We propose new iterative schemes for finding the common element of the set of common fixed points of countable family of nonexpansive mappings, the set of solutions of the variational inequality problem for relaxed cocoercive and Lipschitz continuous, the set of solutions of system of variational inclusions problem, and the set of solutions of equilibrium problems in a real Hilbert space by using the viscosity approximation method. We prove strong convergence theorem under some parameters. The results in this paper unify and generalize some well-known results in the literature.

1. Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H. A mapping S of C into itself is called nonexpansive if ∥Sx − Sy∥≤∥x − y∥ for all x, y ∈ C. We denote by F(S) the set of fixed points of S; that is, F(S) = {x ∈ C : Sx = x}. If C ⊂ H is nonempty, closed and convex and let S : C → C be a nonexpansive mapping, then F(S) is closed and convex and F(S) ≠ ∅, when C is bounded; see, for example, [1, 2]. The metric projection, P_C, onto a given nonempty, closed and convex subset C, satisfies the nonexpansive with F(P_C) = C. A mapping B : C → C is called monotone if 〈Bx − By, x − y〉≥0 for all x, y ∈ C. A mapping B : C → C is called β-inverse-strongly monotone if there exists a constant β > 0 such that 〈Bx − By, x − y〉≥β∥x − y∥², for all x, y ∈ C. A mapping B : C → C is called relaxed (ϕ, ω)-cocoercive if there exists ϕ, ω > 0 such that

()

A mapping B : C → C is said to be ξ-Lipschitz continuous if there exists ξ ≥ 0 such that

()

Let B : H → H be a single-valued nonlinear mapping and M : H → 2^H a multivalued mapping. The variational inclusion problem is to find

such that

()

where θ is the zero vector in H. The set of solutions of problem (1.3) is denoted by I(B, M). If M = ∂ψ_C, where C is a nonempty closed convex subset of H and ∂ψ_C : H → [0, +∞] is the indicator function of C; that is,

()

then, the variational inclusion problem (1.3) is equivalent to the variational inequality problems denoted by VI (C, B) which is to find

such that

()

In 2003, Takahashi and Toyoda [3] to find x^* ∈ F(S)∩VI (C, B) introduced the following iterative scheme:

()

where B is a β-inverse-strongly monotone mapping, {α_n} is a sequence in (0, 1), and {ξ_n} is a sequence in (0,2β). They showed that if F(S)∩VI (C, B) is nonempty, then the sequence {x_n} generated by (1.6) converges weakly to some x^* ∈ F(S)∩VI (C, B).

In 2008, Zhang et al. [4] to find x^* ∈ F(S)∩I(M, B). They introduced the following new iterative scheme:

()

where J_M,λ = (I + λM) ⁻¹ is the resolvent operator associated with M and a positive number λ, {α_n} is a sequence in the interval [0,1].

Let F be a bifunction of C × C into ℝ, where ℝ is the set of real numbers. The equilibrium problem for F : C × C → ℝ is to find

such that

()

The set of solutions of (1.8) is denoted by EP (F). Many problems in applied sciences, such as monotone inclusion problems, variational inequality problems, saddle point problems, Nash equilibria in noncooperative games, as well as certain fixed-point problems reduce to finding some element to EP (F) in Hilbert and Banach spaces (see [5–14]).

Given any r > 0. The operator T_r : H → C defined by

()

is called the resolvent of F (see [5, 6]).

It is shown in [6] that, under suitable hypotheses on F (to be stated precisely in Section 2), T_r : H → C is single valued and firmly nonexpansive and satisfies

()

Using this result, for finding an element of F(S)∩VI (C, B)∩EP (F), Su et al. [15] introduced the following iterative scheme by the viscosity approximation method in Hilbert spaces:

()

where f : C → C is a contraction (i.e., ∥f(x) − f(y)∥ ≤ ψ∥x − y∥, for all x, y ∈ C and 0 ≤ ψ < 1) and {α_n}⊂(0,1), ξ_n ⊂ (0,2β), and r_n ⊂ (0, ∞) satisfy some appropriate conditions. Furthermore, they prove {x_n} converges strongly to the same point x^* ∈ F(S)∩VI (C, B)∩EP (F), where x^* = P_{F(S)∩VI(C,B)∩EP(F)}f(x^*).

