Approximating fixed points of amenable semigroup and infinite family of nonexpansive mappings and solving systems of variational inequalities and systems of equilibrium problems

We introduce an iterative scheme for finding a common element of the set of solutions for systems of equilibrium problems and systems of variational inequalities and the set of common fixed points for an infinite family and left amenable semigroup of nonexpansive mappings in Hilbert spaces. The results presented in this paper mainly extend and improved some well-known results in the literature.

Mathematics Subject Classification (2000): 47H09; 47H10; 47H20; 43A07; 47J25.

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H.

Let A: C → H be a nonlinear mapping. The classical variational inequality problem is to fined x ∈ C such that

The set of solution of (1) is denoted by VI(C, A), i.e.,

Recall that the following definitions:

(1) A is called monotone if

(2) A is called α-strongly monotone if there exists a positive constant α such that

(3) A is called μ-Lipschitzian if there exist a positive constant μ such that

(4) A is called α-inverse strongly monotone, if there exists a positive real number α > 0

such that

It is obvious that any α-inverse strongly monotone mapping B is -Lipschitzian.

(5) A mapping T : C → C is called nonexpansive if ∥ Tx - Ty ∥≤∥ x - y ∥ for all x, y ∈ C. Next, we denote by Fix(T) the set of fixed point of T.

(6) A mapping f : C → C is said to be contraction if there exists a coefficient α ∈ (0, 1) such that

(7) A set-valued mapping U : H → 2^H is called monotone if for all x, y ∈ H, f ∈ Ux and g ∈Uy imply 〈x - y, f - g〉 ≥ 0.

(8) A monotone mapping U : H → 2^H is maximal if the graph G(U) of U is not properly contained in the graph of any other monotone mapping.

It is known that a monotone mapping U is maximal if and only if for (x, f) ∈ H × H, 〈x - y, f - g〉 ≤ 0 for every (y, g) ∈ G(U) implies that f ∈ Ux. Let B be a monotone mapping of C into H and let N_Cx be the normal cone to C at x ∈ C, that is, N_Cx = {y ∈ H : 〈x - z, y〉 ≤ 0, ∀z ∈ C} and define

Then U is the maximal monotone and 0 ∈ Ux if and only if x ∈ VI(C, B); see [1].

Let F be a bi-function of C×C into ℝ, where ℝ is the set of real numbers. The equilibrium problem for F : C × C → ℝ is to determine its equilibrium points, i.e the set

Let be a family of bi-functions from C × C into ℝ. The system of equilibrium problems for is to determine common equilibrium points for , i.e the set

Numerous problems in physics, optimization, and economics reduce into finding some element of EP(F). Some method have been proposed to solve the equilibrium problem; see, for instance [2-5]. The formulation (3), extend this formalism to systems of such problems, covering in particular various forms of feasibility problems [6,7].

Given any r > 0 the operator defined by

is called the resolvent of F, see [3]. It is shown [3] that under suitable hypotheses on F (to be stated precisely in Sect. 2), is single- valued and firmly nonexpansive and

satisfies

Using this result, in 2007, Yao et al. [8], proposed the following explicit scheme with respect to W-mappings for an infinite family of nonexpansive mappings:

They proved that if the sequences {α_n}, {β_n}, {γ_n} and {r_n} of parameters satisfy appropriate conditions, then, the sequences {x_n} and both converge strongly to the unique , where . Their results extend and improve the corresponding results announced by Combettes and Hirstoaga [3] and Takahashi and Takahashi [5].

Very recently, Jitpeera et al. [9], introduced the iterative scheme based on viscosity and Cesàro mean

where B : C → H is β-inverse strongly monotone, φ: C → ℝ ∪ {∞} is a proper lower semi-continuous and convex function, Tⁱ : C → C is a nonexpansive mapping for all i = 1, 2, ..., n, {α_n}, {β_n}, {δ_n} ⊂ (0, 1), {λ_n} ⊂ (0, 2β) and {r_n} ⊂ (0, ∞) satisfy the following conditions

(i) lim_n→∞ α_n = 0, ,

(ii) lim_n→∞ δ_n = 0

(iii) 0 < lim inf_n→∞ β_n ≤ lim sup_n→∞ β_n < 1.

