Abstract
The purpose of this paper is to introduce and consider a new hybrid shrinking projection algorithm for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of a system of variational inequality problems, the set of solutions of a system of optimization problems, the common fixed point set of a uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings in reflexive Banach spaces. Strong convergence theorems have been proved under the appropriate conditions. The main innovative points in this paper are as follows: (1) the notion of the uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings is presented and the useful conclusions are given; (2) the relative examples of the uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings are given in classical Banach spaces l 2 and L 2 ; (3) the hybrid shrinking projection method presented in this paper modified some mistakes in the recent result of Ugwunnadi et al. (Fixed Point Theory Appl. 2014:231, 2014. These new results improve and extend the previously known ones in the literature.
Abstract
MSC: 47H05; 47H09; 47H10
Introduction
Let C be a nonempty subset of a real Banach space and T be a mapping from C into itself. We denote by F(T) the set of fixed points of T. Recall that T is said to be asymptotically nonexpansive [] if there exists a sequence {k n } ⊂ [, +∞) with lim n→∞ k n =  such that T n x -T n y ≤ k n x -y , ∀x, y ∈ C, n ≥ .
It is well known that T is said to be nonexpansive if
Tx -Ty ≤ x -y , ∀x, y ∈ C.
It is well known that T is said to be nonexpansive if
In the framework of Hilbert spaces, Takahashi In , Bregman [] discovered an elegant and effective technique for using of the socalled Bregman distance function (see Section ) in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman's technique is applied in various ways in order to design and analyze not only iterative algorithms for solving feasibility and optimization problems, but also algorithms for solving variational inequalities, for approximating equilibria, and for computing fixed points of nonlinear mappings.
It is well known that T is said to be nonexpansive if
Many authors have studied iterative methods for approximating fixed points of mappings of nonexpansive type with respect to the Bregman distance; see [-] . In [] , the author introduced a new class of nonlinear mappings which is an extension of asymptotically quasi-nonexpansive mappings with respect to the Bregman distance in the intermediate sense and proved the strong convergence theorems for asymptotically quasinonexpansive mappings with respect to Bregman distances in the intermediate sense by using the shrinking projection method.
It is well known that T is said to be nonexpansive if
Recently Motivated and inspired by the above works, in  Ugwunnadi et al.
It is well known that T is said to be nonexpansive if
[] proved a new strong convergence theorem for a finite family of closed quasi-Bregman strictly pseudocontractive mappings and a system of equilibrium problems in a real reflexive Banach space.
It is well known that T is said to be nonexpansive if
The purpose of this paper is to introduce and consider a new hybrid shrinking projection algorithm for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of a system of variational inequality problems, the set of solutions of a system of optimization problems, the common fixed point set of a uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings in reflexive Banach spaces. Strong convergence theorems have been proved under the appropriate conditions. The main innovative points in this paper are as follows: () the notion of uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings is presented and the useful conclusions are given; () the relative examples of the uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings are given in classical Banach spaces l  and L  ; () the hybrid shrinking projection method presented in this paper modified some mistakes in the recent result of Ugwunnadi et al. [] . These new results improve and extend the previously known ones in the literature.
Preliminaries
Throughout this paper, we assume that E is a real reflexive Banach space with the dual space of E * and ·, · is the pairing between E and E * .
Preliminaries
Let f : E → (-∞, +∞] be a function. The effective domain of f is defined by
Preliminaries
When dom f = ∅, we say that f is proper. We denote by int dom f the interior of the effective domain of f . We denote by ran f the range of f . The function f is said to be strongly coercive if
Preliminaries
Given a proper and convex function f : E → (-∞, +∞], the subdifferential of f is a mapping ∂f : E → E * defined by
Preliminaries
The Fenchel conjugate function of f is the convex function f
Preliminaries
We know that x * ∈ ∂f (x) if and only if (ii) f is strongly coercive.
Preliminaries
Let f : E → (-∞, +∞] be a convex function and x ∈ int dom f . For any y ∈ E, we define the right-hand derivative of f at x in the direction y by
Preliminaries
The function f is said to be Gâteaux differentiable at x if the limit (.) exists for any y. In this case, the gradient of f at x is the function ∇f (x) :
Preliminaries
for all y ∈ E. The function f is said to be Gâteaux differentiable if it is Gâteaux differentiable at each x ∈ int dom f . If the limit (.) is attained uniformly in y = , then the function f is said to be Fréchet differentiable at x. The function f is said to be uniformly Fréchet differentiable on a subset C of E if the limit (.) is attained uniformly for x ∈ C and y = . This fact, when combined with conditions (L) and (L), implies the following equalities:
Preliminaries
Conditions (L) and (L) imply that the functions f and f * are strictly convex on the interior of their respective domains. In [], authors gave an example of the Legendre function.