In this paper, motivated and inspired by the above facts, we introduce a new iterative scheme for finding a common element of the set of solutions of the variational inequalities for μ-Lipschitz continuous and relaxed (ϕ, ω)-cocoercive mapping, the set of solutions to the variational inclusion for family of α-inverse strongly monotone mappings, the set of fixed points of a countable family of nonexpansive mappings, and the set of solutions of an equilibrium problem in a real Hilbert space by using the viscosity approximation method. Strong convergence results are derived under suitable conditions in a real Hilbert space.

2. Preliminaries

In this section, we will recall some basic notations and collect some conclusions that will be used in the next section.

Let H be a real Hilbert space whose inner product and norm are denoted by 〈·, ·〉 and ∥·∥, respectively. We denote strong convergence of {x_n} to x ∈ H by x_n → x and weak convergence by x_n⇀x. Let C be nonempty closed convex subset of H. Recall that for all x ∈ H there exists a unique nearest point in C to x denoted P_Cx; that is, ∥x − P_Cx∥≤∥x − y∥, for all y ∈ C. The mapping P_C is nonexpansive; that is, ∥P_Cx − P_Cy∥≤∥x − y∥, for all x, y ∈ H. The mapping P_C is firmly nonexpansive; that is, ∥P_Cx − P_Cy∥² ≤ 〈P_Cx − P_Cy, x − y〉, for all x, y ∈ H. It is well known that

()

A set-valued mapping M : H → 2^H is called monotone if, for all x, y ∈ H, f ∈ Mx and g ∈ My imply 〈x − y, f − g〉≥0. A monotone mapping M : H → 2^H is called maximal, if its graph of any Graph (M) : = {(x, f) ∈ H × H | f ∈ M(x)} of M is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping M is maximal if and only if for all (x, f) ∈ H × H, 〈x − y, f − g〉 ≥ 0, for all (y, g)∈ Graph (M) (the graph of mapping M) implies that f ∈ Mx.

Definition 2.1. Let M : H → 2^H be a multivalued maximal monotone mapping; then the set-valued mapping J_M,λ : H → H defined by

()

is called the resolvent operator associated with M, where λ is any positive number and I is the identity mapping.

Lemma 2.2 (see [16].)Let M : H → 2^H be a maximal monotone mapping and let B : H → H be a Lipschitz continuous mapping. Then the mapping M + B : H → 2^H is a maximal monotone mapping.

Lemma 2.3 (see [16], [17].)

(1)
The resolvent operator J_M,λ is single valued and nonexpansive for all λ > 0; that is,

()

(2)
The resolvent operator J_M,λ is 1-inverse-strongly monotone; that is,

()

Lemma 2.4 (see [17].)

(1)
Let is a solution of problem (1.3) if and only if for all λ > 0; that is,

()

(2)
If λ ∈ [0,2β], then I(B, M) is a closed convex subset in H.

Lemma 2.5 (see [18].)Each Hilbert space H satisfies Opial’s condition; that is, for any sequence {x_n} ⊂ H with x_n⇀x, the inequality

()

holds for each y ∈ H with y ≠ x.

Lemma 2.6 (see [19].)Let {x_n} and {z_n} be bounded sequences in a Banach space E, and let {β_n} be a sequence in [0,1] with 0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1. Suppose x_n+1 = (1 − β_n)z_n + β_nx_n for all integers n ≥ 1 and limsup _n→∞(∥z_n+1 − z_n∥−∥x_n+1 − x_n∥) ≤ 0. Then, lim _n→∞∥z_n − x_n∥ = 0.

Lemma 2.7 (see [20].)Assume {a_n} is a sequence of nonnegative real numbers such that

()

where {b_n} is a sequence in (0,1) and {δ_n} is a sequence in ℝ such that

(1)
,
(2)
limsup _n→∞δ_n/b_n ≤ 0 or .

Then lim _n→∞a_n = 0.