(iv) {λ_n} ⊂ [a, b] ⊂ (0, 2β) and lim inf_n→∞ | λ_n+1 - λ_n |= 0,

(v) lim inf_n→∞ r_n > 0 and lim inf_n→∞ | r_n+1 - r_n |= 0.

They show that if is nonempty, then the sequence {x_n} converges strongly to the z = P_θ(I - A + γf )z which is the unique solution of the variational inequality

In this paper, motivated and inspired by Yao et al. [8,10-15], Lau et al. [16], Jitpeera et al. [9], Kangtunyakarn [17] and Kim [18], Atsushiba and Takahashi [19], Saeidi [20], Piri [21-23] and Piri and Badali [24], we introduce the following iterative scheme for finding a common element of the set of solutions for a system of equilibrium problems for a family of equilibrium bi-functions, systems of variational inequalities, the set of common fixed points for an infinite family ψ = {T_i, i = 1, 2, ...} of nonexpansive mappings and a left amenable semigroup φ = {T_t : t ∈ S} of nonexpansive mappings, with respect to W-mappings and a left regular sequence {μ_n} of means defined on an appropriate space of bounded real-valued functions of the semigroup

where A: C → H be β-inverse monotone map and B : C → H be δ-inverse monotone map. We prove that under mild assumptions on parameters like that in Yao et al. [8], the sequences {x_n} and converge strongly to , where .

Compared to the similar works, our results have the merit of studying the solutions of systems of equilibrium problems, systems of variational inequalities and fixed point problems of amenable semigroup of nonexpansive mappings. Consequence for nonnegative integer numbers is also presented.

Let S be a semigroup and let B(S) be the space of all bounded real valued functions defined on S with supremum norm. For s ∈ S and f ∈ B(S), we define elements l_sf and r_sf in B(S) by

Let X be a subspace of B(S) containing 1 and let X* be its topological dual. An element μ of X* is said to be a mean on X if ∥ μ ∥ = μ(1) = 1. We often write μ_t(f(t)) instead of μ(f) for μ ∈ X* and f ∈ X. Let X be left invariant (respectively right invariant), i.e., l_s(X) ⊂ X (respectively r_s(X) ⊂ X) for each s ∈ S. A mean μ on X is said to be left invariant (respectively right invariant) if μ(l_sf) = μ(f) (respectively μ(r_sf) = μ(f)) for each s ∈ S and f ∈ X. X is said to be left (respectively right) amenable if X has a left (respectively right) invariant mean. X is amenable if X is both left and right amenable. As is well known, B(S) is amenable when S is a commutative semigroup, see [25]. A net {μ_α} of means on X is said to be strongly left regular if

for each s ∈ S, where is the adjoint operator of l_s.

Let S be a semigroup and let C be a nonempty closed and convex subset of a reflexive Banach space E. A family φ = {T_t : t ∈ S} of mapping from C into itself is said to be a nonexpansive semigroup on C if T_t is nonexpansive and T_ts = T_tT_s for each t, s ∈ S. By Fix(φ) we denote the set of common fixed points of φ, i.e.

Lemma 2.1. [25]Let S be a semigroup and C be a nonempty closed convex subset of a reflexive Banach space E. Let φ = {T_t : t ∈ S} be a nonexpansive semigroup on H such that {T_tx : t ∈ S} is bounded for some x ∈ C, let X be a subspace of B(S) such that 1 ∈ X and the mapping t → 〈T_tx, y*〉 is an element of X for each x ∈ C and y* ∈ E*, and μ is a mean on X. If we write T_μx instead of ∫ T_txdμ(t), then the followings hold.

(i) T_μ is nonexpansive mapping from C into C.

(ii) T_μx = x for each x ∈ Fix(φ).

(iii) for each x ∈ C.

Let C be a nonempty subset of a Hilbert space H and T : C → H a mapping. Then T is said to be demiclosed at v ∈ H if, for any sequence {x_n} in C, the following implication holds:

where → (respectively ⇀) denotes strong (respectively weak) convergence.

Lemma 2.2. [26]Let C be a nonempty closed convex subset of a Hilbert space H and suppose that T : C → H is nonexpansive. then, the mapping I - T is demiclosed at zero.