Preliminaries
is called the Bregman distance with respect to f (cf.
Preliminaries
[]). In general, the Bregman distance is not a metric since it is not symmetric and does not satisfy the triangle inequality. However, it has the following important property, which is called the three point identity (cf.
Preliminaries
[]): for any x ∈ dom f and y, z ∈ int dom f ,
Preliminaries
bounded, then the sequence {x n } is also bounded.
Proposition . ([]) Let f : E → (-∞, +∞] be a Legendre function. Then the following statements hold:
Let f : E → (-∞, +∞] be a convex function on E which is Gâteaux differentiable on int dom f . The function f is said to be totally convex at a point x ∈ int dom f if its modulus of total convexity at
Proposition . ([]) Let f : E → (-∞, +∞] be a Legendre function. Then the following statements hold:
is positive whenever t > . The function f is said to be totally convex when it is totally convex at every point of int dom f . The function f is said to be totally convex on bounded sets if, for any nonempty bounded set B ⊂ E, the modulus of total convexity of f on B, v f (B, t) is positive for any t > , where
Proposition . ([]) Let f : E → (-∞, +∞] be a Legendre function. Then the following statements hold:
We remark in passing that f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets; see [, ] .
Proposition . ([]) Let f : E → (-∞, +∞] be a convex function whose domain contains at least two points. If f is lower semi-continuous, then f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets.

Proposition . ([])
Let f : E → R be a totally convex function. If x ∈ E and the sequence {D f (x n , x)} is bounded, then the sequence {x n } is also bounded.
Proposition . ([])
Let f : E → [, +∞) be a convex function on E which is Gâteaux differentiable on int dom f . The function f is said to be sequentially consistent (cf.
Proposition . ([])
[]) if for any two sequences {x n } and {y n } in int dom f and dom f , respectively, such that the first one is bounded,
Proposition . ([]) A function f : E → [, +∞) is totally convex on bounded subsets of E if and only if it is sequentially consistent.
Let C be a nonempty, closed, and convex subset of E. Let f : E → (-∞, +∞] be a convex function on E which is Gâteaux differentiable on int dom f . The Bregman projection proj
Proposition . ([]) A function f : E → [, +∞) is totally convex on bounded subsets of E if and only if it is sequentially consistent.
Let E be a Banach space with dual E * . We denote by J the normalized duality mapping from E to  E * defined by
Proposition . ([]) A function f : E → [, +∞) is totally convex on bounded subsets of E if and only if it is sequentially consistent.
where ·, · denotes the generalized duality pairing. It is well known that if E is smooth, then J is single-valued.
Proposition . ([]) Let f : E → R be an admissible, strongly coercive, and strictly convex function. Let C be a nonempty, closed, and convex subset of
(i) If E is a Hilbert space, then the Bregman projection is reduced to the metric projection onto C. (ii) If E is a smooth Banach space, then the Bregman projection is reduced to the generalized projection C (x) which is defined by
Proposition . ([]) Let f : E → R be an admissible, strongly coercive, and strictly convex function. Let C be a nonempty, closed, and convex subset of
where φ is the Lyapunov functional (cf.
