Let H be a real Hilbert space with inner product 〈·, ·〉, and induced norm || · ||. A mapping T: H → H is called nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for all x, y ∈ H. The set of the fixed points of T is denoted by Fix(T): = {x ∈ H: Tx = x}.

Iterative theory and methods for nonlinear mappings and variational inequalities have recently been applied to solve convex minimization problems, zero point problems and many others; see, e.g., 123456789 and references therein.

The viscosity approximation method was first introduced by Moudafi 10. Starting with an arbitrary initial x₀ ∈ H, define a sequence {x_n} generated by:

x n + 1 = ε n 1 + ε n f ( x n ) + 1 1 + ε n T x n , ∀ n ≥ 0 ,

where f is a contraction with a coefficient α ∈ [0,1) on H, i.e., ||f(x) - f(y)|| ≤ α||x - y|| for all x, y ∈ H, and {ε_n} is a sequence in (0,1) satisfying the following given conditions:

(1) lim_n→∞ ε_n = 0;

(2) ∑ n = 0 ∞ ε n = ∞ ;

(3) lim n → ∞ ( 1 ε n − 1 ε n + 1 ) = 0 .

It is proved that the sequence {x_n} generated by (1.1) converges strongly to the unique solution x* ∈ C(C: = Fix(T)) of the variational inequality:

⟨ ( I - f ) x * , x - x * ⟩ ≥ 0 , ∀ x ∈ F i x ( T ) .

In 1, Xu proved that the sequence {x_n} defined by the below process started with an arbitrary initial x₀ ∈ H:

x n + 1 = α n b + ( I - α n A ) T x n , ∀ n ≥ 0 ,

converges strongly to the unique solution of the minimization problem (1.3) provided the the sequence {α_n} satisfies certain conditions:

min x ∈ C 1 2 ⟨ A x , x ⟩ - ⟨ x , b ⟩ ,

where C is the set of fixed points set of T on H and b is a given point in H.

In 2, Marino and Xu combined the iterative method (1.2) with the viscosity approximation method (1.1) and considered the following general iterative method:

x n + 1 = α n γ f ( x n ) + ( I - α n A ) T x n , ∀ n ≥ 0 .

It is proved that if the sequence {α_n} satisfies appropriate conditions, the sequence {x_n} generated by (1.4) converges strongly to the unique solution of the variational inequality:

⟨ ( γ f - A ) x ̃ , x - x ̃ ⟩ ≤ 0 , ∀ x ∈ C ,

or equivalently x ̃ = P F i x ( T ) ( I - A + γ f ) x ̃ , where C is the fixed point set of a nonexpansive mapping T.

In 11, Maingé considered the viscosity approximation method (1.1), and expanded the strong convergence to quasi-nonexpansive mappings in Hilbert space. Motivated by Marino and Xu 2 and Maingé 11, we consider the following iterative process:

x 0 = x ∈ H a r b i t r a r i l y c h o s e n , x n + 1 = α n γ f ( x n ) + ( I - α n A ) T ω x n , ∀ n ≥ 0 ,

where T_ω = (1 - ω)I + ωT, and T is a quasi-nonexpansive mapping. Under some appropriate conditions on ω and {α_n}, we obtain strong convergence over the class of quasi-nonexpansive mappings in Hilbert spaces. Our result is more general than Maingé's 11 conclusion, and also extends the iterative method (1.4) to quasi-nonexpansive mappings.

Throughout this article, we write x_n ⇀ x to indicate that the sequence {x_n} converges weakly to x. x_n → x implies that the sequence {x_n} converges strongly to x. The following lemmas are useful for our article.

The following identities are valid in a Hilbert space H: for each x,y ∈ H, t ∈ [0, 1]

(i) ||x + y||² ≤ ||x||² + 2〈y, x + y〉;

(ii) ||(1 - t)x + ty||² = (1 - t)||x||² + t||y||² - (1 - t) t||x - y||²;

(iii) ⟨ x , y ⟩ = - 1 2 x - y 2 + 1 2 x 2 + 1 2 y 2 .

