Main result

3.3 Split generalized equilibrium problem with multiple output sets

3.3.1 Main result

In this section, we present a modified viscosity-type algorithm for approximating a common element of the set of solution of the split generalized equilibrium problem with multiple output sets and the common fixed point problem for an infinite family of multivalued demicontractive mappings in real Hilbert spaces.

LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Fori= 1,2, ..., N, let C_ibe nonempty closed convex subset of Hilbert spacesH_i and letA_i :H→H_ibe bounded linear operators. Let F, ϕ : C×C → R, F_i, ϕ_i : C_i ×C_i → R be bifunctions satisfying assumptions (B1)-(B7) in Assumption 3.3.2 and for each j ∈ N, let S_j : H → CB(H) be a family of multivalued demicontractive mappings with constant k_j ∈(0,1) such that I−S_j is demiclosed at zero andS_j(p) = {p} for eachj ∈N, p∈ ∩^∞_j=1Fix(S_j).Suppose the solution set denoted by

Γ :=

∞

\

j=1

Fix(S_j)∩GEP(F, ϕ)∩

N

\

i=1

A⁻¹_i (GEP(F_i, ϕ_i))

̸=∅. (3.57) Letg :H →Hbe aρ-contraction with constantρ∈(0,1). Let{α_n},{δ_n},{µ_n},{γ_n,j}, j ∈ N, be sequences in (0,1), and {ϕ_n,i} is a sequence of positive real numbers for each i= 0,1,2, . . . , N and n ≥0. Let {x_n} be a sequence generated as follows:

Algorithm 3.3.5.

Step 0: For any x0 ∈H, let H0 =H, T0 =I^H, F0 =F, ϕ0 =ϕ and set n= 0.

Step 1: Compute

v_n=

N

X

i=0

β_i,n

x_n−τ_i,nA^∗_i I^Hi−T_r^(F_i ^i,ϕi) A_ix_n

. (3.58)

Step 2: Compute

y_n =γ_n,0v_n+

∞

X

j=1

γ_n,jz_n^j, (3.59)

where z_n^j ∈S_jv_n, j= 1,2, ....

Step 3: Compute

x_n+1 =α_ng(x_n) +δ_nx_n+µ_ny_n, n∈N. (3.60) Update:

τ_i,n=θ_i,n ∥ I^Hi −Tr^(Fi^i,ϕi)

A_ix_n∥²

∥A^∗_i I^Hi −Tr^(Fi^i,ϕi)

A_ix_n∥²+ϕ_i,n. (3.61) Set n =n+ 1 and go to Step 1.

The following assumptions are needed in order to establish the strong convergence result for Algorithm 3.3.5:

(A1) lim

n→∞α_n = 0 andP∞

n=0α_n=∞, and α_n+δ_n+µ_n= 1, µ_n⊂[a, b]⊂(0,1);

(A2) P∞

j=0γ_n,j = 1,lim inf

n→∞ γ_n,j(γ_n,0−k)>0 for eachj = 1,2, . . . ,wherek := sup_j≥1{k_j}<

1;

(A3) {β_i,n} ⊂[c, d]⊂(0,1) such that PN

i=0β_i,n= 1,{θ_i,n} ⊂[e, f]⊂(0,2);

(A4) r_i >0 for each i= 1,2, . . . , N,maxi=0,1,...,N{sup_n{ϕ_i,n}}=K <∞.

Lemma 3.3.6. The sequence {x_n} generated by Algorithm 3.3.5 is bounded.

Proof. Letp∈Γ,we get A_ip=Tr^(Fi^i,ϕi)A_ip, for eachi= 0,1, ..., N. Thus, by the convexity of the function ∥ · ∥² we obtain

∥v_n−p∥² =∥

N

X

i=0

β_i,n

x_n−τ_i,nA^∗_i I^Hi−T_r^(F_i ^i,ϕi) A_ix_n

−p∥²

≤

N

X

i=0

β_i,n∥x_n−τ_i,nA^∗_i I^Hi−T_r^(F_i ^i,ϕi)

A_ix_n−p∥². (3.62)

Applying the firmly nonexpansiveness of I^Hi−Tr^(Fi ^i,ϕi) for each i= 0,1, ..., N, we obtain

∥x_n−τ_i,nA^∗_i I^Hi−T_r^(Fi,ϕi)

i

A_ix_n−p∥²

=∥x_n−p∥²+τ_i,n² ∥A^∗_i I^Hi−T_r^(F_i ^i,ϕi)

