Main result

5.3 Two-level variational inequality and fixed point problems involving pseu-

5.3.1 Main result

0 2 4 6 8 10 12 14 Iteration number (n)

10^-2 10^-1 10⁰

Errors

Appendix 8.1 Appendix 8.2 Appendix 8.3 Appendix 8.4 Algorithm 3.2

0 5 10 15

Iteration number (n) 10^-2

10^-1 10⁰

Errors

Appendix 8.1 Appendix 8.2 Appendix 8.3 Appendix 8.4 Algorithm 3.2

0 2 4 6 8 10 12 14

Iteration number (n) 10^-3

10^-2 10^-1 10⁰

Errors

Appendix 8.1 Appendix 8.2 Appendix 8.3 Appendix 8.4 Algorithm 3.2

0 5 10 15

Iteration number (n) 10^-2

10^-1 10⁰

Errors

Appendix 8.1 Appendix 8.2 Appendix 8.3 Appendix 8.4 Algorithm 3.2

Figure 5.2: Top left: Case I; Top right: Case II; Bottom left: Case III; Bottom right:

Case IV.

5.3 Two-level variational inequality and fixed point

(S₃) Given I as the identity operator on H, let T : H → H be a uniformly continuous ϱ−demimetric operator with (I −T) demiclosed on zero. Also, let 0 < h ≤ t_n ≤ (1−ϱ) for all n≥1.

(S₄) The operatorf :H →H is an L−contraction with L∈(0,¹₂).

(S₅) The solution set of the variational inequality is nonempty. That is, V I(C, A) ̸= ∅.

Also, Ω̸=∅ where Ω :={x∈V I(C, A) : T(x) = x}.

(S₆) Let {α_n}n≥1 and {γ_n}n≥1 be positive sequences such that α_n ∈ (0,1), lim

n→∞α_n = 0, P

n≥1

α_n =∞, and lim

n→∞

γn

αn = 0.

We introduce and study the following algorithm:

Algorithm 5.3.1. Calculation of the sequence {x_n}.

Initialization: Let τ₀ >0, µ∈(0,1), β∈(0,2)

Iterative Steps: Given the current iterate x_n, calculate x_n+1 as follows:

Step 1: Given x_n−1, x_n with n≥1, choose θ_n such that 0≤θ_n≤θ¯_n where θ¯_n=

(min{_n−1+ϵⁿ⁻¹ ,_∥x ^γn

n−xn−1∥}, if x_n̸=xn−1 n−1

n−1+ϵ, otherwise

Step 2: Compute

w_n =x_n+θ_n(x_n−x_n−1) yn=PC(wn−τnAwn).

Step 3: Compute z_n=w_n−βη_nd_n where

d_n:=w_n−y_n−τ_n(Aw_n−Ay_n) and

η_n:=

(_⟨w

n−y_n,dn⟩

∥dn∥² , if d_n ̸= 0,

0, otherwise.

Step 4: Compute

v_n=z_n−t_n(z_n−T z_n),

xn+1 =αnf(vn) + (1−αn)vn. (5.42)

Update

τ_n+1 =

(min{µ_∥Aw^{∥wn−yn∥}

n−Ayn∥, τ_n},if Aw_n ̸=Ay_n,

τ_n, otherwise. (5.43)

Step 5: Set n :=n+ 1 and go to Step 1.

Remark 5.3.2. The sequence {τ_n} generated by (5.43) is a non-increasing sequence and

n→∞lim τ_n =τ ≥min{τ₀,µ L}.

Moreover, we have that

∥Aw_n−Ay_n∥ ≤ µ

τ_n+1∥w_n−y_n∥, ∀n≥0.

We begin with the following lemmas which are critical in obtaining our strong convergence result.

Lemma 5.3.3. Let {x_n}n≥0 be a sequence in H iteratively generated by Algorithm 5.3.1.

Assume that the conditions (S₁)−(S₆) hold, then {x_n} is bounded.

