    2015, 2015(special): 446-454. doi: 10.3934/proc.2015.0446

## High order periodic impulsive problems

 1 College of the Bahamas, School of Mathematics, Physics and Technologies, Department of Mathematics, Oakes Field Campus, Nassau 2 School of Sciences and Technology. Department of Mathematics, University of Évora, Research Center in Mathematics and Applications of the University of Évora, (CIMA-UE), Rua Romão Ramalho, 59, 7000-671 Évora

Received  September 2014 Revised  January 2015 Published  November 2015

The theory of impulsive problem is experiencing a rapid development in the last few years. Mainly because they have been used to describe some phenomena, arising from different disciplines like physics or biology, subject to instantaneous change at some time instants called moments. Second order periodic impulsive problems were studied to some extent, however very few papers were dedicated to the study of third and higher order impulsive problems.
The high order impulsive problem considered is composed by the fully nonlinear equation \begin{equation*} u^{\left( n\right) }\left( x\right) =f\left( x,u\left( x\right) ,u^{\prime }\left( x\right) ,...,u^{\left( n-1\right) }\left( x\right) \right) \end{equation*} for a. e. $x\in I:=\left[ 0,1\right] ~\backslash ~\left\{ x_{1},...,x_{m}\right\}$ where $f:\left[ 0,1\right] \times \mathbb{R} ^{n}\rightarrow \mathbb{R}$ is $L^{1}$-Carathéodory function, along with the periodic boundary conditions \begin{equation*} u^{\left( i\right) }\left( 0\right) =u^{\left( i\right) }\left( 1\right) ,         i=0,...,n-1, \end{equation*} and the impulsive conditions \begin{equation*} \begin{array}{c} u^{\left( i\right) }\left( x_{j}^{+}\right) =g_{j}^{i}\left( u\left( x_{j}\right) \right) ,        i=0,...,n-1, \end{array} \end{equation*} where $g_{j}^{i},$ for $j=1,...,m,$are given real valued functions satisfying some adequate conditions, and $x_{j}\in \left( 0,1\right) ,$ such that $0 = x_0 < x_1 <...< x_m < x_{m+1}=1.$
The arguments applied make use of the lower and upper solution method combined with an iterative technique, which is not necessarily monotone, together with classical results such as Lebesgue Dominated Convergence Theorem, Ascoli-Arzela Theorem and fixed point theory.
Citation: João Fialho, Feliz Minhós. High order periodic impulsive problems. Conference Publications, 2015, 2015 (special) : 446-454. doi: 10.3934/proc.2015.0446
##### References:
  Z. Benbouziane, A. Boucherif and S. Bouguima, Existence result for impulsive third order periodic boundary value problems,, Appl. Math. Comput., 206 (2008), 728. Google Scholar  A. Cabada and J. Tomeček, Extremal solutions for nonlinear functional $\phi$-Laplacian impulsive equations,, Nonlinear Anal., 67 (2007), 827. Google Scholar  W. Ding, J. Mi and M. Han, Periodic boundary value problems for the first order impulsive functional differential equations,, Appl. Math. Comput., 165 (2005), 433. Google Scholar  J. Fialho and F. Minhós, Fourth order impulsive periodic boundary value problems,, Differential Equations and Dynamical Systems, (2013). Google Scholar  Z. He and J. Yu, Periodic boundary value problem for first-order impulsive ordinary differential equations., J. Math. Anal. Appl., 272 (2002), 67. Google Scholar  R. Liang and J. Shen, Periodic boundary value problem for the first order impulsive functional differential equations., J. Comput. Appl. Math., 202 (2007), 498. Google Scholar  Z. Luo and Z. Jing, Periodic boundary value problem for first-order impulsive functional differential equations., Comput. Math. Appl., 55 (2008), 2094. Google Scholar  I. Rachůnková, M. Tvrdý, Existence results for impulsive second-order periodic problems., Nonlinear Anal., 59 (2004), 133. Google Scholar  X. Wang and J. Zhang, Impulsive anti-periodic boundary value problem of first-order integro-differential equations., J. Comput. Appl. Math., 234 (2010), 3261. Google Scholar  H. Wu and Y. Liu, Periodic boundary value problems of fourth order impulsive differential equations,, 2011 International Symposium on IT in Medicine and Education (ITME), 2 (2011). Google Scholar  G. Ye, X. Zhou and L. Huang, Periodic Boundary Value Problems for Nonlinear Impulsive Differential Equations of Mixed Type,, Intelligent System Design and Engineering Application (ISDEA), (2012). Google Scholar