Proposition . ([]) Let f : E → R be an admissible, strongly coercive, and strictly convex function. Let C be a nonempty, closed, and convex subset of
[]) defined by (ii) The vector x * is the unique solution z of the variational inequality
Proposition . ([]) Let f : E → R be an admissible, strongly coercive, and strictly convex function. Let C be a nonempty, closed, and convex subset of
(iii) The vector x * is the unique solution z of the inequality
Proposition . ([]) Let f : E → R be an admissible, strongly coercive, and strictly convex function. Let C be a nonempty, closed, and convex subset of
In recent years, the following notions have been presented by some authors. A point p ∈ C is said to be asymptotic fixed point of a map T if there exists a sequence {x n } in C which converges weakly to p such that lim n→∞ x n -Tx n = . We denote by F(T) the set of asymptotic fixed points of T. A point p ∈ C is said to be strong asymptotic fixed point [] of a mapping T if there exists a sequence {x n } in C which converges strongly to p such that lim n→∞ x n -Tx n = . We denote by F(T) the set of strong asymptotic fixed points of T. Let f : E → R, a mapping T : C → C is said to be Bregman relatively
Proposition . ([]) Let f : E → R be an admissible, strongly coercive, and strictly convex function. Let C be a nonempty, closed, and convex subset of
for all x ∈ C and p ∈ F(T). In [] quasi-Bregman relatively nonexpansive is called left quasi-Bregman relatively nonexpansive. A mapping T : C → C is said to be right quasi-
Proposition . ([]) Let f : E → R be an admissible, strongly coercive, and strictly convex function. Let C be a nonempty, closed, and convex subset of
In [] , authors presented the definition of quasi-Bregman strictly pseudocontractive mapping. In this paper, we extend this definition to the quasi-Bregman pseudocontractive mapping as follows.
Proposition . ([]) Let f : E → R be an admissible, strongly coercive, and strictly convex function. Let C be a nonempty, closed, and convex subset of
Definition . Let C be a nonempty, closed, and convex subset of E and f : E → (-∞, +∞] be an admissible function. Let T be a mapping from C into itself with a nonempty fixed point set F(T). The mapping T is said to be quasi-Bregman k-pseudocontractive if there exists a constant k ∈ [, +∞) such that
Proposition . ([]) Let f : E → R be an admissible, strongly coercive, and strictly convex function. Let C be a nonempty, closed, and convex subset of
If k ∈ [, ), the mapping T is said to be quasi-Bregman strictly pseudocontractive. If k = , the mapping T is said to be quasi-Bregman pseudocontractive. The mapping T is said to be Bregman quasi-nonexpansive if
Proposition . ([]) Let f : E → R be an admissible, strongly coercive, and strictly convex function. Let C be a nonempty, closed, and convex subset of
In this paper, we will use the following definition.
Definition . ([]
) Let C be a nonempty, closed, and convex subset of E. Let {T n } be a sequence of mappings from C into itself with a nonempty common fixed point set F = ∞ n= F(T n ). {T n } is said to be uniformly closed if for any convergent sequence {z n } ⊂ C such that T n z n -z n →  as n → ∞, the limit of {z n } belongs to F. 
Definition . ([]
Let E be a real Banach space with the dual E * and C be a nonempty closed convex subset of E. Let A : C → E * be a nonlinear mapping and F : C × C → R be a bifunction. Then consider the following generalized equilibrium problem of finding u ∈ C such that
Definition . ([]
The set of solutions of (.) is denoted by EP, i.e.,
Definition . ([]
Whenever A ≡ , ϕ(x) ≡ , problem (.) is equivalent to finding u ∈ C such that
Definition . ([]
which is called the equilibrium problem. The set of its solutions is denoted by EP(F).
Definition . ([]
which is called the variational inequality of Browder type. The set of its solutions is denoted by VI(C, A).
Definition . ([]
which is called the convex optimization problem. The set of its solutions is denoted by MIN(ϕ). Problem (.) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games and others; see, e.g., [, ] .
Definition . ([]
In order to solve the equilibrium problem for finding an element x ∈ C such that
Definition . ([]
let us assume that F : C × C → (-∞, +∞) satisfies the following conditions []:
e., F(x, y) + F(y, x) ≤ , for all x, y ∈ C, (A) for all x, y, z ∈ C, lim sup t↓ F(tz + ( -t)x, y) ≤ F(x, y), (A) for all x ∈ C, F(x, ·) is convex and lower semi-continuous.
For r > , we define a mapping K r : E → C as follows:
e., F(x, y) + F(y, x) ≤ , for all x, y ∈ C, (A) for all x, y, z ∈ C, lim sup t↓ F(tz + ( -t)x, y) ≤ F(x, y), (A) for all x ∈ C, F(x, ·) is convex and lower semi-continuous.
for all x ∈ E. The following two lemmas were proved in []. 
Lemma . Let E be a reflexive Banach space and let f : E → R be a Legendre function. Let C be a nonempty, closed, and convex subset of E and let F : C × C → R be a bifunction satisfying (A)-(A). For r > , let T r : E → C be the mapping defined by (.). Then dom T r = E.