Lemma 2.3. [27]For a given x ∈ H, y ∈ C,

It is well known that P_C is a firmly nonexpansive mapping of H onto C and satisfies

Moreover, P_C is characterized by the following properties: P_Cx ∈ C and for all x ∈ H, y ∈ C,

It is easy to see that (7) is equivalent to the following inequality

Using Lemma 2.3, one can see that the variational inequality (1) is equivalent to a fixed point problem. It is easy to see that the following is true:

Lemma 2.4. [28]Let {x_n} and {y_n} be bounded sequences in a Banach space E and let {α_n} be a sequence in [0, 1] with . Suppose x_n+1 = α_nx_n+(1-α_n)y_n for all integers n ≥ 0 and

Then, .

Let F : C × C → ℝ be a bi-function. Given any r > 0, the operator defined by

is called the resolvent of F, see [3]. The equilibrium problem for F is to determine its equilibrium points, i.e., the set

Let be a family of bi-functions from C × C into ℝ. The system of equilibrium problems for is to determine common equilibrium points for . i.e, the set

Lemma 2.5. [3]Let C be a nonempty closed convex subset of H and F : C × C → ℝ satisfy

(A₁) F (x, x) = 0 for all x ∈ C,

(A₂) F is monotone, i.e, F(x, y) + F(y, x) ≤ 0 for all x, y ∈ C,

(A₃) for all x, y, z ∈ C, lim_t→0 F(tz + (1 - t)x, y) ≤ F (x, y),

(A₄) for all x ∈ C, y → F(x, y) is convex and lower semi-continuous.

Given r > 0, define the operator , the resolvent of F, by

Then,

(1) is single valued,

(2) is firmly nonexpansive, i.e, for all x, y ∈ H,

(3) ,

(4) EP(F) is closed and convex.

Let T₁, T₂, ... be an infinite family of mappings of C into itself and let λ₁, λ₂, ... be a real numbers such that 0 ≤ λ_i < 1 for every i ∈ ℕ. For any n ∈ ℕ, define a mapping W_n of C into C as follows:

Such a mapping W_n is called the W-mapping generated by T₁, T₂, ..., T_n and λ₁, λ₂, ..., λ_n.

Lemma 2.6. [29]Let C be a nonempty closed convex subset of a Hilbert space H, {T_i : C → C} be an infinite family of nonexpansive mappings with , {λ_i} be a real sequence such that 0 < λ_i ≤ b < 1, ∀i ≥ 1. Then

(1) W_n is nonexpansive and for each n ≥ 1,

(2) for each x ∈ C and for each positive integer j, the limit lim_n→∞ U_n,j exists.

(3) The mapping W : C → C defined by

is a nonexpansive mapping satisfying and it is called the W-mapping generated by T₁, T₂, ... and λ₁, λ₂, ....

Lemma 2.7. [30]Let C be a nonempty closed convex subset of a Hilbert space H, {T_i : C → C} be a countable family of nonexpansive mappings with ,{λ_i} be a real sequence such that 0 < λ_i ≤ b < 1, ∀i ≥ 1. If D is any bounded subset of C, then

Lemma 2.8. [31]Let {a_n} be a sequence of nonnegative real numbers such that

where {b_n} and {c_n} are sequences of real numbers satisfying the following conditions:

(i) {b_n} ⊂ [0, 1], ,

(ii) either or .

Then, .

Lemma 2.9. [32]Let (E, 〈., .〉) be an inner product space. Then for all x, y, z ∈ E and α, β, γ, ∈ [0, 1] such that α + β + γ = 1, we have

Notation Throughout the rest of this paper the open ball of radius r centered at 0 is denoted by B_r. For a subset A of H we denote by the closed convex hull of A. For ε > 0 and a mapping T : D → H, we let F_ϵ(T; D) be the set of ϵ-approximate fixed points of T, i.e., F_ϵ(T ; D) = {x ∈ D :∥ x - Tx ∥ ≤ ϵ}. Weak convergence is denoted by ⇀ and strong convergence is denoted by →.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H, A: C → H a β-inverse strongly monotone, B : C → H a γ-inverse strongly monotone, S a semigroup and φ = {T_t : t ∈ S} be a nonexpansive semigroup from C into C such that . Let X be a left invariant subspace of B(S) such that 1 ∈ X, and the function t → 〈T_tx, y〉 is an element of X for each x ∈ C and y ∈ H, {μ_n} a left regular sequence of means on X such that lim_n→∞ ∥μ_n+1 - μ_n∥ = 0. Let be a finite family of bi-functions from C × C into ℝ which satisfy (A₁)-(A₄) and an infinite family of nonexpansive mappings of C into C such that for each i ∈ ℕ and . Let {α_n}, {β_n}, {γ_n} and {η_n} be a sequences in (0, 1). Let {ζ_n} a sequence in (0, 2β), {δ_n} a sequence in (0, 2γ), be sequences in (0, ∞) and {λ_n} a sequence of real numbers such that 0 < λ_n ≤ b < 1. Assume that,

