Hybrid Projection Algorithm for Two Countable Families of Hemirelatively Nonexpansive Mappings and Applications

Two countable families of hemirelatively nonexpansive mappings are considered based on a hybrid projection algorithm. Strong convergence theorems of iterative sequences are obtained in an uniformly convex and uniformly smooth Banach space. As applications, convex feasibility problems, equilibrium problems, variational inequality problems, and zeros of maximal monotone operators are studied.

Observe that every -uniformly convex is uniformly convex.One should note that no Banach space is -uniformly convex for 1 <  < 2. It is well known that   (  ) or    is uniformly convex if  ≥ 2 and 2-uniformly convex if 1 <  ≤ 2; see [8] for more details.
In particular, if  = 2,   is called the normalized duality mapping.If  is a Hilbert space, then   = , where  is the identity mapping.In this paper, We denote by  the normalized duality mapping.It is known that the duality mapping  has the following properties: (i) if  is smooth, then  is single valued; (ii) if  is strictly convex, then  is one to one; (iii) if  is reflexive, then  is surjective; (iv) if  is uniformly smooth, then  is uniformly normto-norm continuous on each bounded subset of ; (v) if  * is uniformly convex, then  is uniformly continuous on bounded subsets of  and  is single valued and also one to one (see [9][10][11][12]).
Let  be a smooth Banach space.Consider the function defined by It is obvious from the definition of the function  that We also know that (, ) = 0 if and only if  =  (see [13]).Moreover, if  is a Hilbert space, (12) reduces to (, ) = ‖ − ‖ 2 , for any ,  ∈ .
Let  be a closed convex subset of , and let  be a mapping from  into itself.We denote by () the set of fixed points of .A point  in  is said to be an asymptotic fixed point of  [14] if  contains a sequence {  } which converges weakly to  such that the strong lim  → ∞ (  −   ) = 0.The set of asymptotic fixed points of  will be denoted by F().A point  in  is said to be a strong asymptotic fixed point of  [14] if  contains a sequence {  } which converges strong to  such that lim  → ∞ ‖  −  ‖ = 0.The set of strong asymptotic fixed points of  will be denoted by F().
Let  :  →  be a mapping, and recall the following definition: Remark 1. From the definitions, it is obvious that a relatively nonexpansive mapping is a weak relatively nonexpansive mapping, and a weak relatively nonexpansive mapping is a hemi-relatively nonexpansive mapping, but the converse is not true.
Next, we give an example which is a closed hemirelatively nonexpansive mapping.
Example 2. Let Π  be the generalized projection from a smooth, strictly convex, and reflexive Banach space  onto a nonempty closed convex subset  ⊂ .Then Π  is a relatively nonexpansive mapping, and then it is also a closed hemirelatively nonexpansive mapping.
In 2005, Matsushita and Takahashi [13] obtained strong convergence theorems for a single relatively nonexpansive mapping in a uniformly convex and uniformly smooth Banach space .To be more precise, they proved the following theorem.
Theorem MT (see Matsushita and Takahashi [13,Theorem 3.1]).Let  be precisely a uniformly convex and uniformly smooth Banach space and  a nonempty closed convex subset of , and let  be a relatively nonexpansive mapping from  into itself, and let {  } be a sequence of real numbers such that 0 ≤   < 1 and lim sup  → ∞   < 1. Suppose that {  } is given by where  is the duality mapping on .If () is nonempty, then {  } converges strongly to Π () , where Π () is the generalized projection from  onto ().
Since then, algorithms constructed for solving the same equilibrium problem, variational inequality problems, and fixed point of relatively nonexpansive mappings (or weak relatively nonexpansive mappings or hemi-relatively nonexpanisve mappings) have been further developed by many authors.For a part of works related to these problems, please see [4,[15][16][17][18], and for the hybrid algorithm projection methods for these problems, please see  and the references therein.
Motivated and inspired by the results in the literature, in this paper we focus our attention on finding a common fixed point of two countable families of hemi-relatively nonexpansive mappings (we shall give the definition of a countable family of hemi-relatively nonexpansive mappings in the next section) by using a simple hybrid algorithm.Furthermore, we will give some applications of our main result in equilibrium problems, variational inequality problems, and convex feasibility problems.

Preliminaries
Let  be a closed convex subset of , and let {  } ∞ =0 be a countable family of mappings from  into itself.We denote by  the set of common fixed points of {  } ∞ =0 .That is,  = ⋂ ∞ =0 (  ), where (  ) denote the set of fixed points of   , for all  ∈ N ∪ {0}.
A point  ∈  is said to be an asymptotic fixed point of Now, we introduce the definition of countable family of hemi-relatively nonexpansive mappings which is more general than countable family of relatively nonexpansive mappings and countable family of weak relatively nonexpansive mappings.

