11, we obtain the required assertion.If in Theorem12, we assume that Ai, for i=2,…,N, are continuous monotone mappings, then Ai′s are hemicontinuous, and hence we get the following corollary.Corollary 13.
LetC be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space E with E* as its dual. Let A1:C→E* be a maximal monotone mapping, and let A2,A3,…,AN:E→E* be bounded and continuous monotone mappings. Assume that F:=(A1+A2+⋯+AN)-1(0) is nonempty. Let {xn} be a sequence generated by
(36)x0=w∈C,chosenarbitrarily,xn+1=J-1(αnJw+(1-αn)J(J+rnA)-1Jxn),∀n≥0,
where A=A1+A2+⋯+AN, αn∈(0,1) and {rn} a sequence of (0,∞) satisfying: limn→∞αn=0, ∑n=1∞‍αn=∞, and limn→∞rn=∞. Then, {xn} converges strongly to p=ΠF(w).If in Theorem12, we assume that Ai, for i=2,…,N, are uniformly continuous monotone mapping, then Ai′s are bounded and hemicontinuous, and hence we get the following corollary.Corollary 14.
LetC be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space E with E* as its dual. Let A1:C→E* be a maximal monotone mapping, and let A2,A3,…,AN:E→E* be monotone uniformly continuous mappings. Assume that F:=(A1+A2+⋯+AN)-1(0) is nonempty. Let {xn} be a sequence generated by
(37)x0=w∈C,chosenarbitrarily,xn+1=J-1(αnJw+(1-αn)J(J+rnA)-1Jxn),∀n≥0,
where A=A1+A2+⋯+AN, αn∈(0,1) and {rn} a sequence of (0,∞) satisfying: limn→∞αn=0, ∑n=1∞‍αn=∞, and limn→∞rn=∞. Then, {xn} converges strongly to p=ΠF(w).If in Theorem12 we assume that Ai≡0, for i=2,…,N, then we get the following corollary.Corollary 15.
LetC be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space E. Let A:C→E* be a maximal monotone mapping. Assume that F:=A-1(0) is nonempty. Let {xn} be a sequence generated by
(38)x0=w∈C,chosenarbitrarily,xn+1=J-1(αnJw+(1-αn)J(J+rnA)-1Jxn),∀n≥0,
where αn∈(0,1) and {rn} a sequence of (0,∞) satisfying: limn→∞αn=0, ∑n=1∞‍αn=∞, and limn→∞rn=∞. Then, {xn} converges strongly to p=ΠF(w).IfE=H, a real Hilbert space, then E is smooth and uniformly convex real Banach space. In this case, J=I, identity map on H and ΠC=PC, projection mapping from H onto C. Thus, the following corollaries follow from Theorems 11 and 12.Corollary 16.
LetC and D be nonempty, closed, and convex subsets of a real Hilbert space H. Assume that C∩int(D)≠∅. Let A1:C→H, and let A2,A3,…,AN:D→H be maximal monotone mappings. Assume that F:=(A1+A2+⋯+AN)-1(0) is nonempty. Let {xn} be a sequence generated by
(39)x0=w∈C,chosenarbitrarily,xn+1=αnw+(1-αn)(I+rnA)-1xn,∀n≥0,
where A=A1+A2+⋯+AN, αn∈(0,1) and {rn} a sequence of (0,∞) satisfying: limn→∞αn=0, ∑n=1∞‍αn=∞, and limn→∞rn=∞. Then, {xn} converges strongly to p=PF(w).Corollary 17.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let A1:C→H be a maximal monotone mapping, and let A2,A3,…,AN:H→H be bounded, hemicontinuous, and monotone mappings. Assume that F:=(A1+A2+⋯+AN)-1(0) is nonempty. Let {xn} be a sequence generated by
(40)x0=w∈C,chosenarbitrarily,xn+1=αnw+(1-αn)(I+rnA)-1xn,∀n≥0,
where A=A1+A2+⋯+AN, αn∈(0,1) and {rn} a sequence of (0,∞) satisfying: limn→∞αn=0, ∑n=1∞‍αn=∞, and limn→∞rn=∞. Then, {xn} converges strongly to p=PF(w).Corollary 18.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let A1:C→H be a maximal monotone mapping, and let A2,A3,…,AN:H→H be uniformly continuous monotone mappings. Assume that F:=(A1+A2+⋯+AN)-1(0) is nonempty. Let {xn} be a sequence generated by
(41)x0=w∈C,chosenarbitrarily,xn+1=αnw+(1-αn)(I+rnA)-1xn,∀n≥0,
where A=A1+A2+⋯+AN, αn∈(0,1) and {rn} a sequence of (0,∞) satisfying: limn→∞αn=0, ∑n=1∞‍αn=∞, and limn→∞rn=∞. Then, {xn} converges strongly to p=PF(w).


## 4. Application

In this section, we study the problem of finding a minimizer of a continuously Fréchet differentiable convex functional in Banach spaces. The followings are deduced from Theorems11 and 12.Theorem 19.
LetC and D be a nonempty, closed, and convex subsets of a smooth and uniformly convex real Banach space E. Let C∩int(D)≠∅. Let f be a continuously Fréchet differentiable convex functional, and let ∇f be maximal monotone on C. Let g be a continuously Fréchet differentiable convex functional, and let ∇g be maximal monotone on D. Assume that F:=(∇f+∇g)-1(0)={z∈E:f(z)+g(z)=infy∈E{f(y)+g(y)}}≠∅. Let {xn} be a sequence generated by
(42)x0∈Cchosenarbitrarily,xn+1=J-1(αnJw+(1-αn)J(J+rn(∇f+∇g))-1Jxn),
where αn∈(0,1) and {rn} a sequence of (0,∞) satisfying: limn→∞αn=0, ∑n=1∞‍αn=∞, and limn→∞rn=∞. Then, {xn} converges strongly to an element of F.Theorem 20.
LetC be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space. Let f be a continuously Fréchet differentiable convex functional, and let ∇f be maximal monotone on C. Let g be a continuously Fréchet differentiable convex functional, and let ∇g be bounded, hemicontinuous, and monotone on E with F:=(∇f+∇g)-1(0)={z∈E:f(z)+g(z)=infy∈E{f(y)+g(y)}}≠∅. Let {xn} be a sequence generated by
(43)x0∈Cchosenarbitrarily,xn+1=J-1(αnJw+(1-αn)J(J+rn(∇f+∇g))-1Jxn),
where αn∈(0,1) and {rn} a sequence of (0,∞) satisfying: limn→∞αn=0, ∑n=1∞‍αn=∞, and limn→∞rn=∞. Then, {xn} converges strongly to an element of F.Remark 21.
Our results provide strong convergence theorems for finding a zero of a finite sum of monotone mappings in Banach spaces and hence extend the results of Rockafellar [11], Kamimura and Takahashi [9], and Lions and Mercier [6].



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