show all references

##### References:
  Z. Benbouziane, A. Boucherif and S. Bouguima, Existence result for impulsive third order periodic boundary value problems,, Appl. Math. Comput., 206 (2008), 728. Google Scholar  A. Cabada and J. Tomeček, Extremal solutions for nonlinear functional $\phi$-Laplacian impulsive equations,, Nonlinear Anal., 67 (2007), 827. Google Scholar  W. Ding, J. Mi and M. Han, Periodic boundary value problems for the first order impulsive functional differential equations,, Appl. Math. Comput., 165 (2005), 433. Google Scholar  J. Fialho and F. Minhós, Fourth order impulsive periodic boundary value problems,, Differential Equations and Dynamical Systems, (2013). Google Scholar  Z. He and J. Yu, Periodic boundary value problem for first-order impulsive ordinary differential equations., J. Math. Anal. Appl., 272 (2002), 67. Google Scholar  R. Liang and J. Shen, Periodic boundary value problem for the first order impulsive functional differential equations., J. Comput. Appl. Math., 202 (2007), 498. Google Scholar  Z. Luo and Z. Jing, Periodic boundary value problem for first-order impulsive functional differential equations., Comput. Math. Appl., 55 (2008), 2094. Google Scholar  I. Rachůnková, M. Tvrdý, Existence results for impulsive second-order periodic problems., Nonlinear Anal., 59 (2004), 133. Google Scholar  X. Wang and J. Zhang, Impulsive anti-periodic boundary value problem of first-order integro-differential equations., J. Comput. Appl. Math., 234 (2010), 3261. Google Scholar  H. Wu and Y. Liu, Periodic boundary value problems of fourth order impulsive differential equations,, 2011 International Symposium on IT in Medicine and Education (ITME), 2 (2011). Google Scholar  G. Ye, X. Zhou and L. Huang, Periodic Boundary Value Problems for Nonlinear Impulsive Differential Equations of Mixed Type,, Intelligent System Design and Engineering Application (ISDEA), (2012). Google Scholar
  John Baxley, Mary E. Cunningham, M. Kathryn McKinnon. Higher order boundary value problems with multiple solutions: examples and techniques. Conference Publications, 2005, 2005 (Special) : 84-90. doi: 10.3934/proc.2005.2005.84  Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851  Alberto Cabada, João Fialho, Feliz Minhós. Non ordered lower and upper solutions to fourth order problems with functional boundary conditions. Conference Publications, 2011, 2011 (Special) : 209-218. doi: 10.3934/proc.2011.2011.209  Feliz Minhós, A. I. Santos. Higher order two-point boundary value problems with asymmetric growth. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 127-137. doi: 10.3934/dcdss.2008.1.127  Angelo Favini, Yakov Yakubov. Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 595-614. doi: 10.3934/dcdss.2011.4.595  João Fialho, Feliz Minhós. The role of lower and upper solutions in the generalization of Lidstone problems. Conference Publications, 2013, 2013 (special) : 217-226. doi: 10.3934/proc.2013.2013.217  Alberto Boscaggin, Fabio Zanolin. Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 89-110. doi: 10.3934/dcds.2013.33.89  Inara Yermachenko, Felix Sadyrbaev. Types of solutions and multiplicity results for second order nonlinear boundary value problems. Conference Publications, 2007, 2007 (Special) : 1061-1069. doi: 10.3934/proc.2007.2007.1061  Ana Maria Bertone, J.V. Goncalves. Discontinuous elliptic problems in $R^N$: Lower and upper solutions and variational principles. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 315-328. doi: 10.3934/dcds.2000.6.315  Olga A. Brezhneva, Alexey A. Tret’yakov, Jerrold E. Marsden. Higher--order implicit function theorems and degenerate nonlinear boundary-value problems. Communications on Pure & Applied Analysis, 2008, 7 (2) : 293-315. doi: 10.3934/cpaa.2008.7.293  Massimo Tarallo, Zhe Zhou. Limit periodic upper and lower solutions in a generic sense. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 293-309. doi: 10.3934/dcds.2018014  John R. Graef, Lingju Kong, Bo Yang. Positive solutions of a nonlinear higher order boundary-value problem. Conference Publications, 2009, 2009 (Special) : 276-285. doi: 10.3934/proc.2009.2009.276  G. Infante. Positive solutions of nonlocal boundary value problems with singularities. Conference Publications, 2009, 2009 (Special) : 377-384. doi: 10.3934/proc.2009.2009.377  John R. Graef, Lingju Kong, Qingkai Kong, Min Wang. Positive solutions of nonlocal fractional boundary value problems. Conference Publications, 2013, 2013 (special) : 283-290. doi: 10.3934/proc.2013.2013.283  John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83  Johnny Henderson, Rodica Luca. Existence of positive solutions for a system of nonlinear second-order integral boundary value problems. Conference Publications, 2015, 2015 (special) : 596-604. doi: 10.3934/proc.2015.0596  John R. Graef, Lingju Kong. Uniqueness and parameter dependence of positive solutions of third order boundary value problems with $p$-laplacian. Conference Publications, 2011, 2011 (Special) : 515-522. doi: 10.3934/proc.2011.2011.515  J. R. L. Webb, Gennaro Infante. Semi-positone nonlocal boundary value problems of arbitrary order. Communications on Pure & Applied Analysis, 2010, 9 (2) : 563-581. doi: 10.3934/cpaa.2010.9.563  John R. Graef, Lingju Kong, Min Wang. Existence of multiple solutions to a discrete fourth order periodic boundary value problem. Conference Publications, 2013, 2013 (special) : 291-299. doi: 10.3934/proc.2013.2013.291  Pavel Gurevich. Periodic solutions of parabolic problems with hysteresis on the boundary. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1041-1083. doi: 10.3934/dcds.2011.29.1041

Impact Factor:

## Metrics

• PDF downloads (17)
• HTML views (0)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]