Definition 5. Countable family of mappings
Remark 6.From Definitions 3-5, one has the following facts.
(2) Countable family of hemi-relatively nonexpansive mappings, which do not need the restriction )), is more general than countable family of relatively nonexpansive mappings (or countable family of weak relatively nonexpansive mappings).Next we give an example which is a countable family of hemi-relatively nonexpansive mappings but not a countable family of relatively nonexpansive mappings.
Example 7. Let  be any smooth Banach space and  0 = (1 + 1/)   0 ̸ = 0 any element of .Define a countable family of mappings   :  →  as follows: for all  ≥ 1, Then {  } ∞ =1 is a countable family of hemi-relatively nonexpansive mappings but not a countable family of relatively nonexpansive mappings.
Proof.First, it is obvious that   has a unique fixed point 0; that is, (  ) = {0} for all  ≥ 1.In addition, one easily sees that This implies that for all  ∈ ⋂ ∞ =1 (  ).It follows from the above inequality that for all  ∈ ⋂ ∞ =1 (  ) and  ∈ .Hence, {  } ∞ =1 is a countable family of hemi-relatively nonexpansive mappings.On the other hand, letting from the definition of   , one has which implies that ‖  −     ‖ → 0 and In what follows, we will need the following lemmas.

Main Results
Now, we give our main results in this paper.
Theorem 11.Let  be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space .Let {  }, {  } be two uniformly closed countable families of hemirelatively nonexpansive mappings from  into itself such that For a point  0 ∈  chosen arbitrarily, let {  } be a sequence generated by the following iterative algorithm: where the sequences   =     .Then the sequence {  } converges strongly to a point  = Π F  0 , where Π F is the generalized projection from  onto F.
Proof.We first show that  +1 is closed and convex.
Remark 12. Theorem 11 improves Theorem 3.15 of Zhang et al. [49] in the following senses: (1) from the class of a countable family of weak relatively nonexpansive mappings to the one of a countable family of hemi-relatively nonexpansive mappings; (2) from a single countable family of mappings to two countable families of mappings.
When   =  in (32), we can obtain the following corollary immediately.

Corollary 13.
Let  be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex Banach space .Let {  } be a uniformly closed countable family of hemi-relatively nonexpansive mappings from  into itself such that For a point  0 ∈  chosen arbitrarily, let {  } be a sequence generated by the following iterative algorithm: Then the sequence {  } converges strongly to a point  = Π F  0 , where Π F is the generalized projection from  onto F.
Remark 14.We notice that if {  } is a countable family of weak relatively nonexpansive mappings, Corollary 13 is still held.Therefore, Corollary 13 extends and improves Theorem 3.15 in [49].

Applications to Convex Feasibility Problems
In this section, we consider the following convex feasibility problem (CFP): where  ∈ N ∪ {0}, and {  } ∞ =0 is an intersecting closed convex subset sequence of a Banach space .This problem is a frequently appearing problem in diverse areas of mathematical and physical sciences.There is a considerable investigation on (CFP) in the framework of Hilbert spaces which captures applications in various disciplines such as image restoration [50][51][52][53], computer tomography [54], and radiation therapy treatment planning [55].In computer tomography with limited data, in which an unknown image has to be reconstructed from a priori knowledge and from measured results, each piece of information gives a constraint which in turn gives rise to a convex set   to which the unknown image should belong (see [56]).
Using Theorem 11, we discuss the convex feasibility problems as an application.

Theorem 15. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex
For a point  0 ∈  chosen arbitrarily, let {  } be a sequence generated by the following iterative algorithm: where the sequences   = Π Ω *    .Then the sequence {  } converges strongly to a point  = Π Ω  0 , where Π Ω is the generalized projection from  onto Ω.
Proof.From Lemma 9, we easily have that {Π Ω  } and {Π Ω *  } are two countable families of hemi-relatively nonexpansive mappings.In view of the continuity of Π Ω  and Π Ω *  , we have that {Π Ω  } and {Π Ω *  } are two uniformly closed countable families of hemi-relatively nonexpansive mappings.Thus, by using Theorem 11, we have that the sequence {  } converges strongly to a point  = Π Ω  0 .This completes the proof.
If we only consider a countable family of nonempty closed convex subset of , the following corollary can be obtained by using Theorem 15.

Corollary 16.
Let  be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex Banach space .
Let {Ω  } ∞ =0 be a countable family of nonempty closed convex subset of  such that For a point  0 ∈  chosen arbitrarily, let {  } be a sequence generated by the following iterative algorithm: Then the sequence {  } converges strongly to a point  = Π Ω  0 , where Π Ω is the generalized projection from  onto Ω.