Proof. Letx^∗ ∈VI(C, A). We obtain from the definition of w_n that

∥w_n−x^∗∥=∥x_n+θ_n(x_n−x_n−1)−x^∗∥

≤ ∥xn−x^∗∥+θn∥xn−xn−1∥

=∥x_n−x^∗∥+α_nθn

α_n∥x_n−xn−1∥.

Now from Step 1, observe that θ_n∥x_n−xn−1∥ ≤γ_n, ∀ n≥1, which implies that θ_n

α_n∥xn−xn−1∥ ≤ γ_n

α_n →0, as n → ∞. (5.44)

Thus, there exists K₁ >0 such that θ_n

α_n∥x_n−x_n−1∥ ≤K₁, ∀ n ≥1, (5.45) and so,

∥w_n−x^∗∥ ≤ ∥x_n−x^∗∥+α_nK₁, ∀n≥1. (5.46) Since y_n=P_C(w_n−τ_nAw_n), we obtain from properties of projection that,

⟨w_n−τ_nAw_n−y_n, y_n−x^∗⟩ ≥0. (5.47) Also, since x^∗ ∈ V I(C, A) and y_n ∈ C, we have that ⟨Ax^∗, y_n−x^∗⟩ ≥ 0. By the pseu- domonotonicity of A, we get that,

⟨Ay_n, y_n−x^∗⟩ ≥0. (5.48)

Since τ_n>0, we have that

⟨τ_nA(y_n), y_n−x^∗⟩ ≥0. (5.49) Using (5.47) and (5.49), we obtain that;

⟨y_n−x^∗, w_n−y_n−τ_n(Aw_n−Ay_n)⟩ ≥0. (5.50) Equivalently,

⟨y_n−x^∗, d_n⟩ ≥0. (5.51)

Observe from Step 3 and (5.51) that

⟨w_n−x^∗, d_n⟩=⟨w_n−y_n, d_n⟩+⟨y_n−x^∗, d_n⟩

≥ ⟨wn−yn, dn⟩. (5.52)

Using Algorithm 5.3.1, (5.46) and (5.52), we get that (using the fact that∥d_n∥ ̸= 0),

∥z_n−x^∗∥² =∥w_n−βη_nd_n−x^∗∥²

=∥w_n−x^∗∥²+β²η_n²∥d_n∥²−2βη_n⟨w_n−x^∗, d_n⟩

≤ ∥w_n−x^∗∥²+β²η_n⟨w_n−y_n, d_n⟩ −2βη_n⟨w_n−y_n, d_n⟩

=∥w_n−x^∗∥²−(2−β)βη_n⟨w_n−y_n, d_n⟩

=∥w_n−x^∗∥²−(2−β)β∥η_nd_n∥²

≤ ∥wn−x^∗∥²− 1

β(2−β)∥wn−zn∥²

≤ ∥x_n−x^∗∥²− 1

β(2−β)∥w_n−z_n∥²+α_nK₁. (5.53) Using Algorithm 5.3.1, (5.53) and the definition of T, we obtain that,

∥v_n−x^∗∥² =∥z_n−t_n(z_n−T z_n)−x^∗∥²

=∥z_n−x^∗∥²+t²_n∥z_n−T z_n∥²−2t_n⟨z_n−T z_n, z_n−x^∗⟩

≤ ∥zn−x^∗∥²−tn(1−ϱ−tn)∥zn−T zn∥²

≤ ∥x_n−x^∗∥²− 1

β(2−β)∥w_n−z_n∥²−t_n(1−ϱ−t_n)∥z_n−T z_n∥²+α_nK₁. (5.54) Again, using Algorithm 7.2.3, (5.54), and the fact thatf is a contraction, we obtain that;

