A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems

Algorithm 1.1

where A is a monotone and a Lipschitz continuous operator and λ ∈ 0 , 1 L . Clearly, the TEgM requires one projection to be computed per iteration and hence has an advantage in computing projection over the EgM.

Algorithm 1.2

(SEgM)

Algorithm 1.3

Let x 1 ∈ H , λ 1 > 0 and μ ∈ ( 0 , 1 ) . Compute x n + 1 as follows:

Step 1. Calculate

Step 2. Compute

where T n = { x ∈ H : ⟨ x n − λ n f x n − y n , x − y n ⟩ ≤ 0 } .

Step 3. Compute

and

Set n ≔ n + 1 and go to Step 1.

Theorem 1.4

Let A : H 1 → H 2 be a bounded linear operator and f : H 1 → H 1 and g : H 2 → H 2 , be, respectively, α 1 - and α 2 -inverse strongly monotone operators with α ≔ min { α 1 , α 2 } . Assume that SVIP (equations (8) and (9)) is consistent, τ ∈ 0 , 1 L with L being the spectral radius of the operator A ∗ A , λ ∈ ( 0 , 2 α ) , and suppose that for all x ∗ solving SVIP (equations (8) and (9)),

Then the sequence { x n } generated by (10) converges weakly to a solution of SVIP (equations (8) and (9)).

Theorem 1.5

Let H 1 and H 2 be real Hilbert spaces, and let C be a nonempty closed convex subset of H 1 . Let A : H 1 → H 2 be a bounded linear operator such that A ≠ 0 , and S : H 2 → H 2 be a nonexpansive mapping. Let f : C → H 1 be a monotone and L-Lipschitz continuous mapping. Suppose that Γ = { z ∈ VI ( C , f ) : A z ∈ F ( S ) } ≠ ∅ and the sequences { x n } is defined for arbitrary x 1 ∈ C by (13), where { τ n } ⊂ [ a , b ] for some a , b ∈ 0 , 1 ‖ A ‖ 2 and { λ n } ⊂ [ c , d ] for some c , d ∈ 0 , 1 L . Then { x n } converges weakly.

Algorithm 1.6

Initialization: Let γ > 0 , l , μ ∈ ( 0 , 1 ) and x 1 ∈ H 1 be given arbitrary.

Iterative Steps: Calculate x n + 1 as follows:

Step 1. Compute

where 0 < a ≤ τ n ≤ b < 1 ‖ A ‖ 2 , T β = β I + ( 1 − β ) S , and λ n is chosen to be the largest λ ∈ { γ , γ ł , γ ł 2 , … } satisfying

Step 2. Compute

z n = y n − λ n ( f y n − f w n ) .

Set n ≔ n + 1 and go back to Step 1.

Theorem 1.7

Let { x n } be a sequence generated by Algorithm 1.6. Assume that lim n → ∞ α n = 0 and ∑ n = 1 ∞ α n = + ∞ . Then { x n } converges strongly to z = P Γ g ( z ) , where Γ = { z ∈ VI ( C , f ) : A z ∈ F ( S ) } ≠ ∅ .

Definition 2.1

Let T : C → C be a mapping. Then, T is called

1. L -Lipschitz continuous, if there exists a constant L > 0 such that

   ‖ T x − T y ‖ ≤ L ‖ x − y ‖ ∀ x , y ∈ C ;

   If L ∈ [ 0 , 1 ) , then T is a contraction mapping and it is nonexpansive if L = 1 ;

2. quasi-nonexpansive, if F ( T ) ≠ ∅ and

   ‖ T x − p ‖ ≤ ‖ x − p ‖ ∀ x ∈ C and p ∈ F ( T ) ;

3. k -strictly pseudocontractive mapping, if there exists κ ∈ [ 0 , 1 ) such that

   ‖ T x − T y ‖ 2 ≤ ‖ x − y ‖ 2 + κ ‖ ( I − T ) x − ( I − T ) y ‖ 2 ∀ x , y ∈ C ;

   if k = 1 , then T is pseudocontractive;