Lemma . Let E be a reflexive Banach space and let f : E → R be a convex, continuous, and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E and let F : C × C → R be a bifunction satisfying (A)-(A). For r > , let T r : E → C be the mapping defined by (.). Then the following statements hold: (i) T r is single-valued. (ii) T r is a firmly nonexpansive-type mapping, i.e., for all x, y ∈ E, T r x -T r y, ∇f (T r x) -∇f (T r y) ≤ T r x -T r y, ∇f (x) -∇f (y) . (iii) F(T r ) = F(T r ) = EP(F). (iv) EP(F) is closed and convex.
(v) D f (p, T r x) + D f (T r x, x) ≤ D f (p, x), ∀p ∈ EP(F), ∀x ∈ E.
Lemma . Let E be a reflexive Banach space and let f : E → R be a convex, continuous, and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E and let F : C × C → R be a bifunction satisfying (A)-(A). Let A : C → E
K r (x) = z ∈ C : G(z, y) +  r y -z, ∇f (z) -∇f (x) ≥ , ∀y ∈ C , where G(x, y) = ϕ(y) -ϕ(x) + F(x, y) + Ax, y -x .
Then the following statements hold: (i) K r is single-valued. (ii) K r is a firmly nonexpansive-type mapping, i.e., for all x, y ∈ E,
K r x -K r y, ∇f (K r x) -∇f (K r y) ≤ K r x -K r y, ∇f (x) -∇f (y) . (iii) F(K r ) = F(K r ) = EP. (iv) EP is closed and convex. (v) D f (p, K r x) + D f (K r x, x) ≤ D f (p, x), ∀p ∈ EP(F), ∀x ∈ E. Proof Let G(x, y) = ϕ(y) -ϕ(x) + F(x, y) + Ax, y -x , ∀x, y ∈ C.
It is easy to show that G(x, y) satisfies conditions (A)-(A). Replacing F(x, y) by G(x, y) in
Lemma ., we can get the conclusions.
It is easy to show that G(x, y) satisfies conditions (A)-(A). Replacing F(x, y) by G(x, y) in
From [] we have the following conclusion.
Theorem . Let E be a p-uniformly convex Banach space with p ≥ . Then for all x, y
where J p is the generalized duality mapping from E into E * and /c is the p-uniformly convexity constant of E.
Theorem . Let E be a p-uniformly convex Banach space with p ≥ . Then for all x, y
From Theorem ., we know that the generalized duality mapping J p : E → E * is a monotone operator. It is well known that if E is also smooth and -uniformly convex, the normalized duality mapping J = J  : E → E * is a single-valued monotone operator.
Main results
We now prove the following theorem. 
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
for j = , , , . . . , m, and {α n }, {β j,n } are sequences satisfying lim sup n→∞ α n < , {r n } is a sequence satisfying lim inf n→∞ r n > . Then {x n } converges to q = P f F x  .
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Proof We divide the proof into six steps.
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Step . We show that C n is closed and convex for all n ≥ . Let
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Since C  = C is closed and convex, it is sufficient to prove that the sets D n , E j,n are closed and convex for all n ≥ . We show that D n is closed and convex for all n ≥ . We rewrite D n as follows:
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
From the above expression, we know that D n is closed and convex for all n ≥ . Next we show that E j,n is closed and convex for all n ≥ , j = , , , . . . , m. We rewrite E j,n as follows:
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
From the above expression, we know that E j,n is closed and convex for all n ≥ , j = , , , . . . , m. Therefore C n is closed and convex for all n ≥ .
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Step . We show that F ⊂ C n for all n ≥ . Note that
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
where
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
from (.) and (.) we know that p ∈ C n+ , which implies F ⊂ C n for all n ≥ .
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Step . We show that {x n } converges to a point p ∈ C. Since x n = P f C n x  and C n+ ⊂ C n , then we get
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Therefore {D f (x n , x  )} is nondecreasing. On the other hand, by Proposition ., we have
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
for all p ∈ F ⊂ C n and for all n ≥ . Therefore, D f (x n , x  ) is also bounded. This together with (.) implies that the limit of {D f (x n , x  )} exists. Put
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
From Proposition ., we have, for any positive integer m, that
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
for all n ≥ . This together with (.) implies that
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
holds uniformly for all m. Therefore, we get that
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
holds uniformly for all m. Then {x n } is a Cauchy sequence, therefore there exists a point p ∈ C such that x n → p.