∥x_n+1−x^∗∥² =∥α_nf(v_n) + (1−α_n)v_n−x^∗∥²

≤α_n∥f(v_n)−f(x^∗) +f(x^∗)−x^∗∥²+ (1−α_n)∥v_n−x^∗∥²

≤2α_n[L∥v_n−x^∗∥²+∥f(x^∗)−x^∗∥²] + (1−α_n)∥v_n−x^∗∥²

= [1−α_n(1−2L)]∥v_n−x^∗∥² + 2α_n∥f(x^∗)−x^∗∥²

≤[1−αn(1−2L)]∥xn−x^∗∥²+αn(1−2L) 2

1−2L∥f(x^∗)−x^∗∥²

−tn[1−αn(1−2L)](1−ϱ−tn)∥zn−T zn∥²

− 1

β(2−β)[1−α_n(1−2L)]∥w_n−z_n∥²

+αn[1−αn(1−2L)]αnK1 (5.55)

Using the assumptions S₃ and S₆, and for some K₂ >0, we have that;

∥xn+1−x^∗∥² ≤[1−αn(1−2L)] ∥xn−x^∗∥²+K2

+αn(1−2L) 2

1−2L∥f(x^∗)−x^∗∥². (5.56) If we define M := max

∥x₀−x^∗∥²+K₂,_1−2L² ∥f(x^∗)−x^∗∥² , then it is easy to see that

∥x_n+1−x^∗∥² ≤M for all n≥0. (5.57) Thus, {xn}^∞_n=1 is bounded. Consequently, {wn},{Awn},{yn},{Ayn},{zn},{T zn}, and {v_n} are all bounded. ■

Lemma 5.3.4. Suppose that conditions S₁ and S₂ hold, and {x_n}^∞_n=1 is a sequence gen- erated by Algorithm 5.3.1, then, there exists n0 ∈N such that

∥wn−yn∥² ≤

1 +µ_τ^τn

n+1

2

h

1−µ_τ^τn

n+1

βi2∥zn−wn∥² ∀ n≥n0. (5.58) Proof. From Algorithm 3.1, we easily see that,

∥d_n∥=∥w_n−y_n−τ_n(Aw_n−Ay_n)∥ ≥ ∥w_n−y_n∥ −τ_n∥Aw_n−Ay_n∥

≥ ∥wn−yn∥ −µ τ_n

τ_n+1∥wn−yn∥

=

1−µ τ_n τn+1

∥w_n−y_n∥. (5.59)

But lim

n→∞

1−µ_τ^τn

n+1

= 1−µ >0. So, there exists n0 ∈N such that 1−µ τ_n

τ_n+1 > 1−µ

2 ∀n ≥n₀.

Therefore, for each n≥n₀, we have that ∥d_n∥> ^1−µ₂ ∥w_n−y_n∥>0.

Moreover,

∥d_n∥ ≤ ∥w_n−y_n∥+τ_n∥Aw_n−Ay_n∥

≤ ∥w_n−y_n∥+µ τn

τ_n+1∥w_n−y_n∥

=

1 +µ τ_n τ_n+1

∥wn−yn∥ ∀ n ≥n0. (5.60) Thus, for each n≥n₀, we obtain that,

∥d_n∥² ≤

1 +µ τn

τ_n+1 2

∥w_n−y_n∥². (5.61)

Also, from Algorithm 5.3.1, we get that,

⟨w_n−y_n, d_n⟩=⟨w_n−y_n, w_n−y_n−τ_n(Aw_n−Ay_n)⟩

=∥w_n−y_n∥²−τ_n⟨w_n−y_n, Aw_n−Ay_n⟩

≥

1−µ τ_n τ_n+1

∥w_n−y_n∥². (5.62)

So, for each n≥n₀, we obtain from (5.61) and (5.62);

η_n = ⟨w_n−y_n, d_n⟩

∥d_n∥² ≥

1−µ_τ^τn

n+1

1 +µ_τ^τn

n+1

2. (5.63)

Also, from Algorithm 5.3.1 and (5.62), we obtain that;

η_n∥d_n∥² =⟨w_n−y_n, d_n⟩ ≥

1−µ τ_n τ_n+1

∥w_n−y_n∥². (5.64)