A_ix_n∥²

−2τ_i,n⟨A^∗_i I^Hi −T_r^(F_i ^i,ϕi)

A_ix_n, x_n−p⟩

=∥x_n−p∥²+τ_i,n² ∥A^∗_i I^Hi−T_r^(F_i ^i,ϕi)

A_ix_n∥²

−2τ_i,n⟨ I^Hi−T_r^(F_i ^i,ϕi)

A_ix_n, A_ix_n−A_ip⟩

=∥x_n−p∥²−2τ_i,n⟨ I^Hi−T_r^(F_i ^i,ϕi)

A_ix_n− I^Hi−T_r^(F_i ^i,ϕi)

A_ip, A_ix_n−A_ip⟩

+τ_i,n² ∥A^∗_i I^Hi−T_r^(F_i ^i,ϕi)

Aixn∥²

≤ ∥x_n−p∥²−2τ_i,n∥ I^Hi−T_r^(F_i ^i,ϕi)

A_ix_n∥² +τ_i,n² ∥A^∗_i I^Hi −T_r^(F_i^i,ϕi)

Aixn∥²+ϕi,n

=∥x_n−p∥²−θ_i,n(2−θ_i,n) ∥ I^Hi −Tr^(Fi^i,ϕi)

A_ix_n∥⁴

∥A^∗_i I^Hi −Tr^(Fi^i,ϕi)

A_ix_n∥² +ϕ_i,n, (3.63) which follows from (3.62) and (3.63) that

∥v_n−p∥² ≤ ∥x_n−p∥²−

N

X

i=0

β_i,nθ_i,n(2−θ_i,n) ∥ I^Hi −Tr^(Fi ^i,ϕi)

A_ix_n∥⁴

∥A^∗_i I^Hi −Tr^(Fi ^i,ϕi)

A_ix_n∥²+ϕ_i,n. (3.64) Now, from Algorithm3.3.5, Lemma2.5.12and the fact thatS_j is demicontractive for each j ∈N we have that

∥y_n−p∥² =∥γ_n,0v_n+

∞

X

j=1

γ_n,jz_n^j −p∥²

=∥γ_n,0(v_n−p) +

∞

X

j=1

γ_n,j(z_n^j −p)∥²

=γ_n,0∥v_n−p∥²+

∞

X

j=1

γ_n,j∥z^j_n−p∥²−

∞

X

j=1

γ_n,0γ_n,j∥v_n−z_n^j∥²

≤γ_n,0∥v_n−p∥²+

∞

X

j=1

γ_n,jH²(S_jv_n, S_jp)−

∞

X

j=1

γ_n,0γ_n,j∥v_n−z_n^j∥²

≤γn,0∥vn−p∥²+

∞

X

j=1

γn,j[∥vn−p∥²+kdist(vn, Sjvn)²]−

∞

X

j=1

γn,0γn,j∥vn−z_n^j∥²

≤γ_n,0∥v_n−p∥²+

∞

X

j=1

γ_n,j[∥v_n−p∥²+k∥v_n−z_n^j∥²]−

∞

X

j=1

γ_n,0γ_n,j∥v_n−z_n^j∥²

=∥v_n−p∥²−

∞

X

j=1

γ_n,j(γ_n,0−k)∥v_n−z_n^j∥².

It follows from (3.64) that

∥y_n−p∥² ≤ ∥x_n−p∥²−

∞

X

j=1

γ_n,j(γ_n,0−k)∥v_n−z_n^j∥²

−

N

X

i=0

β_i,nθ_i,n(2−θ_i,n) ∥ I^Hi−Tr^(Fi ^i,ϕi)

Aixn∥⁴

∥A^∗_i I^Hi−Tr^(Fi ^i,ϕi)

A_ix_n∥²+ϕ_i,n, (3.65) and it follows from the conditions (A2)-(A4), that

∥y_n−p∥² ≤ ∥x_n−p∥². (3.66)

Further, by applying (3.66) we have

∥x_n+1−p∥=∥α_ng(x_n) +δ_nx_n+µ_ny_n−p∥

≤α_n∥g(x_n)−p∥+δ_n∥x_n−p∥+µ_n∥y_n−p∥

≤α_n ∥g(x_n)−g(p)∥+∥g(p)−p∥

+δ_n∥x_n−p∥+µ_n∥y_n−p∥

≤α_n ρ∥x_n−p∥+∥g(p)−p∥

+δ_n∥x_n−p∥+µ_n∥x_n−p∥

= (ρα_n+δ_n+µ_n)∥x_n−p∥+α_n∥g(p)−p∥

= 1−α_n(1−ρ)

∥x_n−p∥+ α_n(1−ρ)∥g(p)−p∥

1−ρ

≤maxn

∥x_n−p∥,∥g(p)−p∥

1−ρ o

. It therefore follows by induction that

∥x_n+1−p∥ ≤maxn

∥x₁−p∥,∥g(p)−p∥

1−ρ o

.

