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The concepts of

Well-posedness plays a crucial role in the stability theory for optimization problems, which guarantees that, for an approximating solution sequence, there exists a subsequence which converges to a solution. The study of well-posedness for scalar minimization problems started from Tykhonov [

In this paper, we are interested in investigating four classes of well-posednesses for a mixed quasi variational-like inequality problem. The paper is organized as follows. In Section

Throughout this paper, without other specification, let

Denote by

A sequence

there exists a sequence

there exists a sequence

(MQVLI) is said to be

A sequence

(MQVLI) is said to be L-

It is worth noting that if

In order to investigate the

We recall the notion of Mosco convergence [

If

It is easy to see that a sequence

We will use the usual abbreviations usc and lsc for “upper semicontinuous” and “lower semicontinuous”, respectively. For any

Let

there exists

The following lemmas play important role in this paper.

Let

According to the

Next prove (ii) _{1}. Hence, according to the Sion Minimax Theorem [

Finally, for each

Let

The necessity is easy to get; next we start to prove the sufficiency. Let

In this section, we investigate some metric characterizations of

For any

Let the same assumptions be as in Lemma

(MQVLI) is

Moreover, if

We only prove (a). The proof of (b) is similar and is omitted here. Suppose that (MQVLI) is

Conversely, let

Let

for every converging sequence

the map

The proof of the above theorem relies on the following lemma.

Let the same assumptions be made as in Theorem

Since

For each

Assumption (ii) applied to the constant sequence

Thus, for every

The necessity follows from Theorem

Let

Now, we present a result in which assumption (ii) and the monotonicity of

Let

the multifunction

the map

The proof of the above theorem relies on the following lemma.

Let the assumptions be as in Theorem

Since

As in Lemma

The necessity follows from Theorem

We have analogous results for L-

Let

the multifunction

for every converging sequence

the map

Let the same assumptions be as in Theorem

Since

Let

Assume that (

Conversely, assume that the problem is L-

Let

the multifunction

the map

Then (MQVLI) is L-

Let the same assumptions be as in Theorem

Since

It follows from the Lower Semi-Mosco convergence of

Assume that (

In this section, we investigate some metric characterizations of

Let

Let

Let the same assumptions be as in Lemma

(MQVLI) is

Moreover, if

We only prove (a). The proof of (b) is similar and is omitted here. Assume that (MQVLI) is

For the converse, let

Let the same assumptions be as in Theorem

Assume that (MQVLI) is

Conversely, assume that (

Now we show that

It follows from (

Since

Since

Let the same assumptions be as in Theorem

Assume that (MQVLI) is L-

Conversely, assume that (

Now we show that

It follows from (

Since

For each

(i) It is easy to see that if

(ii) The proof methods of Theorems

The authors would like to express their thanks to the referee for helpful suggestions. This research was supported by the National Natural Science Foundation of China (Grants 10771228 and 10831009), the Natural Science Foundation of Chongqing (Grant no. CSTC, 2009BB8240), and the Research Project of Chongqing Normal University (Grant 08XLZ05).