Strong Convergence of a Monotone Projection Algorithm in a Banach Space

In this paper, a common solution problem is investigated based on a Bregman projection. Strong convergence of the monotone projection algorithm for monotone operators and bifunctions is obtained in a reflexive Banach space.


Introduction
Fixed point theory as an important branch of nonlinear analysis theory has been applied in the study of nonlinear phenomena. Lots of problems arising in economics, engineering, and physics can be studied based on fixed point techniques; see  and the references therein. Many wellknown problems can be studied by using algorithms which are iterative in their nature. As an example, in computer tomography with limited data, each piece of information implies the existence of a convex set in which the required solution lies. The problem of finding a point in the intersection ∩ =1 , where ≥ 1 is some positive integer, is then of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such point. The well-known convex feasibility problem which captures applications in various disciplines such as image restoration and radiation therapy treatment planning is to find a point in the intersection of common fixed point sets of a family of nonlinear mappings see, for example, [22][23][24][25][26][27][28].
Interest in monotone operators stems mainly from their firm connection with equations of evolution which is an important class of nonlinear operators. It is known that many physically significant problems can be modelled by initial value problems of the form: where is an monotone operator in an appropriate Banach space. Typical examples, where such evolution equations occur, can be found in the heat, wave, or Schrödinger's equations. If ( ) is dependent on , then (1) is reduced to whose solutions correspond to the equilibrium points of the system (1). Consequently, considerable research efforts have been devoted, especially within the past 40 years or so, to methods for finding approximate solutions (when they exist) of (2). An early fundamental result in the theory of monotone operators, due to Browder [29], states that the initial value problem (1) is solvable if is locally Lipschitz and accretive on . The Krasnoselskii-Mann iterative algorithm is efficient for treating zero points of monotone operators. However, it is weak convergence only. In many disciplines, including economics, image recovery, and control theory, problems arise in infinite dimension spaces. In such problems, strong convergence (norm convergence) is often much more desirable than weak convergence for it translates the physically tangible property that the energy ‖ − ‖ of the error between the iterate and the solution eventually becomes arbitrarily small. The importance of strong convergence is also underlined, where a convex function is minimized via the proximal-point algorithm; it is shown that the rate of convergence of the value sequence { ( )} is better when { } converges strongly than when it converges weakly. Such properties have a direct impact when the process is executed directly in the underlying infinite dimensional space. Projection methods which were first introduced by Haugazeau [ The Scientific World Journal have been considered for treating zero points of monotone operators. The advantage of projection methods is that strong convergence of iterative sequences can be guaranteed without any compact assumptions.
In this paper, we study common zero points of a family of maximal monotone operators and common solutions of a system of equilibrium problems based on the Bregman projection. Strong convergence of the monotone projection algorithm is obtained in a reflexive Banach space.