Thus, from (5.64) and Algorithm 5.3.1,

∥w_n−y_n∥² ≤

1−µ τ_n τn+1

−1

η_n∥d_n∥²

=

1−µ τ_n τ_n+1

−1

∥βη_nd_n∥² 1 β²

1 η_n

=

1−µ τ_n τ_n+1

−1

∥z_n−w_n∥² 1 β²

1

η_n. (5.65)

Thus, using (5.63) in (5.65), we obtain that,

∥w_n−y_n∥² ≤

1 +µ_τ^τn

n+1

2

h

1−µ_τ^τn

n+1

βi2∥z_n−w_n∥² ∀ n ≥n₀.

■

Lemma 5.3.5. Let {x_n} be sequence generated by Algorithm5.3.1 under Assumptions S.

Suppose that there exists a subsequence {x_nb} of {x_n} which converges weakly to x¯ ∈ H and lim

b→∞∥w_nb −y_nb∥= 0, then x¯∈V I(C, A).

Proof. Clearly, from Algorithm 5.3.1 and AssumptionS₆, we obtain that,

∥wn−xn∥=αn

θ_n

α_n∥xn−xn−1∥ ≤ γ_n

α_n →0 as n→ ∞. (5.66) Since {x_n} is bounded, there exists a subsequence {x_nb} of {x_n} which converges weakly to ¯x∈H. From (2.9), we have that;

⟨w_nb−τ_nbAw_nb −y_nb, x−y_nb⟩ ≤0, ∀ x∈C.

Rearranging, we obtain that, 1

τ_nb⟨w_nb −y_nb, x−y_nb⟩ ≤ ⟨Aw_nb, x−y_nb⟩, ∀ x∈C.

Thus, 1 τn_b

⟨w_nb −y_nb, x−y_nb⟩+⟨Aw_nb, y_nb −w_nb⟩ ≤ ⟨Aw_nb, x−w_nb⟩, ∀ x∈C. (5.67) Fixx∈C. Since{wn_b}is bounded andAis Lipschitz continuous, then{Awn_b}is bounded.

Also, using the assumption that lim

b→∞∥w_nb−y_nb∥= 0 and, taking lim inf of (5.67) asb→ ∞, we get

lim inf

b→∞ ⟨Awn_b, x−wn_b⟩ ≥0. (5.68) By assumptions that lim

b→∞∥w_nb−y_nb∥= 0, and A is Lipschitz continuous, we get that

b→∞lim ∥Aw_nb −Ay_nb∥= 0. (5.69) It is easy to see that,

⟨Ay_nb, x−y_nb⟩=⟨Ay_nb −Aw_nb, x−w_nb⟩+⟨Aw_nb, x−w_nb⟩+⟨Ay_nb, w_nb −y_nb⟩ (5.70) Using (5.68) and (5.69) in (5.70), we get that;

lim inf

b→∞ ⟨Ay_nb, x−y_nb⟩ ≥0. (5.71) Let us choose a decreasing sequence of positive numbers {β_b} with lim

b→∞β_b = 0. For each b ∈N, we denote byN_b the smallest positive number such that

⟨Ay_nj, x−y_nj⟩+β_b ≥0 ∀ j ≥N_b. (5.72) Since {β_b} is decreasing, then the sequence {N_b} is increasing. Furthermore, for each b ∈ N, since {y_nb} ⊂ C, we assume thatAy_Nb ̸= 0 (else, y_Nb is a solution of of VI(C,A)).

Set

m_Nb = Ay_Nb

∥AyN_b∥²,

where ⟨Ay_Nb, m_Nb⟩= 1 for eachb ∈N. From (5.72), for eachb ∈N, we obtain that,

⟨Ay_Nb, x+β_bm_Nb−y_Nb⟩ ≥0. (5.73) Since A is pseudomonotone, we obtain that,

⟨A(x+β_bm_Nb), x+β_bm_Nb−y_Nb⟩ ≥0.

