On split feasibility problem for finite families of equilibrium and fixed point problems in Banach spaces

Definition 1.1

[5] Let f : E → ( − ∞ , + ∞ ] be a convex and Gâteaux differentiable function. The function D f : E × E → [ 0 , + ∞ ) defined by

is called the Bregman distance with respect to f , where x , y ∈ E .

Definition 1.2

Let C be a nonempty, closed, and convex subset of a real Banach space E . Let { T i } i = 1 N be a finite family of Bregman quasi-asymptotically nonexpansive mappings. For n ∈ N , we define a mapping K n : C → C as follows:

The mapping K n defined in (3) is called a Bregman K -mapping generated by T 1 , T 2 , T 3 , … , T N and { α n , i } ∈ ( 0 , 1 ) for i = 1 , 2 , 3 , … , N .

Theorem 1.3

Let C be a nonempty, closed, and convex subset of a reflexive Banach space E and f : E → R be a super coercive Legendre function which is bounded, uniformly Frechet differentiable, and totally convex on bounded subset of E. Let for i = 1 , 2 , … , N , g i : C × C → R be a bifunction satisfying (L1)–(L5). Assume that for each 1 ≤ r ≤ M , T r : C B ( C ) be a multivalued Bregman relatively nonexpansive mapping, such that Γ = ( ∩ r = 1 M F ( T r ) ) ∩ ( ∩ i = 1 N EP ( g i ) ) ≠ ∅ . Suppose that { x n } is a sequence generated by x 1 ∈ C and

where C B ( C ) denotes the family of a nonempty, closed, and convex subset of C and { α n } , { β n , r } , { λ n } , and { u n } satisfy the following conditions:

1. { α n } ⊂ ( 0 , 1 ) , lim n → ∞ α n = 0 , ∑ n = 1 ∞ α n = ∞ .

2. { β n , r } ⊂ ( 0 , 1 ) , ∑ r = 0 M β n , r = 1 , liminf n → ∞ β n , 0 β n , r > 0 for all 1 ≤ r ≤ M and n ∈ N .

3. { λ n } ⊂ [ a , b ] ⊂ ( 0 , p ) , where p = min 1 c 1 , 1 c 2 , c 1 = max 1 ≤ i ≤ N c i , 1 , c 2 = max 1 ≤ i ≤ N c i , 2 , and c i , 1 , c i , 2 are the Bregman-Lipschitz coefficients of g i for all 1 ≤ i ≤ N .

4. { u n } ⊂ E , lim n → ∞ u n = u for some u ∈ E .

Definition 2.1

A function f : E → R is said to be strongly coercive if

Lemma 2.2

[11] Let g be a bifunction satisfying conditions (L1) and (L3)–(L5), then E P ( g ) is closed and convex.

Lemma 2.3

[23] Let E be a Banach space, s > 0 be a constant, and ρ s be the gauge of uniform convexity of f, where f : E → R is a convex function which is uniformly convex on bounded subsets of E. Then,

1. For any x , y ∈ B s and α ∈ ( 0 , 1 ) , we have

   f ( α x + ( 1 − α ) y ) ≤ α f ( x ) + ( 1 − α ) f ( y ) − α ( 1 − α ) ρ s ( ‖ x − y ‖ ) .

2. For any x , y ∈ B s ,

   ρ s ( ‖ x − y ‖ ) ≤ D f ( x , y ) .

   Here, B s ≔ { z ∈ E : ‖ z ‖ ≤ s } .

Lemma 2.4

[24] Let E be a reflexive Banach space, f : E → R be a strongly coercive Bregman function, and V be a function defined by

The following assertions also hold:

Lemma 2.5

[6] Let E be a reflexive Banach space with the dual E ∗ and C be a nonempty, closed, convex, and bounded subset of E . Let f : E → ( − ∞ , + ∞ ] be strongly coercive, Legendre, uniformly Fréchet differentiable, and totally convex function that is bounded on bounded subsets of E. Let { T i } i = 1 N be a finite family of continuous Bregman quasi-asymptotically nonexpansive mappings of C into itself such that ∩ i = 1 N F ( T i ) ≠ ∅ . Let { α n , i } be a real sequence in (0, 1) such that liminf n → ∞ α n , i > 0 ∀ i ∈ 1 , 2 , … , N . Let K n be the Bregman K-mapping generated by T 1 , T 2 , T 3 , … , T N and { α n , i } , ∀ i = 1 , 2 , … , N . Then,

1. D f ( x ∗ , K n x ) ≤ ( 1 + t n ) D f ( x ∗ , x ) ∀ x ∗ ∈ F ( K n ) ;

2. F ( K n ) = ∩ i = 1 N F ( T i ) ;

3. K n is a closed mapping.

Lemma 2.6

[24] Let E be a Banach space and f : E → R be a Gâteaux differentiable function that is uniformly convex on bounded subsets of E. Let { x n } n ∈ N and { y n } n ∈ N be bounded sequences in E. Then,

Lemma 2.7

[11] Let C be a nonempty, closed, and convex subset of a reflexive Banach space E and f : E → R be a Legendre and super coercive function. Suppose that g : C × C → R is a bifunction satisfying ( L 1 ) – ( L 4 ) . For arbitrary sequence { x n } ⊂ C and { λ n } ⊂ ( 0 , + ∞ ) , let { w n } and { z n } be sequences generated by

Then, for all x ∗ ∈ E P ( g ) , we have that

where c 1 and c 2 are the Bregman-Lipschitz coefficients of g.

