A new approach to BVPs with state-dependent impulses
© Rachůnková and Tomeček; licensee Springer. 2013
Received: 24 October 2012
Accepted: 15 January 2013
Published: 11 February 2013
The paper deals with the second-order Dirichlet boundary value problem with one state-dependent impulse
Proofs of the main results contain a new approach to boundary value problems with state-dependent impulses which is based on a transformation to a fixed point problem of an appropriate operator in the space . Sufficient conditions for the existence of solutions to the problem are given here. The presented approach can be extended to more impulses and to other boundary conditions.
Keywordsimpulsive differential equation state-dependent impulses Dirichlet problem second-order ODE
Most papers in the literature on impulsive boundary value problems concern the case with fixed moments of impulsive effects. Papers dealing with state-dependent impulses, called also impulses at variable times, focus their attention on initial value problems or periodic problems. Such papers investigate the existence, stability or asymptotic properties of solutions of initial value problems [4–8] or solvability of autonomous periodic problems [9, 10] and nonautonomous ones [11–15]. We can also find papers investigating other boundary value problems with state-dependent impulses through some initial value problems for multi-valued maps [16, 17].
Under assumptions (4)-(8), we prove the solvability of problem (1)-(3). In particular, we transform problem (1)-(3) to a fixed point problem for a proper operator in the space . This approach can be also used for other types of boundary conditions and it can be easily extended to more impulses.
It is well-known that the mentioned normed spaces are Banach spaces. Recall that for , a function satisfies the Carathéodory conditions on (we write ) if
is measurable for all ,
is continuous for a.e. ,
for each compact set , there exists a function such that for a.e. and each .
We say that is a solution of problem (1)-(3), if z is continuous on , there exists unique such that , and have absolutely continuous first derivatives, z satisfies equation (1) for a.e. and fulfills conditions (2), (3).
Therefore, σ is strictly decreasing on and hence it has exactly one root in . □
where fulfills (9). The next lemma provides an important result about the continuity of which is fundamental for our approach.
Lemma 2 The functional is continuous on .
for each . By Lemma 1 and the continuity of , it follows that for . □
Lemma 3 The operator ℱ is compact on .
as uniformly w.r.t. . Therefore, converges to x in . Similar arguments can be applied to the sequence .
holds for all . Consequently, is relatively compact in by the Arzelà-Ascoli theorem. □
is a solution of problem (1)-(3).
Therefore, σ is strictly decreasing on , which yields for . Consequently, is a unique point in satisfying (20).
3 Main result
Here, using the Leray-Schauder degree theory, we prove our main result about the solvability of problem (1)-(3). To this end, we will need the following lemma on a priori estimates.
. Since , it follows that and therefore , , and . There are two possibilities as follows.
Then , which yields .
which is a contradiction.
For , the solution of (21) is , and it clearly belongs to Ω. □
Theorem 6 Assume (4)-(8). Then the operator ℱ has a fixed point in Ω.
has a solution in Ω. This solution is a fixed point of the operator ℱ. □
Proof From Theorem 6 it follows that there exists a fixed point of the operator ℱ. Lemma 4 yields that the function z defined in (17) (with ) is a solution of problem (1)-(3). Estimates (25) follow from (17) and from the definitions of Ω and (cf. (12) and (8)). □
In this section we demonstrate that Theorem 7 can be applied to sublinear, linear and superlinear problems.
Example 9 (Sublinear problem)
we can check that conditions (8) are satisfied in both cases. Therefore, by Theorem 7, the corresponding problem (1)-(3) has at least one solution.
Note that (27) shows that γ need not be monotonous.
Example 10 (Linear problem)
If (28) holds, then for any sufficiently large K, condition (7) is satisfied. By (8), we have , and problem (1)-(3) has a solution for any γ satisfying (8). Consequently, if γ is given by (26) or (27), problem (1)-(3) is solvable.
satisfies (30) as well. Put, for example, , . Then we get that for inequality (30) holds. Consequently, (8) gives and the corresponding problem (1)-(3) is solvable for any γ satisfying (8). In particular, γ given by (26) or (27) can be considered in this case as well.
Dedicated to Jean Mawhin on the occasion of his 70th birthday.
