Main result

3.2 Pseudomonotone equilibrium and common fixed point problems

3.2.1 Main result

Chapter 3

Split Generalized Equilibrium

Problems and Fixed Point Problems in Banach Spaces.

3.1 Introduction

In this chapter, we present our results on psuedomonotone equilibrium problem and com- mon fixed point problem of Bregman quasi-nonexpansive mappings in the framework of p-uniformly convex and uniformly smooth Banach spaces. Furthermore, we present our results on split generalized equilibrium problem with multiple output sets and common fixed point problem for an infinite family of multivalued demicontractive mappings in real Hilbert spaces. A review of some important works that motivate our study in this chapter was done in Section 2.4.7.

3.2 Pseudomonotone equilibrium and common fixed

Assumption 3.2.1.

1. E is a p-uniformly convex real Banach space, which is also uniformly smooth.

2. C is a nonempty, closed and convex subset of E.

3. f :C×C →R is a bifunction satisfying A₁−A₄ of Assumption A.

4. Dj : E → E, j = 1,2, . . . , m is a finite family of Bregman quasi-nonexpansive mappings such that I−D_j is demiclosed at zero for each j = 1,2, . . . , m.

5. The solution set Ω :=EP(f, C)∩Tm

j=1Fix(Dj)̸=∅.

In addition, let {γ_n}, {ϵ_n}, {η_n,j}^m_j=0 be positive sequences satisfying the following con- ditions:

(C1) {γ_n} ∈(0,1), lim_n→∞γ_n = 0, P∞

n=0γ_n =∞.

(C2) {ϵ_n} ∈(0,1) such that lim

n→∞

ϵn

γn = 0.

(C3) {ηn,j} ∈(0,1), Pm

j=0ηn,j = 1 and lim infn→∞ηn,0ηn,j >0.

We now present the main algorithm. The sequence {x_n} is generated as follows:

Algorithm 3.2.2. Calculation of the sequence {xn}.

Initialization: Choose x₀ ∈C, u ∈E, ρ₀ >0, µ∈(0,1), θ≥3 and set n= 1.

Step 1. Given the iterates xn−1 and x_n (n ≥1), we choose θ_n such that 0≤θ_n ≤θ¯_n, where θ¯_n:=







min _n−1

n+θ−1,_∥x ^ϵn

n−xn−1∥ , if x_n̸=xn−1 n−1

n+θ−1, otherwise.

(3.1)

Set

w_n=J⁻¹

J(x_n) +θ_n(J(xn−1)−J(x_n))

. (3.2)

Step 2: Compute

u_n = arg min

z∈C{ρ_nf(w_n, z) + ∆_p(z, w_n)}.

If w_n =u_n, set u_n =v_n and go to Step 4. Else, do Step 3.

Step 3: Compute

v_n= arg min

z∈ζn

{ρ_nf(u_n, z) + ∆_p(z, w_n)},

where ζ_n={z ∈E :⟨J(w_n)−ρ_nφ_n−J(u_n), z−u_n⟩ ≤0}, and φ_n ∈∂₂f(w_n, u_n)satisfying J(w_n)−ρ_nφ_n−J(u_n)∈N_C(u_n).

Step 4: Compute

z_n =J⁻¹ η_n,0J(v_n) +

m

X

j=1

η_n,jJ(D_jv_n)

! . Step 5: Calculate x_n+1 and ρ_n+1 as follows:

x_n+1 =J⁻¹(γ_nJ(u) + (1−γ_n)J(z_n)), and

ρ_n+1 =







min{ρ_n,_f(w^{µ(∆p(un,wn)+∆p(vn,un))}

n,vn)−f(wn,un)−f(un,vn)}

if f(wn, vn)−f(wn, un)−f(un, vn)>0 ρn, otherwise.

(3.3)

Set n =n+ 1 and revert to Step 1.

