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

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.

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

H. Zegeye 1 and N. Shahzad 2

Recommended by Yisheng Song

1, Departement of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana
2, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 15 February 2013; Revised 30 March 2013; Accepted 31 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

Let C be a nonempty subset of a real Banach space E with dual ^{E *} . A mapping A : C [arrow right] ^{E *} is said to be monotone if for each x , y ∈ C , the following inequality holds: [figure omitted; refer to PDF] A monotone mapping A ⊂ E × ^{E *} is said to be maximal monotone if 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 Kacurovski[ibreve] [4]. The notion of monotone in the context of variational methods for nonlinear operator equations was also used by Va[ibreve]nberg and Kacurovski[ibreve] [5]. The central problem is to iteratively find a zero of a finite sum of monotone mappings _{A 1} , _{A 2} , ... , _{A N} in a Banach space E , namely, a solution to the inclusion problem [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] can be formulated as (2) when the governing maximal monotone F is of the form F : = _{A 1} + _{A 2} + ... + _{A N} (see, e.g., [6]). In addition, optimization problems often need [7] to solve a minimization problem of the form [figure omitted; refer to PDF] where _{f i} , i = 1,2 , ... , N are proper lower semicontinuous convex functions from E to the extended real line R - : = ( - ∞ , ∞ ] . If in (2), we assume that _{A i} : = ∂ _{f i} , for i = 1,2 , ... , N , where ∂ _{f i} is the subdifferential operator of _{f i} 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 equation 0 ∈ A x in a Hilbert space H is the proximal point algorithm : _{x 1} = x ∈ H and [figure omitted; refer to PDF] where _{r n} ⊂ ( 0 , ∞ ) and _{J r} = ( I + r A ^{) - 1} for all r > 0 . This algorithm was first introduced by Martinet [10]. In 1976, Rockafellar [11] proved that if _{liminf n [arrow right] ∞ r n} > 0 and ^{A - 1} ( 0 ) ...0; ∅ , then the sequence { _{x n} } 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 _{J r} = ( I + r A ^{) - 1} for all r > 0 , the sequence { _{x n} } defined by [figure omitted; refer to PDF] where { _{α n} } ⊂ [ 0,1 ] and { _{r n} } ⊂ ( 0 , ∞ ) satisfy certain conditions, called Halpern type, converges strongly to a point in ^{A - 1} ( 0 ) .

In a reflexive Banach space E and for a maximal monotone mapping A : E [arrow right] ^{2 E *} , Reich and Sabach [19] proved that the sequence { _{x n} } defined by [figure omitted; refer to PDF] where _{λ n} > 0 and pro ^{j C f} 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 { _{v n} } by the recursion [figure omitted; refer to PDF] where ^{J λ T} denotes the resolvent of a monotone mapping T ; that is, ^{J λ T} : = ( I + λ T ^{) - 1} . They proved that the nonlinear Douglas-Rachford algorithm (8) converges weakly to a point v , a solution of the inclusion, [figure omitted; refer to PDF] 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

Let E 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 [figure omitted; refer to PDF] 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]).

The modulus of convexity of E is the function _{δ E} : ( 0,2 ] [arrow right] [ 0,1 ] defined by [figure omitted; refer to PDF] E is called uniformly convex if and only if _{δ E} ( ... ) > 0 , for every ... ∈ ( 0,2 ] (see [26]).

Lemma 1 (see [27]).

Let E be a smooth, strictly convex, and reflexive Banach space. Let C be a nonempty closed convex subset of E , and let A : C ⊂ E [arrow right] ^{E *} be a monotone mapping. Then, A is maximal if and only if R ( J + r A ) = ^{E *} , for all r > 0 , where J is the normalized duality mapping from E into ^{2 E *} defined, for each x ∈ E , by [figure omitted; refer to PDF] where Y9; · , · YA; 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]).

Let E be a reflexive with ^{E *} as its dual. Let A : D ( A ) ⊆ E [arrow right] ^{E *} , and let B : D ( B ) ⊆ E [arrow right] ^{E *} be maximal monotone mappings. Suppose that D ( A ) ∩ int D ( B ) ...0; ∅ . Then, A + B is a maximal monotone mapping.

Lemma 3 (see [28]).

