Nonlocal Boundary Value Problem for Nonlinear Impulsive -Integrodifference Equation
A nonlinear impulsive integrodifference equation within the frame of -quantum calculus is investigated by applying using fixed point theorems. The conditions for existence and uniqueness of solutions are obtained.
Recently, by introducing and applying the fractional difference operators to real world problems (see, e.g., [1–7] and the references therein) we revitalized the importance of the quantum calculus . However the real world phenomena are usually described by complex model based involving different types of operators. In this way we hope to understand deeper the dynamics of complex or hypercomplex systems and to reveal their hidden aspects.
On this line of thought in this paper, we study the existence and uniqueness of solutions for nonlinear -integrodifference equation with nonlocal boundary condition and impulses: where , are -derivatives and -integrals , respectively. , , , , , , , , where and denote the right and the left limits of at , respectively.
Let us set , , and introduce the space: with the norm . Then, is a Banach space.
For convenience, let us recall some basic concepts of -calculus .
For and , we define the -derivatives of a real valued continuous function as Higher order -derivatives are given by The -integral of a function is defined by provided the series converges. If and is defined on the interval , then Observe that For , the following reversing order of -integration holds
Note that if and in (3) and (5), then , , where and are the well-known -derivative and -integral of the function defined by
Lemma 1. For given , the function is a solution of the impulsive -integrodifference equation if and only if satisfies the -integral equation
Proof. Let be a solution of -difference equation (10). For , applying the operator on both sides of , we have
Similarly, for , applying the operator on both sides of , then In view of , it holds
Repeating the above process, we can get
Using the boundary value condition given in (10), it follows
Conversely, assume that satisfies the impulsive -integral equation (11); applying on both sides of (11) and substituting in (11), then (10) holds. This completes the proof.
3. Main Results
Letting , in view of Lemma 1, we introduce an operator as By reversing the order of integration, we obtain Then, the impulsive -integrodifference equation (1) has a solution if and only if the operator equation has a fixed point.
In order to prove the existence of solutions for (1), we need the following known result .
Theorem 2. Let be a Banach space. Assume that is a completely continuous operator and the set is bounded. Then has a fixed point in .
Theorem 3. Assume the following.There exist nonnegative bounded functions such that for any , .There exist positive constants such that for any , .Then problem (1) has at least one solution provided
Proof. Firstly, we prove the operator is completely continuous. Clearly, continuity of the operator follows from the continuity of , , , and . Let be bounded. Then , ; there exist positive constants such that , , , . Thus
This implies .
Furthermore, for any , satisfying , we have As , the right hand side of the above inequality tends to zero. Thus, is relatively compact. As a consequence of Arzela Ascoli's theorem, is a compact operator. Therefore, is a completely continuous operator.
Define the set .
Next, we show is bounded. Let ; then , . For any , by conditions and , we have which implies So, the set is bounded. Thus, Theorem 2 ensures the impulsive -integrodifference equation (1) has at least one solution.
Corollary 4. Assume the following.There exist nonnegative constants such that
for any , , .
Then problem (1) has at least one solution.
Theorem 5. Assume the following.There exist nonnegative bounded functions and such that
for .There exist positive constants such that
for and ..
Then problem (1) has a unique solution.
Proof. Denote , . For , by and , we have
As by , then . Therefore, is a contractive map. Thus, the conclusion of the Theorem 5 follows by Banach contraction mapping principle.
Consider the following nonlinear -integrodifference equation with impulses Obviously, , , + , , , and .
By a simple calculation, we can get
Take , , , , and . Then all conditions of Theorem 3 hold. By Theorem 3, nonlinear impulsive -integrodifference (31) has at least one solution.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the Natural Science Foundation for Young Scientists of Shanxi Province, China (2012021002-3).
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