Abstract
In this article, we establish some impulsive differential and impulsive integral inequalities for integral jump conditions. The new jump conditions for impulse effects are related to the integral conditions of the past state. Two examples are given to illustrate the advantage of our results.
2010 Mathematics Subject Classification: 34A37; 34A40.
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1 Introduction
In [1], Lakshmikantham et al. developed a famous impulsive differential inequality given as Theorem A below.
Lakshmikantham et al. assume that 0 ≤ t0 < t1 < t2 <⋯, limk→∞t k = ∞, R+ = [0, +∞) and I ⊂ R. They define PC(R+, I) = {u: R+ → I; u(t) is continuous for t ≠ t k , and u(0+), , and exist, and and PC1(R+,I) = {u ∈ PC(R+, I): u'(t) is continuous everywhere for t ≠ t k , and u'(0+), and exist, and .
Theorem A. Assume that
(H0) the sequence {t k } satisfies 0 ≤ t0 < t1 < t2 < ⋯, limk→∞t k = ∞;
(H1) m ∈ PC1[R+, R] and m(t) is left-continuous at t k , k = 1, 2,...;
(H2) for k = 1, 2,..., t ≥ t0,
where q, p ∈ C[R+, R], d k ≥ 0 and b k are constants.
Then,
Impulsive differential and impulsive integral inequalities play an important role in the study of the theory of impulsive differential equations (see [1–4]). In recent years, many authors have used impulsive (differential or integral) inequalities to investigate properties of solutions of various impulsive problems, such as existence, uniqueness, boundedness, stability, asymptotic behavior, and oscillation etc. (see, for example [5–39]). There are many good results on the impulsive differential and impulsive integral inequalities (see for example [40–48]). However, most of these articles deal with jump conditions at impulse point t k depending on the left-hand limit m(t k ) or a time-delay value, m(t k -τ), τ > 0. Our main goal is to extend the theory of impulsive differential and impulsive integral inequalities to include integral jump conditions.
In the present article, we will show that Theorem A can be generalized to obtain differential inequalities for integral jump conditions by replacing the inequality in (1.2) by the inequality in (1.4).
where 0 ≤ σ k ≤ τ k ≤ t k - tk-1. We note that if c k = 0 for all k = 1, 2,..., then condition (1.4) reduces to condition (1.2). If d k = 0, c k ≠ 0 and 0 ≤ σ k < τ k ≤ t k -tk-1, k = 1, 2,..., then condition (1.4) means that the bound of the jump condition at t k is a functional of past states on the interval (t k - τ k , t k - σ k ] before the impulse point t k . Moreover, we establish some new impulsive integral inequalities with integral jump conditions.
At the end of this article, we will show some applications of our results to prove a maximum principle and the boundedness of solutions for impulsive problems.
2 Main results
Denote l = max{k: t ≥ t k , k = 1, 2,...}. Now we are in the position to state and prove our results.
Theorem 2.1. Let (H 0 ) and (H1) hold. Suppose that p, q ∈ C[R+, R] and for k = 1, 2,..., t ≥ t0,
where c k , d k ≥ 0, 0 ≤ σ k ≤ τ k ≤ t k - tk-1and b k are constants.
Then,
Proof. From (2.1) we have that
for t ∈ [t0, t1]. Integrating (2.4) from t0 to t for t ∈ [t0, t1], we get
Hence (2.3) is valid on [t0, t1]. Assume that (2.3) holds for t ∈ [t0, t n ] for some integer n > 1. Then, for t ∈ [t n , tn+1], it follow from (2.1) and (2.5) that
Now using (2.2) and (2.6), we have
By the principle of mathematical induction, (2.7) can be expressed as
Set
Substituting (2.9), (2.10) into (2.8), we get that for t ∈ [t n , tn+1]
Hence,
for t n ≤ t ≤ tn+1. Therefore, the estimate (2.3) holds for t0 ≤ t ≤ tn+1. This completes the proof.
Remark 2.2. If c k = 0 for all k = 1, 2,..., then Theorem 2.1 reduces to Theorem A.
Corollary 2.3. Let (H0) and (H1) hold. Suppose that p, q ∈ C[R+, R] and for k = 1, 2,..., t ≥ t0,
where c k ≥ 0, 0 ≤ σ k ≤ τ k ≤ t k - tk-1and b k are constants.
