We need the following result for the existence of solutions of a certain variational inequality, which is slightly an improvement of Theorem 3.1 of Tian [12].

**Theorem T2**.

*Let H be a Hilbert space, C be a closed convex subset of H such that C* ±

*C* ⊂

*C, and T* :

*C* →

*C be a nonexpansive mapping with F*(

*T*) ≠ ∅.

*Let F* :

*C* →

*C be a κ-Lipschitzian and η-strongly monotone operator with κ >* 0

*and η >* 0.

*Let f* :

*C* →

*C be a contraction with the contractive constant α* ∈ (0, 1).

*Let*
,

*and τ <* 1.

*Let x*_{
t
}*be a fixed point of a contraction St* ∋

*x* α

*tγf* (

*x*) + (

*I - tμF* )

*Tx for t* ∈ (0, 1)

*and*
.

*Then* {

*x*_{
t
}}

*converges strongly to a fixed point*
*of T as t* → 0,

*which solves the following variational inequality:**Equivalently, we have*
.

Now, we study the strong convergence result for a general iterative scheme (IS).

**Theorem 3.1**. *Let H be a Hilbert space, C be a closed convex subset of H such that C* ± *C* ⊂ *C, and T* : *C* → *H be a k-strictly pseudo-contractive mapping with F*(*T*) ≠ ∅ *for some* 0 ≤ *k <* 1. *Let F* : *C* → *C be a κ-Lipschitzian and η-strongly monotone operator with κ >* 0 *and η >* 0. *Let f* : *C* → *C be a contraction with the contractive constant α* ∈ (0, 1). *Let*
,
*and τ <* 1. *Let f*{*α*_{
n
}} *and* {*β*_{
n
}} *be sequences in* (0, 1) *which satisfy the conditions:*

(C1) lim_{n→∞ }
*α*
_{
n
}= 0;

(C2)
;

(B) 0 *<* lim inf_{n→∞ }
*β*
_{
n
}≤ lim sup_{n→∞ }
*βn* < 1.

*Let* {

*x*_{
n
}}

*be a sequence in C generated by**where S* :

*C* →

*H is a mapping defined by Sx* =

*kx* + (1 -

*k*)

*Tx and P*_{
C
}*is the metric projection of H onto C. Then* {

*x*_{
n
}}

*converges strongly to q* ∈

*F*(

*T*),

*which solves the following variational inequality:***Proof**. First, from the condition (C1), without loss of generality, we assume that *α*_{
n
}*τ <* 1,
and *α*_{
n
}*<* (1 - *β*_{
n
}) for *n* ≥ 0.

We divide the proof several steps:

**Step 1**. We show that

for all

*n* ≥ 0 and all

*p* ∈

*F*(

*T*) =

*F*(

*S*). Indeed, let

*p* ∈

*F*(

*T*). Then from Lemma 2.9, we have

Using an induction, we have
. Hence, {*x*
_{
n
}} is bounded, and so are {*f*(*x*
_{
n
})}, {*P*
_{
C
}
*Sx*
_{
n
}} and {*FP*
_{
C
}
*Sx*
_{
n
}}.

**Step 2**. We show that lim

_{n→∞}||

*x*_{n+1 }-

*x*_{
n
}|| = 0. To this show, define

Observe that from the definition of

*z*
_{
n
},

From the condition (C1) and (B), it follows that

Hence, by Lemma 2.7, we have

**Step 3**. We show that lim

_{n→∞}||

*x*_{
n
}*- P*_{
C
}*Sx*_{
n
}|| = 0. Indeed, since

So, from the conditions (C1) and (B) and Step 2, it follows that

where

*q* = lim

_{t→0 }
*x*
_{
t
} being

*x*
_{
t
} =

*tγf*(

*x*
_{
t
}) + (

*I - tμF* )

*P*
_{
C
}
*Sx*
_{
t
} for 0

*< t <* 1 and

. We note that from Lemmas 2.2 and 2.3 and Theorem T2,

*q* ∈

*F*(

*T*) =

*F*(

*S*) and

*q* is a solution of a variational inequality

To show this, we can choose a subsequence

of {

*x*
_{
n
}} such that

Since {

*x*
_{
n
}} is bounded, there exists a subsequence

of

which converges weakly to

*w*. Without loss of generality, we can assume that

. Since ||

*x*
_{
n
}
*- P*
_{
C
}
*Sx*
_{
n
}|| → 0 by Step 3, we obtain

*w* =

*P*
_{
C
}
*Sw*. In fact, if

*w* ≠

*P*
_{
C
}
*Sw*, then, by Opial condition,

which is a contradiction. Hence

*w* =

*P*
_{
C
}
*Sw*. Since

*F*(

*P*
_{
C
}
*S*) =

*F*(

*S*), from Lemma 2.3, we have

*w* ∈

*F*(

*T*). Therefore, from (3.1), we conclude that

**Step 5**. We show that lim

_{n→∞}||

*x*_{
n
}*- q*|| = 0, where

*q* = lim

_{t→0 }*x*_{
t
} being

*x*_{
t
} =

*tγf* (

*xt*) + (

*I - tμF*)