Lemma 2.8. Let H be a real Hilbert space. Then hold the following identities:

(i)
∥tx+(1−t)y∥² = t∥x∥² + (1 − t)∥y∥² − t(1 − t)∥x−y∥², ∀ t ∈ [0,1], ∀ x, y ∈ H,
(ii)
∥x+y∥² ≤ ∥x∥² + 2〈y, x + y〉, ∀ x, y ∈ H.

Lemma 2.9 (see [21].)Let C be a nonempty closed subset of a Banach space, and let {S_n} be a sequence of mappings of C into itself. Suppose that . Then, for each y ∈ C, {S_ny} converges strongly to some point of C. Moreover, let S be a mapping of C into itself defined by

()

Then lim _n→∞sup {∥Sz − S_nz∥:z ∈ C} = 0.

For solving the equilibrium problem for a bifunction F : C × C → ℝ, let us assume that F satisfies the following conditions:

(A1)
F(x, x) = 0 for all x ∈ C;
(A2)
F is monotone, that is, F(x, y) + F(y, x) ≤ 0, ∀ x, y ∈ C;
(A3)
for each x, y, z ∈ C, lim _t↓0F(tz + (1 − t)x, y) ≤ F(x, y);
(A4)
for each x ∈ C, y ↦ F(x, y) is convex and lower semicontinuous.

Lemma 2.10 (see [5].)Let C be a nonempty closed convex subset of H, and let F be a bifunction of C × C into ℝ satisfying (A1)–(A4). Let r > 0 and x ∈ H. Then, there exists z ∈ C such that

()

Lemma 2.11 (see [6].)Assume that F : C × C → ℝ satisfies (A1)–(A4). For r > 0 and x ∈ H, define a mapping T_r : H → C as follows:

()

for all x ∈ H. Then, the following hold:

(i)
T_r is single valued;
(ii)
T_r is firmly nonexpansive; that is, for any ;
(iii)
F(T_r) = EP (F);
(iv)
EP (F) is closed and convex.

Lemma 2.12 (see [22].)Let H be a Hilbert space and M a maximal monotone on H. Then, the following holds:

()

where J_M,r = (I + rM) ⁻¹ and J_M,s = (I + sM) ⁻¹.

3. Main Results

In this section, we will use the viscosity approximation method to prove a strong convergence theorem for finding a common element of the set of fixed points of a countable family of nonexpansive mappings, the set of solutions of the variational inequality problem for relaxed cocoercive and Lipschitz continuous mappings, the set of solutions of system of variational inclusions, and the set of solutions of equilibrium problem in a real Hilbert space.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H, and let B : C → H be relaxed (ϕ, ω)-cocoercive and μ-Lipschitz continuous with ω > ϕμ², for some ϕ, ω, μ > 0. Let 𝒢 = {G_k : k = 1,2, 3, …, N} be a finite family of β-inverse strongly monotone mappings from C into H, and let F be a bifunction from C × C → ℝ satisfying (A1)–(A4). Let f : C → C be a contraction with coefficient ψ (0 ≤ ψ < 1), and let {S_n} be a sequence of nonexpansive mappings of C into itself such that

()

Let the sequences {x_n} and {y_n} be generated by

()

where {α_n}, {β_n}, {γ_n}⊂(0,1) and {ξ_n}, {r_n}⊂(0, ∞) satisfy the following conditions:

(C1)
α_n + β_n + γ_n = 1,
(C2)
,
(C3)
0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1,
(C4)
{ξ_n}⊂[a, b] for some a, b with 0 ≤ a ≤ b ≤ 2(ω − ϕμ²)/μ² and lim _n→∞|ξ_n+1 − ξ_n| = 0,
(C5)
and lim _n→∞|λ_k,n+1 − λ_k,n| = 0, for each k ∈ {1,2, …, N},
(C6)
liminf _n→∞r_n > 0 and lim _n→∞|r_n+1 − r_n| = 0.

Suppose that

for any bounded subset K of C. Let S be a mapping of C into itself defined by Sy = lim _n→∞S_ny for all y ∈ C and suppose that

. Then, the sequences {x_n} and {y_n} converge strongly to the same point x^* ∈ Ω, where x^* = P_Ωf(x^*).

Proof. First, we prove that the mapping P_Ωf : H → C has a unique fixed point.

