Convergence analysis

3.3 Common solution of the variational inequality and the fixed point problems 59

3.3.2 Convergence analysis

Lemma 3.3.5. Let {x_n} be a sequence generated by Algorithm 3.3.2 under Assumption 3.3.1. Then

∥z_n−p∥² ≤ ∥w_n−p∥²−

1−λ²_nµ² λ²_n+1

∥y_n−w_n∥².

Proof. Letp∈Γ. Sinceyn=PCn(wn−λnwn),we obtain by the characteristic property of PC_n that ⟨y_n−w_n+λ_nAw_n, y_n−p⟩ ≤0 and this implies that

⟨y_n−w_n, y_n−p⟩ ≤ −λ_n⟨Aw_n, y_n−p⟩. (3.3.4) Also, from the definition of z_n inStep 4, and Lemma2.1.1, we have

∥z_n−p∥² ≤ ∥y_n−λ_n(Ay_n−Aw_n)−p∥²

=∥y_n−p∥²+λ²_n∥Ay_n−Aw_n∥²−2λ_n⟨Ay_n−Aw_n, y_n−p⟩

=∥w_n−p∥²+∥y_n−w_n∥²+ 2⟨y_n−w_n, w_n−p⟩+λ²_n∥Ay_n−Aw_n∥²

−2λ_n⟨Ay_n−Aw_n, y_n−p⟩

=∥wn−p∥²+∥yn−wn∥²+λ²_n∥Ayn−Awn∥²−2⟨yn−wn, yn−wn⟩ + 2⟨y_n−w_n, y_n−p⟩

−2λ_n⟨Ay_n−Aw_n, y_n−p⟩

=∥w_n−p∥²− ∥y_n−w_n∥²+λ²_n∥Ay_n−Aw_n∥²+ 2⟨y_n−w_n, y_n−p⟩

−2λ_n⟨Ay_n−Aw_n, y_n−p⟩ (3.3.5)

From (3.3.4) and (3.3.5), we obtain

∥z_n−p∥² ≤ ∥w_n−p∥²− ∥y_n−w_n∥²+λ²_n∥Ay_n−Aw_n∥²−2λ_n⟨Aw_n, y_n−p⟩

−2λ_n⟨Ay_n−Aw_n, y_n−p⟩

=∥w_n−p∥²− ∥y_n−w_n∥²+λ²_n∥Ay_n−Aw_n∥²−2λ_n⟨Ay_n, y_n−p⟩

≤ ∥w_n−p∥²− ∥y_n−w_n∥²+λ²_n µ²

λ²_n+1∥y_n−w_n∥²−2λ_n⟨Ay_n, y_n−p⟩

≤ ∥w_n−p∥²−

1−λ²_nµ² λ²_n+1

∥y_n−w_n∥². (3.3.6)

Consider the limit

n→∞lim

1− λ²_nµ² λ²_n+1

= 1−µ² >0.

Hence there exists n₀ ≥ 0 such that for all n ≥ n₀, we have that

1− ^λ_λ²ⁿ2^µ2 n+1

> 0. Thus, from (3.3.6), we have that

∥z_n−p∥ ≤ ∥w_n−p∥.

Lemma 3.3.6. Let {x_n} be a sequence generated by Algorithm 3.3.2 under Assumption 3.3.1. Then, {x_n} is bounded.

Proof. First, we show that P_Γ(I −D+γ₁f) is a contraction of H. For all x, y ∈ H, we have

∥PΓ(I−D+γ1f)x−PΓ(I −D+γ1f)y∥ ≤ ∥(I−D+γ1f)x−(I−D+γ1f)y∥

≤ ∥(I−D)x−(I −D)y∥+γ₁∥f x−f y∥

≤(1−γ₂)∥x−y∥+γ₁ρ∥x−y∥

= (1−(γ₂−γ₁ρ))∥x−y∥.

Hence, P_Γ(I −D+γ₁f) is a contraction. Let p ∈ Γ. Then from the definition of w_n in Step 2, we have

∥w_n−p∥=∥x_n+α_n(x_n−xn−1)−p∥

=∥x_n−p∥+α_n∥x_n−xn−1∥.

