Let *F* be a *k*-Lipschitzian and *η*-strongly monotone operator on *H* with 0 < *k*, *T* : *H* → *H* be a nonexpansive mapping. Let *t* ∈ (0,*η*/*k*^{2}) and , and consider a mapping *S*_{t} on *H* defined by

It is easy to see that *S*_{
t
} is a contraction. Indeed, from Lemma 2.5, we have

for all *x*, *y* ∈ *H*. Hence, it has a unique fixed point, denoted as *x*_{
t
}, which uniquely solves the fixed point equation

**Theorem 3.1**. *Let H be a real Hilbert space. Let T* : *H* → *H be a nonexpansive mapping such that F* (*T* ) ≠ ∅,. *Let F be a k-Lipschitzian and η-strongly monotone operator on H with* 0 < *η* ≤ *k*. *For each t* ∈ (0, *η*/*k*^{2}), *let the net* {*x*_{
t
}} *be generated by (3.1). Then, as t* → 0, *the net* {*x*_{
t
}} *converges strongly to a fixed point x** of *T which solves the variational inequality:*

*Proof*. We first show the uniqueness of a solution of the variational inequality (3.2), which is indeed a consequence of the strong monotonicity of *F*. Suppose *x** ∈ *F* (*T* ) and both are solutions to (3.2), then

and

Adding up (3.3) and (3.4) yields

The strong monotonicity of *F* implies that and the uniqueness is proved. Later, we use *x** ∈ *F* (*T* ) to denote the unique solution of (3.2).

Next, we prove that {*x*_{
t
}} is bounded. Take *u* ∈ *F* (*T* ), from (3.1) and using Lemma 2.5, we have

that is,

Observe that

From *t* → 0, we may assume, without loss of generality, that , where is a arbitrarily small positive number. Thus, we have is continuous, . Therefore, we obtain

From (3.5) and (3.6), we have that {*x*_{
t
}} is bounded and so is {*Fx*_{
t
}}.

On the other hand, from (3.1), we obtain

To prove that *x*_{
t
} → *x**. For a given *u* ∈ *F* (*T* ), using Lemma 2.5, we have

Therefore,

From , we have and . Observe that, if *x*_{
t
} ⇀ *u*, we have .

Since {*x*_{
t
}} is bounded, we see that if {*t*_{
n
}} is a sequence in such that *t*_{
n
} → 0 and , then by (3.8), we see . Moreover, by (3.7) and using Lemma 2.1, we have . We next prove that solves the variational inequality (3.2). From (3.1) and *u* ∈ *F* (*T* ), we have

that is,

Now replacing *t* in (3.9) with *t*_{
n
} and letting *n* → _{∞}, we have

That is is a solution of (3.2), hence by uniqueness. In a nutshell, we have shown that each cluster point of {*x*_{
t
}} (at t → 0) equals *x**. Therefore, *x*_{
t
} → *x** as *t* → 0.

**Theorem 3.2**. *Let H be a real Hilbert space*. *Let F be a k-Lipschitzian and η-strongly monotone operator on H with* 0 < *η* ≤ *k*. *Let* *be an infinite family of nonexpansive mappings such that* *and W*_{
n
} *be a W-mapping defined by (2.3). Let* {*λ*_{
n
}} *be a sequence in* [0, ∞) *and* {*α*_{
n
}} *be a sequence in* [0,1], ε *be a arbitrarily small positive number. Assume that the control conditions (C2)*, (*C* 1)', and (*C* 5)' hold for {*λ*_{
n
}} and {*α*_{
n
}},

(*C* 1)': , ∀*n* ≥ *n*_{0} *for some integer n*_{0} ≥ 0, *and*

(*C* 5) ': 0 < γ ≤ lim inf_{n→∞}*α*_{
n
} lim sup_{n→∞}*α*_{
n
} < 1 *for some* γ ∈ (0, 1).

*For x*_{0} ∈ *H arbitrarily, let the sequence* {*y*_{
n
}} *be generated by (1.2)*. *Then*,

*where*
*solves the variational inequality*

*Proof*. On the one hand, suppose that *λ*_{
n
}*F*(*x*_{
n
}) → 0(*n* → ∞). We proceed with the following steps:

*Step 1*. We claim that {*x*_{
n
}} is bounded. In fact, let , from (1.2), (*C* 1)' and using Lemma 2.5, we have

∀*n* ≥ *n*_{0} for some integer *n*_{0} ≥ 0, where . Then, from (1.2) and (3.10), we obtain

By induction, we have

∀*n* ≥ *n*_{0} for some integer *n*_{0} ≥ 0, where .Therefore, {*x*_{
n
}} is bounded. We also obtain that {*y*_{
n
}}, {*W*_{
n
}*y*_{
n
}} and {*Fx*_{
n
}} are bounded.

