High order periodic impulsive problems

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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

Abstract
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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.

Keywords: periodic boundary value problems, differential equations with impulses, Nagumo condition, Higher order problems, lower and upper solutions..

Mathematics Subject Classification: Primary: 34B15, 34A37; Secondary: 34B10, 34K4.

Citation: João Fialho, Feliz Minhós. High order periodic impulsive problems. Conference Publications, 2015, 2015 (special) : 446-454. doi: 10.3934/proc.2015.0446

References:

[1]	Z. Benbouziane, A. Boucherif and S. Bouguima, Existence result for impulsive third order periodic boundary value problems,, Appl. Math. Comput., 206 (2008), 728.
[2]	A. Cabada and J. Tomeček, Extremal solutions for nonlinear functional $\phi $-Laplacian impulsive equations,, Nonlinear Anal., 67 (2007), 827.
[3]	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.
[4]	J. Fialho and F. Minhós, Fourth order impulsive periodic boundary value problems,, Differential Equations and Dynamical Systems, (2013).
[5]	Z. He and J. Yu, Periodic boundary value problem for first-order impulsive ordinary differential equations., J. Math. Anal. Appl., 272 (2002), 67.
[6]	R. Liang and J. Shen, Periodic boundary value problem for the first order impulsive functional differential equations., J. Comput. Appl. Math., 202 (2007), 498.
[7]	Z. Luo and Z. Jing, Periodic boundary value problem for first-order impulsive functional differential equations., Comput. Math. Appl., 55 (2008), 2094.
[8]	I. Rachůnková, M. Tvrdý, Existence results for impulsive second-order periodic problems., Nonlinear Anal., 59 (2004), 133.
[9]	X. Wang and J. Zhang, Impulsive anti-periodic boundary value problem of first-order integro-differential equations., J. Comput. Appl. Math., 234 (2010), 3261.
[10]	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).
[11]	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).

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References:

[1]	Z. Benbouziane, A. Boucherif and S. Bouguima, Existence result for impulsive third order periodic boundary value problems,, Appl. Math. Comput., 206 (2008), 728.
[2]	A. Cabada and J. Tomeček, Extremal solutions for nonlinear functional $\phi $-Laplacian impulsive equations,, Nonlinear Anal., 67 (2007), 827.
[3]	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.
[4]	J. Fialho and F. Minhós, Fourth order impulsive periodic boundary value problems,, Differential Equations and Dynamical Systems, (2013).
[5]	Z. He and J. Yu, Periodic boundary value problem for first-order impulsive ordinary differential equations., J. Math. Anal. Appl., 272 (2002), 67.
[6]	R. Liang and J. Shen, Periodic boundary value problem for the first order impulsive functional differential equations., J. Comput. Appl. Math., 202 (2007), 498.
[7]	Z. Luo and Z. Jing, Periodic boundary value problem for first-order impulsive functional differential equations., Comput. Math. Appl., 55 (2008), 2094.
[8]	I. Rachůnková, M. Tvrdý, Existence results for impulsive second-order periodic problems., Nonlinear Anal., 59 (2004), 133.
[9]	X. Wang and J. Zhang, Impulsive anti-periodic boundary value problem of first-order integro-differential equations., J. Comput. Appl. Math., 234 (2010), 3261.
[10]	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).
[11]	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).

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