Also, from Step 2, we observe that α_n∥x_n−xn−1∥ ≤τ_n,∀n≥1, which implies that αn

θ_n∥xn−xn−1∥ ≤ τn

θ_n →0, as n→ ∞. (3.3.7)

Hence, there exists M1 >0 such that α_n

θ_n∥x_n−x_n−1∥ ≤M₁, ∀n ≥1. (3.3.8)

This implies that ∥w_n−p∥ ≤ ∥x_n−p∥+θ_nM₁,∀n≥1 and hence

∥z_n−p∥ ≤ ∥w_n−p∥ ≤ ∥x_n−p∥+θ_nM₁. (3.3.9) From Step 5, we have

∥T_nz_n−p∥ ≤ ∥(1−β_n)z_n+β_nW_nz_n−p∥

≤(1−β_n)∥z_n−p∥+β_n∥W_nz_n−p∥

≤(1−βn)∥zn−p∥+βn∥zn−p∥

=∥z_n−p∥ (3.3.10)

From (3.3.9) and (3.3.10), we obtain for all n ≥n₀,

∥x_n+1−p∥ ≤ ∥θ_nγ₁f x_n+ (1−θ_nD)T_nz_n−p∥

=∥θ_n(γ₁f x_n−Dp) + (1−θ_nD)(T_nz_n−p)∥

≤θn∥γ1f xn−Dp∥+ (1−θnγ2)∥Tnzn−p∥

≤θ_n∥γ₁(f x_n−f p) + (γ₁f p−Dp)∥+ (1−θ_nγ₂)∥z_n−p∥

≤θ_nγ₁ρ∥x_n−p∥+θ_n∥γ₁f p−Dp∥+ (1−θ_nγ₂)(∥x_n−p∥+θ_nM₁)

≤(1−θ_n(γ₂−γ₁ρ))∥x_n−p∥+θ_n∥γ₁f p−Dp∥+θ_nM₁

= (1−θ_n(γ₂−γ₁ρ))∥x_n−p∥+θ_n(γ₂−γ₁ρ)∥γ₁f p−Dp∥+M₁ γ₂−γ₁ρ

≤maxn

∥x_n−p∥,∥γ₁f p−Dp∥+M₁ γ₂−γ₁ρ

o

≤maxn

∥x_n0 −p∥,∥γ1f p−Dp∥+M1

γ₂ −γ₁ρ

o

Hence, the sequence{x_n}is bounded. Consequently,{w_n},{y_n}and{z_n}are also bounded.

Lemma 3.3.7. Assume that {w_n} and {y_n} are sequences generated by Algorithm 3.3.2 such that

n→∞lim ||w_n −y_n|| = 0. If {w_nj} converges weakly to some xˆ ∈ H as j → ∞, then xˆ ∈ V I(C, A).

Proof. Since w_nj ⇀ x,ˆ then by the hypothesis of the lemma it follows that y_nj ⇀ xˆ as j → ∞. Also, since y_nj ∈ C_nj, then by the definition of C_n we get

h(w_nj) +⟨ξ_nj, y_nj −w_nj⟩ ≤0. (3.3.11) By the boundedness of {wn} and by condition (A3), there exists a constant M > 0 such that||ξ_nj|| ≤M for all j ≥0.Then, from (3.3.11) we obtainh(w_nj)≤M||w_nj−y_nj|| → 0 as j → ∞,and this in turn implies that lim inf

j→∞ h(w_nj)≤0.Applying condition (A2), we haveh(ˆx)≤lim inf

j→∞ h(w_nj)≤0.This implies that ˆx∈ C.From Lemma2.4.1, we obtain

⟨y_nj −w_nj +λ_njAw_nj, z−y_nj⟩ ≥0, ∀ z ∈ C ⊆ C_nj.

