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Wellposedness for a class of nonlinear degenerate parabolic equations
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, (CIMAUE), Rua Romão Ramalho, 59, 7000671 Évora 
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( n1\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,...,n1, \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,...,n1, \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, AscoliArzela Theorem and fixed point theory.
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. Google Scholar 
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A. Cabada and J. Tomeček, Extremal solutions for nonlinear functional $\phi $Laplacian impulsive equations,, Nonlinear Anal., 67 (2007), 827. Google Scholar 
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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 
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show all references
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. Google Scholar 
[2] 
A. Cabada and J. Tomeček, Extremal solutions for nonlinear functional $\phi $Laplacian impulsive equations,, Nonlinear Anal., 67 (2007), 827. Google Scholar 
[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. Google Scholar 
[4] 
J. Fialho and F. Minhós, Fourth order impulsive periodic boundary value problems,, Differential Equations and Dynamical Systems, (2013). Google Scholar 
[5] 
Z. He and J. Yu, Periodic boundary value problem for firstorder impulsive ordinary differential equations., J. Math. Anal. Appl., 272 (2002), 67. Google Scholar 
[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. Google Scholar 
[7] 
Z. Luo and Z. Jing, Periodic boundary value problem for firstorder impulsive functional differential equations., Comput. Math. Appl., 55 (2008), 2094. Google Scholar 
[8] 
I. Rachůnková, M. Tvrdý, Existence results for impulsive secondorder periodic problems., Nonlinear Anal., 59 (2004), 133. Google Scholar 
[9] 
X. Wang and J. Zhang, Impulsive antiperiodic boundary value problem of firstorder integrodifferential equations., J. Comput. Appl. Math., 234 (2010), 3261. Google Scholar 
[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). Google Scholar 
[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). Google Scholar 
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