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

and Applied Analysis 3 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. Let E be a reflexive, strictly convex, and smooth Banach space, and letC be a nonempty closed and convex subset ofE. The generalized projectionmapping, introduced byAlber [29], is a mapping Π C : E → C that assigns an arbitrary point x ∈ E to the minimizer, x, of φ(⋅, x) over C; that is, Π C x = x, where x is the solution to the minimization problem φ (x, x) = min {φ (y, x) , y ∈ C} . (14) 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 φ(x n , y n ) → 0, as n → ∞, then x n − y n → 0, as n → ∞. 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 ⟨z − x 0 , Jx − Jx 0 ⟩ ≤ 0, ∀z ∈ C. (15) We make use of the function V : E × E∗ → R defined by V (x, x ∗ ) = ‖x‖ 2 − 2 ⟨x, x ∗ ⟩ + ‖x‖ 2 , ∀x ∈ E, x ∗ ∈ E,


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: ⟨ − ,  − ⟩ ≥ 0. ( 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  −1 (0) is closed and convex (see [1] for more details).Monotone mappings were introduced by Zarantonello [2], Minty [3], and Kačurovskiȋ [4].The notion of monotone in the context of variational methods for nonlinear operator equations was also used by Vaȋnberg and Kačurovskiȋ [5].The central problem is to iteratively find a zero of a finite sum of monotone mappings  1 ,  2 , . . .,   in a Banach space , namely, a solution to the inclusion problem 0 ∈ ( 1 +  2 + ⋅ ⋅ ⋅ +   ) . ( 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  :=  1 +  2 + ⋅ ⋅ ⋅ +   (see, e.g., [6]).In addition, optimization problems often need [7] to solve a minimization problem of the form min where   ,  = 1, 2, . . .,  are proper lower semicontinuous convex functions from  to the extended real line  := (−∞, ∞].If in (2), we assume that   :=   , for  = 1, 2, . . ., , 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][9][10][11][12]).A well-known method for solving the equation 0 ∈  in a Hilbert space  is the proximal point algorithm:  1 =  ∈  and where   ⊂ (0,∞) and   = ( + ) −1 for all  > 0. This algorithm was first introduced by Martinet [10].In 1976, Rockafellar [11] proved that if lim inf  → ∞   > 0 and  −1 (0) ̸ = 0, then the sequence {  } defined by ( 5) converges weakly to an element of  −1 (0).Later, many researchers have studied the convergence of the sequence defined by (5) in Hilbert spaces; see, for instance, [8,[12][13][14][15][16][17][18] and the references therein.
In 2000, Kamimura and Takahashi [9] proved that for a maximal monotone mapping  in a Hilbert spaces  and   = ( + ) −1 for all  > 0, the sequence {  } defined by where {  } ⊂ [0, 1] and {  } ⊂ (0, ∞) satisfy certain conditions, called Halpern type, converges strongly to a point in  −1 (0).In a reflexive Banach space  and for a maximal monotone mapping  :  → 2  * , Reich and Sabach [19] proved that the sequence {  } defined by where   > 0 and proj   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  −1 (0).
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  } by the recursion where    denotes the resolvent of a monotone mapping ; that is,    := ( + ) −1 .They proved that the nonlinear Douglas-Rachford algorithm (8) converges weakly to a point V, 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.

Preliminaries
Let  be a Banach space and let () = { ∈  : ‖‖ = 1}.Then, a Banach space  is said to be smooth provided that the limit lim exists for each ,  ∈ ().The norm of  is said to be uniformly smooth if the limit ( 10) is attained uniformly for (, ) in () × () (see [1]).
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  > 0, where  is the normalized duality mapping from  into 2  * 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 .
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.
In the sequel, we will make use of the following lemmas.
But (,   ) ≤ (,    +1 ), for all  ∈ ; thus, we obtain that   → .Therefore, from the above two cases, we can conclude that {  } converges strongly to , and the proof is complete.
Proof.By Lemma 3, we have that  =  1 +  2 + ⋅ ⋅ ⋅ +   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  = 2, . . ., , are continuous monotone mappings, then     are hemicontinuous, and hence we get the following corollary.
If  = , a real Hilbert space, then  is smooth and uniformly convex real Banach space.In this case,  = , identity map on  and Π  =   , projection mapping from  onto .Thus, the following corollaries follow from Theorems 11 and 12.

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 .
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].