Since A is monotone, we have

0≤ ⟨y_nj −w_nj, z−y_nj⟩+λ_nj⟨Aw_nj, z−y_nj⟩

=⟨y_nj−w_nj, z−y_nj⟩+λ_nj⟨Aw_nj, z−w_nj⟩+λ_nj⟨Aw_nj, w_nj −y_nj⟩

≤ ⟨y_nj −w_nj, z−y_nj⟩+λ_nj⟨Az, z−w_nj⟩+λ_nj⟨Aw_nj, w_nj−y_nj⟩.

Letting j → ∞, and since lim

j→∞||y_nj −w_nj||= 0, we have

⟨Az, z−x⟩ ≥ˆ 0, ∀ z ∈ C.

Applying Lemma 3.2.2, we have that ˆx∈V I(C, A).

Lemma 3.3.8. Let {xn} be a sequence generated by Algorithm 3.3.2 under Assumption 3.3.1. Then,

∥x_n+1−p∥² ≤(1−η_n)∥x_n−p∥² +η_nh θ_nγ₂²

2(γ₂−γ₁ρ)M₃+ 3M₂(1−θ_nγ₂)² 2(γ₂−γ₁ρ)

α_n

θ_n||x_n−xn−1||

+ 1

(γ2−γ1ρ)⟨γ₁f p−Dp, x_n+1−p⟩i

− (1−θ_nγ₂)² (1−θnγ1ρ)

h

1− λ²_nµ² λ²_n+1

∥y_n−w_n∥² +β_n(1−β_n)∥W_nz_n−z_n∥²i

, where η_n = ^2θ_(1−θ^{n(γ2−γ1ρ)}

nγ1ρ) .

Proof. Let p ∈ Γ. Then from Step 2, by applying the Cauchy-Schwartz inequality and Lemma 2.1.1, we obtain

||w_n−p||² =||x_n+α_n(x_n−xn−1)−p||²

=||x_n−p||²+α²_n||x_n−xn−1||²+ 2α_n⟨x_n−p, x_n−xn−1⟩

≤ ||x_n−p||²+α²_n||x_n−xn−1||²+ 2α_n||x_n−xn−1||||x_n−p||

=||x_n−p||²+α_n||x_n−x_n−1||(α_n||x_n−x_n−1||+ 2||x_n−p||)

≤ ||xn−p||²+ 3M2αn||xn−xn−1||

=||x_n−p||²+ 3M₂θ_nα_n

θ_n||x_n−xn−1||, (3.3.12) where M₂ := sup_n∈

N{||x_n−p||, α_n||x_n−x_n−1||}>0.

Now, applying the last inequality Lemma 2.1.1 and (3.3.6), we have

∥T_nz_n−p∥² =∥(1−β_n)(z_n−p) +β_n(W_nz_n−p)∥²

≤(1−β_n)∥z_n−p∥²+β_n∥W_nz_n−p∥² −β_n(1−β_n)∥W_nz_n−z_n∥²

≤(1−βn)∥zn−p∥²+βn∥zn−p∥² −βn(1−βn)∥Wnzn−zn∥²

=∥z_n−p∥²−β_n(1−β_n)∥W_nz_n−z_n∥²

≤ ∥wn−p∥²−

1−λ²_nµ² λ²_n+1

∥yn−wn∥²−βn(1−βn)∥Wnzn−zn∥²

≤ ||x_n−p||²+ 3M₂θ_nαn

θ_n||x_n−xn−1|| −

1− λ²_nµ² λ²_n+1

∥y_n−w_n∥²

−β_n(1−β_n)∥W_nz_n−z_n∥². (3.3.13)

Also, by applying Lemma 2.1.1 and (3.3.13), we have

∥x_n+1−p∥² =∥θ_nγ₁f x_n+ (1−θ_nD)T_nz_n−p∥²

=∥θ_n(γ₁f x_n−Dp) + (1−θ_nD)(T_nz_n−p)∥²

≤(1−θ_nγ₂)²∥T_nz_n−p∥²+ 2θ_n⟨γ₁f x_n−Dp, x_n+1−p⟩

≤(1−θ_nγ₂)²∥T_nz_n−p∥²+ 2θ_nγ₁⟨f x_n−f p, x_n+1−p⟩

+ 2θn⟨γ1f p−Dp, xn+1−p⟩

≤(1−θ_nγ₂)²h

∥x_n−p∥²+ 3M₂θ_nαn

θ_n||x_n−xn−1||

−

1− λ²_nµ² λ²_n+1

∥y_n−w_n∥²−β_n(1−β_n)∥W_nz_n−z_n∥²i + 2θ_n⟨γ₁f x_n−f p, x_n+1−p⟩+ 2θ_n⟨γ₁f p−Dp, x_n+1−p⟩