*P*_{
C
}*Sx*_{
t
} for 0

*< t <* 1 and

, and

*q* is a solution of a variational inequality

Indeed, from (IS), we have

Applying Lemmas 2.8 and 2.9, we have

where

*M* = sup{||

*x*
_{
n
} -

*q*||2 :

*n* ≥ 0},

and

From the conditions (C1) and (C2) and Step 4, it is easy to see that *λ*
_{
n
} → 0,
, and lim sup_{n→∞ }
*δ*
_{
n
}≤ 0. Hence, by Lemma 2.7, we conclude *x*
_{
n
} → *q* as *n* → ∞. This completes the proof. □

**Remark 3.1**. (1) Theorem 3.1 extends and develops Theorem 3.2 of Tian [12] from a nonexpansive mapping to a strictly pseudo-contractive mapping together with removing the condition (C3)
.

(2) Theorem 3.1 also generalizes Theorem 2.1 of Jung [6] as well as Theorem 2.1 of Cho et al. [5] and Theorem 3.4 of Marino and Xu [10] from a strongly positive bounded linear operator *A* to a *κ*-Lipschitzian and *η*-strongly monotone operator *F*.

(3) Theorem 3.1 also improves the corresponding results of Halpern [13], Moudafi [8], Wittmann [14] and Xu [9] as some special cases.

**Theorem 3.2**. *Let H be a Hilbert space, C be a closed convex subset of H such that C* ± *C* ⊂ *C, and T*_{
i
} : *C* → *H be a k*_{
i
}*-strictly pseudo-contractive mapping for some* 0 ≤ *k*_{
i
}*<* 1 *and*
. *Let F* : *C* → *C be a κ-Lipschitzian and η-strongly monotone operator with κ >* 0 *and η >* 0. *Let f* : *C* → *C be a contraction with the contractive constant α* ∈ (0, 1). *Let*
,
*and τ* < 1. *Let* {*α*_{
n
}} *and* {*βn*} *be sequences in* (0, 1) *which satisfy the conditions*.

(C1) lim_{n→∞ }
*α*
_{
n
}= 0;

(C2)
;

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

*Let* {

*x*_{
n
}}

*be a sequence in C generated by**where S* :

*C* →

*H is a mapping defined by*
*with k* = max{

*k*_{
i
} : 1 ≤

*i* ≤

*N*}

*and* {

*η*_{
i
}}

*is a positive sequence such that*
*and P*_{
C
}*is the metric projection of H onto C. Then* {

*x*_{
n
}}

*converges strongly to q* ∈

*F*(

*T*),

*which solves the following variational inequality:***Proof**. Define a mapping *T* : *C* → *H* by
. By Lemmas 2.4 and 2.5, we conclude that *T* : *C* → *H* is a *k*-strictly pseudo-contractive mapping with *k* = max{*k*_{
i
} : 1 ≤ *i* ≤ *N*} and
. Then the result follows from Theorem 3.1 immediately. □

As a direct consequence of Theorem 3.2, we have the following result for nonexpansive mappings (that is, 0-strictly pseudo-contractive mappings).

**Theorem 3.3**. *Let H be a Hilbert space, C be a closed convex subset of H such that C* ± *C* ⊂ *C*,
*be a finite family of nonexpansive mappings with*
. *Let F* : *C* → *C be a κ-Lipschitzian and η-strongly monotone operator with κ >* 0 *and η >* 0. *Let f* : *C* → *C be a contraction with the contractive constant α* ∈ (0, 1). *Let*
,
*and τ <* 1. *Let* {*α*_{
n
}} *and* {*βn*} *be sequences in* (0, 1) *which satisfy the conditions*.

(C1) lim_{n→∞ }
*α*
_{
n
}= 0;

(C2)
;

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

*Let* {

*x*_{
n
}}

*be a sequence in C generated by**where*
*is a positive sequence such that*
*and P*_{
C
}*is the metric projection of H onto C. Then* {

*x*_{
n
}}

*converges strongly to a common fixed point q of*
,

*which solves the following variational inequality:***Remark 3.2**. (1) Theorems 3.2 and 3.3 also generalize Theorems 2.2 and 2.4 of Jung [6] from a strongly positive bounded linear operator *A* to a *κ*-Lipschitzian and *η*-strongly monotone operator *F*.

(2) Theorems 3.2 and 3.3 also improve and complement the corresponding results of Cho et al. [5] by removing the condition (C3)
together with using a *κ*-Lipschitzian and *η*-strongly monotone operator *F*.

(3) As in [19], we also can establish the result for a countable family {*T*
_{
i
}} of *k*
_{
i
}-strict pseudo-contractive mappings with 0 ≤ *k*
_{
i
}
*<* 1.