## Advance in Nonlinear Analysis: Algorithm, Convergence and Applications

View this Special IssueResearch Article | Open Access

H. Zegeye, N. Shahzad, "Proximal Point Algorithms for Finding a Zero of a Finite Sum of Monotone Mappings in Banach Spaces", *Abstract and Applied Analysis*, vol. 2013, Article ID 232170, 7 pages, 2013. https://doi.org/10.1155/2013/232170

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

**Academic Editor:**Yisheng Song

#### 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.

#### 1. Introduction

Let be a nonempty subset of a real Banach space with dual . A mapping is said to be *monotone *if for each , the following inequality holds:
A monotone mapping 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 is maximal monotone mapping, then 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 in a Banach space , namely, a solution to the inclusion problem 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 can be formulated as (2) when the governing maximal monotone is of the form (see, e.g., [6]). In addition, optimization problems often need [7] to solve a minimization problem of the form where , are proper lower semicontinuous convex functions from to the extended real line . If in (2), we assume that , for , where is the subdifferential operator of 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 in a Hilbert space is the *proximal point algorithm*: and
where and for all . This algorithm was first introduced by Martinet [10]. In 1976, Rockafellar [11] proved that if and , then the sequence defined by (5) converges *weakly* to an element of . 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 in a Hilbert spaces and for all , the sequence defined by
where and satisfy certain conditions, called *Halpern* type, converges *strongly* to a point in .

In a reflexive Banach space and for a maximal monotone mapping , Reich and Sabach [19] proved that the sequence defined by
where and is the Bergman projection of on to a closed and convex subset induced by a well-chosen convex function , converges *strongly* to a point in .

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 by the recursion
where denotes the resolvent of a monotone mapping ; that is, . They proved that the nonlinear Douglas-Rachford algorithm (8) converges *weakly* to a point , a solution of the inclusion,
for 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 be a Banach space and let . Then, a Banach space is said to be *smooth* provided that the limit
exists for each . The norm of is said to be *uniformly smooth* if the limit (10) is attained uniformly for in (see [1]).

The *modulus of convexity* of is the function defined by
is called *uniformly convex* if and only if , for every (see [26]).

Lemma 1 (see [27]). * Let be a smooth, strictly convex, and reflexive Banach space. Let be a nonempty closed convex subset of , and let be a monotone mapping. Then, is maximal if and only if , for all , where is the normalized duality mapping from into defined, for each , by
**
where denotes the generalized duality pairing between members of and . We recall that is smooth if and only if is single valued (see [1]). If , a Hilbert space, then the duality mapping becomes the identity map on . *

Lemma 2 (see [27]). *Let be a reflexive with as its dual. Let , and let be maximal monotone mappings. Suppose that . Then, is a maximal monotone mapping.*

Lemma 3 (see [28]). * Let be a reflexive with as its dual. Let be maximal monotone mapping, and let be monotone mappings such that , is hemicontinuous (i.e., continuous from the segments in to the weak star topology in ) and carries bounded sets into bounded sets. Then, is maximal monotone mapping. *

Let be a smooth Banach space with dual . Let the Lyapunov function , introduced by Alber [29], be defined by where is the normalized duality mapping from into . If , a Hilbert space, then (13) reduces to , for .

Let be a reflexive, strictly convex, and smooth Banach space, and let be a nonempty closed and convex subset of . The *generalized projection mapping*, introduced by Alber [29], is a mapping that assigns an arbitrary point to the minimizer, , of over ; that is, , where is the solution to the minimization problem
We know the following lemmas.

Lemma 4 (see [23]). *Let be a real smooth and uniformly convex Banach space, and let and be two sequences of . If either or is bounded and , as , then , as . *

Lemma 5 (see [29]). * Let be a convex subset of a real smooth Banach space , and let . Then if and only if
*

We make use of the function defined by studied by Alber [29]. That is, , for all and .

