In this section, we establish an existence theorem for solution of generalized strongly nonlinear implicit quasi-variational inequality problems and convergence of the iterative sequences generated by (2.18). First, we give some definitions.

Definition 3.1.

A mapping is said to be generalized pseudo-contractive if there exists a constant such that

It is easy to check that (3.1) is equivalent to

For in (3.1), we get the usual concept of pseudo-contractive of , introduced by Browder and Petryshyn [10], that is,

Definition 3.2.

Let and be the mappings. The mapping is said to be as follows.

(i)Generalized pseudo-contractive with respect to in the first argument of , if there exists a constant such that

(ii)Lipschitz continuous with respect to the first argument of if there exists a constant such that

In a similar way, we can define Lipschitz continuity of N with respect to the second and third arguments.

(iii) is also said to be Lipschitz continuous if there exists a constant such that

Definition 3.3.

Let be the mappings. A mapping is said to be the generalized -pseudo-contractive with respect to the second argument of , if there exists a constant such that

Definition 3.4.

Let be a set-valued mapping such that for each , is a nonempty closed convex subset of . The projection is said to be Lipschitz continuous if there exists a constant such that

Remark 3.5.

In many important applications, has the following form:

where is a single-valued mapping and a nonempty closed convex subset of . If is Lipschitz continuous with constant , then from Lemma 2.3, is Lipschitz continuous with Lipschitz constant .

Now, we give the main result of this paper.

Theorem 3.6.

Let be a real Hilbert space and a set-valued mapping with nonempty closed convex values. Let be the Lipschitz continuous mappings with positive constants and respectively. Let be the mapping such that and are Lipschitz continuous with positive constants and respectively. A trimapping is generalized pseudo-contractive with respect to in the first argument of with constant and generalized -pseudo-contractive with respect to in the second argument of with constant , Lipschitz continuous with respect to the first, second, and third arguments with positive constants respectively. Suppose that is Lipschitz continuous with constant . Let , and be the three bounded sequences in and , , , , , , , and are sequences in satisfying the following conditions:

(1)

(2)

(3)

If the following conditions hold:

where and .

Then there exists a unique satisfying the generalized strongly nonlinear implicit quasi-variational inequality (2.1) and as , where is the three-step iteration process with errors defined as follows:

for .

Proof.

We first prove that the generalized strongly nonlinear implicit quasi-variational inequality (2.1) has a unique solution. By Lemma 2.1, it is sufficient to prove the mapping defined by

has a unique fixed point in .

Let be two arbitrary points in . From Lemma 2.2 and Lipschitz continuity of and , we have

Since is generalized pseudo-contractive with respect to in the first argument of and Lipschitz continuous with respect to first argument of and also is Lipschitz continuous, we have

Again since is generalized -pseudo-contractive with respect to in the second argument of and Lipschitz continuous with respect to second argument of and is Lipschitz continuous, we have

It follows from (3.13)–(3.16) that

where

From (3.10), we know that and so has a unique fixed point , which is a unique solution of the generalized strongly nonlinear implicit quasi-variational inequality (2.1).

Now we prove that converges to . In fact, it follows from (3.11) and that

From (3.17) and (3.19), it follows that

Similarly, we have

Again,

Let

Then and

Similarly, we deduce from (3.21) the following:

From the above inequalities, we get

where

Since , it follows from conditions (1) and (3) that

Therefore,

From (3.29)-(3.31) and Lemma 2.4, we know that converges to the solution . This completes the proof.

Remark 3.7.

We now deduce Theorem 3.6 in the direction of Ishikawa iteration.

Theorem 3.8.

Let be a real Hilbert space and a set-valued mapping with the nonempty closed convex values. Let and be the same as in Theorem 3.6. Suppose that is Lipschitz continuous with constant . Let and be the two bounded sequences in and , , , , and be six sequences in satisfying the following conditions:

(1)

(2)

(3)

If the following conditions holds:

Then there exists a unique satisfying the generalized strongly nonlinear implicit quasi-variational inequality (2.1) and as , where is the Ishikawa iteration process with errors defined as follows:

for .

Remark 3.9.

We can also deduce Theorem 3.6 in the direction of (2.16).

Theorem 3.10.

Let and be the same as in Theorem 3.6. Let be a bounded sequence in and , and be three sequences in satisfying the following conditions:

(1) for ,

(2),

(3)and

If the conditions of (3.10) hold, then there exists a unique satisfying the generalized strongly nonlinear implicit quasi-variational inequality (2.1) and as , where is the Mann iterative process with errors defined as follows:

for .

Our results can be further improved in the direction of (2.25).

Theorem 3.11.

Let be a real Hilbert space and a set-valued mapping with nonempty closed convex values. Let be the Lipschitz continuous mapping with respect to positive constants and respectively. Let be the mapping such that and be Lipschitz continuous with respect to positive constants and respectively. A trimapping is generalized pseudo-contractive with respect to map in first argument of with constant and generalized -pseudo-contractive with respect to in the second argument of with constant , Lipschitz continuous with respect to first, second, and third arguments with positive constants , respectively. Suppose that is a Lipschitz continuous with positive constant . Let , and be three bounded sequences in satisfying the conditions (1)–(3) of Theorem 3.6. If the conditions of (3.10) hold for , then there exists a unique satisfying (2.2) and as , where is the three step iteration process with errors defined as follows:

for .

Now, we deduce Theorem 3.6 for three step iterative process in terms of (2.10).

Theorem 3.12.

Let and be the same as in Theorem 3.6. Let , , , , and be six sequences in satisfying conditions:

(1) for

(2),

(3)

If the conditions of (3.10) hold, then there exists satisfying (2.1) and as , where the three-step iteration process is defined by

for .

Next, we state the results in terms of iterations (2.10) and (2.25).

Theorem 3.13.

Let and be the same as in the Theorem 3.11. Let , , , , and be six sequences in satisfying conditions (1)–(3) of Theorem 3.6. If the conditions of (3.10) hold for , then there exists satisfying the generalized strongly nonlinear implicit quasi-variational inequality (2.2) and as , where the three-step iteration process is defined by

for .

Remark 3.14.

Theorem 3.13 can also be deduce for Ishikawa and Mann iterative process.