Lemma 2.1. 2 Let H be a Hilbert space H. Given x ∈ H, C is a closed convex subset of H, f : H → H is a contraction with coefficient 0 < α < 1, and A is a strongly positive linear bounded operator with coefficient γ ̄ . Then for 0 < γ < γ ̄ / α ,

⟨ x - y , ( A - γ f ) x - ( A - γ f ) y ⟩ ≥ ( γ ̄ - γ α ) x - y 2 , ∀ x , y ∈ H .

That is, A - γ f is strongly monotone with coefficient γ ̄ - γ α .

Lemma 2.2. 2 Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficient γ ̄ > 0 and 0 < ρ ≤ ||A||^-1. Then I - ρ A ≤ 1 - ρ γ ̄ .

Lemma 2.3. 11 Let T_ω: = (1 - ω)I + ωT, with T being a quasi-nonexpansive mapping on H, F i x ( T ) ≠ 0̸ , and ω ∈ (0, 1]. Then the following statements are reached:

(a1) Fix(T) = Fix(T_ω);

(a2) T_ω is quasi-nonexpansive;

(a3) ||T_ωx - q||² ≤ ||x - q||² - ω(1 - ω)||Tx - x||² for all x ∈ H and q ∈ Fix(T);

(a4) ⟨ x - T ω x , x - q ⟩ ≥ ω 2 x - T x 2 for all x ∈ H and q ∈ Fix(T).

Remark 2.4. (a4) was revised by Wongchan and Saejung 12 (Proposition 2).

Lemma 2.5. 13 Let {Γ_n} be a sequence of real numbers that does not decrease at infinity, in the sense that there exist a subsequence { Γ n j } j ≥ 0 of {Γ_n} which satisfies Γ n j < Γ n j + 1 for all j ≥ 0. Also consider the sequence of integers { τ ( n ) } n ≥ n 0 defined by

τ ( n ) = max { k ≤ n | Γ k < Γ k + 1 } .

Then {τ(n)}n≥n0 is a nondecreasing sequence verifying lim_n→∞ τ(n) = ∞ and for all n ≥ n₀, it holds that Γ_τ(n) < Γ_τ(n)+1 and we have

Γ n ≤ Γ τ ( n ) + 1 .

Recall the metric projection P_K form a Hilbert space H to a closed convex subset K of H is defined: for each x ∈ H, there exists a unique element P_Kx ∈ K such that

x - P K x : = inf { x - y : y ∈ K } .

Lemma 2.6. Let K be a closed convex subset of H. Given x ∈ H, and z ∈ K, z = P_Kx, if and only if there holds the inequality:

⟨ x - z , y - z ⟩ ≤ 0 , ∀ y ∈ K .

Lemma 2.7. If x* is the solution of the variational inequality (1.5) with demi-closedness of T and {y_n} ∈ H is a bounded sequence such that ||Ty_n - y_n|| → 0, then

lim inf n → ∞ 〈 ( A − γ f ) x * , y n − x * 〉 ≥ 0.

Proof. We assume that there exists a subsequence { y n j } of {y_n} such that y n j ⇀ ỹ . From the given conditions ||Ty_n - y_n|| → 0 and T: H → H demi-closed, we have that any weak cluster point of {y_n} belongs to the fixed point set Fix(T). Hence, we conclude that ỹ ∈ F i x ( T ) , and also have that

lim inf n → ∞ 〈 ( A − γ f ) x * , y n − x * 〉 = lim j → ∞ 〈 ( A − γ f ) x * , y n j − x * 〉 .

Recalling the (1.5), we immediately obtain

lim inf n → ∞ 〈 ( A − γ f ) x * , y n − x * 〉 = 〈 ( A − γ f ) x * , y ˜ − x * 〉 ≥ 0.

This completes the proof.

Let H be a real Hilbert space, let A be a bounded linear operator on H, and let T be a quasi-nonexpansive mapping on H, and f is a contraction with coefficient α; that is ||f (x) - f(y)|| ≤ α||x - y|| for all x, y ∈ H. Assume the set Fix(T) of fixed points of T is nonempty and we note that Fix(T) is closed and convex (see 14 for more general results).