Let E be a reflexive with ^{E *} as its dual. Let A : D ( A ) ⊆ E [arrow right] ^{E *} be maximal monotone mapping, and let B : D ( B ) ⊆ E [arrow right] ^{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.

Let E be a smooth Banach space with dual ^{E *} . Let the Lyapunov function [varphi] : E × E [arrow right] ... , introduced by Alber [29], be defined by [figure omitted; refer to PDF] where J is the normalized duality mapping from E into ^{2 E *} . If E = H , a Hilbert space, then (13) reduces to [varphi] ( x , y ) = ^{|| x - y || 2} , for x , y ∈ H .

Let E 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 _{Π C} : E [arrow right] C that assigns an arbitrary point x ∈ E to the minimizer, x - , of [varphi] ( · , x ) over C ; that is, _{Π C} x = x - , where x - is the solution to the minimization problem [figure omitted; refer to PDF] We know the following lemmas.

Lemma 4 (see [23]).

Let E be a real smooth and uniformly convex Banach space, and let { _{x n} } and { _{y n} } be two sequences of E . If either { _{x n} } or { _{y n} } is bounded and [varphi] ( _{x n} , _{y n} ) [arrow right] 0 , as n [arrow right] ∞ , then _{x n} - _{y n} [arrow right] 0 , as n [arrow right] ∞ .

Lemma 5 (see [29]).

Let C be a convex subset of a real smooth Banach space E , and let x ∈ E . Then _{x 0} = _{Π C} x if and only if [figure omitted; refer to PDF]

We make use of the function V : E × ^{E *} [arrow right] ... defined by [figure omitted; refer to PDF] studied by Alber [29]. That is, V ( x , y ) = [varphi] ( x , ^{J - 1 x *} ) , for all x ∈ E and ^{x *} ∈ ^{E *} .

In the sequel, we will make use of the following lemmas.

Lemma 6 (see [29]).

Let E be a reflexive strictly convex and smooth Banach space with ^{E *} as its dual. Then, [figure omitted; refer to PDF] for all x ∈ E and ^{x *} , ^{y *} ∈ ^{E *} .

Lemma 7 (see [30]).

Let E 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 _{Q r} be the resolvent of A defined by _{Q r} = ( J + r A ^{) - 1} J , for r > 0 and { _{r n} } a sequence of ( 0 , ∞ ) such that _{lim n [arrow right] ∞ r n} = ∞ . If { _{x n} } is a bounded sequence of C such that _{Q r n x n} ... z , then z ∈ ^{A - 1} ( 0 ) .

Lemma 8 (see [31]).

Let E 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 _{Q r} be the resolvent of A defined by _{Q r} = ( J + r A ^{) - 1} J , for r > 0 . Then, for each r > 0 [figure omitted; refer to PDF] for all p ∈ ^{A - 1} ( 0 ) and x ∈ C .

Lemma 9 (see [32]).

Let { _{a n} } be a sequence of nonnegative real numbers satisfying the following relation: [figure omitted; refer to PDF] where { _{α n} } ⊂ ( 0,1 ) and { _{δ n} } ⊂ R satisfying the following conditions: _{lim n [arrow right] ∞ α n} = 0 , ^{∑ n = 1 ∞} ... _{α n} = ∞ , and _{limsup n [arrow right] ∞ δ n} ...4; 0 . Then, _{lim n [arrow right] ∞ a n} = 0 .

Lemma 10 (see [33]).

Let { _{a n} } be the sequences of real numbers such that there exists a subsequence { _{n i} } of { n } such that _{a n i} < _{a n i + 1} , for all i ∈ N . Then, there exists a nondecreasing sequence { _{m k} } ⊂ N such that _{m k} [arrow right] ∞ , and the following properties are satisfied by all (sufficiently large) numbers k ∈ N : [figure omitted; refer to PDF] In fact, _{m k} = max { j ...4; k : _{a j} < _{a j + 1} } .

3. Main Result

Theorem 11.

Let C 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 ) ...0; ∅ . Let _{A 1} : C [arrow right] ^{E *} and _{A 2} , _{A 3} , ... , _{A N} : D [arrow right] ^{E *} be maximal monotone mappings. Assume that F : = ( _{A 1} + _{A 2} + ... + _{A N} ^{) - 1} ( 0 ) is nonempty. Let { _{x n} } be a sequence generated by [figure omitted; refer to PDF] where A = _{A 1} + _{A 2} + ... + _{A N} , _{α n} ∈ ( 0,1 ) and { _{r n} } a sequence of ( 0 , ∞ ) satisfying: _{lim n [arrow right] ∞ α n} = 0 , ^{∑ n = 1 ∞} ... _{α n} = ∞ , and _{lim n [arrow right] ∞ r n} = ∞ . Then, { _{x n} } converges strongly to p = _{Π F} ( w ) .