*Step 2*. We claim that lim_{n→∞}||*x*_{n+1}- *x*_{
n
}|| = 0. To this end, define *x*_{n+1}= (1 - *α*_{
n
})*x*_{
n
}+ *α*_{
n
}*z*_{
n
}. We observe that

From (2.3), for , we have

where *M*_{2} = sup{2 ||*y*_{
n
} - *u*||, *n* ≥ 0}. By (1.2) and (3.12), we have

Substituting (3.13) into (3.11), we have

that is,

Observing *λ*_{
n
}*F*(*x*_{
n
}) → 0(*n* → ∞) and 0 < *ξ*_{
i
} ≤ *b* < 1, it follows that

By (*C* 5)' and using Lemma 2.2, we have lim_{n→∞}||*z*_{
n
} - *x*_{
n
}|| = 0. Therefore,

*Step 3*. We claim that lim_{n→∞}||*x*_{
n
} - *W*_{
n
}*x*_{
n
}|| = 0. Observe that

that is,

*Step 4*. We claim that lim_{n→∞}||*x*_{
n
} - *Wx*_{
n
}|| = 0. Indeed, we have

By (3.15), (3.16) and using Lemma 2.6, we obtain

*Step 5*. We claim that lim sup_{n→∞}〈*Fx**, *x** - *x*_{
n
}〉 ≤ 0, where *x** = lim_{n→∞}*x*_{
t
}and *x*_{
t
}defined by *x*_{
t
}= *W*[(1 - *tF*)*x*_{
t
}]. Since *x*_{
n
}is bounded, there exists a subsequence of {*x*_{
n
}} which converges weakly to *ω*. From Step 4, we obtain . From Lemma 2.1, we have *ω* ∈ *F*(*W*). Hence, by Theorem 3.1, we have

*Step 6*. We claim that {*x*_{
n
}} converges strongly to . From (1.2), we have

∀*n* ≥ *n*_{0} for some integer *n*_{0} ≥ 0, where *M*_{3} = 2||*Fx**||. For every *n* ≥ *n*_{0}, put and *δ*_{
n
} = 2*M*_{1}〈*x** - *x*_{
n
}, *Fx**〉 +*M*_{1}*M*_{3} ||*λ*_{
n
}*Fx*_{
n
}||. It follows that

It is easy to see that and lim sup_{n→∞}*δ*_{
n
} ≤ 0. Hence, by Lemma 2.3, the sequence {*x*_{
n
}} converges strongly to .

Observe that

It follows that the sequence {*y*_{
n
}} converges strongly to . From *x** = lim_{t→0}*x*_{
t
}and Theorem 3.1, we have *x** is the unique solution of the variational inequality: 〈*Fx**, *x** - *u*〉 ≤ 0, .

On the other hand, suppose that as *n* → ∞, where solves the variational inequality:

From (1.2), we have

that is, . Again from (1.2), we obtain that

Since and , we get *λ*_{
n
}*F*(*x*_{
n
}) → 0. This completes the proof.

**Remark 3.3**. *It is clear that condition (C1)' is strictly weaker than condition (C1). In the meantime, condition (C5)' is also strictly weaker than condition (C5)*.

**Corollary 3.4**. *(Yao et al. [5, Theorem 3.2]). Let H be a real Hilbert space. Let F : H* → *H be k-Lipschitzian and η-strongly monotone operator with k* ∈ [1, ∞) *and η* ∈ (0, 1). *Let* *be an infinite family of nonexpansive mappings such that* and {*W*_{
n
}} *be W-mapping defined by (2.3). Let* {*λ*_{
n
}} *be a sequence in* [0, ∞) *and* {*α*_{
n
}} *be a sequence in* [0,1]. *Assume that*

*(C1)* lim_{n→1}*λ*_{
n
} = 0;

*(C2)* ;

*(C5)* *for some γ* > 0.

*Then, the sequence* {*x*_{
n
}} *and* {*y*_{
n
}} *generated by (1.2) converge strongly to* , *which solves the following variational inequality* 〈*Fx**, *x** - *x*〉 ≤ 0, .

*Proof*. Since lim_{n→∞}*λ*_{
n
} = 0, it is easy to see that , ∀*n* ≥ *n*_{0} for some integer *n*_{0} ≥ 0. Without loss of generality, we assume that , ∀*n* ≥ *n*_{0} for some integer *n*_{0} ≥ 0. Repeating the same argument as in the proof of Theorem 3.2, we know that {*x*_{
n
}} is bounded, and so are the sequence {*y*_{
n
}} and {*F*(*x*_{
n
})}. Therefore, we have *λ*_{
n
}*F*(*x*_{
n
}) → 0.

From for some *γ* > 0, we have 0 < *γ* ≤ lim inf_{n→∞}*α*_{
n
} ≤ lim sup_{n→∞}*α*_{
n
} < 1 for some *γ* ∈ (0, 1). Therefore, all conditions of Theorem 3.2 are satisfied. Hence, using Theorem 3.2, we have that {*y*_{
n
}} converges strongly to which solves the following variational inequality 〈*Fx**, *x** - *x*〉 ≤ 0, . It follows from (3.17) that {*x*_{
n
}} also converges strongly to . This completes the proof.