≤(1−θ_nγ₂)²∥x_n−p∥²+ 3M₂(1−θ_nγ₂)²θ_nα_n

θ_n||x_n−x_n−1||

−(1−θ_nγ₂)²

1−λ²_nµ² λ²_n+1

∥y_n−w_n∥²−(1−θ_nγ₂)²β_n(1−β_n)∥W_nz_n−z_n∥² + 2θ_nγ₁⟨f x_n−f p, x_n+1−p⟩+ 2θ_n⟨γ₁f p−Dp, x_n+1−p⟩

≤(1−θ_nγ₂)²∥x_n−p∥²+ 3M₂(1−θ_nγ₂)²θ_nα_n θn

||x_n−xn−1||

−(1−θnγ2)²

1−λ²_nµ² λ²_n+1

∥yn−wn∥²

−(1−θnγ2)²βn(1−βn)∥Wnzn−zn∥²+ 2θnγ1ρ∥xn−p∥∥xn+1−p∥

+ 2θ_n⟨γ₁f p−Dp, x_n+1−p⟩

≤(1−θ_nγ₂)²∥x_n−p∥²+ 3M₂(1−θ_nγ₂)²θ_nα_n

θ_n||x_n−xn−1||

−(1−θ_nγ₂)²h

1− λ²_nµ² λ²_n+1

∥y_n−w_n∥²+β_n(1−β_n)∥W_nz_n−z_n∥²i +θ_nγ₁ρ

∥x_n−p∥²+∥x_n+1−p∥²

+ 2θ_n⟨γ₁f p−Dp, x_n+1−p⟩

= ((1−θ_nγ₂)²+θ_nγ₁ρ)∥x_n−p∥²+θ_nγ₁ρ∥x_n+1−p∥² + 3M2(1−θnγ2)²θn

α_n

θ_n||xn−xn−1||+ 2θn⟨γ1f p−Dp, xn+1−p⟩

−(1−θ_nγ₂)²h

1− λ²_nµ² λ²_n+1

∥y_n−w_n∥²+β_n(1−β_n)∥W_nz_n−z_n∥²i .

From this, we obtain

∥x_n+1−p∥² ≤ (1−2θ_nγ₂+ (θ_nγ₂)²+θ_nγ₁ρ)

(1−θnγ1ρ) ∥x_n−p∥²

+ θn

(1−θ_nγ₁ρ) h

3M2(1−θnγ2)²αn

θ_n||xn−xn−1||+ 2⟨γ1f p−Dp, xn+1−p⟩i

− (1−θnγ2)² (1−θ_nγ₁ρ)

h

1−λ²_nµ² λ²_n+1

∥y_n−w_n∥²+β_n(1−β_n)∥W_nz_n−z_n∥²i

= (1−2θ_nγ₂+θ_nγ₁ρ)

(1−θ_nγ₁ρ) ∥x_n−p∥²+ (θ_nγ₂)²

(1−θ_nγ₁ρ)∥x_n−p∥²

+ θ_n

(1−θ_nγ₁ρ) h

3M₂(1−θ_nγ₂)²α_n

θ_n||x_n−x_n−1||+ 2⟨γ₁f p−Dp, x_n+1−p⟩i

− (1−θ_nγ₂)² (1−θ_nγ₁ρ)

h

1−λ²_nµ² λ²_n+1

∥y_n−w_n∥²+β_n(1−β_n)∥W_nz_n−z_n∥²i

≤

1− 2θ_n(γ₂−γ₁ρ) (1−θ_nγ₁ρ)