Proof.

Observe that by Lemma 2, we have that _{A 2} + _{A 3} + ... + _{A N} is maximal monotone. In addition, since C ∩ int ( D ) ...0; ∅ , the same lemma implies that A = _{A 1} + _{A 2} + ... + _{A N} is maximal monotone. Now, let p = _{Π F} ( w ) , and let _{w n} : = _{Q r n x n} : = ( J + _{r n} A ^{) - 1} J _{x n} . Then, we have that _{x n + 1} = ^{J - 1} ( _{α n} J w + ( 1 - _{α n} ) J _{w n} ) , and since p ∈ ^{A - 1} ( 0 ) , from Lemma 8, we get that [figure omitted; refer to PDF] Now from (21), property of [varphi] , and (22) we get that [figure omitted; refer to PDF] Thus, by induction, [figure omitted; refer to PDF] which implies that { _{x n} } is bounded. In addition, using Lemma 6 and property of [varphi] , we obtain that [figure omitted; refer to PDF] Furthermore, using property of [varphi] and the fact that _{α n} [arrow right] 0 , as n [arrow right] ∞ , imply that [figure omitted; refer to PDF] which implies from Lemma 4 that [figure omitted; refer to PDF] Now, following the method of proof of Lemma 3 . 2 of Maing'e [33], we consider two cases.

Case 1. Suppose that there exists _{n 0} ∈ ... such that { [varphi] ( p , _{x n} ) } is nonincreasing for all n ...5; _{n 0} . In this situation, { [varphi] ( p , _{x n} ) } is convergent. Since { _{x n + 1} } is bounded and E is reflexive, we choose a subsequence { _{x n i + 1} } of { _{x n + 1} } such that _{x n i + 1} ... z and _{limsup n [arrow right] ∞} Y9; _{x n + 1} - p , J w - J p YA; = _{lim i [arrow right] ∞} Y9; _{x n i + 1} - p , J w - J p YA; . Then, from (27), we get that [figure omitted; refer to PDF] Thus, by Lemma 7, we get that z ∈ ^{A - 1} ( 0 ) , and hence z ∈ F = ( _{A 1} + _{A 2} + ... + _{A N} ^{) - 1} ( 0 ) . Therefore, by Lemma 5, we immediately obtain that _{limsup n [arrow right] ∞} Y9; _{x n + 1} - p , J w - J p YA; = _{lim i [arrow right] ∞} Y9; _{x n i + 1} - p , J w - J p YA; = Y9; z - p , J w - J p YA; ...4; 0 . It follows from Lemma 9 and (25) that [varphi] ( p , _{x n} ) [arrow right] 0 , as n [arrow right] ∞ . Consequently, _{x n} [arrow right] p .

Case 2. Suppose that there exists a subsequence { _{n i} } of { n } such that [figure omitted; refer to PDF] for all i ∈ ... . Then, by Lemma 10, there exist a nondecreasing sequence { _{m k} } ⊂ ... such that _{m k} [arrow right] ∞ , satisfying [figure omitted; refer to PDF] Thus, following the method of proof of Case 1, we obtain that [figure omitted; refer to PDF] Then, from (25), we have that [figure omitted; refer to PDF]

Now, inequalities (30) and (32) imply that [figure omitted; refer to PDF] In particular, since _{α m k} > 0 , we get [figure omitted; refer to PDF] Then, from (31), we obtain [varphi] ( p , _{x m k} ) [arrow right] 0 , as k [arrow right] ∞ . This together with (32) gives [varphi] ( p , _{x m k + 1} ) [arrow right] 0 , as k [arrow right] ∞ . But [varphi] ( p , _{x k} ) ...4; [varphi] ( p , _{x m k + 1} ) , for all k ∈ N ; thus, we obtain that _{x k} [arrow right] p . Therefore, from the above two cases, we can conclude that { _{x n} } converges strongly to p , and the proof is complete.

Theorem 12.