In fact, since f : C → C is a contraction with ψ ∈ [0,1) and P_Ωf : H → Ω is also a contraction, we obtain

()

Therefore, there exists a unique element x^* ∈ C such that x^* = P_Ωf(x^*), where

()

Now, we prove that (I − ξ_nB) is nonexpansive.

Indeed, for any x, y ∈ C, since B : C → H is a μ-Lipschitz continuous and relaxed (ϕ, ω)-cocoercive mappings with ω > ϕμ² and ξ_n ≤ 2(ω − ϕμ²)/μ², we obtain

()

Setting

()

thus,

()

which implies that

()

Hence (I − ξ_nB) is nonexpansive.

We divide the proof of Theorem 3.1 into five steps.

Step 1. We show that the sequence {x_n} is bounded.

Now, let and if is a sequence of mappings defined as in Lemma 2.11, then , and let . So, we have

()

For k ∈ {1,2, …, N} and for any positive integer number n, we define the operator

as follows:

()

for all n, we get

. On the other hand, since G_k : C → H is β-inverse strongly monotone and λ_k,n ⊂ [c, d]⊂(0,2β), then

is nonexpansive. Thus

is nonexpansive. From Lemma 2.4(1), we have

. It follows that

()

Setting v_n = P_C(y_n − ξ_nBy_n) and I − ξ_nB is a nonexpansive mapping, we obtain

()

From (3.2) and (3.12), we deduce that

()

It follows from induction that

()

Therefore, {x_n} is bounded and hence so are {v_n}, {y_n}, {u_n}, {By_n}, and {S_nv_n}.

Step 2. We claim that lim _n→∞∥x_n+1 − x_n∥ = 0.

By the definition of T_r, and , we get

()

Taking y = u_n+1 in (3.15) and y = u_n in (3.16), we have

()

and hence

()

So, from (A2) we have

()

and hence

()

Without loss of generality, let us assume that there exists a real number c such that r_n > c > 0 for all n ∈ ℕ. Then, we have

()

and hence

()

where M₁ = sup {∥u_n − x_n∥:n ∈ ℕ}.

Notice from Lemma 2.12 that

()

where M₂ is an appropriate constant such that

()

Since I − ξ_nB is nonexpansive mappings, we have the following estimates:

()

Substituting (3.23) into (3.25), we obtain

()

Indeed, define x_n+1 = (1 − β_n)z_n + β_nx_n for all n ∈ ℕ. It follows that

()

Thus, we have

()

Now, compute

()

Combining (3.28) and (3.29), we have

()

It follows that

()

This together with conditions (C1)–(C6) and lim _n→∞sup {∥S_n+1z − S_nz∥:z ∈ {v_n}} = 0 implies that

()

Hence, by Lemma 2.6, we obtain ∥z_n − x_n∥→0 as n → ∞. It then follows that

()

By (3.26), we also have

()

Step 3. We claim that lim _n→∞∥Sv_n − v_n∥ = 0.

Since {G_k : k = 1,2, 3, …, N} is β-inverse strongly monotone mappings, by the choice of {λ_k,n} for given and k ∈ {0,1, 2, …, N − 1}, we also have

()

Form (3.13), we have

()

It follows that

()

By condition (C2), (3.33), and liminf _n→∞γ_n > 0, we obtain

()

From Lemma 2.3(2) and as I − λ_k+1,nG_k+1 is nonexpansive, we have

()

which yields that

()

Substituting (3.40) into (3.36), we obtain

()

It follows that

()

By condition (C2), (3.33), (3.38), and liminf _n→∞γ_n > 0, we obtain

()

For

, we obtain

()

On the other hand, we have

()

It follows that

()

It now follows from the last inequality, conditions (C2), (3.33), and liminf _n→∞γ_n > 0 that

()

Since P_C is firmly nonexpansive, we have

()

which yields that

()

Substituting (3.49) into (3.45), we obtain

()

It follows that

()

By condition (C2), (3.33), (3.47), and liminf _n→∞γ_n > 0, we obtain

()

On the other hand, in the light of Lemma 2.11(ii)

is firmly nonexpansive; so we have

()

which implies that

()