∥x_n−p∥² + 2θ_n(γ₂−γ₁ρ)

(1−θ_nγ₁ρ)

h θ_nγ₂²

2(γ₂−γ₁ρ)M₃+3M₂(1−θ_nγ₂)² 2(γ₂−γ₁ρ)

α_n

θ_n||x_n−xn−1||

+ 1

(γ₂−γ₁ρ)⟨γ1f p−Dp, xn+1−p⟩i

− (1−θnγ2)² (1−θ_nγ₁ρ)

h

1−λ²_nµ² λ²_n+1

∥yn−wn∥²+βn(1−βn)∥Wnzn−zn∥²i , where M₃ = sup{∥x_n−p∥² :n ∈N} and thus we obtain the desired conclusion.

We are now in the position to give the main theorem for Algorithm 3.3.2.

Theorem 3.3.9. Let {x_n} be a sequence generated by Algorithm 3.3.2 under Assumption 3.3.1. Suppose that {W_n} is the sequence defined by (3.2.1). Then, the sequence {x_n} converges strongly to a point x^∗ ∈ Γ, where x^∗ = P_Γ(I−D+γ₁f)x^∗ is a solution of the variational inequality

⟨(D−γ₁f)x^∗, x^∗−x⟩ ≤0, ∀x∈Γ.

Proof. Since x^∗ =P_Γ(I−D+γ₁f)x^∗, we obtain from Lemma3.3.8 that

∥xn+1−x^∗∥² ≤(1−ηn)∥xn−x^∗∥² +η_nh θ_nγ₂²

2(γ₂−γ₁ρ)M₃+3M₂(1−θ_nγ₂)² 2(γ₂−γ₁ρ)

α_n

θ_n||x_n−x_n−1|| (3.3.14)

+ 1

(γ₂−γ₁ρ)⟨γ₁f x^∗−Dx^∗, x_n+1−x^∗⟩i .

Now, we claim that the sequence {∥x_n−x^∗∥} converges to zero. To show this, by Lemma 2.5.36 it suffices to show that lim sup

k→∞

⟨γ₁f x^∗−Dx^∗, x_nk+1−x^∗⟩ ≤0 for every subsequence

{∥x_nk−x^∗∥} of {∥x_n−x^∗∥}satisfying lim inf

k→∞ (∥x_nk+1−x^∗∥ − ∥x_nk −x^∗∥)≥0. (3.3.15) Suppose that {∥x_nk−x^∗∥}is a subsequence of {∥x_n−x^∗∥}such that (3.3.15) holds. From Lemma 3.3.8, we obtain

(1−θ_nkγ₂)² (1−θn_kγ1ρ)

1−λ²_n

kµ² λ²_nk+1

∥y_nk−w_nk∥² ≤(1−η_nk)∥x_nk −x^∗∥²− ∥x_nk+1−x^∗∥² +η_nkh θ_nkγ₂²

2(γ₂−γ₁ρ)M₃ + 3M₂(1−θ_nkγ₂)²

2(γ₂ −γ₁ρ) α_n

θ_nk||xn_k −xn_k−1||

+ 1

(γ2−γ1ρ)⟨γ₁f x^∗−Dx^∗, x_nk+1−x^∗⟩i .

By (3.3.15) and the fact that lim

k→∞ηn_k = 0 (since lim

k→∞θn_k = 0), we obtain (1−θ_nkγ₂)²

(1−θ_nkγ₁ρ)

1− λ²_n

kµ² λ²_nk+1

∥y_nk −w_nk∥² →0, ask → ∞.