4. monotone, if

   ⟨ T x − T y , x − y ⟩ ≥ 0 ∀ x , y ∈ C ;

5. α -inverse strongly monotone ( α -ism) (or α -co-coercive), if there exists α > 0 such that

   ⟨ T x − T y , x − y ⟩ ≥ α ‖ T x − T y ‖ 2 ∀ x , y ∈ C ;

   if α = 1 , T is firmly nonexpansive;

6. β -strongly monotone, if there exists β > 0 such that

   ⟨ T x − T y , x − y ⟩ ≥ β ‖ x − y ‖ 2 ∀ x , y ∈ C ;

7. pseudomonotone, if

   ⟨ T x , y − x ⟩ ≥ 0 ⇒ ⟨ T y , y − x ⟩ ≥ 0 ∀ x , y ∈ C ;

8. α -averaged, if T = ( 1 − α ) I + α S , where α ∈ ( 0 , 1 ) and S : C → C is nonexpansive, see [29];

9. uniformly continuous, if for every ε > 0 , there exists δ = δ ( ε ) > 0 such that

   ‖ T x − T y ‖ < ε whenever ‖ x − y ‖ < δ ∀ x , y ∈ C .

Definition 2.2

An operator T : C → C is said to be quasi-pseudocontractive, if F ( T ) ≠ ∅ and

Clearly, the class of quasi-pseudocontractive mappings includes the class of pseudo-contractive mappings with nonempty fixed points set, and it contains several other classes of mappings.

Lemma 2.3

[30] Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Given x ∈ H and z ∈ C . Then, z = P C x ⇔ ⟨ x − z , z − y ⟩ ≥ 0 ∀ y ∈ C .

Lemma 2.4

[25] Let { α n } be a sequence of nonnegative real numbers satisfying

where { α n } , { λ n } , and { δ n } satisfy the following conditions:

1. { α n } ⊂ [ 0 , 1 ] , ∑ n = 0 ∞ α n = ∞ ;

2. limsup n → ∞ λ n ≤ 0 ;

3. δ n ≥ 0 ( n ≥ 0 ) , ∑ n = 0 ∞ δ n < ∞ .

Lemma 2.5

[34] Let H be a real Hilbert space and S : H → H be a mapping with L ≥ 1 . Denote

If 0 < Φ < η < 1 1 + 1 + L 2 , then the following hold:

1. F ( S ) = F ( S ( ( I − η ) I + η T ) ) = F ( T ) ;

2. If I − S is demiclosed at zero, then I − T is also demiclosed at zero;

3. In addition, if S : H → H is quasi-pseudocontractive mapping, then the mapping T is quasi-nonexpansive.

Lemma 2.6

[25,35] Let H be a real Hilbert space, then, for all x , y ∈ H and β ∈ R , the following hold:

1. ‖ β x + ( 1 − β ) y ‖ 2 = β ‖ x ‖ 2 + ( 1 − β ) ‖ y ‖ 2 − β ( 1 − β ) ‖ x − y ‖ 2 ;

2. ‖ x + y ‖ 2 ≤ ‖ x ‖ 2 + 2 ⟨ y , x − y ⟩ ;

3. ‖ x + y ‖ 2 = ‖ x ‖ 2 + 2 ⟨ x , y ⟩ + ‖ y ‖ 2 .

Lemma 2.7

[36] Let { Γ n } be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence { Γ n j } j ≥ 0 of { Γ n } such that

Furthermore, consider the sequence of integers { τ ( n ) } n ≥ n 0 defined by

Then, { Γ n } n ≥ n 0 is a nondecreasing sequence such that τ ( n ) → ∞ as n → 0 , and for all n ≥ n 0 , the following hold:

Lemma 2.8

[37] Suppose { λ n } and { θ n } are two nonnegative real sequences such that

If ∑ n = 1 ∞ ϕ n < ∞ , then lim n → ∞ λ n exists.