Then,
The following corollary will be used in our examples. For convenience, we set
Corollary 2.4. Let (H0) and (H1) hold. Suppose that q ∈ C[R+, R], and for k = 1, 2,..., t ≥ t0,
where p ≠ 0, c k ≥ 0, 0 ≤ σ k ≤ τ k ≤ t k - tk-1and b k are constants.
Then,
for t ≥ t0where A k , B k are defined by (2.14), (2.15), respectively.
Proof. By using Corollary 2.3 and reversing the order of double integration, we have the required result.
Corollary 2.5. Let (H0) and (H1) hold. Suppose that q ∈ C[R+, R], and for k = 1, 2,...,t ≥ t0,
where, c k ≥ 0, 0 ≤ σ k ≤ τ k ≤ t k - tk-1and b k are constants.
Then,
Proof. By setting p(t) ≡ 0 and d k = 1(k = 1, 2,...) in Theorem 2.1 and reversing the order of double integration, we have the required result.
Next, we give an application of Theorem 2.1 to the determination of a bound for the solutions of impulsive integral inequalities with integral jump conditions.
Theorem 2.6. Assume that (H0) and (H1) hold. Suppose that p ∈ C[R+, R+] and for k = 1, 2,...
where α k , β k ≥ 0, 0 ≤ σ k ≤ τ k ≤ t k - tk-1and C are constants. Then
Proof. Defining a function v(t) by the right side of (2.22), we have
Since m(t) ≤ v(t), we get
Applying Theorem 2.1, we obtain
which results in (2.23).
Theorem 2.7 . Assume that (H0) and (H1) hold. Suppose that p ∈ C[R+, R+], h ∈ PC[R+, R] and for k = 1, 2,...
where α k , β k ≥ 0 and 0 ≤ σ k ≤ τ k ≤ t k - tk-1are constants.
Then,
Proof. Setting
and from the fact that m(t) ≤ h(t) + v(t), we obtain
Using Theorem 2.1 together with m(t) ≤ h(t) + v(t), we then obtain the estimate (2.25).
Remark 2.8. If α k = 0 for all k = 1, 2,..., then Theorem 2.6 and Theorem 2.7 are reduced to the Theorems 1.5.1 and 1.5.2 in [1], respectively.
3 Some examples
In this section, two applications of impulsive differential and impulsive integral inequalities with integral jump conditions are given.
Corollary 3.1. Assume that u ∈ PC1[J, R] satisfies
where M > 0, a ∈ C[R+, R+], 0 < t1 < t2 < ⋯ < t n < T. c k ≥ 0, 0 ≤ σ k ≤ τ k ≤ t k - tk-1, k = 1, 2,..., n.
Suppose in addition that
(D1)
(D2)
(D3) .
Then u(t) ≤ 0 for t ∈ [0, T].
Proof. By Corollary 2.4 for t ∈ [0, T] we can write that
where
and
Condition (D2) implies that for k = 1, 2,..., n. Then, it is sufficient to show that u(0) ≤ 0. For t = T we have
By the conditions (D1) and (D3), we see that
which implies that u(0) ≤ 0.
Corollary 3.2. Let v ∈ PC1[R+, R] such that
where f ∈ C(R × R, R), I k ∈C(R, R), 0 ≤ t0 < t1 < t2 < ⋯, limk→∞t k = ∞, 0 ≤ σ k ≤ τ k ≤ t k - tk-1, k = 1, 2,.... Assume that
(D4) there exists a constant N > 0, such that
(D5) there exist constants L k ≥ 0 such that
Then the following inequality is valid
Proof. The solution v(t) of problem (3.2) satisfies the equation
From the hypothesis (D4), (D5) it follows for t ≥ t0 that
Hence Theorem 2.6 yields the estimate
Therefore, the inequality (3.3) holds for t ≥ t0 and the proof is complete.
References
Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.
Bainov DD, Simeonov PS: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific & Technical, Harlow; 1993.
Bainov DD, Simeonov PS: Impulsive Differential Equations: Asymptotic Properties of the Solutions. World Scientific, Singapore; 1995.
Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore; 1995.
Bainov DD, Hristova SG: The method of quasilinearization for the periodic boundary value problem for systems of impulsive differential equations. Appl Math Comput 2001, 117: 73–85. 10.1016/S0096-3003(99)00156-3
Bonotto EM, Gimenes LP, Federson M: Oscillation for a second-order neutral equation with impulses. Appl Math Comput 2009, 215: 1–15. 10.1016/j.amc.2009.04.039
Cui BT, Han M, Yang H: Some sufficient conditions for oscillation of impulsive delay hyperbolic systems with Robin boundary conditions. J Comput Appl Math 2005, 180: 365–375. 10.1016/j.cam.2004.11.006
Ding W, Xing Y, Han M: Anti-periodic boundary value problems for first order impulsive functional differential equations. Appl Math Comput 2007, 186: 45–53. 10.1016/j.amc.2006.07.087
Franco D, Nieto JJ: First-order impulsive ordinary differential equations with anti-periodic and nonlinear boundary conditions. Nonlinear Anal 2000, 42: 163–173. 10.1016/S0362-546X(98)00337-X
Franco D, Nieto JJ: Maximum principles for periodic impulsive first order problems. J Comput Appl Math 1998, 88: 149–159. 10.1016/S0377-0427(97)00212-4
Fu X, Zhang L: Forced oscillation for impulsive hyperbolic boundary value problems with delay. Appl Math Comput 2004, 158: 761–780. 10.1016/j.amc.2003.08.148
Gimenes LP, Federson M: Oscillation by impulses for a second-order delay differential equation. Comput Math Appl 2006, 52: 819–828. 10.1016/j.camwa.2006.06.001
He Z, Ge W: Oscillations of second-order nonlinear impulsive ordinary differential equations. J Comput Appl Math 2003, 158: 397–406. 10.1016/S0377-0427(03)00474-6
He Z, Ge W: Periodic boundary value problem for first order impulsive delay differential equations. Appl Math Comput 1999, 104: 51–63. 10.1016/S0096-3003(98)10059-0
He Z, He X: Monotone iterative technique for impulsive integro-differential equations with periodic boundary conditions. Comput Math Appl 2004, 48: 73–84. 10.1016/j.camwa.2004.01.005
He Z, He X: Periodic boundary value problems for first order impulsive integro-differential equations of mixed type. J Math Anal Appl 2004, 296: 8–20. 10.1016/j.jmaa.2003.12.047
He Z, Yu J: Periodic boundary value problem for first-order impulsive ordinary differential equations. J Math Anal Appl 2002, 272: 67–78. 10.1016/S0022-247X(02)00133-6
He Z, Yu J: Periodic boundary value problem for first-order impulsive functional differential equations. J Comput Appl Math 2002, 138: 205–217. 10.1016/S0377-0427(01)00381-8
Hristova SG, Kulev GK: Quasilinearization of a boundary value problem for impulsive differential equations. J Comput Appl Math 2001, 132: 399–407. 10.1016/S0377-0427(00)00442-8
Huang M: Oscillation criteria for second order nonlinear dynamic equations with impulses. Comput Math Appl 2010, 59: 31–41. 10.1016/j.camwa.2009.03.039
Jiao J, Chen L, Li L: Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations. J Math Anal Appl 2008, 337: 458–463. 10.1016/j.jmaa.2007.04.021
Li J: Periodic boundary value problems for second-order impulsive integro-differential equations. Appl Math Comput 2008, 198: 317–325. 10.1016/j.amc.2007.08.079
Li Q, Liang H, Zhang Z, Yu Y: Oscillation of second order self-conjugate differential equation with impulses. J Comput Appl Math 2006, 197: 78–88. 10.1016/j.cam.2005.10.035
Li J, Shen J: Periodic boundary value problems for delay differential equations with impulses. J Comput Appl Math 2006, 193: 563–573. 10.1016/j.cam.2005.05.037
Li J, Shen J: Periodic boundary value problems for impulsive integro-differential equations of mixed type. Appl Math Comput 2006, 183: 890–902. 10.1016/j.amc.2006.06.037
Li WN: On the forced oscillation of solutions for systems of impulsive parabolic differential equations with several delays. J Comput Appl Math 2005, 181: 46–57. 10.1016/j.cam.2004.11.016
Li WN, Han M: Oscillation of solutions for certain impulsive vector parabolic differential equations with delays. J Math Anal Appl 2007, 326: 363–371. 10.1016/j.jmaa.2006.03.005
Liu H, Li Q: Asymptotic behavior of second-order impulsive differential equations, Electron. J Diff Equ 2011, 33: 1–7.
