De la Vallée Poussin inequality for impulsive differential equations
-
Sibel Doğru Akgöl
and
Abdullah Özbekler
Abstract
The de la Vallée Poussin inequality is a handy tool for the investigation of disconjugacy, and hence, for the oscillation/nonoscillation of differential equations. The results in this paper are extensions of former those of Hartman and Wintner [Quart. Appl. Math. 13 (1955), 330–332] to the impulsive differential equations. Although the inequality first appeared in such an early date for ordinary differential equations, its improved version for differential equations under impulse effect never has been occurred in the literature.
In the present study, first, we state and prove a de la Vallée Poussin inequality for impulsive differential equations, then we give some corollaries on disconjugacy. We also mention some open problems and finally, present some examples that support our findings.
(Communicated by Michal Fečkan)
Acknowledgement
The authors would like to express their sincere gratitude to the Anonymous Referees for their valuable comments and helpful suggestions.
References
[1] AGARWAL, R. P. — JLELI, M. — SAMET, B.: On de la Vallée Poussin-type inequalities in higher dimension and applications, Appl. Math. Lett. 86 (2018), 264–269.10.1016/j.aml.2018.07.015Search in Google Scholar
[2] AGARWAL, R. P. — ÖZBEKLER, A.: Disconjugacy via Lyapunov and Vallée-Poussin-type inequalities for forced differential equations, Appl. Math. Comput. 265 (2015), 456–468.10.1016/j.amc.2015.05.038Search in Google Scholar
[3] AKHMETOV, M. — SEJILOVA, R.: The control of the boundary value problem for linear impulsive integro-differential equations, J. Math. Anal. Appl. 236 (1999), 312–326.10.1006/jmaa.1999.6428Search in Google Scholar
[4] BÁŇA, L. — DOŠLÝ, O.: De la Vallée Poussin-type inequality and eigenvalue problem for generalized half-linear differential equation, Arch. Math. (Brno) 50(4) (2014), 193–203.10.5817/AM2014-4-193Search in Google Scholar
[5] DOŠLÝ, O. — LOMTATIDZE A.: Disconjugacy and disfocality criteria for singular half-linear second order differential equations, Ann. Polon. Math. 72 (1999), 273–284.10.4064/ap-72-3-273-284Search in Google Scholar
[6] FERREIRA, R. A. C.: A de la Vallée Poussin Type Inequality on Time Scales, Results Math. 73(3) (2018), Art. 88.10.1007/s00025-018-0851-4Search in Google Scholar
[7] FERREIRA, R. A. C.: Fractional de la Vallée Poussin Inequalities, 22(3) (2019), 917–930.10.7153/mia-2019-22-62Search in Google Scholar
[8] GUSEINOV, G. SH. — ZAFER, A.: Stability criterion for second order linear impulsive differential equations with periodic coefficients, Math. Nachr. 281(9) (2008), 1273–1282.10.1002/mana.200510677Search in Google Scholar
[9] HARTMAN, P. — WINTNER A.: On an oscillation criterion of de la Vallée Poussin, Quart. Appl. Math. 13 (1955), 330–332.10.1090/qam/73773Search in Google Scholar
[10] POUSSIN, D. V.: Sur l’équation différentielle linéqire du second order. Détermination d’une intégrale par deux valuers assignés. Extension aux équasions d’ordre n, J. Math. Pures Appl. 8 (1929), 125–144.Search in Google Scholar
© 2021 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular Papers
- Quasi-decompositions and quasidirect products of Hilbert algebras
- Residuation in finite posets
- On a problem in the theory of polynomials
- Fekete-Szegö problem for starlike functions connected with k-Fibonacci numbers
- Mapping properties of the Bergman projections on elementary Reinhardt domains
- Lemniscate-like constants and infinite series
- On the Oscillation of second order nonlinear neutral delay differential equations
- Oscillation theorems for certain second-order nonlinear retarded difference equations
- De la Vallée Poussin inequality for impulsive differential equations
- Hs-Boundedness of a class of Fourier Integral Operators
- Dynamical behavior of a P-dimensional system of nonlinear difference equations
- Some inequalities for exponentially convex functions on time scales
- Impact of different types of non linearity on the oscillatory behavior of higher order neutral difference equations
- The sine extended odd Fréchet-G family of distribution with applications to complete and censored data
- A new two-parameter lifetime distribution with flexible hazard rate function: Properties, applications and different method of estimations
- Simulations of nonlinear parabolic PDEs with forcing function without linearization
- An existence level for the residual sum of squares of the power-law regression with an unknown location parameter
- A relationship between the category of chain MV-algebras and a subcategory of abelian groups
Articles in the same Issue
- Regular Papers
- Quasi-decompositions and quasidirect products of Hilbert algebras
- Residuation in finite posets
- On a problem in the theory of polynomials
- Fekete-Szegö problem for starlike functions connected with k-Fibonacci numbers
- Mapping properties of the Bergman projections on elementary Reinhardt domains
- Lemniscate-like constants and infinite series
- On the Oscillation of second order nonlinear neutral delay differential equations
- Oscillation theorems for certain second-order nonlinear retarded difference equations
- De la Vallée Poussin inequality for impulsive differential equations
- Hs-Boundedness of a class of Fourier Integral Operators
- Dynamical behavior of a P-dimensional system of nonlinear difference equations
- Some inequalities for exponentially convex functions on time scales
- Impact of different types of non linearity on the oscillatory behavior of higher order neutral difference equations
- The sine extended odd Fréchet-G family of distribution with applications to complete and censored data
- A new two-parameter lifetime distribution with flexible hazard rate function: Properties, applications and different method of estimations
- Simulations of nonlinear parabolic PDEs with forcing function without linearization
- An existence level for the residual sum of squares of the power-law regression with an unknown location parameter
- A relationship between the category of chain MV-algebras and a subcategory of abelian groups