Preliminaries
Let be a real Banach space, * the dual space of , and a nonempty subset of . Let be a bifunction from × to R, where R denotes the set of real numbers. Recall the following equilibrium problem. Find ∈ such that We use EP( ) to denote the solution set of the equilibrium problem (3). That is, Given a mapping : → * , let Then, ∈ EP( ) if and only if is a solution of the following variational inequality. Find such that In order to study the solution of the equilibrium problem (3), we assume that satisfies the following conditions: ∈ int Dom( ) and ∈ , we denote by 0 ( , ) the righthand derivative of at in the direction ; that is, The function is said to be Gâteaux differentiable at if lim → 0 + (( ( + ) − ( ))/ ) exists for any . In this case, 0 ( , ) coincides with ∇ ( ), the value of the gradient ∇ of at . The function is said to be Gâteaux differentiable if it is Gâteaux differentiable everywhere. The function is said to be Fréchet differentiable at if the limit is attained uniformly in ‖ ‖ = 1. The function is said to be Fréchet differentiable if it is Fréchet differentiable everywhere. It is well known that if a continuous convex function : → R is Gâteaux differentiable, then ∇ is norm-to-weak * continuous. If : → R is Fréchet differentiable, then ∇ is norm-to-norm continuous; for more details, see [31] and the references therein; for more details, see [32] and the references therein. The function is said to be strongly coercive [32] if lim ‖ ‖ → ∞ ( ( )/‖ ‖) = ∞. It is said to be bounded on bounded subsets of if ( ) is bounded for each bounded subset of . Let be a reflexive Banach space. For any proper, lower semicontinuous, and convex function: : → (−∞, +∞], the conjugate function * of is defined by It is well known that ( ) + * ( * ) ≥ ⟨ , * ⟩ for all ( , * ) ∈ × * . It is also known that ( , * ) ∈ , where is the subdifferential of , is equivalent to If is a proper, lower semicontinuous, and convex function, then * is a proper, weak * lower semicontinuous, and convex function. Next, we recall some facts about Bregman distance. Let : → R be a convex and Gâteaux differentiable function. Then, Bregman distance corresponding to is the function : × → R defined by It is clear that ( , ) ≥ 0 for all , ∈ . If is smooth and ( ) = ‖ ‖ 2 for all ∈ , we obtain that ∇ ( ) = 2 , where is the generalized duality mapping. If is a nonempty, closed and convex subset of a reflexive Banach space and is a strongly coercive Bregman function, then for each ∈ , there exists a unique 0 ∈ such that ( 0 , ) = min ∈ ( , ). Bregman projection Proj from onto is defined by Proj = 0 for all ∈ . It is also well known [32] that Proj has the following property: The Scientific World Journal 3 The function is called the gauge of the uniform convexity of . If : → R is a convex function which is uniformly convex on bounded subsets, then for all , ∈ and ∈ (0, 1), where is the gauge of the uniform convexity of .
Let : → be a mapping. In this paper, we use ( ) to denote the fixed point set of . is said to be closed; if for any sequence { } ⊂ such that lim → ∞ = 0 and lim → ∞ = 0 , then 0 = 0 . Let : → R be a proper, lower semicontinuous and convex function. Recall that : → is said to be Bregman quasi-nonexpansive, if and only if A point in is said to be an asymptotic fixed point of if and only if contains a sequence { }, which converges weakly to such that lim → ∞ ‖ − ‖ = 0. The set of asymptotic fixed points of will be denoted bỹ( ). is said to be Bregman relatively nonexpansive if and only if̃( ) = ( ) ̸ = 0 and ( , ) ≤ ( , ) for all ∈ and ∈ ( ).
Let be a reflexive, strictly convex, and smooth Banach space, and let be a maximal monotone operator from to * . From Rockafellar [33], we find that > 0 and ∈ ; there exists a unique ∈ ( ) such that ∇ ∈ ∇ + . If = , then we can define a single-valued mapping : → Dom( ) by = (∇ + ) −1 ∇ and such a is called the resolvent of . We know that is closed and −1 (0) = ( ) for all > 0. From [34], we know that : → Dom( ) is a Bregman quasi-nonexpansive.
In this paper, we study a common solution problem based on Bregman projections. Strong convergence of the monotone projection algorithm for monotone operators and bifunctions is obtained in a reflexive Banach space.
In order to introduce our main results, we also need the following lemmas.
Lemma 1 (see [34]). Let be a Banach space and let : → R be a Gâteaux differentiable function which is uniformly convex on bounded subsets of . Let { } and { } be bounded sequences in . Then, Lemma 2 (see [35]). Let be a reflexive Banach space and let : → R be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of . Let be a nonempty, closed, and convex subset of . Let : → be a Bregman relatively nonexpansive mapping. Then, ( ) is closed and convex.