Luo J: Oscillation of hyperbolic partial differential equations with impulses. Appl Math Comput 2002, 133: 309–318. 10.1016/S0096-3003(01)00217-X
Nieto JJ, Rodriguez-Lopez R: New comparison results for impulsive integro-differential equations and applications. J Math Anal Appl 2007, 328: 1343–1368. 10.1016/j.jmaa.2006.06.029
Pandian S, Purushothaman G: Asymptotic behavior of solutions of higher order nonlinear delay impulsive differential equations with damping. Int J Pure Appl Math 2011, 72: 401–414.
Peng M: Oscillation caused by impulses. J Math Anal Appl 2001, 255: 163–176. 10.1006/jmaa.2000.7218
Peng M: Oscillation criteria for second-order impulsive delay difference equations. Appl Math Comput 2003, 146: 227–235. 10.1016/S0096-3003(02)00539-8
Peng M: Oscillation theorems of second-order nonlinear neutral delay difference equations with impulses. Comput Math Appl 2002, 44: 741–748. 10.1016/S0898-1221(02)00187-6
Shen J: New maximum principles for first-order impulsive boundary value problems. Appl Math Lett 2003, 16: 105–112. 10.1016/S0893-9659(02)00151-9
Stamova IM: Lyapunov method for boundedness of solutions of nonlinear impulsive functional differential equations. Appl Math Comput 2006, 177: 714–719. 10.1016/j.amc.2005.09.107
Wang P, Wu Y: Oscillation criteria for impulsive parabolic differential equations of neutral type. Int J Pure Appl Math 2004, 14: 505–514.
Zhang C, Feng W, Yang F: Oscillations of higher order nonlinear functional differential equations with impulses. Appl Math Comput 2007, 190: 370–381. 10.1016/j.amc.2007.01.029
Zhang C, Feng W, Yang J, Huang M: Oscillations of second order impulses nonlinear FDE with forcing term. Appl Math Comput 2008, 198: 271–279. 10.1016/j.amc.2007.08.033
Ale SO, Oyelami BO, Sesay MS: Cone-valued impulsive differential and integrodifferential inequalities. Electron J Diff Equ 2005, 66: 1–14. 2005
Deng S, Prather C: Generalization of an impulsive nonlinear singular Gronwall-Bihari inequality with delay. J Inequal Pure Appl Math 2008., 9(2): Art. 34
Hristova SG: Nonlinear delay integral inequalities for piecewise continuous functions and applications. J Inequal Pure Appl Math 2004., 5(4): Art. 88
Li J: On some new impulsive integral inequalities. J Inequal Appl 2008., 312395(8):
Peng Y, Kang Y, Yuan M, Huang R, Yang L: Gronwall-type integral inequalities with impulses on time scales. Adv Diff Equ 2011., 2011(26):
Tatar NE: An impulsive nonlinear singular version of the Gronwall-Bihari inequality. J Inequal Appl 2006., 84561(12):
Wang H, Ding C: A new nonlinear impulsive delay differential inequality and its applications. J Inequal Appl 2011., 2011(11):
Wang WS, Li Z: A new class of impulsive integral inequalities and its application. IEEE 2011, 1897–1899. 2011 International Conference on Multimedia Technology
Yan J: Stability theorems of perturbed linear systems with impulse effect. Portuga Math 1996, 53: 43–51.
Acknowledgements
The authors thank the referees for several useful remarks and interesting comments. This research was supported by the Centre of Excellence in Mathematics, Thailand.
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Thiramanus, P., Tariboon, J. Impulsive differential and impulsive integral inequalities with integral jump conditions. J Inequal Appl 2012, 25 (2012). https://doi.org/10.1186/1029-242X-2012-25
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DOI: https://doi.org/10.1186/1029-242X-2012-25