Consequently, we have

k→∞lim ∥y_nk −w_nk∥= 0. (3.3.16) Following similar argument, from Lemma 3.3.8 we have

k→∞lim ∥W_nkz_nk −z_nk∥= 0. (3.3.17) From the definition of zn_k inStep 4 and (3.3.16), we have

∥z_nk −w_nk∥=∥y_nk −λ_nk(Ay_nk−Aw_nk)−w_nk∥

≤ ∥y_nk −w_nk∥+λ_nk∥Ay_nk −Aw_nk∥

≤ ∥y_nk −w_nk∥+λ_nk µ

λ_nk+1∥w_nk−y_nk∥

=

1 + λ_nkµ λ_nk+1

∥y_nk −w_nk∥ →0, ask → ∞, which implies that

k→∞lim ∥z_nk−w_nk∥= 0. (3.3.18) From (3.3.16) and (3.3.18) we have

k→∞lim ∥y_nk −z_nk∥= 0. (3.3.19)

Also, from (3.3.17), (3.3.18) and (3.3.19), we get

k→∞lim ∥w_nk−y_nk∥= 0 lim

k→∞∥W_nkz_nk−w_nk∥= 0, lim

k→∞∥W_nkz_nk −y_nk∥= 0.

(3.3.20) Now, from Step 2 and by Remark3.3.3, we get

k→∞lim ∥w_nk −x_nk∥= lim

k→∞α_nk∥x_nk −x_nk−1∥= 0. (3.3.21) From (3.3.18), (3.3.20) and (3.3.21), we obtain

k→∞lim ∥z_nk −x_nk∥= 0, lim

k→∞∥W_nkz_nk−x_nk∥= 0. (3.3.22) By applying (3.3.17), we have

∥T_nkz_nk−z_nk∥=∥(1−β_nk)z_nk+β_nkW_nkz_nk−z_nk∥ (3.3.23)

≤(1−β_nk)∥z_nk−z_nk∥+β_nk∥W_nkz_nk −z_nk∥ →0, k → ∞.

Now, by using (3.3.18), (3.3.21) and (3.3.23) we obtain

k→∞lim ||T_nkz_nk −w_nk||= 0, lim

k→∞||T_nkz_nk−x_nk||= 0. (3.3.24) Consequently, by applying the fact that lim

k→∞θ_nk = 0 we get

∥x_nk+1−x_nk∥=∥θ_nkγ₁f x_nk + (1−θ_nkD)T_nkz_nk−x_nk∥

=∥(θn_kγ1f xn_k−θn_kDxn_k) + (1−θn_kD)(Tn_kzn_k−xn_k)∥

≤θ_nk∥γ₁f x_nk −Dx_nk∥+ (1−θ_nkγ₂)∥T_nkz_nk −x_nk∥ →0, k → ∞.

Hence

k→∞lim ∥x_nk+1−x_nk∥= 0. (3.3.25) Now, we show that w_ω(x_n) ⊂ ∩^∞_i=1F(S_i) = F(W). Let z ∈ w_ω(x_n) and suppose that z /∈F(W),that is, W z ̸=z.From (3.3.22), we have thatw_ω(x_n) =w_ω(z_n) and by Lemma 2.1.15 we have

lim inf

k→∞ ||zn_k −z||<lim inf

k→∞ |zn_k −W z|||

≤lim inf

k→∞ {||z_nk−W z_nk||+||W z_nk −W z||}

≤lim inf

k→∞ {||z_nk−W z_nk||+||z_nk −z||}. (3.3.26) Since x_nk ∈K for all k≥1 and w_ω(x_n) =w_ω(z_n), we obtain

||W z_nk −z_nk|| ≤ ||W z_nk −W_nkz_nk||+||W_nkz_nk −z_nk||

≤ sup

x∈K

||W x−W_nkx||+||W_nkz_nk−z_nk||.

By applying Lemma 3.2.5 and (3.3.17), we have lim

k→∞||W z_nk −z_nk||= 0. Combining this with (3.3.26) yields

lim inf

k→∞ ||z_nk−z||<lim inf

k→∞ ||z_nk −z||,

which is a contradiction. Hence, we have z ∈F(W) =∩^∞_i=1F(S_i), i.e., w_ω(x_n)⊂F(W) = T∞

i=1F(S_i).

Next, we show thatwω(xn)⊂V I(C, A).By invoking Lemma3.3.7, it follows from (3.3.20) that z ∈V I(C, A).Thus, w_ω(x_n)⊂Γ.

