Abstract
Recently Tariboon and Ntouyas (Adv. Differ. Equ. 2013:282, 2013) introduced the notions of -derivative and -integral of a function on finite intervals. As applications existence and uniqueness results for initial value problems for first- and second-order impulsive -difference equations was proved. In this paper, continuing the study of Tariboon and Ntouyas (Adv. Differ. Equ. 2013:282, 2013), we apply the quantum calculus to initial value problems for impulsive first- and second-order -difference inclusions. We establish new existence results, when the right hand side is convex valued, by using the nonlinear alternative of Leray-Schauder type. Some illustrative examples are also presented.
MSC:34A60, 26A33, 39A13, 34A37.
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1 Introduction and preliminaries
In [1] the notions of -derivative and -integral of a function , have been introduced and their basic properties was proved. As applications, existence and uniqueness results for initial value problems for first- and second-order impulsive -difference equations was proved.
We recall the notions of -derivative and -integral on finite intervals. 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 (1.1), 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 is a continuous function, we call the second-order -derivative provided is -differentiable on with . Similarly, we define higher order -derivative .
The properties of -derivative are discussed in [1].
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 (1.3) reduces to q-integral of a function , defined by for .
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, that is, differential equations involving the impulse effect, appear as a natural description of observed evolution phenomena of several real world problems. For some monographs on the impulsive differential equations we refer to [16–18].
Here, we remark that the classical q-calculus cannot be considered in problems with impulses as the definition of q-derivative fails to work when there are impulse points for some . On the other hand, this situation does not arise for impulsive problems on a q-time scale as the points t and are consecutive points, where is the backward jump operator; see [19]. In [1], quantum calculus on finite intervals, the points t and are considered only in an interval . Therefore, the problems with impulses at fixed times can be considered in the framework of -calculus.
In this paper, continuing the study of [1], we apply -calculus to establish existence results for initial value problems for impulsive first- and second-order -difference inclusions. In Section 3, we consider the following initial value problem for the first-order -difference inclusion:
where , , is a multivalued function, is the family of all nonempty subjects of ℝ, , , and for .
In Section 4, we study the existence of solutions for the following initial value problem for second-order impulsive -difference inclusion:
where and .
We establish new existence results, when the right hand side is convex valued by using the nonlinear alternative of Leray-Schauder type.
The paper is organized as follows. In Section 2, we recall some preliminary facts that we need in the sequel. In Section 3 we establish the existence result for first-order -difference inclusions, while the existence result for second-order -difference inclusions is presented in Section 4. Some illustrative examples are also presented.
2 Preliminaries
In this section we recall some basic concepts of multivalued analysis [20, 21].
For a normed space , let , , and .
A multivalued map is convex (closed) valued if is convex (closed) for all ; is bounded on bounded sets if is bounded in X for all (i.e. ); is called upper semicontinuous (u.s.c.) on X if for each , the set is a nonempty closed subset of X, and if for each open set N of X containing , there exists an open neighborhood of such that ; is said to be completely continuous if is relatively compact for every .
In the sequel, we denote by the space of all continuous functions from with norm . By we denote the space of all functions f defined on such that .
For each , define the set of selections of F by
Definition 2.1 A multivalued map is said to be Carathéodory (in the sense of -calculus) if is upper semicontinuous on J. Further a Carathéodory function F is called -Carathéodory if there exists such that for all on J for each .
We recall the well-known nonlinear alternative of Leray-Schauder for multivalued maps and a useful result regarding closed graphs.
Lemma 2.2 (Nonlinear alternative for Kakutani maps) [22]
Let E be a Banach space, C a closed convex subset of E, U an open subset of C and . Suppose that is a upper semicontinuous compact map. Then either
-
(i)
F has a fixed point in , or
-
(ii)
there is a and with .
Let X be a Banach space. Let be an -Carathéodory multivalued map and let Θ be a linear continuous mapping from to . Then the operator
is a closed graph operator in .
Let , , for . Let = { is continuous everywhere except for some at which and exist and , }. is a Banach space with the norms .
3 First-order impulsive -difference inclusions
In this section, we study the existence of solutions for the first-order impulsive -difference inclusion (1.4).
The following lemma was proved in [1].
Lemma 3.1 If , then for any , , the solution of the problem
is given by
with .
Before studying the boundary value problem (1.4) let us begin by defining its solution.
Definition 3.2 A function is said to be a solution of (1.4) if , , , and there exists such that on J and
Theorem 3.3 Assume that:
(H1) is Carathéodory and has nonempty compact and convex values;
(H2) there exist a continuous nondecreasing function and a function such that
(H3) there exist constants such that , for each ;
(H4) there exists a constant such that
Then the initial value problem (1.4) has at least one solution on J.
Proof Define the operator by
for .
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
Consequently,
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
Observe that
as .
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 (H4), 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 3.4 Let us consider the following first-order initial value problem for impulsive -difference inclusions:
Here , , , , and . We find that and .