(B₁) lim_n→∞ η_n = η ∈ (0, 1), lim_n→∞ α_n = 0 and ,

(B₂) 0 < lim inf_n→∞ β_n ≤ lim sup_n→∞ β_n < 1,

(B₃) α_n + β_n + γ_n = 1,

(B₄) lim_n→∞ | ζ_n+1 - ζ_n |= lim_n→∞ | δ_n+1 - δ_n |= 0,

(B₅) lim inf_n→∞ r_k,n > 0 and lim_n→∞ (r_k,n+1 - r_k,n) = 0 for k ∈ {1, 2, · · ·, M}.

Let f be a contraction of C into itself with coefficient α ∈ (0, 1) and given x₁ ∈ C arbitrarily. If the sequences {x_n}, {y_n} and {z_n} are generated iteratively by x₁ ∈ C and

then, the sequences {x_n}, {y_n} and converge strongly to , which is the unique solution of the system of variational inequalities:

Proof. Since A is a β-inverse strongly monotone map, for any x, y ∈ C, we have

It follows that

Since B is a β-inverse strongly monotone map, repeating the same argument as above, we can deduce that

Let , in the context of the variational inequality problem the characterization of projection (9) implies that p = P_C(p - ζ_nAp) and p = P_C(p - δ_nBp). Using (12) and (13), we get

By taking v_n = P_C(z_n - ζ_nAz_n), w_n = P_C(z_n - δ_nBz_n) and for k ∈ {1, 2, ..., M} and for all n ∈ ℕ, we shall equivalently write scheme (11) as follows:

We shall divide the proof into several steps.

Step 1. The sequence {x_n} is bounded.

Proof of Step 1. Let . Since for each k ∈ {1, 2, ..., M}, is nonexpansive we have

Thus, by Lemmas 2.1, 2.5 and (14), we have

By induction,

Step 2. Let {u_n} be a bounded sequence in H. Then

for every k ∈ {1, 2, ..., M}.

Proof of Step 2. This assertion is proved in [27, Step 2].

Step 3. Let {u_n} be a bounded sequence in H. Then

This assertion is proved in [21, Step 3].

Step 4. lim_n→∞ ∥ x_n+1 - x_n ∥ = 0.

Proof of Step 4. Setting x_n+1 = β_nx_n + (1 - β_n)t_n for all n ≥ 1, we have

Therefore, we have

On the other hand

Observing that , and we get

and

Take y = z_n+1 in (17) and y = z_n in (18), by using (A₂), it follows that

and hence

Thus, we have

Since v_n = P_C(z_n - ζ_nAz_n) and w_n = P_C(z_n - δ_nBz_n), it follows from the definition of {y_n} that

Therefore,

This together with conditions (B₁), (B₄), Steps 2 and 3 imply that

Hence by Lemma 2.4, we obtain lim_n→∞ ∥ t_n - x_n ∥ = 0. Consequently,

Step 5. , ∀k ∈ {0, 1, 2, ..., M - 1}.

Proof of Step 5. Let and k ∈ {1, 2, ..., M - 1}. Since is firmly nonexpansive, we obtain

It follows that

Using Lemma 2.9, (14) and (19), we obtain

Then, we have

It is easily seen that lim inf_n→∞ γ_n > 0. So we have

Step 6. .

Proof of Step 6. Observe that

hence

It follows from conditions (B₁), (B₂) and Step 4, that

Step 7. lim_n→∞ ∥ x_n - T_tx_n ∥ = 0, for all t ∈ S.