Theorem 4. Let be a reflexive Banach space and let : → be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and smooth on bounded subsets of . Let be a nonempty, closed, and convex subset of and let Λ be an index set. Let { } be a positive real number sequence. Let be a bifunction from × to R satisfying (A1)-(A4) and let
: → * be a maximal monotone operator such that Dom ( ) ⊂ for every ∈ Λ. Assume that the common solution set := ∩ ∈Λ −1 (0) ⋂ ∩ ∈Λ ( ) is nonempty. Let { } be a sequence generated in the following manner: where ∇ is the right-hand derivative of , where is some positive real number, for every ∈ Λ. Then, the sequence { } converges strongly to 1 , where is the Bregman projection from onto .
Proof. In view of Lemma 3, we find that the common solution set CSS is closed and convex. Next, we prove that is closed and convex. It suffices to show that, for each fixed but arbitrary ∈ Λ, ( , ) is closed and convex. This can be proved by induction into . It is obvious that (1, ) = is closed and convex. Assume that ( , ) is closed and convex for some ≥ 1. For 1 , 2 ∈ ( +1, ) , we see that 1 , 2 ∈ ( , ) . It follows that = 1 + (1 − ) 2 ∈ ( , ) , where ∈ (0, 1). Notice that The above inequalities are equivalent to That is, where ∈ ( , ) . This implies that ( +1, ) is closed and convex. We conclude that ( , ) is closed and convex. This in turn implies that = ∩ ∈Λ ( , ) is closed and convex. This implies that Proj +1 1 is well defined.
Next, we show that CSS ⊂ . CSS ⊂ 1 = is clear. Suppose that CSS ⊂ ( , ) for some positive integer . For any ∈ CSS ⊂ ( , ) , we see that which shows that ∈ ( +1, ) . This implies that CSS ⊂ ( , ) . This yields that CSS ⊂ ∩ ∈Λ ( , ) . This completes the proof that CSS ⊂ . Next, we show that the sequence { } is bounded. It follows from (12) that This implies that the sequence { ( , 1 )} is bounded. It follows that the sequence { } is also bounded. In view of the construction of the set , we find that ⊂ and = proj 1 ∈ ⊂ for any positive integer ≥ . It follows from (12) that It follows that This shows that the sequence { ( , 1 )} is nondecreasing. Hence, the limit lim → ∞ ( , 1 ) exists. In view of = Proj 1 , we find that The Scientific World Journal 5 This concludes that lim → ∞ ( , 1 ) exists. Let , → ∞ in (26); we find that ( , ) → 0. In view of Lemma 1, we obtain that ‖ − ‖ → 0 as , → ∞. This yields that { } is a Cauchy sequence. Since is closed and convex, we see that there exists an ∈ ; that is, Next, we show that ∈ CSS. It follows from (26) that Therefore, we obtain that Since +1 ∈ +1 , we find that In view of (30), we find that It follows that Notice that It follows from (29) and (34) Since ∇ is uniformly norm-to-norm continuous on any bounded subset of , we find that Notice that It follows from (36) where is the gauge of the uniform convexity of * . It follows that In view of the restrictions on the sequence { ( , ) }, we find from (39) that lim → ∞ ‖∇ ( ) − ∇ ( )‖ = 0. Since ∇ * is uniformly norm-to-norm continuous on bounded subsets of * , we obtain that Since is closed Bregman quasi-nonexpansive, we find that = . This proves that ∈ ∩ ∈Λ −1 (0).
Finally, we prove that = Proj CSS 1 . In view of = Proj 1 , we conclude that Since CSC ⊂ , we arrive at Letting → ∞ in the above inequality, we see that This yields that = Proj CSS 1 . This completes the proof.
If the index set is singleton, then we have the following result.

Corollary 5. Let be a reflexive Banach space and let : →
be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and smooth on bounded subsets of . Let be a nonempty, closed, and convex subset of and let be a positive real number. Let be a bifunction from × to R satisfying (A1)-(A4) and let : → * be a maximal monotone operator such that Dom ( ) ⊂ . Assume that the common solution set := −1 (0) ⋂ ( ) is nonempty. Let { } be a sequence generated in the following manner: 0 ∈ chosen arbitrarily, 1 = , ∈ such that where ∇ is the right-hand derivative of .
is some positive real number. Then, the sequence { } converges strongly to 1 , where is the Bregman projection from onto . Remark 6. Corollary 5 improves Theorem 4 of Qin et al. [24] in the following aspects: (1) the framework of spaces is relaxed; (2) a more general notion of Bregman function is considered instead of the generalized duality mapping; (3) a more general notion of Bregman projection is considered instead of the generalized projection operator; (4) families of operators extend from a finite family to an infinite family.
If the index set is singleton, then we have the following result.