Hence, {x_n} is bounded and consequently,{v_n}, {y_n} and {z_n^j} are all bounded.

It is easy to see that the operatorP_Γ◦g is a contraction. Thus, by the Banach Contraction Principle, there exists a unique point x^∗ ∈ Γ such that x^∗ = P_Γ◦g(x^∗). It follows from the characterization of the projection mapping that

⟨g(x^∗)−x^∗, x−x^∗⟩ ≤0, ∀x∈Γ. (3.67) Lemma 3.3.7. Let {x_n} be a sequence generated by Algorithm3.3.5 and let p∈Γ. Then, under conditions (A1)-(A4) and Assumption 3.3.2 the following inequality holds for all n ∈N:

∥x_n+1−p∥² ≤

1− 2α_n[1−ρ]

(1−α_nρ)

∥x_n−p∥² +2α_n(1−ρ)

(1−αρ)

α_nM₁

2(1−ρ) + 1

(1−ρ)⟨g(p)−p, x_n+1−p⟩

−µ_n(1−α_n) (1−αnρ)

^∞ X

j=1

γ_n,j(γ_n,0−k)∥v_n−z_n^j∥²

+

N

X

i=0

β_i,nθ_i,n(2−θ_i,n)

∥

I^Hi −Tr^(Fi ^i,ϕi)

A_ix_n∥⁴

∥A^∗_i

I^Hi −Tr^(Fi ^i,ϕi)

A_ix_n∥²+ϕ_i,n

Proof. Let p∈Γ. By applying Lemma 2.5.18(iii) and (3.65), we have

∥xn+1−p∥² =∥αng(xn) +δnxn+µnyn−p∥²

≤ ∥δ_n(x_n−p) +µ_n(y_n−p)∥²+ 2α_n⟨g(x_n)−p, x_n+1−p⟩

≤δ_n²∥x_n−p∥² +µ²_n∥y_n−p∥²+ 2δ_nµ_n∥x_n−p∥∥y_n−p∥+ 2α_n⟨g(x_n)−p, x_n+1−p⟩

≤δ_n²∥xn−p∥² +µ²_n∥yn−p∥²+δnµn ∥xn−p∥²+∥yn−p∥²

+ 2αn⟨g(xn)−g(p), xn+1−p⟩

+ 2α_n⟨g(p)−p, x_n+1−p⟩

=δ_n(δ_n+µ_n)∥x_n−p∥²+µ_n(µ_n+δ_n)∥y_n−p∥²+ 2α_n⟨g(x_n)−g(p), x_n+1−p⟩

+ 2α_n⟨g(p)−p, x_n+1−p⟩

≤δ_n(1−α_n)∥x_n−p∥² +µ_n(1−α_n)

∥x_n−p∥²−

∞

X

j=1

γ_n,j(γ_n,0−k)∥v_n−z_n^j∥²

−

N

X

i=0

β_i,nθ_i,n(2−θ_i,n)

∥

I^Hi−Tr^(Fi ^i,ϕi)

A_ix_n∥⁴

∥A^∗_i

I^Hi−Tr^(Fi ^i,ϕi)

A_ix_n∥²+ϕ_i,n

+ 2αnρ∥xn−p∥∥xn+1−p∥+ 2αn⟨g(p)−p, xn+1−p⟩

≤(1−α_n)²∥x_n−p∥²−µ_n(1−α_n) ^∞

X

j=1

γ_n,j(γ_n,0−k)∥v_n−z_n^j∥²

+

N

X

i=0

β_i,nθ_i,n(2−θ_i,n)

∥

I^Hi−Tr^(Fi ^i,ϕi)

A_ix_n∥⁴

∥A^∗_i

I^Hi−Tr^(Fi ^i,ϕi)

A_ix_n∥²+ϕ_i,n

+α_nρ

∥x_n−p∥²+∥x_n+1−p∥² + 2α_n⟨g(p)−p, x_n+1−p⟩

= (1−αn)²+αnρ

∥xn−p∥²+αnρ∥xn+1−p∥² + 2αn⟨g(p)−p, xn+1−p⟩

−µ_n(1−α_n) ^∞

X

j=1

γ_n,j(γ_n,0−k)∥v_n−z_n^j∥²

+

N

X

i=0

βi,nθi,n(2−θi,n)

∥

I^Hi−Tr^(Fi ^i,ϕi)

A_ix_n∥⁴

∥A^∗_i

I^Hi−Tr^(Fi ^i,ϕi)

A_ix_n∥²+ϕ_i,n

.