Proof of Step 7. Let and set and D = {y ∈ H : ∥ y - p ∥ ≤ M₀}, we remark that D is bounded closed convex set, {y_n} ⊂ D and it is invariant under , φ and W_n for all n ∈ ℕ. We will show that

Let ϵ > 0. By [33, Theorem 1.2], there exists δ > 0 such that

Also by [33, Corollary 1.1], there exists a natural number N such that

for all t, s ∈ S and y ∈ D. Let t ∈ S. Since {μ_n} is strongly left regular, there exists N₀ ∈ ℕ such that for n ≥ N₀ and i = 1, 2, ..., N. Then, we have

By Lemma 2.1 we have

It follows from (21), (22), (23) and (24) that

for all y ∈ D and n ≥ N₀. Therefore,

Since ϵ > 0 is arbitrary, we get (20).

Let t ∈ S and ϵ > 0. Then, there exists δ > 0, which satisfies (21). From condition (B₁), (20) and Step 6, there exists N₁ ∈ ℕ such that , for all y ∈ D and for all n ≥ N₁. We note that

for all n ≥ N₁. Therefore, we have

for all n ≥ N₁. This shows that

Since ϵ > 0 is arbitrary, we get lim_n→∞ ∥ x_n - T_t(x_n) ∥ = 0.

Step 8. The weak ω-limit set of {x_n}, ω_ω{x_n}, is a subset of .

Proof of Step 8. Let z ∈ ω_ω{x_n} and let be a subsequence of {x_n} weakly converging to z, we need to show that . Noting Step 5, with no loss of generality, we may assume that . At first, note that by (A₂) and given y ∈ C and k ∈ {1, 2, ..., M}, we have

Step 5 and condition(B₅) imply that

Since , from the lower semi-continuity of F_k+1 on the second variable, we have F_k+1(y, z) ≤ 0 for all y ∈ C and for all k ∈ {0, 1, 2, ..., M - 1}. For t with 0 < t ≤ 1 and y ∈ C, let y_t = ty + (1 - t)z. Since y ∈ C and z ∈ C, we have y_t ∈ C and hence F_k+1(y_t, z) ≤ 0. So from the convexity of F_k+1 on second variable, we have

hence F_k+1(y_t, y) ≥ 0. therefore, we have F_k+1(z, y) ≥ 0 for all y ∈ C and k ∈ {0, 1, 2, ..., M-1}. Therefore .

Since , it follows by Step 7 and Lemma 2.2 that z ∈ Fix(T_t) for all t ∈ S. Therefore, z ∈ Fix(φ). We will show z ∈ Fix(W). Assume z ∉ Fix(W) Since , by our assumption, we have T_iz ∈ Fix(φ),∀i ∈ ℕ and then W_nz ∈ Fix(φ). Hence by Lemma 2.1, , therefore by Lemma 2.5, we get

Now, by (25), Step 6, Lemma 2.6 and Opial's condition, we have

This is a contradiction. So we get .

Now, let us show that z ∈ VI(C, A) ∩ VI(C, B). Observe that,

From (26), we have

which implies that

Therefore, from step 4 and condition B₁, we obtain

On the other hand from (26), we have

which implies that

Therefore, from step 4 and condition B₁, we obtain

From (6) and (12), we have

So we obtain

By using the same method as (29), we have

From (29), (30) and definition of y_n, we have,

By (31), we have

which implies that

and

Therefore, from 0 < lim inf_n→∞ γ_n ≤ lim sup_n→∞ γ_n < 1, condition B₁, step 4, (27) and (28) we get

Let U : H → 2^H be a set-valued mapping is defined by

where N_Cx is the normal cone to C at x ∈ C. Since A is monotone. Thus U is maximal monotone see [1]. Let (x, y) ∈ G(U), hence y - Ax ∈ N_Cx and since v_n = P_C(z_n - ζ_nAz_n) therefore, 〈x - v_n, y - Ax〉 ≥ 0. On the other hand from (7), we have

i.e.,

Therefore, we have

From (32), we get . Noting that and A is -lipschitzian, we obtain

Since U is maximal monotone, we have z ∈ U^-10, and hence z ∈ VI(C, A). Let V : H → 2^H be a set-valued mapping is defined by

where N_Cx is the normal cone to C at x ∈ C. Since B is monotone. Thus U is maximal monotone see [1]. Repeating the same argument as above, we can derive z ∈ VI(C, B). Therefore, .

