In this section, we study the existence of solutions for the second-order impulsive -difference inclusion (1.5).
We recall the following lemma from .
Lemma 4.1 If , then for any , the solution of the problem
is given by
Definition 4.2 A function is said to be a solution of (1.5) if , , , , and there exists such that on J and
Theorem 4.3 Assume that (H1), (H2) hold. In addition we suppose that:
(A1) there exist constants , such that , , for each ;
(A2) there exists a constant such that
Then the initial value problem (1.5) has at least one solution on J.
Proof Define the operator by
We will show that ℋ satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that ℋ is convex for each . This step is obvious since is convex (F has convex values), and therefore we omit the proof.
In the second step, we show that ℋ maps bounded sets (balls) into bounded sets in . For a positive number ρ, let be a bounded ball in . Then, for each , , there exists such that
Then for we have
Now we show that ℋ maps bounded sets into equicontinuous sets of . Let , with , , for some and . For each , we obtain
Obviously the right hand side of the above inequality tends to zero independently of as . Therefore it follows by the Arzelá-Ascoli theorem that is completely continuous.
Since ℋ is completely continuous, in order to prove that it is upper semicontinuous it is enough to prove that it has a closed graph. Thus, in our next step, we show that ℋ has a closed graph. Let , and . Then we need to show that . Associated with , there exists such that, for each ,
Thus it suffices to show that there exists such that, for each ,
Let us consider the linear operator given by
Thus, it follows by Lemma 2.3 that is a closed graph operator. Further, we have . Since , therefore, we have
for some .
Finally, we show there exists an open set with for any and all . Let and . Then there exists with such that, for , we have
Repeating the computations of the second step, we have
Consequently, we have
In view of (A2), there exists M such that . Let us set
Note that the operator is upper semicontinuous and completely continuous. From the choice of U, there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 2.2), we deduce that ℋ has a fixed point which is a solution of the problem (1.4). This completes the proof. □
Example 4.4 Let us consider the following second-order impulsive -difference inclusion with initial conditions:
Here , , , , , , , and . We find that , , and , ; and we have
Let be a multivalued map given by
For , we have
with , . Further, using the condition (A2) we find
which implies . Therefore, all the conditions of Theorem 4.3 are satisfied. So, problem (4.5) with given by (4.6) has at least one solution on .
If is a multivalued map given by
For , we have
Here , , with , . It is easy to verify that . Then, by Theorem 4.3, the problem (4.5) with given by (4.7) has at least one solution on .