-
(a)
Let be a multivalued map given by
(3.4)
For , we have
Thus,
with , . Further, using the condition (H4) we find that . Therefore, all the conditions of Theorem 3.3 are satisfied. So, problem (3.3) with given by (3.4) has at least one solution on .
-
(b)
If is a multivalued map given by
(3.5)
For , we have
Here , , with , . It is easy to verify that . Then, by Theorem 3.3, the problem (3.3) with given by (3.5) has at least one solution on .
4 Second-order impulsive -difference inclusions
In this section, we study the existence of solutions for the second-order impulsive -difference inclusion (1.5).
We recall the following lemma from [1].
Lemma 4.1 If , then for any , the solution of the problem
is given by
with .
Definition 4.2 A function is said to be a solution of (1.5) if , , , , and there exists such that on J and
with .
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
where
Then the initial value problem (1.5) has at least one solution on J.
Proof Define the operator by
for .
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
Consequently,
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
Observe that
as .
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
-
(a)
Let be a multivalued map given by
(4.6)
For , we have
Thus,
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 .
-
(b)
If is a multivalued map given by
(4.7)
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 .
Authors’ information
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
References
Tariboon J, Ntouyas SK: Quantum calculus on finite intervals and applications to impulsive difference equations. Adv. Differ. Equ. 2013., 2013: Article ID 282
Kac V, Cheung P: Quantum Calculus. Springer, New York; 2002.
Bangerezako G: Variational q -calculus. J. Math. Anal. Appl. 2004, 289: 650-665. 10.1016/j.jmaa.2003.09.004
Dobrogowska A, Odzijewicz A: Second order q -difference equations solvable by factorization method. J. Comput. Appl. Math. 2006, 193: 319-346. 10.1016/j.cam.2005.06.009
Gasper G, Rahman M: Some systems of multivariable orthogonal q -Racah polynomials. Ramanujan J. 2007, 13: 389-405. 10.1007/s11139-006-0259-8
Ismail MEH, Simeonov P: q -Difference operators for orthogonal polynomials. J. Comput. Appl. Math. 2009, 233: 749-761. 10.1016/j.cam.2009.02.044
Bohner M, Guseinov GS: The h -Laplace and q -Laplace transforms. J. Math. Anal. Appl. 2010, 365: 75-92. 10.1016/j.jmaa.2009.09.061
El-Shahed M, Hassan HA: Positive solutions of q -difference equation. Proc. Am. Math. Soc. 2010, 138: 1733-1738.
Ahmad B: Boundary-value problems for nonlinear third-order q -difference equations. Electron. J. Differ. Equ. 2011., 2011: Article ID 94
Ahmad B, Alsaedi A, Ntouyas SK: A study of second-order q -difference equations with boundary conditions. Adv. Differ. Equ. 2012., 2012: Article ID 35
Ahmad B, Ntouyas SK, Purnaras IK: Existence results for nonlinear q -difference equations with nonlocal boundary conditions. Commun. Appl. Nonlinear Anal. 2012, 19: 59-72.
Ahmad B, Nieto JJ: On nonlocal boundary value problems of nonlinear q -difference equations. Adv. Differ. Equ. 2012., 2012: Article ID 81
Ahmad B, Ntouyas SK: Boundary value problems for q -difference inclusions. Abstr. Appl. Anal. 2011., 2011: Article ID 292860
Zhou W, Liu H: Existence solutions for boundary value problem of nonlinear fractional q -difference equations. Adv. Differ. Equ. 2013., 2013: Article ID 113
Yu C, Wang J: Existence of solutions for nonlinear second-order q -difference equations with first-order q -derivatives. Adv. Differ. Equ. 2013., 2013: Article ID 124
Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.
Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore; 1995.
Benchohra M, Henderson J, Ntouyas SK 2. In Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York; 2006.
Bohner M, Peterson A: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser Boston, Boston; 2001.
Deimling K: Multivalued Differential Equations. de Gruyter, Berlin; 1992.
Hu S, Papageorgiou N: Handbook of Multivalued Analysis. Vol. I. Theory. Kluwer Academic, Dordrecht; 1997.
Granas A, Dugundji J: Fixed Point Theory. Springer, New York; 2005.
Lasota A, Opial Z: An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 1965, 13: 781-786.
Frigon M: Théorèmes d’existence de solutions d’inclusions différentielles. NATO ASI Series C 472. In Topological Methods in Differential Equations and Inclusions. Edited by: Granas A, Frigon M. Kluwer Academic, Dordrecht; 1995:51-87.
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This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
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Ntouyas, S.K., Tariboon, J. Applications of quantum calculus on finite intervals to impulsive difference inclusions. Adv Differ Equ 2014, 262 (2014). https://doi.org/10.1186/1687-1847-2014-262
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DOI: https://doi.org/10.1186/1687-1847-2014-262