Step 9. There exists a unique x* ∈ C such that

Proof of Step 9. Note that f is a contraction mapping with coefficient α ∈ (0, 1). Then for all x, y ∈ H. Therefore is a contraction of H into itself, which implies that there exists a unique element x* ∈ H such that . at the same time, we note that x* ∈ C. Using Lemma 2.3, we have

We can choose a subsequence of {x_n} such that

Since is bounded, therefore, has subsequence such that . With no loss of generality, we may assume that . Applying Step 8 and (34), we have

Step 10, The sequences {x_n} converges strongly to x*, which is obtained in Steep 9.

Proof of Step 10. We have

Which implies that

where

By Step 9, and condition (B₁), we get lim sup_n→∞ τ_n ≤ 0. Now applying Lemma 2.8 to (35), we conclude that x_n → x*. Consequently, from , we have , for all k ∈ {1, 2, ..., M}.

Corollary 3.2. (see Yao et al. [8]) Let C be a nonempty closed convex subset of a real Hilbert space H, F a bi-functions from C×C into ℝ which satisfy (A₁) - (A₄) and an infinite family of nonexpansive mapping of C into C such that . Let {α_n}, {β_n} and {γ_n} are three sequences in (0, 1) such that α_n + β_n + γ_n = 1 and {r_n} ⊂ (0, ∞). Suppose the following conditions are satisfied:

(B₁) lim_n→∞ α_n = 0 and ,

(B₂) 0 < lim inf_n→∞ β_n ≤ lim sup_n→∞ β_n < 1,

(B₃) lim inf_n→∞ r_n > 0 and lim_n→∞ (r_n+1 - r_n) = 0.

Let f be a contraction of C into itself with coefficient α ∈ (0, 1) and given x₁ ∈ C arbitrarily. Then the sequence {x_n} generated by

converge strongly to , where .

Proof. Take A = B = 0, φ = {I}, F₁ = F and F_k = 0 for k ∈ {2, ..., M} in Theorem 3.1, then we have and . So from Theorem 3.1 the sequence {x_n} converges strongly to , where .

Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H, be a finite family of bi-functions from C × C into ℝ which satisfy (A₁)-(A₄), T a nonexpansive mappings on C such that . Let {α_n}, {β_n} and {γ_n} are three sequences in (0, 1) such that α_n + β_n + γ_n = 1 and be sequences in (0, ∞). Suppose the following conditions are satisfied:

(B₁) lim_n→∞ α_n = 0 and ,

(B₂) 0 < lim inf_n→∞ β_n ≤ lim sup_n→∞ β_n < 1,

(B₃) lim inf_n→∞ r_k,n > 0 and lim_n→∞ (r_k,n+1 - r_k,n) = 0 for k ∈ {1, 2, ..., M}.

Let f be a contraction of H into itself and given x₁ ∈ H arbitrarily. If the sequences {x_n} generated iteratively by

Then, sequences {x_n} and converge strongly to , where .

Proof. Let S = {0, 1, ...}, φ = {Tⁱ : i ∈ S} and T⁰ = I. For f = (x₀, x₁, ...) ∈ B(S), define

Then {μ_n} is a regular sequence of means on B(S) such that lim_n→∞ ∥ μ_n+ - μ_n ∥ = 0; for more details, see [34]. Next for each x ∈ H and n ∈ ℕ, we have

Take A = B = 0, T_i = I for all i ∈ ℕ in Theorem 3.1 then we have y_n = z_n and W_n = I for all n ∈ ℕ. Therefore, it follows from Theorem 3.1 that the sequences {x_n} and converge strongly, as n → ∞ to a point , where .

Remark 3.4. Theorem 3.1 improve [8, Theorem 1.2] in the following aspects.

(a) Our iterative process (11) is more general than Yao et al. process (14) because it can be applied to solving the problem of finding a common element of the set of solutions of systems of equilibrium problems and systems of variational inequalities.

(b) Our iterative process (11) is very diffident from Yao et al. process (14) because there are left amenable semigroup of nonexpansive mappings.

(c) Our method of proof is very different from the on in Yao et al. [8] for example we use Corollary 1.1 and Theorem 1.2 of Bruck [33] fore the proof of Theorem 3.1.

The authors declare that they have no competing interests.

The authors are extremely grateful to the referees for useful suggestions that improved the contents of the paper.

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