 Research Article
 Open Access
 Published:
Multiple Positive Solutions for Nonlinear FirstOrder Impulsive Dynamic Equations on Time Scales with Parameter
Advances in Difference Equations volume 2009, Article number: 830247 (2009)
Abstract
By using the LeggettWilliams fixed point theorem, the existence of three positive solutions to a class of nonlinear firstorder periodic boundary value problems of impulsive dynamic equations on time scales with parameter are obtained. An example is given to illustrate the main results in this paper.
1. Introduction
Let be a time scale, that is, is a nonempty closed subset of . Let be fixed and be points in , an interval denoting time scales interval, that is, Other types of intervals are defined similarly. Some definitions concerning time scales can be found in [1–5].
In this paper, we are concerned with the existence of positive solutions for the following nonlinear firstorder periodic boundary value problem on time scales:
where is a positive parameter, , is rightdense continuous, , and for each and represent the right and left limits of at .
The theory of impulsive differential equations is emerging as an important area of investigation, since it is a lot richer than the corresponding theory of differential equations without impulse effects. Moreover, such equations may exhibit several real world phenomena in physics, biology, engineering, and so forth, (see [6–8]). At the same time, the boundary value problems for impulsive differential equations and impulsive difference equations have received much attention [9–19]. On the other hand, recently, the theory of dynamic equations on time scales has become a new important branch (see, e.g., [1–5]). Naturally, some authors have focused their attention on the boundary value problems of impulsive dynamic equations on time scales [20–27]. In particular, for the firstorder impulsive dynamic equations on time scales
where is a time scale which has at least finitelymany rightdense points, is regressive and rightdense continuous, is given function, . The paper [21] obtained the existence of one solution to problem (1.2) by using the nonlinear alternative of LeraySchauder type.
In [22], Benchohra et al. considered the following impulsive boundary value problem on time scales
They proved the existence of one solution to the problem (1.3) by applying Schaefer's fixed point theorem and the nonlinear alternative of LeraySchauder type.
In [26], Li and Shen studied the problem (1.3). Some existence results to problem (1.3) are established by using a fixed point theorem, which is due to Krasnoselskii and Zabreiko, and the LeggettWilliams fixed point theorem.
In [27], the first author studied the problem (1.1) when . The existence of positive solutions to the problem (1.1) was obtained by means of the wellknown GuoKrasnoselskii fixed point theorem.
Recently, Sun and Li [28] considered the following periodic boundary value problem:
By using the fixed point index, some existence, multiplicity and nonexistence criteria of positive solutions to the problem (1.4) were obtained for suitable .
Motivated by the results mentioned above, in this paper, we shall show that the problem (1.1) has at least three positive solutions for suitable by using the LeggettWilliams fixed point theorem [29]. We note that for the case and problem (1.1) reduces to the problem studied by [30].
In the remainder of this section, we state the following theorem, which are crucial to our proof.
Let be a real Banach space and be a cone. A function is called a nonnegative continuous concave functional if is continuous and
for all and .
Let be constants,
Theorem 1.1 (see [29]).
Let be a completely continuous map and be a nonnegative continuous concave functional on such that Suppose there exist with such that

(i)
and
 (ii)

(iii)
with
Then has at least three fixed points in satisfying
2. Preliminaries
Throughout the rest of this paper, we always assume that the points of impulse are rightdense for each
We define
where is the restriction of to and
Let
with the norm Then X is a Banach space.
Definition 2.1.
A function is said to be a solution of the problem (1.1) if and only if satisfies the dynamic equation
the impulsive conditions
and the periodic boundary condition
Lemma 2.2.
Suppose is continuous, then is a solution of
where
if and only if is a solution of the boundary value problem
Proof.
Since the method is similar to that of in [27, Lemma 3.1], we omit it here.
Lemma 2.3.
Let be defined as Lemma 2.2, then
Proof.
It is obvious, so we omit it here.
Let
where It is not difficult to verify that is a cone in
We define an operator by
By [27, Lemmas 3.3 and 3.4], it is easy to see that is completely continuous.
3. Main Result
Notation 1.
Let
and for we define
Theorem 3.1.
Assume that there exists a number such that the following conditions:
(H_{1}) for
(H_{2}) hold. Then the problem (1.1) has at least three positive solutions for
Proof.
Let it is easy to see that is a nonnegative continuous concave functional on such that
First, we assert that there exists such that is completely continuous.
In fact, by the condition of (H_{2}), there exist and such that
Let if then and we have
Take then the set is a bounded set. According to that is completely continuous, then maps bounded sets into bounded sets and there exists a number such that
If we deduce that is completely continuous. If then from (3.4), we know that for any and hold. Then we have is completely continuous. Take then and are completely continuous.
Second, we assert that and for all
In fact, take so Moreover, for then and we have
Third, we assert that there exist such that if
Indeed, by the condition of (H_{2}), there exist and such that
Then we get
Finally, we assert that if and
To do this, if and then
To sum up, all the hypotheses of Theorem 1.1 are satisfied by taking Hence has at least three fixed points, that is, the problem (1.1) has at least three positive solutions and such that
Corollary 3.2.
Using (H_{3}) instead of (H_{2}) in Theorem 3.1, the conclusion of Theorem 3.1 remains true.
4. Example
Example 4.1.
Let We consider the following problem on
where is a positive parameter, and
Taking then by it is easy to see that So, for all we have Obviously, we have
Therefore, together with Corollary 3.2, it follows that the problem (4.1) has at least three positive solutions for .
References
 1.
Agarwal RP, Bohner M: Basic calculus on time scales and some of its applications. Results in Mathematics 1999,35(12):322.
 2.
Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.
 3.
Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.
 4.