From the fact that lim

k→∞∥x_nk−z_nk∥= 0,we have thatw_ω{x_n}=w_ω{z_n}.By Lemma3.3.6, we have that {x_nk} is bounded which implies that there exists a subsequence {x_{nk j}} of {xn_k}such that xn_{k j} ⇀x¯and

j→∞lim⟨γ₁f x^∗−Dx^∗, x_{nk j} −x^∗⟩= lim sup

k→∞

⟨γ₁f x^∗−Dx^∗, x_nk−x^∗⟩

= lim sup

k→∞

⟨γ₁f x^∗−Dx^∗, zn_k−x^∗⟩.

Since x^∗ =P_Γ(I−D+γ₁f)x^∗, we have that lim sup

k→∞

⟨γ₁f x^∗−Dx^∗, x_nk−x^∗⟩= lim

j→∞⟨γ₁f x^∗−Dx^∗, x_{nk j} −x^∗⟩

=⟨γ₁f x^∗ −Dx^∗,x¯−x^∗⟩

≤0. (3.3.27)

From (3.3.25) and (3.3.27), we have lim sup

k→∞

⟨γ₁f x^∗−Dx^∗, x_nk+1−x^∗⟩= lim sup

k→∞

⟨γ₁f x^∗−Dx^∗, x_nk−x^∗⟩

=⟨γ1f x^∗−Dx^∗,x¯−x^∗⟩

≤0. (3.3.28)

By applying Lemma 2.5.36 to (3.3.14) and using (3.3.28) together with the fact that

n→∞lim θ_n= 0 and lim

n→∞

αn

θn∥x_n−xn−1∥= 0,we have that lim

n→∞∥x_n−x^∗∥= 0.Therefore,{x_n} converges strongly to x^∗.

Next, we propose our second algorithm which is a slight modification of Algorithm (3.3.2).

Algorithm 3.3.3. Inertial method with adaptive step size strategy.

Step 0: Choose sequences {θ_n}^∞_n=1 and {τ_n}^∞_n=1 such that condition (B5) holds and let λ₁ >0, µ∈(0,1), α≥3 and x₀, x₁ ∈ H be given arbitrarily. Set n:= 1.

Step 1: Given the iteratesxn−1 and x_n for eachn≥1,chooseα_nsuch that 0 ≤α_n≤α¯_n, where

¯ α_n:=

(minn

n−1

n+α−1,_∥x ^τn

n−xn−1∥

o

, if x_n ̸=xn−1 n−1

n+α−1, otherwise.

(3.3.29) Step 2: Compute

w_n =x_n+α_n(x_n−x_n−1).

Step 3: Construct the halfspace

C_n ={w∈ H |c(w_n) +⟨ξ_n, w−w_n⟩ ≤0}

and compute

y_n =PCn(w_n−λ_nAw_n).

If y_n =w_n(Aw_n = 0), then setw_n=z_n and go to Step 5. Else go to Step 4.

Step 4: compute

zn =yn−λn(Ayn−Awn).

Step 5: Compute

xn+1 =θnγ1f wn+ (I−θnD)Tnzn, where

T_n = (1−β_n)I+β_nW_n and W_n is the mapping defined by (3.2.1).

Step 6: Compute

λn+1 = (min

n µ||wn−yn||

||Aw_n−Ay_n||, λn

o

, if Awn̸=Ayn

λ_n, otherwise. (3.3.30)

Set n :=n+ 1 and go back to Step 1.

Theorem 3.3.10. Let{x_n}be a sequence generated by Algorithm3.3.3under Assumption 3.3.1. Suppose that {W_n} is the sequence defined by (3.2.1). Then, the sequence {x_n} converges strongly to a point x^∗ ∈ Γ, where x^∗ = P_Γ(I−D+γ₁f)x^∗ is a solution of the variational inequality

⟨(D−γ₁f)x^∗, x^∗−x⟩ ≤0, ∀x∈Γ.

Proof. The proof of this result follows similar argument with the proof of the result of Theorem (3.3.9).

Taking γ₁ = 1 and D =I in Theorem 3.3.9 (where I is the identity mapping), we obtain the following:

Algorithm 3.3.4. Inertial method with adaptive step size strategy.