**Remark 3.5**. *Theorem 3.2 is more general than Theorem 3.2 of Yao et al.* [5]. *The following example shows that all conditions of Theorem 3.2 are satisfied. However, the conditions λ*_{
n
} → 0, *η* ∈ (0, 1) *and* *for some γ* > 0 *in [5, Theorem 3.2] are not satisfied*.

**Example 3.6**. *Let H = R the set of real numbers and T*_{
n
} ≡ *T. Define a nonexpansive mapping T* : *H* → *H and an operator F* : *H* → *H as follows:*

*It is easy to see that F*(*T*) = {0}, *and W*_{
n
}*x* = (1 - *ξ*_{1})*x*, ∀*x* ∈ *R*. *Let* , *we have* , ∀*x* ∈ *R*. *Given sequences* {*α*_{
n
}} *and* , *for all n* ≥ 0. *For an arbitrary x*_{0} ∈ *H*, *let* {*x*_{
n
}} *defined as follows:*

*that is*,

*Observe that for all n* ≥ 0,

*Hence, we have* *for all n* ≥ 0. *This implies that* {*x*_{
n
}} *converges strongly to* . *Since* , *we have that* {*y*_{
n
}} *converges strongly to* .

*Observe that* 〈*F*(0), 0 - *u*〉 ≤ 0, , *that is*, 0 *is the solution of the variational inequality* 〈*Fx**, *x** - *u*〉 ≤ 0, .

*Finally, we have*

*By F*(*x*) = *x, we have η* = *k* = 1. *Furthermore, it is easy to see that the following hold true:*

*(B1)* , ∀*n* ≥ *n*_{0} *for some integer n*_{0} ≥ 0;

*(B2)* ;

*(B3)* *for some constant* .

*Hence, there is no doubt that all conditions of Theorem 3.2 are satisfied. Since* , *η* = 1 *and* , *the conditions that λ*_{
n
} → 0, *for some γ* > 0 *and η* ∈ (0, 1) *of Yao et al. [5, Theorem 3.2] are not satisfied*.

Next, we give a weak convergence theorem for hybrid iterative algorithm (1.2) involving an infinite family of nonexpansive mappings in a Hilbert space.

**Theorem 3.7**. *Let H be a real Hilbert space. Let F* : *H* → *H be k-Lipschitzian and η-strongly monotone operator with* 0 < η ≤ *k*. *Let* *be an infinite family of nonexpansive mappings such that* , *and* {*W*_{
n
}} *be W-mapping defined by (2.3). Let* {*λ*_{
n
}} *and* {*α*_{
n
}} *be two sequences in* (0, 1). *Assume that*

*(A1)* ;

*(A2)* 0 < lim inf_{n→∞}*α*_{
n
} ≤ lim sup_{n→∞}*α*_{
n
} < 1.

*Then, the sequence* {*x*_{
n
}} *and* {*y*_{
n
}} *generated by (1.2) converge weakly to* .

*Proof*. From (A1), we have , ∀*n* ≥ *n*_{0} for some integer *n*_{0} ≥ 0. Repeating the same argument as in the proof of Theorem 3.2, we know that {*x*_{
n
}} is bounded, and so are the sequences {*y*_{
n
}} and {*F*(*x*_{
n
})}. Assuming , we have

where *M*_{4} = sup{||*F*(*x*_{
n
})||^{2}, 2 ||*x*_{
n
} - *p*|| ||*F*(*x*_{
n
})||, *n* ≥ 0}. Since , we have . Therefore, . Utilizing Lemma 2.4, we deduce that lim_{n→∞}||*x*_{
n
}- *p*|| exists. Further-more, from(3.17), we have

Since *λ*_{
n
} → 0, and (A2), it follows from (3.18) that

Utilizing Lemma 2.6, we have

Now, we show that *ω*_{
w
}(*y*_{
n
}) ⊂ *F*(*T*). Indeed, let *x** ∈ *ω*_{
w
}(*y*_{
n
}). Then, there exists a subsequence of {*y*_{
n
}} such that . Since ||*y*_{
n
} - *Wy*_{
n
}|| → 0, by Lemma 2.1, we have .

Next, we show that *ω*_{
w
}(*y*_{
n
}) is a singleton. Indeed, let be another subsequence of {*y*_{
n
}} such that . Then, . If , then, by Opial's property of *H*, we have

This is a contradiction. Therefore, *ω*_{
w
}(*y*_{
n
}) is a singleton. Consequently, {*y*_{
n
}} converges weakly to . From (1.2), we have that {*x*_{
n
}} converges weakly to . This completes the proof.

**Remark 3.8**. *It is worth pointing out that the conditions (C1) and (C2) in [5, Theorem 3.2] are replaced by the one (A1) in Theorem 3.7. It is also worth pointing out that condition (A2) is strictly weaker than the condition (C5). The advantages of there results in this study are that weaker and fewer restrictions are imposed on parameters α*_{
n
}, *λ*_{
n
} *and η*.