Remark 3.2.3. Indeed, from (3.1), it follows that

n→∞lim θ_n

γ_n∥xn−1−xn∥= 0. (3.4)

In fact, we obtain the following:

θn∥xn−1−xn∥ ≤ϵn,

which follows from the above inequality and our hypothesis (C2) that

n→∞lim θ_n γn

∥xn−1−x_n∥ ≤ ϵ_n γn

= 0.

Consequently, by the norm-to-norm continuity of J, it follows that

n→∞lim θn

γ_n∥J xn−1−J x_n∥= 0. (3.5)

Lemma 3.2.4. The sequence{ρ_n}generated by Algorithm3.2.2is well-defined and bounded.

Proof. Evidently, from (3.3), it is clear that that ρ_n+1 ≤ ρ_n ∀ n ∈ N. This means that ρ_n+1 is monotonically decreasing. Furthermore, we obtain fromAssumption A that

c₁∆_g(u_n, w_n) +c₂∆_g(v_n, u_n)≥f(w_n, v_n)−f(w_n, u_n)−f(u_n, v_n).

Hence

µ(∆_g(u_n, w_n) + ∆_g(v_n, u_n))

f(w_n, v_n)−f(w_n, u_n)−f(u_n, v_n) ≥ µ(∆_g(u_n, w_n) + ∆_g(v_n, u_n)) c₁∆_g(u_n, w_n) +c₂∆_g(v_n, u_n)

≥ µ(∆_g(u_n, w_n) + ∆_g(v_n, u_n)) max{c₁, c₂}(∆_g(u_n, w_n) + ∆_g(v_n, u_n))

= µ

max{c₁, c₂}.

Therefore, we conclude that the sequence {ρ_n} is bounded by min{ρ₀,_max{c^µ

1,c2}} > 0, which implies that lim

n→∞ρ_n =ρ >0.

Lemma 3.2.5. Let {x_n} be a sequence generated by Algorithm 3.2.2 such that {x_n} is bounded. Suppose conditions (C1) and (C2) hold, then for all x^∗ ∈Ω we have:

n→∞lim θ_n α_n

△_p(x^∗, xn−1)− △_p(x^∗, x_n)

= 0.

Proof. Let x^∗ ∈Ω. Observe that

∆_p(x^∗, xn−1)−∆_p(x^∗, x_n) = 1

q∥xn−1∥^p− ⟨x^∗, J(xn−1)⟩+1 p∥x^∗∥^p

−1

q∥x_n∥^p− ⟨x^∗, J(x_n)⟩+ 1

p∥x^∗∥^p

= 1 q

∥xn−1∥^p− ∥x_n∥^p

+⟨x^∗, J(x_n)−J(xn−1)⟩

≤ 1

qM∥xn−1−xn∥+∥J(xn)−J(xn−1)∥∥x^∗∥, (3.6) for some constant M >0.

Applying (3.4) and (3.5), we obtain from (3.6) that

n→∞lim θn

γ_n

∆p(x^∗, xn−1)−∆p(x^∗, xn)

≤ lim

n→∞

M q · θn

γ_n∥xn−1−xn∥ +∥x^∗∥θ_n

γ_n∥J(x_n)−J(xn−1)∥

= 0, which is the required result.

Lemma 3.2.6. Suppose Assumptions 3.2.1 hold. The sequence {x_n} generated by Algo- rithm 3.2.2 is bounded.

Proof. Let x^∗ ∈Ω. We obtain from (3.2) that

∆_p(x^∗, w_n) = ∆_p

x^∗, J⁻¹

(1−θ_n)J(x_n) +θ_nJ(xn−1)

≤(1−θ_n)∆_p(x^∗, x_n) +θ_n(x^∗, xn−1). (3.7)

From Algorithm 3.2.2, since v_n∈ζ_n, we obtain that

⟨J(w_n)−ρ_nφ_n−J(u_n), v_n−u_n⟩ ≤0, which implies that

⟨J(w_n)−J(u_n), v_n−u_n⟩ ≤ρ_n⟨φ_n, v_n−u_n⟩. (3.8) Also, since φn ∈∂2f(wn, un),we have that

f(wn, z)−f(wn, un)≥ ⟨φn, z−un⟩ ∀ z ∈E.