Consequently, we obtain

∥x_n+1−p∥² ≤ (1−2αn+α²_n+αnρ)

(1−α_nρ) ∥x_n−p∥²+ 2αn

(1−α_nρ)⟨g(p)−p, x_n+1−p⟩

− µ_n(1−α_n) (1−α_nρ)

^∞ X

j=1

γ_n,j(γ_n,0−k)∥v_n−z_n^j∥²

+

N

X

i=0

β_i,nθ_i,n(2−θ_i,n)

∥

I^Hi−Tr^(Fi ^i,ϕi)

A_ix_n∥⁴

∥A^∗_i

I^Hi−Tr^(Fi ^i,ϕi)

Aixn∥² +ϕi,n

= (1−2α_n+α_nρ)

(1−α_nρ) ∥xn−p∥²+ α²_n

(1−α_nρ)∥xn−p∥²+ 2α_n

(1−α_nρ)⟨g(p)−p, xn+1−p⟩

− µn(1−αn) (1−α_nρ)

^∞ X

j=1

γ_n,j(γ_n,0−k)∥v_n−z_n^j∥²

+

N

X

i=0

β_i,nθ_i,n(2−θ_i,n)

∥

I^Hi−Tr^(Fi ^i,ϕi)

A_ix_n∥⁴

∥A^∗_i

I^Hi−Tr^(Fi ^i,ϕi)

Aixn∥² +ϕi,n

≤

1− 2α_n(1−ρ) (1−α_nρ)

∥x_n−p∥²+2α_n(1−ρ) (1−αρ)

α_nM₁

2(1−ρ)+ 1

(1−ρ)⟨g(p)−p, x_n+1−p⟩

− µ_n(1−α_n) (1−α_nρ)

^∞ X

j=1

γn,j(γn,0−k)∥vn−z_n^j∥²

+

N

X

i=0

βi,nθi,n(2−θi,n)

∥

I^Hi−Tr^(Fi ^i,ϕi)

A_ix_n∥⁴

∥A^∗_i

I^Hi−Tr^(Fi ^i,ϕi)

A_ix_n∥² +ϕ_i,n

,

where M₁ := sup{∥x_n−p∥² :n∈N}. Hence, the proof is complete.

Theorem 3.3.8. Suppose that conditions (A1)-(A4) and Assumption 3.3.2 hold. Then, the sequence {x_n} generated by Algorithm 3.3.5 converges strongly to xˆ ∈ Γ, where xˆ = PΓ◦g(ˆx).

Proof. Let ˆx=P_Ω◦g(ˆx). From Lemma 3.3.7, we obtain

∥x_n+1−x∥ˆ ² ≤

1−2α_n[1−ρ]

(1−α_nρ)

∥x_n−x∥ˆ ² +2α_n(1−ρ)

(1−αρ)

α_nM₁

2(1−ρ) + 1

(1−ρ)⟨g(ˆx)−x, xˆ _n+1−x⟩ˆ

(3.68) Next, we show that the sequence {∥x_n−x∥}ˆ converges to zero. In order to establish this, by Lemma2.5.55, it is enough to show that lim sup_k→∞⟨g(ˆx)−x, xˆ _nk+1−x⟩ ≤ˆ 0 for every subsequence {∥x_nk −x∥}ˆ of {∥x_n−x∥}ˆ satisfying

lim inf

k→∞

∥x_nk+1−x∥ − ∥xˆ _nk −x∥ˆ

≥0. (3.69)

Now, suppose that {∥x_nk −x∥}ˆ is a subsequence of {∥x_n −x∥}ˆ such that (3.69) holds.