Lemma 2.9

[38] Let H be a real Hilbert space and T : H → H be a nonexpansive mapping with F ( T ) ≠ ∅ . If { x n } is a sequence in H converging weakly to x ∗ and if { ( I − T ) x n } converges strongly to y , then ( I − T ) x ∗ = y .

Lemma 2.10

[39] Let C be a nonempty, closed, and convex of real Hilbert space H. Let operator f : C → H be continuous and pseudomonotone. Then, x ∗ is a solution of VI ( C , f ) if and only if ⟨ f x , x − x ∗ ⟩ ≥ 0 ∀ x ∈ C .

Algorithm 3.1

Step 0: Select x 0 , x 1 ∈ H , λ 1 > 0 , μ ∈ ( 0 , 1 ) , and 0 < ϕ 1 ≤ ϕ n ≤ ϕ 2 < 1 and set n = 1 .

Step 1: Given the ( n − 1 )th and nth iterates, choose θ n such that 0 ≤ θ n ≤ θ ˜ n with θ ˜ n defined as follows:

Step 2: Compute

Step 3: Compute

where

Step 4: Compute

Step 5: Compute

where

and

Set n ≔ n + 1 and go to Step 1.

Remark 3.2

1. By conditions (C1) and (C3), it follows from (63) that

   (23) lim n → ∞ θ n ‖ x n − x n − 1 ‖ = 0 and lim n → ∞ θ n α n ‖ x n − x n − 1 ‖ = 0 .

2. Observe that by Lemma 2.5, the mapping T is quasi-nonexpansive and I − T is demiclosed at zero.

Remark 3.3

1. We point out that condition (C2)(a) is a much weaker assumption than the sequential weakly continuity assumption used in several of the existing results in the literature.

2. Observe that while the pseudomonotone operator f is not necessarily Lipschitz, our proposed method does not require any linesearch technique but uses a simple step size rule in (67), which generates a nonmonotonic sequence of step sizes. The step size is constructed such that it reduces the dependence of the algorithm on the initial step size λ 1 .

Lemma 4.1

Let { λ n } be the sequence of step sizes generated by Algorithm 3.1. Then, { λ n } is well defined and lim n → ∞ λ n = λ ∈ min μ N , λ 1 , λ 1 + Ψ , where Ψ = ∑ n = 1 ∞ ψ n and for some N > 0 .

Proof

Since f is uniformly continuous, then by (14) it follows that for any given ε > 0 , there exists K < + ∞ such that ‖ f z n − f y n ‖ ≤ K ‖ z n − y n ‖ + ε . Thus, for the case f z n − f y n ≠ 0 for all n ≥ 1 , we have

where ε = ε 1 ‖ z n − y n ‖ for some ε 1 ∈ ( 0 , 1 ) and N = K + ε 1 . Therefore, by the definition of λ n + 1 , the sequence { λ n } has lower bound min { μ N , λ 1 } and has upper bound λ 1 + Ψ . By Lemma 2.8, the limit lim n → ∞ λ n exists and denoted by λ = lim n → ∞ λ n . Clearly, λ ∈ [ min { μ N , λ 1 } , λ 1 + Ψ ] . □

Lemma 4.2

Let { x n } be a sequence generated by Algorithm 3.1. Then, { x n } is bounded. Furthermore, if lim n → ∞ α n = 0 , then lim n → ∞ ‖ x n + 1 − u n ‖ = 0 .

Proof

By (67), we have

which implies that

Let q ∈ Γ . By applying the triangle inequality, we obtain

Since, by (23), lim n → ∞ θ n α n ‖ x n − x n − 1 ‖ = 0 , then there exists a constant M 1 > 0 such that θ n α n ‖ x n − x n − 1 ‖ ≤ M 1 ∀ n ≥ 1 . Hence, it follows from (25) that

By applying Lemma 2.6(iii) and Cauchy-Schwarz inequality, we obtain

where M ≔ sup n ∈ N { ‖ x n − p ‖ , θ ‖ x n − x n − 1 ‖ } .

Next, by the definition of y n with the firmly nonexpansivity of P C , we obtain