In particular, we have

f(wn, vn)−f(wn, un)≥ ⟨φn, vn−un⟩. (3.9) Using (3.8) and (3.9), we obtain

⟨J(wn)−J(un), vn−un⟩ ≤ρn{f(wn, vn)−f(wn, un)}. (3.10) Furthermore, since vn = arg minz∈ζn{ρnf(un, z) + ∆g(z, wn)}, we obtain from Lemma 2.5.44 that

0∈∂(ρ_nf(u_n, z) + ∆_g(z, w_n)) (v_n) +N_ζn(v_n).

The above expression implies that there exists ¯φ_n ∈ ∂₂f(u_n, v_n) and ϕ ∈ N_ζn(v_n), such that

ρ_nφ¯_n+J(v_n)−J(w_n) +ϕ= 0.

Furthermore, since ϕ∈N_ζn(v_n), for all z ∈ζ_n,we have ⟨ϕ, z−v_n⟩ ≤0.Therefore,

ρ_n⟨φ¯_n, z−v_n⟩ ≥ ⟨J(w_n)−J(v_n), z−v_n⟩, ∀ z ∈ζ_n. (3.11) Moreover, ¯φ_n∈∂₂f(u_n, v_n) implies that

f(u_n, z)−f(u_n, v_n)≥ ⟨φ¯_n, z−v_n⟩ ∀z ∈ζ_n. Thus,

ρ_n(f(u_n, z)−f(u_n, v_n))≥ρ_n⟨φ¯_n, z−v_n⟩ ∀z ∈ζ_n. (3.12) Combining (3.11) and (3.12), we obtain

⟨J(w_n)−J(v_n), z−v_n⟩ ≤ρ_n(f(u_n, z)−f(u_n, v_n)) ∀ z ∈ζ_n. (3.13) Since x^∗ ∈ Ω, then x^∗ ∈ EP(f, C). Observe that EP(f, C) ⊂ C ⊂ ζ_n. Hence, by letting z =x^∗ ∈EP(f, C) in (3.13), we get

⟨J(w_n)−J(v_n), x^∗−v_n⟩ ≤ρ_n(f(u_n, x^∗)−f(u_n, v_n)).

Since f(x^∗, u_n) ≥ 0, then by the pseudo-monotonicity of the bifunction f, we have that f(u_n, x^∗)≤0.Consequently, we have

⟨J(wn)−J(vn), x^∗−vn⟩ ≤ −ρnf(un, vn). (3.14) Adding (3.10) and (3.14) together gives

⟨J(w_n)−J(v_n), x^∗−v_n⟩+⟨J(w_n)−J(u_n), v_n−u_n⟩

≤ρn{f(wn, vn)−f(wn, un)−f(un, vn)}.

Applying Bregman three point identity, (2.13) we have

∆p(x^∗, vn)≤∆p(x^∗, wn)−∆p(un, wn)−∆p(vn, un) +ρ_n{f(w_n, v_n)−f(w_n, u_n)−f(u_n, v_n)}.

Besides, we obtain from the definition of ρ_n that

∆p(x^∗, vn)≤∆p(x^∗, wn)−∆p(un, wn)−∆p(vn, un) (3.15) + ρ_n

ρ_n+1ρ_n+1{f(w_n, v_n)−f(w_n, u_n)−f(u_n, v_n)}

≤∆_p(x^∗, w_n)−∆_p(u_n, w_n)−∆_p(v_n, u_n) + ρ_n

ρ_n+1µ(∆_p(u_n, w_n) + ∆_p(v_n, u_n))

= ∆_p(x^∗, w_n)−

1− ρ_n ρn+1

µ

∆_p(u_n, w_n) + ∆_p(v_n, u_n)

. (3.16)

Since limn→∞

1− _ρ^ρn

n+1µ

= 1−µ >0, then there existsN ∈N such that

1− ρ_n ρ_n+1µ

>0, ∀n≥N.

