Inertial shrinking projection algorithm with self-adaptive step size for split generalized equilibrium and fixed point problems for a countable family of nonexpansive multivalued mappings

Definition 2.1

A single-valued mapping S : C → C is said to be

• nonexpansive, if and only if

  ∥ S x − S y ∥ ≤ ∥ x − y ∥ , ∀ x , y ∈ C ;

• δ -inverse strongly monotone [36], if there exists a positive real number δ such that

  ⟨ x − y , S x − S y ⟩ ≥ δ ∥ S x − S y ∥ 2 , ∀ x , y ∈ C ;

• monotone, if and only if

  ⟨ y − x , S y − S x ⟩ ≥ 0 , ∀ x , y ∈ C .

A subset K of H is called proximal if for each x ∈ H , there exists y ∈ K such that

We denote by C B ( C ) , C C ( C ) , K ( C ) , and P ( C ) the families of all nonempty closed bounded subsets of C , nonempty closed convex subset of C , nonempty compact subsets of C , and nonempty proximal bounded subsets of C , respectively. The Pompeiu-Hausdorff metric on C B ( C ) is defined by

for all A , B ∈ C B ( C ) . Let S : C → 2 C be a multivalued mapping. We say that S satisfies the endpoint condition if S p = { p } for all p ∈ F ( S ) . For multivalued mappings S i : C → 2 C ( i ∈ N ) with ⋂ i = 1 ∞ F ( S i ) ≠ ∅ , we say S i satisfies the common endpoint condition if S i ( p ) = { p } for all i ∈ N , p ∈ ⋂ i = 1 ∞ F ( S i ) . We recall some basic and useful definitions on multivalued mappings.

Definition 2.2

A multivalued mapping S : C → C B ( C ) is said to be nonexpansive if

The class of nonexpansive multivalued mappings contains the class of nonexpansive single-valued mappings. If S is a nonexpansive single-valued mapping on a closed convex subset of a Hilbert space, then F ( S ) is closed and convex. The closedness of F ( S ) can easily be extended to the multivalued case. However, the convexity of F ( S ) cannot be extended (see, e.g., [37]). But, if S is a nonexpansive multivalued mapping which satisfies the endpoint condition, then F ( S ) is always closed and convex as shown by the following result:

Lemma 2.3

[38] Let C be a nonempty closed convex subset of a real Hilbert space H . Let S : C → C B ( C ) be a nonexpansive multivalued mapping with F ( S ) ≠ ∅ and S p = { p } for each p ∈ F ( S ) . Then, F ( S ) is a closed and convex subset of C .

Lemma 2.4

[40] In a real Hilbert space H , the following inequalities hold for all x , y ∈ H :

1. ∥ x + y ∥ 2 ≤ ∥ x ∥ 2 + 2 ⟨ y , x + y ⟩ ;

2. ∥ x + y ∥ 2 = ∥ x ∥ 2 + 2 ⟨ x , y ⟩ + ∥ y ∥ 2 ;

3. ∥ x − y ∥ 2 = ∥ x ∥ 2 − 2 ⟨ x , y ⟩ + ∥ y ∥ 2 .

Lemma 2.5

[41] Let H be a Hilbert space, { x n } be a sequence in H , and α 1 , α 2 , … , α N be real numbers such that ∑ i = 1 N α i = 1 . Then,

Lemma 2.6

[42] Let H be a Hilbert space, and let { x n } be a sequence in H . Let u , v ∈ H be such that lim n → ∞ ∥ x n − u ∥ and lim n → ∞ ∥ x n − v ∥ exist. If { x n k } and { x m k } are subsequences of { x n } that converge weakly to u and v respectively, then u = v .

Lemma 2.7

[43] Let C be a nonempty closed convex subset of a real Hilbert space H . Given x , y , z ∈ H and a real number α , the set { u ∈ C : ∥ y − u ∥ 2 ≤ ∥ x − u ∥ 2 + ⟨ z , u ⟩ + α } is closed and convex.