Then,

lim inf

k→∞

∥x_nk+1−x∥ˆ ²− ∥x_nk −x∥ˆ ²

= lim inf

k→∞

h(∥x_nk+1−x∥ − ∥xˆ _nk −x∥)ˆ (∥x_nk+1−x∥ˆ +∥x_nk −x∥)ˆ i

≥0. (3.70)

Again, from Lemma 3.3.7 we have

µ_nk(1−α_nk) (1−α_nkρ)

∞

X

j=1

γn_k,j(γn_k,0−k)∥vn_k −z_n^j_k∥² ≤

1− 2α_nk[1−ρ]

(1−α_nkρ)

∥xn_k −x∥ˆ ²

− ∥x_nk+1−x∥ˆ ²+ 2α_n(1−ρ) (1−αρ)

h α_nkM₁

2(1−ρ) + 1

(1−ρ)⟨g(ˆx)−x, xˆ _nk+1−x⟩ˆ i .

By applying (3.70) together with condition (A1), we obtain µ_nk(1−α_nk)

(1−α_nkρ)

∞

X

j=1

γ_nk,j(γ_nk,0−k)∥v_nk −z_n^j

k∥² →0, k → ∞.

Consequently, we have

k→∞lim ∥v_nk −z^j_n

k∥= 0, j = 1,2, ... (3.71)

It then follows that

k→∞lim dist(v_nk, S_jv_nk)≤ lim

k→∞∥v_nk −z_n^j

k∥= 0, j = 1,2, ... (3.72) By similar argument, we obtain from Lemma 3.3.7 that

k→∞limµ_nk ^N

X

i=0

β_i,nkθ_i,nk(2−θ_i,nk) ∥ I^Hi−Tr^(Fi ^i,ϕi)

A_ix_nk∥⁴

∥A^∗_i I^Hi −Tr^(Fi ^i,ϕi)

Aixn_k∥²+ϕi,n_k

= 0

∀i= 0,1,2, ..., N.

By condition (A3), we have

∥ I^Hi−Tr^(Fi ^i,ϕi)

A_ix_nk∥⁴

∥A^∗_i I^Hi −Tr^(Fi ^i,ϕi)

A_ix_nk∥²+ϕ_i,nk →0, k → ∞, ∀ i= 0,1,2, ..., N. (3.73)

It follows from the boundedness of the operators A_i, the nonexpansivity of the mappings Tr^(Fi ^i,ϕi), and the boundedness of the sequence {x_nk} that

L := maxi=0,1,...,N{sup_n{∥A^∗_i I^Hi − Tr^(Fi ^i,ϕi)

Aixn_k∥²}} < ∞. Therefore, it follows from (A4), that

∥ I^Hi −Tr^(Fi^i,ϕi)

Aixn_k∥⁴

∥A^∗_i I^Hi−Tr^(Fi ^i,ϕi)

A_ix_nk∥²+ϕ_i,nk ≥ ∥ I^Hi −Tr^(Fi^i,ϕi)

Aixn_k∥⁴

L+K .

Using the last inequality together with (3.73), we have

k→∞lim ∥ I^Hi−T_r^(F_i ^i,ϕi)

Aixn_k∥= 0, ∀ i= 0,1,2, ..., N. (3.74) Furthermore, we obtain from (3.58) that

k→∞lim ∥vn_k −xn_k∥=∥

N

X

i=0

βi,n_kτi,n_kA^∗_i I^Hi −T_r^(F_i ^i,ϕi)

Aixn_k∥. (3.75) Applying (3.74) together with (A3), it follows from the last inequality that

k→∞lim ∥v_nk −x_nk∥= 0. (3.76) Also, from (3.59) and (3.71), we have that

∥y_nk −v_nk∥=∥γ_nk,0(v_nk −v_nk) +

∞

X

j=1

γ_nk,j(z^j_n

k −v_nk)∥

≤

∞

X

j=1

γ_nk,j∥v_nk −z^j_n

k∥ →0, k→ ∞. (3.77)

Observe that from (3.71) and (3.77), we have

∥y_nk −z_n^j

k∥ ≤ ∥y_nk −v_nk∥+∥v_nk−z_n^j

k∥ →0, k → ∞, ∀ j = 1,2, ... (3.78) It follows from (3.76) and (3.77) that

∥x_nk−y_nk∥ ≤ ∥x_nk−v_nk∥+∥v_nk−y_nk∥ →0, k→ ∞. (3.79) Consequently, by applying condition (A1) we have

∥x_nk+1 −x_nk∥=∥α_nkg(x_nk) +δ_nkx_nk +µ_nky_nk−x_nk∥

≤α_nk∥g(x_nk)−x_nk∥+δ_nk∥x_nk−x_nk∥+µ_nk∥y_nk −x_nk∥ →0, k → ∞.