Lemma 2.8

[25] If dom f contains at least two points, then the function f is totally convex on bounded sets if and only if the function f is sequentially consistent.

Lemma 2.9

[26] Let f : E → R be a Gâteaux differentiable and totally convex function. If x 0 ∈ E and the sequence { D f ( x n , x 0 ) } is bounded, then the sequence { x n } is also bounded.

Definition 2.10

Let E be a reflexive Banach space and C be a nonempty, closed, and convex subset of E . A Bregman projection of x ∈ intdom f onto C ⊂ intdom f is the unique vector P C f ( x ) ∈ C satisfying

Lemma 2.11

[27] Let C be a nonempty, closed, and convex subset of a reflexive Banach space E and x ∈ E . Let f : E → R be a Gâteaux differentiable and totally convex function. Then,

1. z = P C f ( x ) if and only if ⟨ ∇ f ( x ) − ∇ f ( z ) , y − z ⟩ ≤ 0 , ∀ y ∈ C .

2. D f ( y , P C f ( x ) ) + D f ( P C f ( x ) , x ) ≤ D f ( y , x ) , ∀ y ∈ C .

Lemma 2.12

[28] Let C be a nonempty, closed, and convex subset of a reflexive Banach space E and f : C → R be a convex and subdifferentiable function on C. Then, f attains its minimum at x ∈ C if and only if 0 ∈ ∂ f ( x ) + N C ( x ) , where N C ( x ) is the normal cone of C at x , that is,

Lemma 2.13

[29] If f and g are two convex functions on E such that there is a point x 0 ∈ dom f ∩ dom g where f is continuous, then

Theorem 3.1

Let C and Q be nonempty, closed, convex, and bounded subsets of reflexive Banach spaces E 1 and E 2 with duals E 1 ∗ and E 2 ∗ , respectively. Let f 1 : E 1 → ( − ∞ , + ∞ ] and f 2 : E 2 → ( − ∞ , + ∞ ] be strongly coercive Legendre functions that are bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E 1 and E 2 , respectively. For i = 1 , 2 , … , N , let { T i } i = 1 N be a finite family of continuous Bregman quasi-asymptotically nonexpansive mappings of C into itself, and for j = 1 , 2 , … , m , let g j : C × C → R be a bifunction satisfying (L1)–(L5). Let A : E 1 → E 2 be a bounded linear operator and K n be the Bregman K -mapping generated by T 1 , T 2 , T 3 , … , T N with Γ ≔ { x ∗ ∈ ⋂ i = 1 N F ( T i ) ⋂ ⋂ j = 1 m E P ( g j ) : A x ∗ ∈ Q } . Let the sequence { x n } be generated iteratively as follows:

where the step-size ρ n is chosen as follows:

P C n + 1 f 1 is the Bregman projection of E 1 onto C n + 1 and σ n ∈ ( 0 , 1 ) is defined in such a way that ∑ n = 1 ∞ ρ n = ∞ . Assume that the sequences { α n } , { λ n } , and { μ n } satisfy the following conditions:

1. liminf n → ∞ α n ( 1 − α n ) > 0 ;

2. { λ n } ⊂ [ b , d ] ⊂ ( 0 , p ) , where p = min 1 c 1 , 1 c 2 ; where c 1 = max 1 ≤ j ≤ m c j , 1 , c 2 = max 1 ≤ j ≤ m c j , 2 , where c j , 1 and c j , 2 are the Bregman-Lipschitz coefficients;

3. μ n = ( 1 − α n ) t n sup x ∗ ∈ Γ D f 1 ( x ∗ , x n ) , where t n ∈ ( 0 , 1 ) .

Proof

We divide our proof into steps.

Step 1: First, we show that Γ is closed and convex. Indeed, from Lemma 2.5 (ii), we have that F ( K n ) is closed and convex. Also, from Lemma 2.2, we have that E P ( g j ) is closed and convex for j = 1 , 2 , … , m . Consequently, Γ is well defined. Next, we prove by induction that C n is closed and convex. We have from (7) that C 1 = C is closed and convex, which implies that C 1 is closed and convex. Now, suppose that C k is closed and convex for each k ∈ N , then we have that

It follows that C k + 1 is closed and convex. Therefore, we conclude that C n is closed and convex for all n ≥ 1 . Thus, iterative scheme (7) is well-defined.

Step 2: We prove that Γ ⊂ C n for all n ≥ 1 .

Clearly, Γ ⊂ C 1 = C . Assume that Γ ⊂ C k for some k ∈ N x ∗ ∈ Γ , then x ∗ ∈ C k .

Now, using (7), condition (ii) of (7), and Lemma 2.7, we have that