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

Lemma 6 (see [29]). *Let be a reflexive strictly convex and smooth Banach space with as its dual. Then,
**
for all and . *

Lemma 7 (see [30]). *Let be a smooth and strictly convex Banach space, be a nonempty closed convex subset of , and be a maximal monotone mapping. Let be the resolvent of defined by , for and a sequence of such that . If is a bounded sequence of such that , then .*

Lemma 8 (see [31]). * Let be a smooth and strictly convex Banach space, be a nonempty closed convex subset of , and be a maximal monotone mapping, and is nonempty. Let be the resolvent of defined by , for . Then, for each **
for all and . *

Lemma 9 (see [32]). * Let be a sequence of nonnegative real numbers satisfying the following relation:
**
where and satisfying the following conditions: , , and . Then, . *

Lemma 10 (see [33]). *Let be the sequences of real numbers such that there exists a subsequence of such that , for all . Then, there exists a nondecreasing sequence such that , and the following properties are satisfied by all (sufficiently large) numbers :
**
In fact, . *

#### 3. Main Result

Theorem 11. *Let and be nonempty, closed and convex subsets of a smooth and uniformly convex real Banach space with as its dual. Assume that . Let and be maximal monotone mappings. Assume that is nonempty. Let be a sequence generated by
**
where , and a sequence of satisfying: , , and . Then, converges strongly to . *

*Proof. * Observe that by Lemma 2, we have that is maximal monotone. In addition, since , the same lemma implies that is maximal monotone. Now, let , and let . Then, we have that , and since , from Lemma 8, we get that
Now from (21), property of , and (22) we get that
Thus, by induction,
which implies that is bounded. In addition, using Lemma 6 and property of , we obtain that
Furthermore, using property of and the fact that , as , imply that
which implies from Lemma 4 that
Now, following the method of proof of Lemma of Maing’e [33], we consider two cases.*Case **1.* Suppose that there exists such that is nonincreasing for all . In this situation, is convergent. Since is bounded and is reflexive, we choose a subsequence of such that and . Then, from (27), we get that
Thus, by Lemma 7, we get that , and hence . Therefore, by Lemma 5, we immediately obtain that . It follows from Lemma 9 and (25) that , as . Consequently, .*Case **2.* Suppose that there exists a subsequence of such that
for all . Then, by Lemma 10, there exist a nondecreasing sequence such that , satisfying
Thus, following the method of proof of Case 1, we obtain that
Then, from (25), we have that

Now, inequalities (30) and (32) imply that
In particular, since , we get
Then, from (31), we obtain , as . This together with (32) gives , as . But , for all ; thus, we obtain that . Therefore, from the above two cases, we can conclude that converges strongly to , and the proof is complete.

Theorem 12. *Let be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space with as its dual. Let be maximal monotone mapping, and let be bounded and hemicontinuous monotone mappings. Assume that is nonempty. Let be a sequence generated by
**
where , and is a sequence of satisfying: , , and . Then, converges strongly to . *

*Proof. * By Lemma 3, we have that 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 , for , are continuous monotone mappings, then are hemicontinuous, and hence we get the following corollary.

Corollary 13. *Let be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space with as its dual. Let be a maximal monotone mapping, and let be bounded and continuous monotone mappings. Assume that is nonempty. Let be a sequence generated by
**
where , and a sequence of satisfying: , , and . Then, converges strongly to . *

If in Theorem 12, we assume that , for , are uniformly continuous monotone mapping, then are bounded and hemicontinuous, and hence we get the following corollary.

Corollary 14. *Let be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space with as its dual. Let be a maximal monotone mapping, and let be monotone uniformly continuous mappings. Assume that is nonempty. Let be a sequence generated by
**
where , and a sequence of satisfying: , , and . Then, converges strongly to . *

If in Theorem 12 we assume that , for , then we get the following corollary.

Corollary 15. *Let be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space . Let be a maximal monotone mapping. Assume that is nonempty. Let be a sequence generated by
**
where and a sequence of satisfying: , , and . Then, converges strongly to . *

If , a real Hilbert space, then is smooth and uniformly convex real Banach space. In this case, , identity map on and , projection mapping from onto . Thus, the following corollaries follow from Theorems 11 and 12.