Form (3.45), we have

()

It follows that

()

By condition (C2), (3.33), and liminf _n→∞γ_n > 0, we obtain

()

Observe that

()

By condition (C2) and (3.33), we have

()

Since

()

from (3.57) and (3.59), we have

()

Form (3.55), we have

()

It follows that

()

By condition (C2), (3.33), and liminf _n→∞γ_n > 0, we obtain

()

Since

()

from (3.57) and (3.64), we have

()

Furthermore, by the triangular inequality we also have

()

From (3.52), (3.61), and (3.66), we have

()

Applying Lemma 2.9 and (3.68), we have

()

Step 4. We claim that limsup _n→∞〈f(x^*) − x^*, x_n − x^*〉 ≤ 0.

Indeed, we choose a subsequence of {v_n} such that

()

Without loss of generality, let

. From ∥Sv_n − v_n∥→0, we obtain

. Then, (3.70) reduces to

()

In order to show 〈f(x^*) − x^*, z − x^*〉 ≤ 0, it suffices to show that

()

Firstly, we will show

Assume z ∉ F(S). By Opial’s theorem (Lemma 2.5) and ∥Sv_n − v_n∥→0, we have

()

This is a contradiction. Thus, we obtain z ∈ F(S).

Next, we will show that z ∈ VI (C, B).

Let

()

Since B is relaxed (ϕ, ω)-cocoercive, μ-Lipschitz continuous with ω > ϕμ², we obtain

()

which yields that B is monotone. Then T is maximal monotone (see [23]). Let (w₁, w₂) ∈ G(T). Since w₂ − Bw₁ ∈ N_C(w₁) and v_n ∈ C, we have 〈w₁ − v_n, w₂ − Bw₁〉 ≥ 0. On the other hand, from v_n = P_C(y_n − ξ_nBy_n), we have

()

that is,

()

Therefore, we obtain

()

Noting that

and B is relaxed (ϕ, ω)-cocoercive and (3.78), we obtain

()

Since T is maximal monotone, we have z ∈ T⁻¹0, and hence z ∈ VI (C, B).

Now, we will show that .

For this purpose, let k ∈ {1,2, 3, …, N} and G_k is β-inverse strongly monotone, G_k is an 1/β-Lipschitz continuous monotone mapping. From Lemma 2.2, we know that M_k + G_k is maximal monotone. Let (v, g) ∈ G(M_k + G_k); that is, g − G_kv ∈ M_k(v). On the other hand, since , we have

()

that is,

()

By virtue of the maximal monotonicity of M_k + G_k, we have

()

and so

()

From

, we also obtain that

and {G_k : k = 1,2, 3, …, N} are Lipschitz continuous; we have

()

Since M_k + G_k is maximal monotone, we have θ ∈ (M_k + G_k)(z); that is,

Finally, we will show that z ∈ EP (F).

Since , we have

()

If follows from (A2) that

()

and hence

()

Since

and

, it follows by (A4) that F(y, z) ≤ 0 for all y ∈ H. For t with 0 < t ≤ 1 and y ∈ H, let y_t = ty + (1 − t)z. Since y ∈ H and z ∈ H, we have y_t ∈ H, and hence F(y_t, z) ≤ 0. So, from (A1) and (A4) we have

()

and hence F(y_t, y) ≥ 0. From (A3), we have F(z, y) ≥ 0 for all y ∈ H and hence z ∈ EP (F). Therefore, it follows that z ∈ Ω.

Since x^* = P_Ωf(x^*), we have

()

On the other hand, we have

()

Since ∥x_n+1 − x_n∥→0 as n → ∞ and (3.91), we have

()

Step 5. We claim that lim _n→∞∥x_n − x^*∥ = 0.

Indeed, from (3.2) and (3.12), we obtain

()

which implies that

()

where b_n = α_n(1 − ψ²) and δ_n = 2α_n〈f(x^*) − x^*, x_n+1 − x^*〉. It is easy to see that b_n → 0,

, and limsup _n→∞δ_n/b_n ≤ 0. Applying Lemma 2.7 to (3.93), we conclude that

()

Consequently, also {y_n} converges strongly to x^*. The proof is now complete.