Thus,

⟨Ax, x−y_Nb⟩ ≥ ⟨Ax−A(x+β_bm_Nb), x+β_bm_Nb−y_Nb⟩ −β_b⟨Ax, m_Nb⟩. (5.74) Since {x_nb} converges weakly to ¯x as b→ ∞, then, using (5.66) and the assumption that

b→∞lim ∥wn_b −yn_b∥ = 0, we easily obtain that yn_b ⇀ x¯ as b → ∞. By sequentially weakly continuity of A, we have that Ay_nb ⇀ A¯x as b → ∞. Assume that A¯x̸= 0 (otherwise, ¯x is a solution), by the sequentially weakly lower semicontinuity of norm, we have that

0<∥A¯x∥ ≤lim inf

b→∞ ∥Ay_nb∥.

Since {y_Nb} ⊂ {y_nb}and β_b →0 as b→ ∞, we have that, 0≤lim sup

b→∞

∥β_bm_Nb∥= lim sup

b→∞

βb

∥Ay_nb∥

≤ lim sup_b→∞βb

lim infb→∞∥Ay_nb∥ ≤ 0

∥A¯x∥ = 0.

Thus, ∥β_bm_Nb∥ → 0 as b → ∞. Hence, taking the limit as b → ∞ in (5.74), we obtain that,

⟨Ax, x−x⟩ ≥¯ 0.

Since x∈H is arbitrary, we obtain from Lemma 2.5.10 that ¯x∈V I(C, A)■

Lemma 5.3.6. Let {x_n}n≥0 be a sequence in H iteratively generated by Algorithm 5.3.1.

Assume that ζ_n = (1−L)α_n, and assumptions S₁−S₆ hold, then

∥x_n+1−x^∗∥² ≤(1−ζ_n)h

∥x_n−x^∗∥²+α_nK₁i

+ζ_n 2

1−L⟨f(x^∗)−x^∗, x_n+1−x^∗⟩. (5.75) Proof. From Algorithm 5.3.1, (5.54), Lemma 2.5.18 and Assumption S₆, we obtain that,

∥xn+1−x^∗∥² =∥αnf(vn) + (1−αn)vn−x^∗∥²

=∥α_n[f(v_n)−f(x^∗)] + (1−α_n)[v_n−x^∗] +α_n[f(x^∗)−x^∗]∥²

≤ ∥α_n[f(v_n)−f(x^∗)] + (1−α_n)[v_n−x^∗]∥²+ 2α_n⟨x_n+1−x^∗, f(x^∗)−x^∗⟩

≤[1−(1−L)α_n]∥v_n−x^∗∥²+ 2α_n⟨x_n+1−x^∗, f(x^∗)−x^∗⟩

≤[1−(1−L)α_n]∥x_n−x^∗∥²+ 2α_n⟨x_n+1−x^∗, f(x^∗)−x^∗⟩ + [1−(1−L)α_n]α_nK₁− 1

β(2−β)[1−(1−L)α_n]∥w_n−z_n∥²

−t_n(1−ϱ−t_n)[1−(1−L)α_n]∥z_n−T z_n∥².

So, using assumption S₄, we obtain that,

∥x_n+1−x^∗∥² ≤[1−(1−L)α_n]∥x_n−x^∗∥²+ [1−(1−L)α_n]α_nK₁ + (1−L)α_n 2

1−L⟨f(x^∗)−x^∗, x_n+1−x^∗⟩ (5.76) If we let ζ_n= (1−L)α_n, then we obtain from (5.76) that,

∥x_n+1−x^∗∥² ≤(1−ζ_n)h

∥x_n−x^∗∥²+α_nK₁i

+ζ_n 2

1−L⟨f(x^∗)−x^∗, x_n+1−x^∗⟩. (5.77)

■

Theorem 5.3.7. Let {xn}n≥0 be a sequence in H iteratively generated by Algorithm 5.3.1. Assume that the conditions (S₁) − (S₆) hold, then {x_n} converges strongly to x^∗ =P_Ω(f(x^∗)).