Throughout this article, we assume that A is strongly positive; that is, there exist a constant γ̄>0 such that ⟨ A x , x ⟩ ≥ γ ̄ x 2 , for all x ∈ H. Let 0 < γ < γ ̄ / α .

Theorem 3.1. Starting with an arbitrary chosen x₀ ∈ H, let the sequence {x_n} be generated by

x n + 1 = α n γ f ( x n ) + ( I - α n A ) T ω x n ,

where the sequence {α_n} ⊂ (0,1) satisfies lim_n→∞ α_n = 0, and ∑ n = 0 ∞ α n = ∞ . Also ω ∈ ( 0 , 1 2 ) , T_ω: = (1 - ω)I + ωT with two conditions on T:

(C1) ||Tx - q|| ≤ ||x - q|| for any x ∈ H, and q ∈ Fix(T); this means that T is a quasi-nonexpansive mapping;

(C2) T is demiclosed on H; that is: if {y_k} ∈ H, y_k ⇀ z, and (I - T)y_k → 0, then z ∈ Fix(T).

Then {x_n} converges strongly to the x* ∈ Fix(T) which is the unique solution of the VIP:

⟨ ( γ f - A ) x * , x - x * ⟩ ≤ 0 , ∀ x ∈ F i x ( T ) .

Remark 3.2. Equivalently, from the VIP (3.2), we have

x * = P F i x ( T ) ∘ ( I - A + γ f ) x * .

Proof. First we show that {x_n} is bounded.

Take any p ∈ Fix(T), from Lemma 2.3 (a3), we have

x n + 1 - p = α n γ f ( x n ) + ( I - α n A ) T ω x n - p = α n γ ( f ( x n ) - f ( p ) ) + α n ( γ f ( p ) - A p ) + ( I - α n A ) ( T ω x n - p ) ≤ α n γ α f ( x n ) - f ( p ) + α n γ f ( p ) - A p + ( 1 - α n γ ̄ ) x n - p ≤ ( 1 - α n ( γ ̄ - γ α ) ) x n - p + α n γ f ( p ) - A p .

By induction

x n - p ≤ max x 0 - p , γ f ( p ) - A p γ ̄ - γ α , ∀ n ≥ 0 .

Hence {x_n} is bounded, so are the {f(x_n)} and {A(x_n)}.

Let x* = P_Fix(T) o(I - A + γf)x* From (3.1), we have

x n + 1 - x n + α n ( A x n - γ f ( x n ) ) = ( I - α n A ) ( T ω x n - x n ) .

Since x* ∈ Fix(T), from (a4), and together with (3.5), we obtain

⟨ x n + 1 - x n + α n ( A x n - γ f ( x n ) ) , x n - x * ⟩ = ⟨ ( I - α n A ) ( T ω x n - x n ) , x n - x * ⟩ = ( 1 - α n ) ⟨ T ω x n - x n , x n - x * ⟩ + α n ⟨ ( I - A ) ( T ω x n - x n ) , x n - x * ⟩ ≤ - ω 2 ( 1 - α n ) x n - T x n 2 + ω α n ⟨ ( I - A ) ( T - I ) x n , x n - x * ⟩ ,

it follows from the previous inequality that

- ⟨ x n - x n + 1 , x n - x * ⟩ ≤ - α n ⟨ ( A - γ f ) x n , x n - x * ⟩ - ω 2 ( 1 - α n ) x n - T x n 2 + ω α n ⟨ ( I - A ) ( T - I ) x n , x n - x * ⟩ .

From (iii), we obviously have

⟨ x n - x n + 1 , x n - x * ⟩ = - 1 2 x n + 1 - x * 2 + 1 2 x n - x * 2 + 1 2 x n + 1 - x n 2 .

Set Γ n : = 1 2 x n - x * 2 , and combine with (3.6), it follows that

Γ n + 1 - Γ n - 1 2 x n + 1 - x n 2 ≤ - α n ⟨ ( A - γ f ) x n , x n - x * ⟩ - ω 2 ( 1 - α n ) x n - T x n 2 + ω α n ⟨ ( I - A ) ( T - I ) x n , x n - x * ⟩ .

Now, we calculate ||x_n+1 - x_n||.