Hence, we have

∆_p(x^∗, v_n)≤∆_p(x^∗, w_n), ∀n ≥N. (3.17) Also, from Algorithm 3.2.2 and the fact thatD_j is quasi-nonexpansive for eachj ∈N, we have

∆_p(x^∗, z_n) = ∆_p x^∗, J⁻¹

η_n,0J(v_n) +

m

X

j=1

η_n,jJ(D_jv_n) !

≤ηn,0∆p(x^∗, vn) +

m

X

j=1

ηn,j∆p(x^∗, Djvn)

≤η_n,0∆_p(x^∗, v_n) +

m

X

j=1

η_n,j∆_p(x^∗, v_n)

= ∆_p(x^∗, v_n). (3.18)

Therefore, we get

∆_p(x^∗, x_n+1) = ∆_p

x^∗, J⁻¹

γ_nJ(u) + (1−γ_n)J(z_n)

≤γn∆p(x^∗, u) + (1−γn)∆p(x^∗, zn) (3.19)

≤γ_n∆_p(x^∗, u) + (1−γ_n)∆_p(x^∗, v_n)

≤γ_n∆_p(x^∗, u) + (1−γ_n)∆_p(x^∗, w_n)

≤γ_n∆_p(x^∗, u) + (1−γ_n)[(1−θ_n)∆_p(x^∗, x_n) +θ_n(x^∗, xn−1)]

≤max{∆_p(x^∗, u)},max{∆_p(x^∗, x_n),∆_p(x^∗, xn−1)} (3.20) ...

≤max{∆_p(x^∗, u)},max{∆_p(x^∗, x_N),∆_p(x^∗, x_N−1)}.

Therefore, {∆_p(x^∗, x_n)} is bounded and by Lemma 2.5.34, the sequence {x_n} is also bounded. Consequently, {z_n}, {v_n} {u_n} and {w_n} are all bounded.

Lemma 3.2.7. Suppose we letr = sup_n∈N{∥J(v_n)∥, ∥J(D_jv_n)∥} and let W_q :E^∗ →R be the gauge of uniform convexity of the conjugate function g^∗. It follows that

∆_p(x^∗, z_n)≤∆_p(x^∗, v_n)− W_q(η_n,j)

q g ∥J(v_n)−J(D_jv_n)∥

, (3.21)

where W_q(η_n,j) = (η_n,0)^qPm

j=1η_n,j +η_n,0(Pm

j=1η_n,j)^q. Proof. Indeed, we obtain from Algorithm 3.2.2 that:

∆_p(x^∗, z_n) = ∆_p x^∗, J⁻¹

η_n,0J(v_n) +

m

X

j=1

η_n,jJ(D_jv_n) !

=V_p x^∗, η_n,0J(v_n) +

m

X

j=1

η_n,jJ(D_jv_n)

! .

Furthermore, we get from Lemma 2.5.45 and Lemma 2.5.46that

∆_p(x^∗, z_n) = 1

p∥x^∗∥^p−η_n,0⟨x^∗, J(v_n)⟩ −

m

X

j=1

η_n,j⟨x^∗, J(D_jv_n)⟩

+ 1

q∥η_n,0J(v_n) +

m

X

j=1

η_n,jJ(D_jv_n)∥^q

≤ 1

p∥x^∗∥^p −η_n,0⟨x^∗, J(v_n)⟩ −

m

X

j=1

η_n,j⟨x^∗, J(D_jv_n)⟩

+ 1

qη_n,0∥J(v_n)∥^q+1 q

m

X

j=1

η_n,j∥J(D_jv_n)∥^q− W_q(η_n,j)