As in [21, Theorem 4.1], we can generate a sequence {S_n} of nonexpansive mappings satisfying condition for any bounded subset K of C by using convex combination of general sequence {T_k} of nonexpansive mappings with a common fixed point.

Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H, and let B : C → H be relaxed (ϕ, ω)-cocoercive and μ-Lipschitz continuous with ω > ϕμ², for some ϕ, ω, μ > 0. Let 𝒢 = {G_k : k = 1,2, 3, …, N} be a finite family of β-inverse strongly monotone mappings from C into H, and let F be a bifunction from C × C → ℝ satisfying (A1)–(A4). Let f : C → C be a contraction with coefficient ψ (0 ≤ ψ < 1), and let be a family of nonnegative numbers with indices n, k ∈ ℕ with k ≤ n such that

()

Let the sequences {x_n} and {y_n} be generated by

()

where {α_n}, {β_n}, {γ_n}⊂(0,1) and {ξ_n}, {r_n}⊂(0, ∞) satisfy the following conditions:

(C1)
α_n + β_n + γ_n = 1,
(C2)
,
(C3)
0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1,
(C4)
{ξ_n} ⊂ [a, b] for some a, b with 0 ≤ a ≤ b ≤ 2(ω − ϕμ²)/μ² and lim _n→∞|ξ_n+1 − ξ_n| = 0,
(C5)
and lim _n→∞|λ_k,n+1 − λ_k,n| = 0, for each k ∈ {1,2, …, N},
(C6)
liminf _n→∞r_n > 0 and lim _n→∞|r_n+1 − r_n| = 0,
(C7)
, and .

Then, the sequences {x_n} and {y_n} converge strongly to the same point x^* ∈ Ω, where x^* = P_Ωf(x^*).

In Theorem 3.1, taking N = 1 and S_n = S, then we have the following corollary.

Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H, and let B : C → H be relaxed (ϕ, ω)-cocoercive and μ-Lipschitz continuous with ω > ϕμ², for some ϕ, ω, μ > 0. Let G be an β-inverse strongly monotone mappings from C into H, and let F be a bifunction from C × C → ℝ satisfying (A1)–(A4). Let f : C → C be a contraction with coefficient ψ (0 ≤ ψ < 1), and let S be a nonexpansive mappings of C into itself such that

()

Let the sequences {x_n} and {y_n} be generated by

()

where {α_n}, {β_n},{γ_n} ⊂ (0,1) and {ξ_n}, {r_n} ⊂ (0, ∞) satisfy the following conditions:

(C1)
α_n + β_n + γ_n = 1,
(C2)
,
(C3)
0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1,
(C4)
{ξ_n} ⊂ [a, b] for some a, b with 0 ≤ a ≤ b ≤ 2(ω − ϕμ²)/μ² and lim _n→∞ | ξ_n+1 − ξ_n | = 0,
(C5)
{λ_n} ⊂ [c, d] ⊂ (0,2β) and lim _n→∞|λ_n+1 − λ_n| = 0,
(C6)
liminf _n→∞r_n > 0 and lim _n→∞|r_n+1 − r_n| = 0.

Then, the sequences {x_n} and {y_n} converge strongly to the same point x^* ∈ Ω, where x^* = P_Ωf(x^*).

Acknowledgments

This research was supported by the Rajamangala University of Technology Rattanakosin Research and Development Institute, the Thailand Research Fund, and the Commission on Higher Education under Grant no. MRG5480206. The second author would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office ofthe Higher Education Commission (under the Project NRU-CSEC no. 54000267) for financial support. The authors are very grateful to the referees for their careful reading, comments, and suggestions which improved the presentation of this article.

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Viscosity Approximation Method for System of Variational Inclusions Problems and Fixed-Point Problems of a Countable Family of Nonexpansive Mappings

Abstract

1. Introduction

2. Preliminaries

3. Main Results

Acknowledgments

References

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Viscosity Approximation Method for System of Variational Inclusions Problems and Fixed-Point Problems of a Countable Family of Nonexpansive Mappings

Abstract

1. Introduction

2. Preliminaries

3. Main Results

Acknowledgments

References

References

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