Corollary 16. *Let and be nonempty, closed, and convex subsets of a real Hilbert space . Assume that . Let , and let be maximal monotone mappings. Assume that is nonempty. Let be a sequence generated by
**
where , and a sequence of satisfying: , , and . Then, converges strongly to . *

Corollary 17. *Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a maximal monotone mapping, and let be bounded, hemicontinuous, and monotone mappings. Assume that is nonempty. Let be a sequence generated by
**
where , and a sequence of satisfying: , , and . Then, converges strongly to . *

Corollary 18. *Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a maximal monotone mapping, and let be uniformly continuous monotone mappings. Assume that is nonempty. Let be a sequence generated by
**
where , and a sequence of satisfying: , , and . Then, converges strongly to . *

#### 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 and be a nonempty, closed, and convex subsets of a smooth and uniformly convex real Banach space . Let . Let be a continuously Fréchet differentiable convex functional, and let be maximal monotone on . Let be a continuously Fréchet differentiable convex functional, and let be maximal monotone on . Assume that . Let be a sequence generated by
**
where and a sequence of satisfying: , , and . Then, converges strongly to an element of . *

Theorem 20. *Let be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space. Let be a continuously Fréchet differentiable convex functional, and let be maximal monotone on . Let be a continuously Fréchet differentiable convex functional, and let be bounded, hemicontinuous, and monotone on with . Let be a sequence generated by
**
where and a sequence of satisfying: , , and . Then, converges strongly to an element of . *

*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.