q g ∥J(v_n)−J(D_jv_n)∥

= 1

pηn,0∥x^∗∥^p+ 1 p

m

X

j=1

ηn,j∥x^∗∥^p−ηn,0⟨x^∗, J(vn)⟩ −

m

X

j=1

ηn,j⟨x^∗, J(Djvn)⟩

+ 1

qη_n,0∥J(v_n)∥^q+1 q

m

X

j=1

η_n,j∥J(D_jv_n)∥^q− Wq(ηn,j)

q g ∥J(v_n)−J(D_jv_n)∥

=ηn,0

1

p∥x^∗∥^p− ⟨x^∗, J(vn)⟩+1

q∥J(vn)∥^q

+

m

X

j=1

η_n,j 1

p∥x^∗∥^p− ⟨x^∗, J(D_jv_n)⟩+1

q∥J(D_jv_n)∥^q

− W_q(η_n,j)

q g ∥J(v_n)−J(D_jv_n)∥

=ηn,0∆p(x^∗, vn) +

m

X

j=1

ηn,j∆p(x^∗, Djvn)−W_q(η_n,j)

q g ∥J(vn)−J(Djvn)∥

. Since D_j is Bregman quasi-nonexpansive for each j ∈N, we obtain

∆_p(x^∗, z_n)≤η_n,0∆_p(x^∗, v_n) +

m

X

j=1

η_n,j∆_p(x^∗, v_n)−W_q(η_n,j)

q g ∥J(v_n)−J(D_jv_n)∥

= ∆_p(x^∗, v_n)− W_q(η_n,j)

q g ∥J(v_n)−J(D_jv_n)∥

.

Next, we prove the strong convergence theorem for the sequence {xn} generated by our proposed algorithm.

Theorem 3.2.8. The iterative sequence {x_n} generated by Algorithm 3.2.2 under As- sumption 3.2.1 strongly converges to x^∗ ∈Ω, where x^∗ =Proj^f_Ω(u).

Proof. Supposex^∗ = Proj^f_Ω(u).Then from Algorithm3.2.2and by applying Lemma2.5.45

(iii), we have

∆_p(x^∗, x_n+1) = ∆_p x^∗, J⁻¹ γ_nJ(u) + (1−γ_n)J(z_n)

=V_p(x^∗, γ_nJ(u) + (1−γ_n)J(z_n))

≤V_p

x^∗, γ_nJ(u) + (1−γ_n)J(z_n)−γ_n J(u)−J(x^∗)

(3.22) +γ_n⟨J(u)−J(x^∗), x_n+1−x^∗⟩

=V_p

x^∗, γ_nJ(x^∗) + (1−γ_n)J(z_n)

+γ_n⟨J(u)−J(x^∗), x_n+1−x^∗⟩

≤γnVp(x^∗, J(x^∗)) + (1−γn)Vp(x^∗, J(zn)) +γn⟨J(u)−J(x^∗), xn+1−x^∗⟩

= (1−γ_n)∆_p(x^∗, z_n) +γ_n⟨J(u)−J(x^∗), x_n+1−x^∗⟩. (3.23) Applying (3.21) in (3.22), we obtain

∆p(x^∗, xn+1)≤(1−γn)

∆p(x^∗, vn)− W_q(η_n,j)

q g ∥J(vn)−J(Djvn)∥

+γ_n⟨J(u)−J(x^∗), x_n+1−x^∗⟩. (3.24) Furthermore, by applying (3.7) and (3.15) in (3.24) we have

∆p(x^∗,xn+1)≤(1−γn)

∆p(x^∗, wn)−

1− ρ_n ρ_n+1µ

∆p(un, wn) + ∆p(vn, un)

−(1−γn)

W_q(η_n,j)

q g(∥J(vn)−J(Djvn)∥)

+γn⟨J(u)−J(x^∗), xn+1−x^∗⟩

≤(1−γn)

(1−θn)∆p(x^∗, xn) +θn∆p(x^∗, xn−1)

−(1−γ_n)