Proof.

From Lemma 5.3.3,{x_n}^∞_n=1 is bounded. Letx^∗ =P_Ω(f(x^∗)). Now, from Algorithm7.2.3, (5.54), and Lemma 2.5.18, we obtain that,

∥x_n+1−x^∗∥² =∥α_nf(v_n) + (1−α_n)v_n−x^∗∥²

≤ ∥α_nf(v_n) + (1−α_n)v_n−x^∗−α_n(f(v_n)−x^∗)∥² + 2α_n⟨f(v_n)−x^∗, x_n+1−x^∗⟩

≤(1−αn)∥vn−x^∗∥²+ 2αn⟨f(vn)−x^∗, xn+1−x^∗⟩

≤(1−α_n)∥x_n−x^∗∥²+ 2α_n⟨f(v_n)−x^∗, x_n+1−x^∗⟩

− 1

β(2−β)(1−α_n)∥w_n−z_n∥²+ (1−α_n)α_nK₁

−(1−α_n)t_n(1−ϱ−t_n)∥z_n−T z_n∥²

≤(1−α_n)∥x_n−x^∗∥²+ 2α_n⟨f(v_n)−x^∗, x_n+1−x^∗⟩

− 1

β(2−β)(1−α_n)∥w_n−z_n∥²+α_nK₁

−(1−α_n)t_n(1−ϱ−t_n)∥z_n−T z_n∥². (5.78) To show that the sequence{∥x_n+1−x^∗∥}converges to zero, we consider two possible cases:

Case 1: Suppose there exists n₀ ∈N such that the real sequence ∥x_n−x^∗∥ is decreasing for all n ≥n0. It then follows that ∥xn−x^∗∥ is convergent. Since {xn} is bounded, then from (5.78), assumptionS₃, and using the fact that α_n→0 as n→ ∞, we obtain that

n→∞lim ∥zn−T zn∥= 0, lim

n→∞∥wn−zn∥= 0. (5.79) From Algorithm 5.3.1, we get that;

∥v_n−z_n∥=t_n∥z_n−T z_n∥ →0 as n→ ∞,

∥x_n+1−v_n∥=α_n∥f(v_n)−v_n∥ →0 as n → ∞. (5.80)

Thus, using (5.66), (5.79), and (5.80), we obtain that,

∥x_n+1−x_n∥ ≤ ∥x_n+1−v_n∥+∥v_n−z_n∥+∥z_n−w_n∥+∥w_n−x_n∥

→0 as n → ∞. (5.81)

From the results above, we can easily deduce that,

∥x_n+1−x_n∥ →0 as n → ∞. (5.82)

Claim:

lim sup

n→∞

D

f(x^∗)−x^∗, x_n−x^∗E

≤0. (5.83)

Proof of Claim: Let{x_nq}_q≥0 be a subsequence of{x_n}_n≥0 such that lim sup

n→∞

D

f(x^∗)−x^∗, xn−x^∗ E

= lim

q→∞

D

f(x^∗)−x^∗, xnq −x^∗ E

(5.84) Since{xnq}q≥0is a bounded sequence inH, there exists a subsequence{xn_qb}b≥0of{xnq}q≥0

which converges weakly to ¯x in H. Hence, x_nqb ⇀ x¯ as b → ∞. For convenience, and WLOG, we will represent {x_nqb}b≥0 by {x_nb}b≥0.

Recall from Lemma 5.3.4,

∥w_nb−y_nb∥² ≤

1 +µ_τ^τnb

nb+1

2

h

1−µ_τ^τnb

nb+1

αi2∥z_nb−w_nb∥² ∀ n_b ≥n₀.

Using (5.79), we have that ∥wn_b −yn_b∥ → 0 as n → ∞. Thus, using Lemma 5.3.5, and the fact that x_nb ⇀x¯ asb → ∞, we have that ¯x∈V I(C, A).

Next, we show that ¯x∈ Ω. From (5.79) and the fact that (I−T) is demiclosed at zero, we can deduce that ¯x∈F(T).