From the given condition: T_ω: = (1 - ω)I + ωT, it is easy to deduce that ||T_ωx_n - x_n|| = ω||x_n - Tx_n||. Thus, it follows from (3.5) that

x n + 1 - x n 2 = α n ( γ f ( x n ) - A x n ) + ( I - α n A ) ( T ω x n - x n ) 2 ≤ 2 α n 2 γ f ( x n ) - A x n 2 + 2 ( 1 - α n γ ̄ ) 2 T ω x n - x n 2 ≤ 2 α n 2 γ f ( x n ) - A x n 2 + 2 ( 1 - α n γ ̄ ) T ω x n - x n 2 ≤ 2 α n 2 γ f ( x n ) - A x n 2 + 2 ω 2 ( 1 - α n γ ̄ ) T x n - x n 2 .

Then from (3.8) and (3.9), we have

Γ n + 1 - Γ n + ω 2 ( 1 - α n ) - ω 2 ( 1 - α n γ ̄ ) x n - T x n 2 ≤ α n α n γ f ( x n ) - A x n 2 - ⟨ ( A - γ f ) x n , x n - x * ⟩ + ω ⟨ ( I - A ) ( T - I ) x n , x n - x * ⟩ .

Finally, we prove x_n → x*. To this end, we consider two cases.

Case 1: Suppose that there exists n₀ such that { Γ n } n ≥ n 0 is nonincreasing, it is equal to Γ _n+1 ≤ Γ_n for all n ≥ n₀. It follows that lim_n→∞ Γ_n exists, so we conclude that

lim n → ∞ ( Γ n + 1 - Γ n ) = 0 .

It follows from (3.10), (3.11) and the fact that lim_n→∞ α_n = 0, we have lim_n→∞ ||x_n -Tx_n|| = 0. Again, from (3.10), we have

- α n [ α n γ f ( x n ) - A x n 2 - ⟨ ( A - γ f ) x n , x n - x * ⟩ + ω ⟨ ( I - A ) ( T - I ) x n , x n - x * ⟩ ] ≤ Γ n - Γ n + 1 .

Then, by ∑n=0∞αn=∞ , we conclude that

lim inf n → ∞ − [ α n ‖ ( γ f − A ) x n ‖ 2 − 〈 ( A − γ f ) x n , x n − x * 〉 + ω 〈 ( I − A ) ( T − I ) x n , x n − x * 〉 ] ≤ 0.

Since {f(x_n)} and {x_n} are both bounded, as well as α_n → 0, and lim_n→∞ ||x_n - Tx_n|| = 0, it follows from (3.13) that

lim inf n → ∞ 〈 ( A − γ f ) x n , x n − x * 〉 ≤ 0.

From Lemma 2.1, it is obvious that

⟨ ( A - γ f ) x n , x n - x * ⟩ ≥ ⟨ ( A - γ f ) x * , x n - x * ⟩ + 2 ( γ ̄ - γ α ) Γ n .

Thus, from (3.14), (3.15) and the fact that lim_n→∞ Γ_n exists, we immediately obtain

lim inf n → ∞ [ 〈 ( A − γ f ) x * , x n − x * 〉 + 2 ( γ ¯ − γ α ) Γ n ]   = 2 ( γ ¯ − γ α ) lim n → ∞ Γ n + lim inf n → ∞ 〈 ( A − γ f ) x * , x n − x * 〉 ≤ 0 ,

or equivalently

2 ( γ ¯ − γ α ) lim n → ∞ Γ n ≤ − lim inf n → ∞ 〈 ( A − γ f ) x * , x n − x * 〉 .

Finally, by Lemma 2.7, we have

2 ( γ ̄ - γ α ) lim n → ∞ Γ n ≤ 0 ,

so we conclude that lim_n→∞ Γ_n = 0, which equivalently means that {x_n} converges strongly to x*.

Case 2: Assume that there exists a subsequence{Γnj}j≥0 of {Γ_n}_n≥0 such that Γnj<Γnj+1 for all j ∈ ℕ. In this case, it follows from Lemma 2.5 that there exists a subsequence {Γ_τ(n)} of {Γ_n} such that Γ_τ(n)+1 > Γ_τ(n), and {τ(n)} is defined as in Lemma 2.5.