1− ρ_n ρn+1

µ

∆_p(u_n, w_n) + ∆_p(v_n, u_n)

−(1−γ_n)W_q(η_n,j)

q g(∥J(v_n)−J(D_jv_n)∥) +γ_n⟨J(u)−J(x^∗), x_n+1−x^∗⟩

= (1−γ_n)∆_p(x^∗, x_n)−(1−γ_n)

1− ρ_n ρ_n+1µ

∆_p(u_n, w_n)∆_p(v_n, u_n)

−(1−γ_n)W_q(η_n,j)

q g(∥J(v_n)−J(D_jv_n)∥) +γ_nΨ_n, (3.25) where Ψ_n:=

θn

γn

∆_p(x^∗, xn−1)−∆_p(x^∗, x_n)

+⟨J(u)−J(x^∗), x_n+1−x^∗⟩ . Thus, from (3.25), we have

(1−γ_n)

1− ρn

ρ_n+1µ ∆_p(u_n, w_n) + ∆_p(v_n, u_n)

≤∆_p(x^∗, x_n)−∆_p(x^∗, x_n+1) +γ_nΨ_n, (3.26) Also, from (3.25), we obtain

(1−γn)W_q(η_n,j)

q g(∥J(vn)−J(Djvn)∥)≤∆p(x^∗, xn)−∆p(x^∗, xn+1) +γnΨn. (3.27)

Moreover, it follows from (3.25) that

∆_p(x^∗, x_n+1)≤(1−γ_n)∆_p(x^∗, x_n) +γ_nΨ_n. (3.28) We now show that the sequence {x_n} converges to x^∗. Let a_n := ∆_p(x_n, x^∗), it is easy to see that (3.28) satisfies the inequality

a_n+1 ≤(1−γ_n)a_n+γ_nΨ_n. To apply Lemma 2.5.55, it is enough to show that lim sup

k→∞

Ψ_nk ≤0 for every subsequence {∆_p(x^∗, x_nk)} of {∆_p(x^∗, x_n)} satisfying the relation

lim inf

k→∞ (∆_p(x^∗, x_nk+1)−∆_p(x^∗, x_nk))≥0. (3.29) Now, we assume that {∆_p(x^∗, x_nk)} is a subsequence of {∆_p(x^∗, x_n)} such that (3.29) holds. Then, from (3.26) and by applying (3.29) and condition (C1), we have that

lim sup

k→∞

(1−γ_nk)

1− ρ_nk ρ_nk+1µ

(∆_p(u_nk, w_nk) + ∆_p(v_nk, u_nk))

≤lim sup

k→∞

∆p(x^∗, xn_k)−∆p(x^∗, xn_k+1) +γn_kΨn_k

= lim sup

k→∞

∆p(x^∗, xn_k)−∆p(x^∗, xn_k+1)

≤ −lim inf

k→∞

∆p(x^∗, xn_k+1)−∆p(x^∗, xn_k)

≤0. (3.30)

It follows that

lim sup

k→∞

1− ρ_nk ρn_k+1

µ

(∆_p(u_nk, w_nk) + ∆_p(v_nk, u_nk)) = 0. (3.31) Since lim

k→∞ρ_nk exists, then lim

n→∞

ρ_nk

ρ_nk+1 = 1. Using the fact that µ ∈ (0,1), it follows from (3.31) that

k→∞lim ∆_p(u_nk, w_nk) = 0; lim

k→∞∆_p(v_nk, u_nk) = 0.

