Abstract
This article studies the existence and uniqueness of solutions for a boundary value problem of nonlinear second-order impulsive -integro-difference equations with separated boundary conditions. Several new results are obtained by applying a variety of fixed point theorems. Some examples are presented to illustrate the results.
MSC:26A33, 39A13, 34A37.
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1 Introduction
In this paper, we study the separated boundary value problem for impulsive -integro-difference equation of the following form:
where , ,
is a continuous function, , for , and for .
The notions of -derivative and -integral on finite intervals were introduced in [1]. For a fixed let be an interval and be a constant. We define -derivative of a function at a point as follows.
Definition 1.1 Assume is a continuous function and let . Then the expression
is called the -derivative of function f at t.
We say that f is -differentiable on provided exists for all . Note that if and in (1.3), then , where is the well-known q-derivative of the function defined by
In addition, we should define the higher -derivative of functions.
Definition 1.2 Let be a continuous function, we call the second-order -derivative provided is -differentiable on with . Similarly, we define higher order -derivative .
The -integral is defined as follows.
Definition 1.3 Assume is a continuous function. Then the -integral is defined by
for . Moreover, if then the definite -integral is defined by
Note that if and , then (1.5) reduces to q-integral of a function , defined by for .
For the basic properties of -derivative and -integral we refer to [1].
The book by Kac and Cheung [2] covers many of the fundamental aspects of the quantum calculus. In recent years, the topic of q-calculus has attracted the attention of several researchers and a variety of new results can be found in the papers [3–15] and the references cited therein.
Impulsive differential equations serve as basic models to study the dynamics of processes that are subject to sudden changes in their states. Recent development in this field has been motivated by many applied problems, such as control theory, population dynamics and medicine. For some recent works on the theory of impulsive differential equations, we refer the interested reader to the monographs [16–18].
In this paper we prove an existence and uniqueness result for the impulsive boundary value problem (1.1) by using Banach’s contraction mapping principle and three existence results by applying Schaefer’s, Krasnoselskii’s fixed point theorems and the Leray-Schauder Nonlinear Alternative. The rest of this paper is organized as follows: In Section 2 we present an auxiliary lemma which is used to convert the impulsive boundary value problem (1.1) into an equivalent integral equation. The main results are given in Section 3, while examples illustrating the results are presented in Section 4.
2 An auxiliary lemma
Let , , for . Let = { is continuous everywhere except for some at which and exist and , }. is a Banach space with the norm .
We now consider the following linear case:
where is a continuous function.
Lemma 2.1 The unique solution of problem (2.1) is given by
with for .
Proof Taking the -integral for the first equation of (2.1), for , we have
which leads to
For we get by -integrating (2.3),
In particular, for , we obtain
For , -integrating (2.1), we have
Using the third condition of (2.1) with (2.4), it follows that
Taking -integral to (2.6) for , we obtain
Applying the second equation of (2.1) with (2.5) and (2.7), we get
Repeating the above process, for , we get
From the first boundary condition of (2.1) (i.e. ) and (2.8), we have
Also, the second boundary condition of (2.1) (i.e. ) and (2.8), yields
From (2.9) and (2.10), we have that
which implies
Substituting constants A and B into (2.8), we obtain (2.2) as requested. □
3 Main results
In view of Lemma 2.1, we define an operator by
It should be noticed that problem (1.1) has solutions if and only if the operator has fixed points.
Our first result is an existence and uniqueness result for the impulsive boundary value problem (1.1) by using the Banach contraction mapping principle.
Let . Further, for convenience we set
and
Theorem 3.1 Assume that:
(H1) The function is continuous and there exist constants such that
for each and , .
(H2) The functions are continuous and there exist constants such that
for each , .
If
where ω is defined by (3.2), and , then the boundary value problem (1.1) has a unique solution on J.
Proof We transform the boundary value problem (1.1) into a fixed point problem, , where the operator is defined by (3.1). By using the Banach contraction mapping principle, we shall show that has a fixed point which is the unique solution of the boundary value problem (1.1).
Let , , and be nonnegative constants such that , , and . By choosing a constant R as
where and defined by (3.3), we will show that , where a ball is defined by . For , we have
which implies that .
For any and for each , we have
which implies that . As , is a contraction. Therefore, by the Banach contraction mapping principle, we find that has a fixed point which is the unique solution of problem (1.1). This completes the proof. □
The second existence result is based on Schaefer’s fixed point theorem.
Theorem 3.2 Assume that:
(H3) is a continuous function and there exists a constant such that
for each and all , .
(H4) The functions are continuous and there exist constants such that
for all , .
Then the boundary value problem (1.1) has at least one solution on J.
Proof We will use Schaefer’s fixed point theorem to prove that , defined by (3.1), has a fixed point. We divide the proof into four steps.
Step 1: Continuity of .
Let be a sequence such that in . Since f is a continuous function on and , are continuous functions on ℝ for , we have
and , for , as .
Then, for each , we get
which gives as . This means that is continuous.
Step 2: maps bounded sets into bounded sets in .
So, let us prove that for any , there exists a positive constant ρ such that for each , we have . For any , we have
Hence, we deduce that .
Step 3: maps bounded sets into equicontinuous sets of .