The authors would like to thank the anonymous referees for their valuable comments and suggestions. This work was supported by the grant Matematické modely a struktury, PrF_ 2012_ 017.
- Bainov D, Simeonov P Pitman Monographs and Surveys in Pure and Applied Mathematics 66. In Impulsive Differential Equations: Periodic Solutions and Applications. Longman, Essex; 1993.Google Scholar
- Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.MATHView ArticleGoogle Scholar
- Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore; 1995.MATHGoogle Scholar
- Afonso SM, Bonotto EM, Federson M, Schwabik Š: Discontinuous local semiflows for Kurzweil equations leading to LaSalle’s invariance principle for differential systems with impulses at variable times. J. Differ. Equ. 2011, 250: 2969-3001. 10.1016/j.jde.2011.01.019MATHMathSciNetView ArticleGoogle Scholar
- Benchohra M, Henderson J, Ntouyas SK, Ouahab A: Impulsive functional differential equations with variable times. Comput. Math. Appl. 2004, 47: 1659-1665. 10.1016/j.camwa.2004.06.013MATHMathSciNetView ArticleGoogle Scholar
- Domoshnitsky A, Drakhlin M, Litsyn E: Nonoscillation and positivity of solutions to first order state-dependent differential equations with impulses in variable moments. J. Differ. Equ. 2006, 228: 39-48. 10.1016/j.jde.2006.05.009MATHMathSciNetView ArticleGoogle Scholar
- Frigon M, O’Regan D: Impulsive differential equations with variable times. Nonlinear Anal. 1996, 26: 1913-1922. 10.1016/0362-546X(95)00053-XMATHMathSciNetView ArticleGoogle Scholar
- Kaul S, Lakshmikantham V, Leela S: Extremal solutions, comparison principle and stability criteria for impulsive differential equations with variable times. Nonlinear Anal. 1994, 22: 1263-1270. 10.1016/0362-546X(94)90109-0MATHMathSciNetView ArticleGoogle Scholar
- Liu L, Sun J: Existence of periodic solution of a harvested system with impulses at variable times. Phys. Lett. A 2006, 360: 105-108. 10.1016/j.physleta.2006.07.080MATHView ArticleGoogle Scholar
- Qi J, Fu X: Existence of limit cycles of impulsive differential equations with impulses at variable times. Nonlinear Anal. 2001, 44: 345-353. 10.1016/S0362-546X(99)00268-0MATHMathSciNetView ArticleGoogle Scholar
- Bajo I, Liz E: Periodic boundary value problem for first order differential equations with impulses at variable times. J. Math. Anal. Appl. 1996, 204: 65-73. 10.1006/jmaa.1996.0424MATHMathSciNetView ArticleGoogle Scholar
- Belley J, Virgilio M: Periodic Duffing delay equations with state dependent impulses. J. Math. Anal. Appl. 2005, 306: 646-662. 10.1016/j.jmaa.2004.10.023MATHMathSciNetView ArticleGoogle Scholar
- Belley J, Virgilio M: Periodic Liénard-type delay equations with state-dependent impulses. Nonlinear Anal. 2006, 64: 568-589. 10.1016/j.na.2005.06.025MATHMathSciNetView ArticleGoogle Scholar
- Frigon M, O’Regan D: First order impulsive initial and periodic problems with variable moments. J. Math. Anal. Appl. 1999, 233: 730-739. 10.1006/jmaa.1999.6336MATHMathSciNetView ArticleGoogle Scholar
- Yong L, Fuzhong C, Zhanghua L: Boundary value problems for impulsive differential equations. Nonlinear Anal. TMA 1997, 29: 1253-1264. 10.1016/S0362-546X(96)00177-0MATHView ArticleGoogle Scholar
- Benchohra M, Graef JR, Ntouyas SK, Ouahab A: Upper and lower solutions method for impulsive differential inclusions with nonlinear boundary conditions and variable times. Dyn. Contin. Discrete Impuls. Syst. 2005, 12: 383-396.MATHMathSciNetGoogle Scholar
- Frigon M, O’Regan D: Second order Sturm-Liouville BVP’s with impulses at variable times. Dyn. Contin. Discrete Impuls. Syst. 2001, 8: 149-159.MATHMathSciNetGoogle Scholar