# American Institute of Mathematical Sciences

2015, 2015(special): 851-860. doi: 10.3934/proc.2015.0851

## On higher order nonlinear impulsive boundary value problems

 1 Departamento de Matemática. Universidade de Évora, Centro de Investigação em Matemática e Aplicaçoes da U.E. (CIMA-UE), Rua Romão Ramalho, 59. 7000-671 Évora 2 Centro de Investigação em Matematica e Aplicações (CIMA-UE), Instituto de Investigação e Formacão Avançada, Universidade de Évora, Rua Romão Ramalho, 59, 7000-671 Évora, Portugal

Received  July 2014 Revised  November 2014 Published  November 2015

This work studies some two point impulsive boundary value problems composed by a fully differential equation, which higher order contains an increasing homeomorphism, by two point boundary conditions and impulsive effects. We point out that the impulsive conditions are given via multivariate generalized functions, including impulses on the referred homeomorphism. The method used apply lower and upper solutions technique together with fixed point theory. Therefore we have not only the existence of solutions but also the localization and qualitative data on their behavior. Moreover a Nagumo condition will play a key role in the arguments.
Citation: 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
##### References:
 [1] R. Agarwal, D. O'Regan, Multiple nonnegative solutions for second order impulsive diferential equations, Appl. Math. Comput. 155 (2000) 51-59. [2] P. Chen, X. Tang, Existence and multiplicity of solutions for second-order impulsive differential equations with Dirichlet problems, Appl. Math. Comput. 218 (2012) 1775-11789. [3] J. Fialho and F. Minhós, High Order Boundary Value Problems: Existence, Localization and Multiplicity Results, Mathematics Research Developments, Nova Science Publishers, Inc. New York, 2014 ISBN: 978-1-63117-707-1. [4] J. R. Graef, L. Kong and F. Minhós, Higher order boundary value problems with phi-Laplacian and functional boundary conditions, Comp. Math. Appl., 61 (2011) 236-249. [5] M. R.Grossinho, F. Minhós and A.I. Santos, A note on a class of problems for a higher-order fully nonlinear equation under one-sided Nagumo-type condition, Nonlinear Anal., 70 (2009) 4027-4038. [6] X. Hao, L. Liu and Y. WU, Positive solutions for nth-order singular nonlocal boundary value problems, Boubd. Value Probl. (2007) 10, Article ID 74517. [7] V. Lakshmikantham, D. Baĭnov and P. Simeonov, Theory of impulsive differential equations. Series in Modern Applied Mathematics, 6. World Scientific Publishing Co., Inc., 1989. [8] X. Liu and D. Guo, Method of upper and lower solutions for second-order impulsive integro-differential equations in a Banach space, Comput. Math. Appl., 38 (1999), 213-223. [9] Y. Liu and D. O'Regan, Multiplicity results using bifurcation techniques for a class of boundary value problems of impulsive differential equations, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 1769-1775. [10] B. Liu and J. Yu, Existence of solution for m-point boundary value problems of second-order differential systems with impulses, Appl. Math. Comput., 125, (2002), 155-175 [11] R. Ma, B. Yang and Z. Wang, Positive periodic solutions of first-order delay differential equations with impulses, Appl. Math. Comput. 219 (2013) 6074-6083. [12] J. Nieto and R. López, Boundary value problems for a class of impulsive functional equations, Comput. Math. Appl. 55 (2008) 2715-2731 [13] J. Nieto and D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Anal. RWA, (2009), 680-690. [14] A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. [15] Y. Tian and W. Ge, Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations, Nonlinear Analysis: Theory, Methods & Applications, 72, (2010), 277-287. [16] J. Xiao, J. Nieto and Z. Luo, Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods, Communications in Nonlinear Science and Numerical Simulation, 17, (2012), 426-432. [17] X. Zhang, M. Feng and W. Ge, Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces, J. Comput. Appl. Math, 233, (2010), 1915-1926 .

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##### References:
 [1] R. Agarwal, D. O'Regan, Multiple nonnegative solutions for second order impulsive diferential equations, Appl. Math. Comput. 155 (2000) 51-59. [2] P. Chen, X. Tang, Existence and multiplicity of solutions for second-order impulsive differential equations with Dirichlet problems, Appl. Math. Comput. 218 (2012) 1775-11789. [3] J. Fialho and F. Minhós, High Order Boundary Value Problems: Existence, Localization and Multiplicity Results, Mathematics Research Developments, Nova Science Publishers, Inc. New York, 2014 ISBN: 978-1-63117-707-1. [4] J. R. Graef, L. Kong and F. Minhós, Higher order boundary value problems with phi-Laplacian and functional boundary conditions, Comp. Math. Appl., 61 (2011) 236-249. [5] M. R.Grossinho, F. Minhós and A.I. Santos, A note on a class of problems for a higher-order fully nonlinear equation under one-sided Nagumo-type condition, Nonlinear Anal., 70 (2009) 4027-4038. [6] X. Hao, L. Liu and Y. WU, Positive solutions for nth-order singular nonlocal boundary value problems, Boubd. Value Probl. (2007) 10, Article ID 74517. [7] V. Lakshmikantham, D. Baĭnov and P. Simeonov, Theory of impulsive differential equations. Series in Modern Applied Mathematics, 6. World Scientific Publishing Co., Inc., 1989. [8] X. Liu and D. Guo, Method of upper and lower solutions for second-order impulsive integro-differential equations in a Banach space, Comput. Math. Appl., 38 (1999), 213-223. [9] Y. Liu and D. O'Regan, Multiplicity results using bifurcation techniques for a class of boundary value problems of impulsive differential equations, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 1769-1775. [10] B. Liu and J. Yu, Existence of solution for m-point boundary value problems of second-order differential systems with impulses, Appl. Math. Comput., 125, (2002), 155-175 [11] R. Ma, B. Yang and Z. Wang, Positive periodic solutions of first-order delay differential equations with impulses, Appl. Math. Comput. 219 (2013) 6074-6083. [12] J. Nieto and R. López, Boundary value problems for a class of impulsive functional equations, Comput. Math. Appl. 55 (2008) 2715-2731 [13] J. Nieto and D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Anal. RWA, (2009), 680-690. [14] A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. [15] Y. Tian and W. Ge, Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations, Nonlinear Analysis: Theory, Methods & Applications, 72, (2010), 277-287. [16] J. Xiao, J. Nieto and Z. Luo, Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods, Communications in Nonlinear Science and Numerical Simulation, 17, (2012), 426-432. [17] X. Zhang, M. Feng and W. Ge, Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces, J. Comput. Appl. Math, 233, (2010), 1915-1926 .
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