Let for some , , be a bounded set of as in Step 2, and let . Then we have
As , the right-hand side of the above inequality (which is independent of x) tends to zero. As a consequence of Steps 1 to 3, together with the Arzelá-Ascoli theorem, we deduce that is completely continuous.
Step 4: We show that the set
is bounded.
Let . Then for some . Thus, for each , by using the computations of Step 2, we have that
This shows that the set E is bounded. As a consequence of Schaefer’s fixed point theorem, we conclude that has a fixed point which is a solution of the impulsive -integro-difference boundary value problem (1.1). □
The third existence result for the impulsive boundary value problem (1.1) is based on Krasnoselskii’s fixed point theorem.
Lemma 3.3 (Krasnoselskii’s fixed point theorem) [19]
Let M be a closed, bounded, convex and nonempty subset of a Banach space X. Let A, B be the operators such that
-
(a)
whenever ;
-
(b)
A is a compact and continuous;
-
(c)
B is a contraction mapping.
Then there exists such that .
For convenience we put
and
Theorem 3.4 Let be a continuous function. Assume that:
(A1) , and .
(A2) There exist constants such that and , , for .
(A3) There exist constants such that and , , for .
If
then boundary value problem (1.1) has at least one solution on J.
Proof We define and choose a suitable constant ρ as
where Λ and are defined by (3.5) and (3.6), respectively. We define the operators Φ and Ψ on as
and
For , we have
Thus, .
For , from (A3), we have
which implies, by (3.7), that Ψ is a contraction mapping.
Continuity of f implies that the operator Φ is continuous. Also, Φ is uniformly bounded on as
Now we prove the compactness of the operator Φ.
We define , for some with and consequently we get
which is independent of x and tends to zero as . Thus, Φ is equicontinuous. So Φ is relatively compact on . Hence, by the Arzelá-Ascoli theorem, Φ is compact on . Thus all the assumptions of Lemma 3.3 are satisfied. So the boundary value problem (1.1) has at least one solution on J. The proof is completed. □
Our final, fourth existence result is based on the Leray-Schauder Nonlinear Alternative.
Lemma 3.5 (Nonlinear alternative for single valued maps) [20]
Let E be a Banach space, C a closed, convex subset of E, U an open subset of C and . Suppose that is a continuous, compact (that is, is a relatively compact subset of C) map. Then either
-
(i)
F has a fixed point in , or
-
(ii)
there is a (the boundary of U in C) and with .
Theorem 3.6 Assume that:
(A4) There exist a continuous nondecreasing function and a continuous function such that
(A5) There exist continuous nondecreasing functions such that
for all , .
(A6) There exists a constant such that
where , and
Then the impulsive boundary value problem (1.1) has at least one solution on J.
Proof First we show that maps bounded sets (balls) into bounded sets in . For a positive number , let be a bounded ball in . Then for we have
Hence, we deduce that .
Next we show that maps bounded sets into equicontinuous sets of .
Let for some , , be a bounded set of as in the previous step, and let . Then we have
The right-hand side of the above inequality, which is independent of x, tends to zero as . From all above and by the Arzelá-Ascoli theorem is completely continuous.
The result will follow from the Leray-Schauder nonlinear alternative (Lemma 3.5) once we have proved the boundedness of the set of all solutions to the equations for some .
Let x be a solution. Thus, for each , we have
This implies by (A4) and (A5) that for each , we have
Consequently, we have
In view of (A6), there exists such that . Let us set
Note that the operator is continuous 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 3.5), we deduce that has a fixed point which is a solution of the problem (1.1). This completes the proof. □
4 Examples
In this section, we will give examples to illustrate our main results.
Example 4.1 Consider the following boundary value problem for the nonlinear second-order impulsive -integro-difference equation:
Here , , for , , , , and . Since
(H1) and (H2) are satisfied with , , , . We can show that
Hence, by Theorem 3.1, boundary value problem (4.1) has a unique solution on .
Example 4.2 Consider the following boundary value problem for the nonlinear second-order impulsive -integro-difference equation:
Here , , for , , , , and . We can show that
Hence, by Theorem 3.2, boundary value problem (4.2) has at least one solution on .
Example 4.3 Consider the following nonlinear second-order impulsive -difference equation with separated boundary condition:
where , , for , , , , and . Since
(A3) is satisfied with , . It is easy to verify that , , and for all , , . Thus (A1) and (A2) are satisfied. We can show that
Hence, by Theorem 3.4, boundary value problem (4.3) has at least one solution on .
Example 4.4 Consider the following nonlinear second-order impulsive -difference equation with separated boundary condition:
Here , , for , , , , , and . Clearly,
Choosing , , , and , we obtain
which implies that . Hence, by Theorem 3.6, boundary value problem (4.4) has at least one solution on .
Authors’ information
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
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Acknowledgements
The research of C Thaiprayoon and J Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand.
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Thaiprayoon, C., Tariboon, J. & Ntouyas, S.K. Separated boundary value problems for second-order impulsive q-integro-difference equations. Adv Differ Equ 2014, 88 (2014). https://doi.org/10.1186/1687-1847-2014-88
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DOI: https://doi.org/10.1186/1687-1847-2014-88