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Step . We show that the limit of {x n } belongs to ∞ n= F(T n ). Since x n+ ∈ C n+ , we have from the definition of C n+ that
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
which implies that lim n→∞ D f (x n+ , y n ) = . Since f is totally convex on bounded subsets of E, f is sequentially consistent (see [] ). It follows that
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
From the uniform continuity of ∇f , we have
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
we obtain that
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Since f is strongly coercive and uniformly convex on bounded subsets of E, f * is uniformly
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Fréchet differentiable on bounded sets. Moreover, f * is bounded on bounded sets, and from (.) we obtain
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Since {T n } is uniformly closed and x n → p, we have p ∈ ∞ n= F(T n ).
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Step . We show that the limit of {x n } belongs to EP j for all j = , , , . . . , m. We have proved that x n → p as n → ∞. Now let us show that p ∈ EP j for any j = , , , . . . , m. Since x n+ ∈ C n+ , we have from the definition of C n+ that
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Since lim n→∞ D f (x n+ , y n ) = , we have
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Since f is totally convex on bounded subsets of E, f is sequentially consistent (see [] ). It follows that Since ∇f is uniformly norm-to-norm continuous on bounded subsets of E, from (.) we have lim n→∞ ∇f (u j,n ) -∇f (y n ) = . From lim inf n→∞ r n >  it follows that
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
By the definition of u j,n := K
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
r n y n , we have
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
We have from (A) that
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Since y → f (x, y) + Ax, y -x is convex and lower semi-continuous, letting n → ∞ in the last inequality, from (A) we have
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
For t, with  < t < , and y ∈ C, let y t = ty + ( -t)p. Since y ∈ C and p ∈ C, then y t ∈ C and hence G j (y t , p) ≤ . So, from (A) we have
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Dividing by t, we have
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Letting t → , from (A) we can get
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
So, p ∈ EP j for all j = , , , . . . , m.
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Step . Finally, we prove that p = P f F x  , from Proposition . we have
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
On the other hand, since x n = P f C n x  and F ⊂ C n for all n, also from Proposition ., we have
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
By the definition of D f (x, y), we know that
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Combining (.), (.), and (.), we know that
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
. Therefore, it follows from the uniqueness of P f F x  that p = P f F x  . This completes the proof.
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
Remark . Theorem . includes the following three special cases.
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
() Take T n ≡ I, ϕ(x) ≡ , F(x, y) ≡ , where I denotes the identity operator, then the iterative sequence {x n } converges strongly to a solution of the system of variational in-
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
In this case, the iterative sequence {x n } is defined by
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
() Take T n ≡ I, ϕ(x) ≡ , A ≡ , where I denotes the identity operator, then the iterative sequence {x n } converges strongly to a solution of the system of equilibrium problems
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
In this case, the iterative sequence {x n } is defined by
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
() Take T n ≡ I, F(x, y) ≡ , A ≡ , where I denotes the identity operator, then the iterative sequence {x n } converges strongly to a solution of the system of convex optimization problems
Theorem . Let C be a nonempty, closed, and convex subset of a real reflexive
In this case, the iterative sequence {x n } is defined by 
Examples
Let E be a Hilbert space and C be a nonempty closed convex and balanced subset of E. Let {x n } be a sequence in C such that x n = r > , {x n } converges weakly to x  = , and
Examples
x n -x m ≥ r >  for all n = m. Define a countable family of mappings {T n } : C → C as follows: for all x ∈ C. On the other hand, for any strong convergent sequence {z n } ⊂ E such that z n → z  and z n -T n z n →  as n → ∞, it is easy to see that there exists a sufficiently large nature number N such that z n = x m for any n, m > N . Then Tz n = -z n for n > N , it follows from z n -T n z n →  that z n →  and hence z n → z  = . That is, z  ∈ F. In fact, we are proving that p ∈ EP j for any j = , , , . . . , m, therefore, if we do not know whether p ∈ EP j , then the above inequalities are not right since if p ∈ EP j , the above inequalities are right. In this paper, we have overcome these shortcomings by modifying the iterative scheme.

{'paper_id': '55475861', 'title': 'New hybrid shrinking projection algorithm for common fixed points of a family of countable quasi-Bregman strictly pseudocontractive mappings with equilibrium and variational inequality and optimization problems', 's2_url': 'https://api.semanticscholar.org/CorpusID:55475861', 'source': 'S2ORC'}