By Lemma 2.5.31, we obtain

k→∞lim ∥u_nk−w_nk∥= 0 = lim

k→∞∥v_nk−u_nk∥. (3.32) Consequently, we have

k→∞lim ∥v_nk−w_nk∥ ≤ lim

k→∞∥v_nk−u_nk∥+ lim

k→∞∥u_nk −w_nk∥= 0. (3.33)

Similarly, from (3.27) and by applying (3.29), we have lim sup

k→∞

(1−γ_nk)W_q(η_nk,j)

q g(∥J(v_nk)−J(D_jv_nk)∥)

≤lim sup

k→∞

∆_p(x^∗, x_nk)−∆_p(x^∗, x_nk+1) +γ_nkΨ_nk

= lim sup

k→∞

∆_p(x^∗, x_nk)−∆_p(x^∗, x_nk+1)

≤ −lim inf

k→∞

∆p(x^∗, xn_k+1)−∆p(x^∗, xn_k)

≤0.

Therefore,

lim sup

k→∞

(1−γ_nk)Wq(ηn_k,j)

q g(∥J(v_nk)−J(D_jv_nk)∥) = 0.

By condition (C3), we have

k→∞lim g∥J(v_nk)−J(D_jv_nk)∥= 0, j = 1,2, . . . , m.

By the property ofg and the fact thatJ⁻¹ is norm-to-norm continuous on bounded subsets of E^∗, we have

k→∞lim ∥v_nk−D_jv_nk∥= 0, j = 1,2, . . . , m. (3.34) Moreover, by applying condition (C3) and (3.34) we obtain

k→∞lim ∥J z_nk−J v_nk∥= lim

k→∞∥η_nk,0J(v_nk) +

m

X

j=1

η_nk,jJ(D_jv_nk)−J(v_nk)∥

≤ lim

k→∞

η_nk,0∥J(v_nk)−J(v_nk)∥+

m

X

j=1

η_nk,j∥J(D_jv_nk)−J(v_nk)∥

= 0.

By the norm-to-norm continuity of J⁻¹ on bounded subsets of E^∗, we get

k→∞lim ∥z_nk−v_nk∥= 0. (3.35) From (3.33) and (3.35), we have

k→∞lim ∥z_nk−w_nk∥= 0. (3.36)

Recall that w_nk =J⁻¹h

J(x_nk) +θ_nk J(x_nk−1)−J(x_nk)i

. Then, by using (3.5) we obtain

k→∞lim ∥J(w_nk)−J(x_nk)∥= lim

k→∞γ_nk · θ_nk

γ_nk∥J(x_nk−1)−J(x_nk)∥= 0. (3.37) By the norm-to-norm continuity of J⁻¹ on bounded subsets of E^∗, we get

k→∞lim ∥wn_k −xn_k∥= 0. (3.38) Then, from (3.32), (3.35), (3.36) and (3.38) we obtain

k→∞lim ∥zn_k −xn_k∥= lim

k→∞∥vn_k −xn_k∥= lim

k→∞∥un_k−xn_k∥= 0. (3.39) Furthermore, from Algorithm 3.2.2, we have

k→∞lim ∥J(xn_k+1)−J(zn_k)∥= lim

k→∞∥γn_kJ(u) + (1−γn_k)J(zn_k)−J(zn_k)∥

= lim

k→∞ γ_nk∥J(u)−J(z_nk)∥+ (1−γ_nk)∥J(z_nk)−J(z_nk)∥

= 0.

By the norm-to-norm continuity of J⁻¹ on bounded subsets of E^∗, we have

k→∞lim ∥x_nk+1−z_nk∥= 0. (3.40) Therefore, from (3.39) and (3.40) we obtain

k→∞lim ∥x_nk+1−x_nk∥= 0. (3.41) Next, since {x_n} is bounded, then w_ω(x_n) is nonempty. Let ˆx ∈ w_ω(x_n) be an arbitrary element. Then, there exists a subsequence{x_nk}of{x_n}which converges weakly to ˆx∈E.

Then, by (3.39) it follows that {u_nk} converges weakly to ˆx∈E. Also, since u_nk = arg min

z∈C{ρ_nkf(w_nk, z) + ∆_p(z, w_nk)}, it follows from Lemma 2.5.44that

0∈∂

ρ_nkf(w_nk, z) + ∆_p(z, w_nk)

(u_nk) +N_C(u_nk), z∈C.