Invoking the (3.10) again, it follows that

Γ τ ( n ) + 1 − Γ τ ( n ) + [ ω 2 ( 1 − α τ ( n ) ) − ω 2 ( 1 − α τ ( n ) γ ¯ ) ] ‖ x τ ( n ) − T x τ ( n ) ‖ 2   ≤ α τ ( n ) [ α τ ( n ) ‖ γ f ( x τ ( n ) ) − A x τ ( n ) ‖ 2 − 〈 ( A − γ f ) x τ ( n ) , x τ ( n ) − x * 〉     + ω 〈 ( I − A ) ( T − I ) x τ ( n ) , x τ ( n ) − x * 〉 ] .

Recalling the fact that Γ_τ(n)+1 > Γ_τ(n), we have

[ ω 2 ( 1 − α τ ( n ) ) − ω 2 ( 1 − α τ ( n ) γ ¯ ) ] ‖ x τ ( n ) − T x τ ( n ) ‖ 2   ≤ α τ ( n ) [ α τ ( n ) ‖ γ f ( x τ ( n ) ) − A x τ ( n ) ‖ 2 − 〈 ( A − γ f ) x τ ( n ) , x τ ( n ) − x * 〉     + ω 〈 ( I − A ) ( T − I ) x τ ( n ) , x τ ( n ) − x * 〉 ] .

From the preceding results, we get the boundedness of {x_n} and α_n → 0, which obviously lead to

lim n → ∞ x τ ( n ) - T x τ ( n ) = 0 .

Hence, combining (3.19) with (3.20), we immediately deduce that

⟨ ( A - γ f ) x τ ( n ) , x τ ( n ) - x * ⟩ ≤ α τ ( n ) γ f ( x τ ( n ) ) - A x τ ( n ) 2 + ω ⟨ ( I - A ) ( T - I ) x τ ( n ) , x τ ( n ) - x * ⟩ .

Again, (3.15) and (3.21) yield

⟨ ( A - γ f ) x * , x τ ( n ) - x * ⟩ + 2 ( γ ̄ - γ α ) Γ τ ( n ) ≤ α τ ( n ) γ f ( x τ ( n ) ) - A x τ ( n ) 2 + ω ⟨ ( I - A ) ( T - I ) x τ ( n ) , x τ ( n ) - x * ⟩ .

Recall that lim_n→∞ α _τ(n) = 0 and (3.20), we immediately have

2 ( γ ̄ - γ α ) lim sup n → ∞ Γ τ ( n ) ≤ - lim inf n → ∞ ⟨ ( A - γ f ) x * , x τ ( n ) - x * ⟩

By Lemma 2.7, we have

lim inf n → ∞ ⟨ ( A - γ f ) x * , x τ ( n ) - x * ⟩ ≥ 0 .

Consider (3.23) again, we conclude that

lim sup n → ∞ Γ τ ( n ) = 0 ,

which means that lim_n→∞ Γ_τ(n) = 0. By Lemma 2.5, it follows that Γ _n ≤ Γ_τ(n), thus, we get lim_n→∞ Γ_n = 0, which is equivalent to x_n → x*.

Corollary 3.3. 11 Let the sequence {x_n} be generated by

x n + 1 = α n f ( x n ) + ( 1 - α n ) T ω x n ,

where the sequence {α_n} ⊂ (0,1) satisfies lim_n→∞ α_n = 0, and ∑n=0∞αn=∞ . Also ω∈(0,12) , and T_ω: = (1 - ω)I + ωT with two conditions on T:

(C1) ||Tx - q|| ≤ ||x - q|| for any x ∈ H, and q ∈ Fix(T); this means that T is a quasi-nonexpansive mapping;

(C2) T is demiclosed on H; that is: if{y_k} ∈ H, y_k ⇀ z, and (I - T)y_k → 0, z ∈ Fix(T).

Then {x_n} converges strongly to the x* ∈ Fix(T) which is the unique solution of the VIP (3.27):

⟨ ( I - f ) x * , x - x * ⟩ ≥ 0 , ∀ x ∈ F i x ( T ) .

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.