Lemma 2.8

[44,45] Let C be a nonempty closed convex subset of a real Hilbert space H , and let P C : H → C be the metric projection. Then,

Assumption 2.9

Let C be a nonempty closed and convex subset of a Hilbert space H 1 . Let F 1 : C × C → R and ϕ 1 : C × C → R be two bifunctions satisfying the following conditions:

1. F 1 ( x , x ) = 0 for all x ∈ C ,

2. F 1 is monotone, that is, F 1 ( x , y ) + F 1 ( y , x ) ≤ 0 for all x , y ∈ C ,

3. F 1 is upper hemicontinuous, that is, for all x , y , z ∈ C , lim t ↓ 0 F 1 ( t z + ( 1 − t ) x , y ) ≤ F 1 ( x , y ) ,

4. for each x ∈ C , y ↦ F 1 ( x , y ) is convex and lower semicontinuous,

5. ϕ 1 ( x , x ) ≥ 0 for all x ∈ C ,

6. for each y ∈ C , x ↦ ϕ 1 ( x , y ) is upper semicontinuous,

7. for each x ∈ C ↦ ϕ 1 ( x , y ) is convex and lower semicontinuous,

Lemma 2.10

[46] Let C be a nonempty closed and convex subset of a Hilbert space H 1 . Let F 1 : C × C → R and ϕ 1 : C × C → R be two bifunctions satisfying Assumption 2.9. Assume that ϕ 1 is monotone. For r > 0 and x ∈ H 1 , define a mapping T r ( F 1 , ϕ 1 ) : H 1 → C as follows:

Lemma 2.11

[47] Let C be a nonempty closed and convex subset of a Hilbert space H 1 . Let F 1 : C × C → R and ϕ 1 : C × C → R be two bifunctions satisfying Assumption 2.9, and let T r ( F 1 , ϕ 1 ) be defined as in Lemma 2.10 for r > 0 . Let x , y ∈ H 1 and r 1 , r 2 > 0 . Then,

Theorem 3.1

Let C and Q be nonempty closed convex subsets of real Hilbert spaces H 1 and H 2 , respectively. Let A : H 1 → H 2 be a bounded linear operator, and let { S i } be a countable family of nonexpansive multivalued mappings of C into C B ( C ) . Let F 1 , ϕ 1 : C × C → R , F 2 , ϕ 2 : Q × Q → R be bifunctions satisfying Assumption 2.9. Let ϕ 1 , ϕ 2 be monotone, ϕ 1 be upper hemicontinuous, and F 2 and ϕ 2 be upper semicontinuous in the first argument. Assume that Ω = ⋂ i = 1 ∞ F ( S i ) ∩ SGEP ( F 1 , ϕ 1 , F 2 , ϕ 2 ) ≠ ∅ and S i p = { p } for each p ∈ ⋂ i = 1 ∞ F ( S i ) . Let x 0 , x 1 ∈ C with C 1 = C , and let { x n } be a sequence generated as follows:

where 0 < a ≤ τ n ≤ b < 1 , { θ n } ⊂ R , { α n , i } ⊂ ( 0 , 1 ) , such that ∑ i = 0 n α n , i = 1 , and { r n } ⊂ ( 0 , ∞ ) . Suppose that the following conditions hold:

1. lim inf n → ∞ r n > 0 ,

2. the limits lim n → ∞ α n , i ∈ ( 0 , 1 ) exist for all i ≥ 0 .

Proof

We divide the proof into several steps as follows:

Step 1: First, we show that { x n } is well-defined for every n ∈ N .

By Lemmas 2.3 and 2.10, we have that SGEP ( F 1 , ϕ 1 , F 2 , ϕ 2 ) and ⋂ i = 1 ∞ F ( S i ) are closed and convex subsets of C . Therefore, the solution set Ω is a closed and convex subset of C . By Lemma 2.7, it then follows that C n + 1 is closed and convex for each n ∈ N . Let p ∈ Ω , then we have p = T r n F 1 , ϕ 1 p and A p = T r n ( F 2 , ϕ 2 ) ( A p ) . Since T r n ( F 1 , ϕ 1 ) is nonexpansive, by Lemma 2.4, we have

By the firmly nonexpansivity of I − T r n ( F 2 , ϕ 2 ) , we get

By substituting (17) into (16), applying the definition of γ n and the condition on τ n , we obtain

Applying Lemma 2.5 and using (19), we have

Also, by applying Lemma 2.4(iii), we get