This means that

ρ_nkφ_nk +J(u_nk)−J(w_nk) +ϕ = 0, where ϕ ∈N_C(u_nk). (3.42) Observe that

⟨ϕ, z−u_nk⟩ ≤0, ∀z ∈C.

Therefore, it follows from (3.42) that

⟨ρ_nkφ_nk, z−u_nk⟩+⟨ϕ, z−u_nk⟩=⟨J(w_nk)−J(u_nk), z−u_nk⟩, Consequently, we have

ρ_nk⟨φ_nk, z−u_nk⟩ ≥ ⟨J(w_nk)−J(u_nk), z−u_nk⟩. (3.43) Furthermore, since φ_n ∈∂₂f(w_nk, u_nk), we have

f(w_nk, z)−f(w_nk, u_nk)≥ ⟨φ_nk, z−u_nk⟩, ∀z ∈C. (3.44) Combining (3.43) and (3.44), we have

ρ_nk

f(w_nk, z)−f(w_nk, u_nk)

≥ ⟨J(w_nk)−J(u_nk), z−u_nk⟩, ∀z ∈C. (3.45) Since lim

k→∞∥w_nk −u_nk∥ = 0 and J is uniformly continuous on bounded subsets of E, we have

k→∞lim ∥J(w_nk)−J(u_nk)∥= 0. (3.46) Consequently, passing limit k → ∞ in (3.45) and by applying condition (A4), (3.38) and (3.46) we obtain

f(ˆx, z)≥0, ∀ z ∈C, which implies that,

ˆ

x∈EP(f, C). (3.47)

Also, since lim

k→∞∥v_nk −D_jv_nk∥= 0, j = 1,2, . . . , m and v_nk ⇀xˆ by (3.39), then by the demiclosedness of I −D_j, we obtain ˆx ∈F(D_j), ∀ j = 1,2, . . . , m. Therefore, it follows that

ˆ x∈

m

\

j=1

F(D_j). (3.48)

Since ˆx∈w_ω(x_n) is arbitrary, then by combining (3.47) and (3.48), we have

w_ω(x_n)⊂EP(f, C)∩

m

\

j=1

F(D_j). (3.49)

To complete the proof, we need to show that lim

k→∞Ψ_nk ≤0. Firstly, we show that lim sup

k→∞

⟨J(u)−J(x^∗), x_nk+1−x^∗⟩ ≤0.

By the boundedness of{x_nk},there exists a subsequence{x_nkj}of{x_nk}such thatx_nkj ⇀ q and

j→∞lim⟨J(u)−J(x^∗), x_nkj −x^∗⟩= lim sup

k→∞

⟨J(u)−J(x^∗), x_nk−x^∗⟩.

Since x^∗ = Proj^f_Ω(u), then by (2.19) and (3.41) we get lim sup

k→∞

⟨J(u)−J(x^∗), x_nk+1 −x^∗⟩

= lim sup

k→∞

⟨J(u)−J(x^∗), x_nk−x^∗⟩+ lim sup

k→∞

⟨J(u)−J(x^∗), x_nk+1−x_nk⟩

= lim

j→∞⟨J(u)−J(x^∗), xn_kj −x^∗⟩

=⟨J(u)−J(x^∗), q−x^∗⟩

≤0. (3.50)

Recall that Ψn= θn

γn(1−γn)(∆p(x^∗, xn−1)−∆p(x^∗, xn)) +⟨J(u)−J(x^∗), xn+1−x^∗⟩ . Now, by Lemma 6.3.6 and (3.50) we have that lim sup_k→∞Ψ_nk ≤ 0. Next, by applying Lemma 2.5.55to (3.28), we conclude that ∆_p(x^∗, x_n)→0 as n→ ∞.By Lemma 2.5.31, it follows that lim∥x_n−x^∗∥= 0.Hence, {x_n}converges strongly to ˆx∈Ω as required.