Let C be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space E with ^{E *} as its dual. Let _{A 1} : C [arrow right] ^{E *} be maximal monotone mapping, and let _{A 2} , _{A 3} , ... , _{A N} : E [arrow right] ^{E *} be bounded and hemicontinuous monotone mappings. Assume that F : = ( _{A 1} + _{A 2} + ... + _{A N} ^{) - 1} ( 0 ) is nonempty. Let { _{x n} } be a sequence generated by [figure omitted; refer to PDF] where A = _{A 1} + _{A 2} + ... + _{A N} , _{α n} ∈ ( 0,1 ) and { _{r n} } is a sequence of ( 0 , ∞ ) satisfying: _{lim n [arrow right] ∞ α n} = 0 , ^{∑ n = 1 ∞} ... _{α n} = ∞ , and _{lim n [arrow right] ∞ r n} = ∞ . Then, { _{x n} } converges strongly to p = _{Π F} ( w ) .

Proof.

By Lemma 3, we have that A = _{A 1} + _{A 2} + ... + _{A N} is maximal monotone, and hence following the method of proof of Theorem 11, we obtain the required assertion.

If in Theorem 12, we assume that _{A i} , for i = 2 , ... , N , are continuous monotone mappings, then ^{A i [variant prime]} s are hemicontinuous, and hence we get the following corollary.

Corollary 13.

Let C be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space E with ^{E *} as its dual. Let _{A 1} : C [arrow right] ^{E *} be a maximal monotone mapping, and let _{A 2} , _{A 3} , ... , _{A N} : E [arrow right] ^{E *} be bounded and continuous monotone mappings. Assume that F : = ( _{A 1} + _{A 2} + ... + _{A N} ^{) - 1} ( 0 ) is nonempty. Let { _{x n} } be a sequence generated by [figure omitted; refer to PDF] where A = _{A 1} + _{A 2} + ... + _{A N} , _{α n} ∈ ( 0,1 ) and { _{r n} } a sequence of ( 0 , ∞ ) satisfying: _{lim n [arrow right] ∞ α n} = 0 , ^{∑ n = 1 ∞} ... _{α n} = ∞ , and _{lim n [arrow right] ∞ r n} = ∞ . Then, { _{x n} } converges strongly to p = _{Π F} ( w ) .

If in Theorem 12, we assume that _{A i} , for i = 2 , ... , N , are uniformly continuous monotone mapping, then ^{A i [variant prime]} s are bounded and hemicontinuous, and hence we get the following corollary.

Corollary 14.

Let C be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space E with ^{E *} as its dual. Let _{A 1} : C [arrow right] ^{E *} be a maximal monotone mapping, and let _{A 2} , _{A 3} , ... , _{A N} : E [arrow right] ^{E *} be monotone uniformly continuous mappings. Assume that F : = ( _{A 1} + _{A 2} + ... + _{A N} ^{) - 1} ( 0 ) is nonempty. Let { _{x n} } be a sequence generated by [figure omitted; refer to PDF] where A = _{A 1} + _{A 2} + ... + _{A N} , _{α n} ∈ ( 0,1 ) and { _{r n} } a sequence of ( 0 , ∞ ) satisfying: _{lim n [arrow right] ∞ α n} = 0 , ^{∑ n = 1 ∞} ... _{α n} = ∞ , and _{lim n [arrow right] ∞ r n} = ∞ . Then, { _{x n} } converges strongly to p = _{Π F} ( w ) .

If in Theorem 12 we assume that _{A i} ...1; 0 , for i = 2 , ... , N , then we get the following corollary.

Corollary 15.

Let C be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space E . Let A : C [arrow right] ^{E *} be a maximal monotone mapping. Assume that F : = ^{A - 1} ( 0 ) is nonempty. Let { _{x n} } be a sequence generated by [figure omitted; refer to PDF] where _{α n} ∈ ( 0,1 ) and { _{r n} } a sequence of ( 0 , ∞ ) satisfying: _{lim n [arrow right] ∞ α n} = 0 , ^{∑ n = 1 ∞} ... _{α n} = ∞ , and _{lim n [arrow right] ∞ r n} = ∞ . Then, { _{x n} } converges strongly to p = _{Π F} ( w ) .

If E = 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 _{Π C} = _{P C} , projection mapping from H onto C . Thus, the following corollaries follow from Theorems 11 and 12.