Step 0: Choose sequences {θn}^∞_n=1 and {τn}^∞_n=1 such that condition (B5) holds and let λ₁ >0, µ∈(0,1), α≥3 and x₀, x₁ ∈ H be given arbitrarily. Set n:= 1.

Step 1: Given the iteratesxn−1 and x_n for eachn≥1,chooseα_nsuch that 0 ≤α_n≤α¯_n, where

¯ α_n:=

(minn

n−1

n+α−1,_∥x ^τn

n−xn−1∥

o

, if x_n ̸=xn−1 n−1

n+α−1, otherwise.

Step 2: Compute

w_n =x_n+α_n(x_n−x_n−1).

Step 3: Construct the halfspace

C_n ={w∈ H |c(w_n) +⟨ξ_n, w−w_n⟩ ≤0}

and compute

y_n =PC_n(w_n−λ_nAw_n).

If y_n =w_n(Aw_n = 0), then setw_n=z_n and go to Step 5. Else go to Step 4.

Step 4: compute

z_n =y_n−λ_n(Ay_n−Aw_n).

Step 5: Compute

x_n+1 =θ_nf x_n+ (I−θ_n)T_nz_n, where

T_n = (1−β_n)I+β_nW_n and W_n is the mapping defined by (3.2.1).

Step 6: Compute

λ_n+1 =

(minn _µ||w

n−yn||

||Awn−Ayn||, λ_no

, if Aw_n̸=Ay_n

λ_n, otherwise.

Set n :=n+ 1 and go back to Step 1.

Corollary 3.3.5. Let {x_n} be a sequence generated by Algorithm 3.3.4under Assumption 3.3.1. Suppose that {W_n} is the sequence defined by (3.2.1). Then, the sequence {x_n} converges strongly to a point x^∗ ∈ Γ, where x^∗ =P_Γ(f)x^∗ is a solution of the variational inequality

⟨(I −f)x^∗, x^∗−x⟩ ≤0, ∀x∈Γ.

Taking γ₁ = 1, D = I (where I is the identity mapping) and S_n = S for all n ≥ 1 in Theorem 3.3.9, we obtain the following:

Algorithm 3.3.6. Inertial method with adaptive step size strategy.

Step 0: Choose sequences {θ_n}^∞_n=1 and {τ_n}^∞_n=1 such that condition (B5) holds and let λ1 >0, µ∈(0,1), α≥3 and x0, x1 ∈ H be given arbitrarily. Set n:= 1.

Step 1: Given the iteratesxn−1 and x_n for eachn≥1,chooseα_nsuch that 0 ≤α_n≤α¯_n, where

¯ α_n:=

(minn

n−1

n+α−1,_∥x ^τn

n−xn−1∥

o

, if x_n ̸=xn−1 n−1

n+α−1, otherwise.

Step 2: Compute

w_n =x_n+α_n(x_n−xn−1).

Step 3: Construct the halfspace

C_n ={w∈ H |c(w_n) +⟨ξ_n, w−w_n⟩ ≤0}

and compute

y_n =PC_n(w_n−λ_nAw_n).

If y_n =w_n(Aw_n = 0), then setw_n=z_n and go to Step 5. Else go to Step 4.

Step 4: compute

z_n =y_n−λ_n(Ay_n−Aw_n).

Step 5: Compute

x_n+1 =θ_nf x_n+ (I−θ_n)T_nz_n, where

T_n= (1−β_n)I+β_nS.

Step 6: Compute

λ_n+1 =

(minn _µ||w

n−yn||

||Awn−Ayn||, λ_no

, if Aw_n̸=Ay_n

λ_n, otherwise.

Set n :=n+ 1 and go back to Step 1.

Corollary 3.3.7. Let {x_n} be a sequence generated by Algorithm 3.3.6under Assumption 3.3.1. Then the sequence {xn} converges strongly to a point x^∗ ∈Γ, where x^∗ = PΓ(f)x^∗ is a solution of the variational inequality

⟨(I −f)x^∗, x^∗−x⟩ ≤0, ∀x∈Γ.