Hilger S: Analysis on measure chainsa unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(12):1856.
 5.
Lakshmikantham V, Sivasundaram S, Kaymakcalan B: Dynamic Systems on Measure Chains, Mathematics and Its Applications. Volume 370. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:x+285.
 6.
Baĭnov DD, Simeonov PS: Systems with Impulse Effect: Stability, Theory and Applications, Ellis Horwood Series: Mathematics and Its Applications. Ellis Horwood, Chichester, UK; 1989:255.
 7.
Bainov DD, Simeonov PS: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific and Technical, Harlow, UK; 1993.
 8.
Lakshmikantham V, Baĭnov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273.
 9.
Agarwal RP, O'Regan D: Multiple nonnegative solutions for second order impulsive differential equations. Applied Mathematics and Computation 2000,114(1):5159. 10.1016/S00963003(99)000740
 10.
He Z, Yu J: Periodic boundary value problem for firstorder impulsive functional differential equations. Journal of Computational and Applied Mathematics 2002,138(2):205217. 10.1016/S03770427(01)003818
 11.
He Z, Zhang X: Monotone iterative technique for first order impulsive difference equations with periodic boundary conditions. Applied Mathematics and Computation 2004,156(3):605620. 10.1016/j.amc.2003.08.013
 12.
Li JL, Shen JH: Existence of positive periodic solutions to a class of functional differential equations with impulses. Mathematica Applicata 2004,17(3):456463.
 13.
Li J, Nieto JJ, Shen J: Impulsive periodic boundary value problems of firstorder differential equations. Journal of Mathematical Analysis and Applications 2007,325(1):226236. 10.1016/j.jmaa.2005.04.005
 14.
Li J, Shen J: Positive solutions for first order difference equations with impulses. International Journal of Difference Equations 2006,1(2):225239.
 15.
Li Y, Fan X, Zhao L: Positive periodic solutions of functional differential equations with impulses and a parameter. Computers & Mathematics with Applications 2008,56(10):25562560. 10.1016/j.camwa.2008.05.007
 16.
Nieto JJ: Basic theory for nonresonance impulsive periodic problems of first order. Journal of Mathematical Analysis and Applications 1997,205(2):423433. 10.1006/jmaa.1997.5207
 17.
Nieto JJ: Impulsive resonance periodic problems of first order. Applied Mathematics Letters 2002,15(4):489493. 10.1016/S08939659(01)00163X
 18.
Nieto JJ: Periodic boundary value problems for firstorder impulsive ordinary differential equations. Nonlinear Analysis: Theory, Methods & Applications 2002,51(7):12231232. 10.1016/S0362546X(01)008896
 19.
Vatsala AS, Sun Y: Periodic boundary value problems of impulsive differential equations. Applicable Analysis 1992,44(34):145158. 10.1080/00036819208840074
 20.
Belarbi A, Benchohra M, Ouahab A: Existence results for impulsive dynamic inclusions on time scales. Electronic Journal of Qualitative Theory of Differential Equations 2005,2005(12):122.
 21.
Benchohra M, Henderson J, Ntouyas SK, Ouahab A: On first order impulsive dynamic equations on time scales. Journal of Difference Equations and Applications 2004,10(6):541548. 10.1080/10236190410001667986
 22.
Benchohra M, Ntouyas SK, Ouahab A: Existence results for second order boundary value problem of impulsive dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2004,296(1):6573. 10.1016/j.jmaa.2004.02.057
 23.
Geng F, Xu Y, Zhu D: Periodic boundary value problems for firstorder impulsive dynamic equations on time scales. Nonlinear Analysis: Theory, Methods & Applications 2008,69(11):40744087. 10.1016/j.na.2007.10.038
 24.
Graef JR, Ouahab A: Extremal solutions for nonresonance impulsive functional dynamic equations on time scales. Applied Mathematics and Computation 2008,196(1):333339. 10.1016/j.amc.2007.05.056
 25.
Henderson J: Double solutions of impulsive dynamic boundary value problems on a time scale. Journal of Difference Equations and Applications 2002,8(4):345356. 10.1080/1026190290017405
 26.
Li J, Shen J: Existence results for secondorder impulsive boundary value problems on time scales. Nonlinear Analysis: Theory, Methods & Applications 2009,70(4):16481655. 10.1016/j.na.2008.02.047
 27.
Wang DB: Positive solutions for nonlinear firstorder periodic boundary value problems of impulsive dynamic equations on time scales. Computers & Mathematics with Applications 2008,56(6):14961504. 10.1016/j.camwa.2008.02.038
 28.
Sun JP, Li WT: Positive solutions to nonlinear firstorder PBVPs with parameter on time scales. Nonlinear Analysis: Theory, Methods & Applications 2009,70(3):11331145. 10.1016/j.na.2008.02.007
 29.
Leggett RW, Williams LR: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana University Mathematics Journal 1979,28(4):673688. 10.1512/iumj.1979.28.28046
 30.
Sun JP, Li WT: Existence and multiplicity of positive solutions to nonlinear firstorder PBVPs on time scales. Computers & Mathematics with Applications 2007,54(6):861871. 10.1016/j.camwa.2007.03.009
Acknowledgment
The authors express their gratitude to the anonymous referee for his/her valuable suggestions.
Author information
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Wang, D., Guan, W. Multiple Positive Solutions for Nonlinear FirstOrder Impulsive Dynamic Equations on Time Scales with Parameter. Adv Differ Equ 2009, 830247 (2009) doi:10.1155/2009/830247
Received:
Accepted:
Published:
Keywords
 Dynamic Equation
 Fixed Point Theorem
 Real Banach Space
 Scale Interval
 Impulsive Differential Equation