Applications to Generalized Mixed Equilibrium Problems
In this section, we apply our main results to prove some strong convergence theorems concerning generalized mixed equilibrium problems in a Banach space .
Let  :  →  * be a mapping.First, we recall the following definition: ( We remark here that an -inverse strongly monotone  is (1/)-Lipschitz continuous.
The following result can be found in Blum and Oettli [1].
(IV) We show that the function   → (, ) is convex and lower semicontinuous for each  ∈ .
Lemma 20 (see Zhang et al. [57]).Let  be a -uniformly convex with  ≥ 0 and uniformly smooth Banach space, and let  be a nonempty closed convex subset of .Let  be a bifunction from  ×  to R satisfying ( 1 )-( 4 ).Let {  } be a positive real sequence such that lim  → ∞   =  > 0. Then the sequence of mappings    is uniformly closed.
Proof.From Lemmas 18 and 20, we learn that {   } and {   } are uniformly closed.And by Lemma 19 (5), one can easily get that {   } and {   } are uniformly closed countable families of hemi-relatively nonexpansive mappings.Notice that if  is -uniformly convex, it must be uniformly convex.Therefore, by using Theorem 11, we can obtain the conclusion of Theorem 21.This completes the proof.

Theorem 22.
Let  be a -uniformly convex with  ≥ 2 and uniformly smooth Banach space, and let  be a nonempty closed convex subset of .Let  :  →  * be a monotone and continuous mappings, let the function  :  → R be convex and lower semicontinuous and let  be a bifunction from × to R satisfying ( 1 )-( 4 ) such that I = GMEP(, , ) ̸ = 0.For a point  0 ∈  chosen arbitrarily, let {  } be a sequence generated by the following iterative algorithm: where   =      and lim  → ∞   = .Then the sequence {  } converges strongly to a point  = Π I  0 , where Π I is the generalized projection from  onto I.
Proof.From Lemmas 18 and 20, we learn that {   } is uniformly closed.And by Lemma 19 (5), one can easily get that {   } is an uniformly closed countable family of hemirelatively nonexpansive mappings.Notice that if  is uniformly convex, it must be uniformly convex.Therefore, by using Corollary 13, we can obtain the conclusion of Theorem 21.This completes the proof.

Corollary 23.
Let  be a -uniformly convex with  ≥ 2 and uniformly smooth Banach space, and let  be a nonempty closed convex subset of .Let  be a bifunction from  ×  to R satisfying ( 1 )-( 4 ) and  :  →  * a monotone and continuous mapping.Suppose that I = VI(, ) ⋂ EP() ̸ = 0.For a point  0 ∈  chosen arbitrarily, let {  } be a sequence generated by the following iterative algorithm: where   =      , V  =      , and lim  → ∞   = .Then the sequence {  } converges strongly to a point  = Π I  0 , where Π I is the generalized projection from  onto I.
Remark 24.By analysis of special cases for generalized mixed equilibrium problem, we can obtain the corresponding results based on Theorems 21 and 22 in sequence.Here, we do not itemize these results.

Applications to Maximal Monotone Operators
Let A be a multivalued operator from  to  * with domain (A) = { ∈  : A ̸ = 0} and range (A) = { ∈  :  ∈ (A)}.An operator A is said to be monotone if A monotone operator A is said to be maximal if its graph (A) = {(, ) :  ∈ A} is not properly contained in the graph of any other monotone operator.It is well known that if A is a maximal monotone operator, then A −1 0 is closed and convex.
The following result is also well known.
Lemma 25 (see Rockafellar [58]).Let  be a reflexive, strictly convex, and smooth Banach space and A a monotone operator from  to  * .Then A is maximal if and only if ( + A) =  * for all  > 0.
Let  be a reflexive, strictly convex, and smooth Banach space and A a maximal monotone operator from  to  * .Using Lemma 25 and the strict convexity of , it follows that, for all  > 0 and  ∈ , there exists a unique   ∈ (A) such that  ∈   + A  . (76) If    =   , then we can define a single-valued mapping   :  → (A) by   = ( + A) −1  and such a   is called the resolvent of A. We know that A −1 0 = (  ) for all  > 0 (see [10,59] for more details).
First, we give an important lemma for this section and remark that the following lemma can be as example of a countable family of hemi-relatively nonexpansive mappings.Lemma 26.Let  be a strictly convex and uniformly smooth Banach space, let A be a maximal monotone operator from  to  * such that A −1 0 is nonempty, and let {  } be a sequence of positive real numbers which is bounded away from 0 such that    = ( +   A) −1 .Then {   } is a uniformly closed countable family of hemi-relatively nonexpansive mappings.
We consider the problem of strong convergence concerning maximal monotone operators in a Banach space.Such a problem has been also studied in [4,13,49].Using Theorem 11, we obtain the following result.where the sequences   =  A     .Then the sequence {  } converges strongly to a point  = Π F  0 , where Π F is the generalized projection from  onto F.
Proof.From Lemma 26, we know that { A   } and { B   } are two uniformly closed countable families of hemi-relatively nonexpansive mappings.Furthermore, applying Theorem 11, one sees that the sequence {  } converges strongly to a point Π F  0 .and Engineering Research Cluster (CSEC-KMUTT) (Grant Project no.NRU56000508).The first author is supported by the Project of Shandong Province Higher Educational Science and Technology Program (Grant no.J13LI51) and the Foundation of Shandong Yingcai University (Grant no.12YCZDZR03).