Therefore, it follows from our argument above that ¯x∈Ω. Thus, we obtain from Lemma 2.5.26 and (5.83) that

lim sup

n→∞

D

f(x^∗)−x^∗, x_n−x^∗E

= lim

q→∞

D

f(x^∗)−x^∗, x_nq −x^∗E

≤0. (5.85) Also, using the fact that ∥x_n+1−x_n∥ →0 as n→ ∞, we get from (5.85) that,

lim sup

n→∞

D

f(x^∗)−x^∗, x_n+1−x^∗E

≤0. (5.86)

Recall that from Lemma 5.3.6, we obtain that,

∥xn+1−x^∗∥² ≤(1−ζn) h

∥xn−x^∗∥²+αnK1

i +ζn

2

1−L⟨f(x^∗)−x^∗, xn+1−x^∗⟩. (5.87)

Thus, by Lemma 2.5.26, it follows that ∥x_n−x^∗∥ converges strongly to zero as n → ∞.

Hence, {x_n} converges to x^∗ ∈Ω. This completes the proof for the first case.

Case 2: Suppose that there exists a subsequence{∥x_nj−x^∗∥}^∞_j=0 of {∥x_n−x^∗∥}_n≥0 such that ∥x_nj − x^∗∥ < ∥x_nj+1 −x^∗∥ for all j ≥ 0, then we obtain by Lemma 2.5.41 that there exists a non-decreasing sequence {m_k}^∞_k=1 ⊂ N such that m_k → ∞ as k → ∞ and

∥x_mk −x^∗∥ ≤ ∥x_mk+1−x^∗∥ for all k ∈ N. Since the sequences {x_mk}^∞_k=1 is bounded, we obtain from (5.66), (5.79), (5.80) (and using the arguments displayed in Case 1) that as k → ∞;

∥w_mk −x_mk∥, ∥w_mk−z_mk∥ →0

∥v_mk −z_mk∥, ∥x_mk+1−v_mk∥ →0. (5.88) Applying (5.88), we obtain that,

∥x_mk+1−x_mk∥ →0 as k → ∞. (5.89) Furthermore, following the arguments used in Case 1, we obtain that

lim sup

k→∞

D

f(x^∗)−x^∗, x_mk−x^∗E

≤0. (5.90)

Again, using (5.89) and the same arguments as case 1, we have that, lim sup

k→∞

D

f(x^∗)−x^∗, x_mk+1−x^∗E

≤0. (5.91)

Again, from Lemma 5.3.6 and using the fact that ∥x_mk −x^∗∥² ≤ ∥x_mk+1 −x^∗∥² for any k ∈N we obtain by rearranging that,

ζ_mk∥x_mk −x^∗∥² ≤ ∥x_mk −x^∗∥²− ∥x_mk+1−x^∗∥²+ζ_mk 2

1−L⟨f(x^∗)−x^∗, x_mk+1−x^∗⟩

+ (1−ζ_mk)α_mkK₁. (5.92)

Applying Lemma 2.5.41, we obtain that,

ζ_mk∥x_k−x^∗∥² ≤ ∥x_mk −x^∗∥²− ∥x_mk+1−x^∗∥²+ζ_mk 2

1−L⟨f(x^∗)−x^∗, x_mk+1−x^∗⟩

+(1−ζ_mk)α_mkK₁. (5.93)

Thus, it follows that lim sup

k→∞

∥x_k−x^∗∥= 0. Hence, lim

k→∞∥x_k−x^∗∥= 0.

Thus, x_n→x^∗ asn → ∞. This completes the proof. ■

Remark 5.3.8. Theorem 5.3.7provides an iterative scheme for approximation of the so- lution of two-level variational inequality and fixed point problem where the A is Lipschitz continuous and pseudo-monotone and T is ϱ−demimetric. The result extends to solu- tion of variational inequality problem of monotone operators since every pseudo-monotone operator is monotone.