Corollary 16.

Let C and D be nonempty, closed, and convex subsets of a real Hilbert space H . Assume that C ∩ int ( D ) ...0; ∅ . Let _{A 1} : C [arrow right] H , and let _{A 2} , _{A 3} , ... , _{A N} : D [arrow right] H be maximal monotone mappings. Assume that F : = ( _{A 1} + _{A 2} + ... + _{A N} ^{) - 1} ( 0 ) is nonempty. Let { _{x n} } be a sequence generated by [figure omitted; refer to PDF] where A = _{A 1} + _{A 2} + ... + _{A N} , _{α n} ∈ ( 0,1 ) and { _{r n} } a sequence of ( 0 , ∞ ) satisfying: _{lim n [arrow right] ∞ α n} = 0 , ^{∑ n = 1 ∞} ... _{α n} = ∞ , and _{lim n [arrow right] ∞ r n} = ∞ . Then, { _{x n} } converges strongly to p = _{P F} ( w ) .

Corollary 17.

Let C be a nonempty, closed, and convex subset of a real Hilbert space H . Let _{A 1} : C [arrow right] H be a maximal monotone mapping, and let _{A 2} , _{A 3} , ... , _{A N} : H [arrow right] H be bounded, hemicontinuous, and monotone mappings. Assume that F : = ( _{A 1} + _{A 2} + ... + _{A N} ^{) - 1} ( 0 ) is nonempty. Let { _{x n} } be a sequence generated by [figure omitted; refer to PDF] where A = _{A 1} + _{A 2} + ... + _{A N} , _{α n} ∈ ( 0,1 ) and { _{r n} } a sequence of ( 0 , ∞ ) satisfying: _{lim n [arrow right] ∞ α n} = 0 , ^{∑ n = 1 ∞} ... _{α n} = ∞ , and _{lim n [arrow right] ∞ r n} = ∞ . Then, { _{x n} } converges strongly to p = _{P F} ( w ) .

Corollary 18.

Let C be a nonempty, closed, and convex subset of a real Hilbert space H . Let _{A 1} : C [arrow right] H be a maximal monotone mapping, and let _{A 2} , _{A 3} , ... , _{A N} : H [arrow right] H be uniformly continuous monotone mappings. Assume that F : = ( _{A 1} + _{A 2} + ... + _{A N} ^{) - 1} ( 0 ) is nonempty. Let { _{x n} } be a sequence generated by [figure omitted; refer to PDF] where A = _{A 1} + _{A 2} + ... + _{A N} , _{α n} ∈ ( 0,1 ) and { _{r n} } a sequence of ( 0 , ∞ ) satisfying: _{lim n [arrow right] ∞ α n} = 0 , ^{∑ n = 1 ∞} ... _{α n} = ∞ , and _{lim n [arrow right] ∞ r n} = ∞ . Then, { _{x n} } converges strongly to p = _{P F} ( 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 Theorems 11 and 12.

Theorem 19.

Let C and D be a nonempty, closed, and convex subsets of a smooth and uniformly convex real Banach space E . Let C ∩ int ( D ) ...0; ∅ . 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 ) = _{inf y ∈ E} { f ( y ) + g ( y ) } } ...0; ∅ . Let { _{x n} } be a sequence generated by [figure omitted; refer to PDF] where _{α n} ∈ ( 0,1 ) and { _{r n} } a sequence of ( 0 , ∞ ) satisfying: _{lim n [arrow right] ∞ α n} = 0 , ^{∑ n = 1 ∞} ... _{α n} = ∞ , and _{lim n [arrow right] ∞ r n} = ∞ . Then, { _{x n} } converges strongly to an element of F .

Theorem 20.

Let C 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 ) = _{inf y ∈ E} { f ( y ) + g ( y ) } } ...0; ∅ . Let { _{x n} } be a sequence generated by [figure omitted; refer to PDF] where _{α n} ∈ ( 0,1 ) and { _{r n} } a sequence of ( 0 , ∞ ) satisfying: _{lim n [arrow right] ∞ α n} = 0 , ^{∑ n = 1 ∞} ... _{α n} = ∞ , and _{lim n [arrow right] ∞ r n} = ∞ . Then, { _{x n} } 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].

Acknowledgments

The authors thank the referee for his comments that considerably improved the paper. The research of N. Shahzad was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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