(3.80) To complete the proof, we need to show that w_ω(x_n) ⊂ Γ. Since the sequence {x_n} is bounded, then w_ω(x_n) is nonempty. Let ¯x∈w_ω(x_n) be an arbitrary element. Then, there

exists a subsequence {x_nk} of {x_n} such that x_nk ⇀x¯ as k → ∞. From (3.76), we have that v_nk ⇀x. Now, from the fact that¯ I −S_j is demiclosed at zero for each j = 1,2, ..., and since from (3.71) lim

k→∞∥v_nk −z_n^j

k∥ →0 as k → ∞for each j = 1,2, ..., we have that

¯

x∈Fix(S_j) for all j = 1,2, ... Hence, ¯x∈ ∩^∞_j=1Fix(S_j). Also, since for each i= 0,1, ..., N, A_i is a bounded linear operator, it follows thatA_ix_nk ⇀ A_ix.¯ Thus, by the demiclosedness principle it follows from (3.74) that A_ix¯ ∈ Fix(T_r^Fi,ϕi

i ) for all i = 0,1, ..., N. Hence, A_ix¯∈TN

i=0(GEP(F_i, ϕ_i)). Consequently, we have ¯x∈Γ,which implies that w_ω(x_n)⊂Γ.

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

j→∞lim⟨g(ˆx)−x, xˆ _nkj −x⟩ˆ = lim sup

k→∞

⟨g(ˆx)−x, xˆ _nk −x⟩.ˆ Since ˆx=PΩ◦g(ˆx),then from (3.80) and (3.67), we have

lim sup

k→∞

⟨g(ˆx)−x, xˆ _nk+1−x⟩ˆ = lim sup

k→∞

⟨g(ˆx)−x, xˆ _nk+1−x_nk⟩ + lim sup

k→∞

⟨g(ˆx)−x, xˆ _nk−x⟩ˆ

= lim sup

j→∞

⟨g(ˆx)−x, xˆ n_kj −x⟩ˆ

=⟨g(ˆx)−x, xˆ ^†−x⟩ ≤ˆ 0. (3.81) Applying Lemma2.5.55to (3.68), and using (3.81) together with the fact that lim_n→∞α_n= 0, we deduce that limn→∞∥x_n−x∥ˆ = 0 as required.

If we take ϕ_i = 0, i = 0,1,2, . . . , N in Theorem 3.3.8, we have the following consequent result for approximating a common solution of the set of solution of split equilibrium problem with multiple output sets and the common fixed point problem for an infinite family of multivalued demicontractive mappings in real Hilbert spaces.

Corollary 3.3.9. Let C be a nonempty closed convex subset of a real Hilbert space H.

For i = 1,2, ..., N, let C_i be nonempty closed convex subset of Hilbert spaces H_i and let F, F_i, and A_i be as defined in Theorem 3.3.8. For each j ∈N, let S_j : H → CB(H) be a family of multivalued demicontractive mappings with constantk_j ∈(0,1)such that I−S_j is demiclosed at zero for eachj ∈N.Suppose the solution set denoted by Γ :=T∞

j=1Fix(S_j)∩ EP(F)∩ TN

i=1A⁻¹_i (EP(F_i))

̸=∅,and conditions (A1)-(A4) and Assumption 3.3.2hold.

Then, the sequence {x_n} generated by the following algorithm converges strongly to xˆ∈Γ, where xˆ=P_Γ◦g(ˆx).

Algorithm 3.3.10.

Step 0: For any x₀ ∈H, let H₀ =H, T₀ =I^H, F₀ =F and set n = 0.

Step 1: Compute

v_n=

N

X

i=0

β_i,n

x_n−τ_i,nA^∗_i I^Hi−T_r^F_iⁱ A_ix_n

. (3.82)

Step 2: Compute

y_n =γ_n,0v_n+

∞

X

j=1

γ_n,jz_n^j, (3.83)

where z_n^j ∈S_jv_n, j= 1,2, ....

Step 3: Compute

x_n+1 =α_ng(x_n) +δ_nx_n+µ_ny_n, n∈N. (3.84) Update:

τ_i,n=θ_i,n ∥ I^Hi −T_r^F_iⁱ

Aixn∥²

∥A^∗_i I^Hi −Tr^Fiⁱ

A_ix_n∥²+ϕ_i,n. Set n =n+ 1 and go to Step 1.