#### References

- W. Takahashi,
*Nonlinear Functional Analysis*, Kindikagaku, Tokyo, Japan, 1988. - E. H. Zarantonello, “Solving functional equations by contractive averaging,” Tech. Rep. 160, Mathematics Research Centre, Univesity of Wisconsin, Madison, Wis, USA, 1960. View at: Google Scholar
- G. J. Minty, “Monotone (nonlinear) operators in Hilbert space,”
*Duke Mathematical Journal*, vol. 29, pp. 341–346, 1962. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. I. Kačurovskiĭ, “On monotone operators and convex functionals,”
*Uspekhi Mathematicheskikh Nauk*, vol. 15, no. 4, pp. 213–215, 1960. View at: Google Scholar - M. M. Vaĭnberg and R. I. Kačurovskiĭ, “On the variational theory of non-linear operators and equations,”
*Doklady Akademii Nauk SSSR*, vol. 129, pp. 1199–1202, 1959. View at: Google Scholar | MathSciNet - P.-L. Lions and B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,”
*SIAM Journal on Numerical Analysis*, vol. 16, no. 6, pp. 964–979, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. L. Combettes and V. R. Wajs, “Signal recovery by proximal forward-backward splitting,”
*Multiscale Modeling & Simulation*, vol. 4, no. 4, pp. 1168–1200, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Brézis and P.-L. Lions, “Produits infinis de résolvantes,”
*Israel Journal of Mathematics*, vol. 29, no. 4, pp. 329–345, 1978. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Kamimura and W. Takahashi, “Approximating solutions of maximal monotone operators in Hilbert spaces,”
*Journal of Approximation Theory*, vol. 106, no. 2, pp. 226–240, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. Martinet, “Régularisation d'inéquations variationnelles par approximations successives,”
*Revue Française d'Informatique et de Recherche Opérationnelle*, vol. 4, pp. 154–158, 1970. View at: Google Scholar | Zentralblatt MATH | MathSciNet - R. T. Rockafellar, “Monotone operators and the proximal point algorithm,”
*SIAM Journal on Control and Optimization*, vol. 14, no. 5, pp. 877–898, 1976. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. V. Solodov and B. F. Svaiter, “Forcing strong convergence of proximal point iterations in a Hilbert space,”
*Mathematical Programming*, vol. 87, no. 1, pp. 189–202, 2000. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. H. Bauschke, E. Matoušková, and S. Reich, “Projection and proximal point methods: convergence results and counterexamples,”
*Nonlinear Analysis: Theory, Methods &Applications*, vol. 56, no. 5, pp. 715–738, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. E. Bruck and S. Reich, “Nonexpansive projections and resolvents of accretive operators in Banach spaces,”
*Houston Journal of Mathematics*, vol. 3, no. 4, pp. 459–470, 1977. View at: Google Scholar | Zentralblatt MATH | MathSciNet - O. Güler, “On the convergence of the proximal point algorithm for convex minimization,”
*SIAM Journal on Control and Optimization*, vol. 29, no. 2, pp. 403–419, 1991. View at: Publisher Site | Google Scholar | MathSciNet - P.-L. Lions, “Une méthode itérative de résolution d'une inéquation variationnelle,”
*Israel Journal of Mathematics*, vol. 31, no. 2, pp. 204–208, 1978. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - O. Nevanlinna and S. Reich, “Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces,”
*Israel Journal of Mathematics*, vol. 32, no. 1, pp. 44–58, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. B. Passty, “Ergodic convergence to a zero of the sum of monotone operators in Hilbert space,”
*Journal of Mathematical Analysis and Applications*, vol. 72, no. 2, pp. 383–390, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Reich and S. Sabach, “A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces,”
*Journal of Nonlinear and Convex Analysis*, vol. 10, no. 3, pp. 471–485, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet - Y. Censor and S. Reich, “Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization,”
*Optimization*, vol. 37, no. 4, pp. 323–339, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Hu and L. Liu, “A new iterative algorithm for common solutions of a finite family of accretive operators,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 6, pp. 2344–2351, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Reich and S. Sabach, “Two strong convergence theorems for a proximal method in reflexive Banach spaces,”
*Numerical Functional Analysis and Optimization*, vol. 31, no. 1–3, pp. 22–44, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,”
*SIAM Journal on Optimization*, vol. 13, no. 3, pp. 938–945, 2002. View at: Publisher Site | Google Scholar | MathSciNet - H. Zegeye and N. Shahzad, “Strong convergence theorems for a common zero for a finite family of $m$-accretive mappings,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 66, no. 5, pp. 1161–1169, 2007. View at: Publisher Site | Google Scholar | MathSciNet - H. Zegeye and N. Shahzad, “Approximating common solution of variational inequality problems for two monotone mappings in Banach spaces,”
*Optimization Letters*, vol. 5, no. 4, pp. 691–704, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Goebel and S. Reich,
*Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings*, vol. 83 of*Monographs and Textbooks in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 1984. View at: MathSciNet - R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,”
*Transactions of the American Mathematical Society*, vol. 149, pp. 75–88, 1970. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. E. Browder, “Nonlinear maximal monotone operators in Banach space,”
*Mathematische Annalen*, vol. 175, pp. 89–113, 1968. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in
*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type*, A. G. Kartsatos, Ed., vol. 178 of*Lecture Notes in Pure and Applied Mathematics*, pp. 15–50, Dekker, New York, NY, USA, 1996. View at: Google Scholar | Zentralblatt MATH | MathSciNet - K. Aoyama, F. Kohsaka, and W. Takahashi, “Proximal point methods for monotone operators in Banach spaces,”
*Taiwanese Journal of Mathematics*, vol. 15, no. 1, pp. 259–281, 2011. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. Kamimura, F. Kohsaka, and W. Takahashi, “Weak and strong convergence theorems for maximal monotone operators in a Banach space,”
*Set-Valued Analysis*, vol. 12, no. 4, pp. 417–429, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H.-K. Xu, “Another control condition in an iterative method for nonexpansive mappings,”
*Bulletin of the Australian Mathematical Society*, vol. 65, no. 1, pp. 109–113, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P.-E. Maingé, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,”
*Set-Valued Analysis*, vol. 16, no. 7-8, pp. 899–912, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2013 H. Zegeye and N. Shahzad. 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.