Multiplicity results for superlinear boundary value problems with impulsive effects
We study the existence and multiplicity of solutions for a nonlinear boundary value problem subject to perturbations of impulsive terms. Under suitable assumptions on the potential of the nonlinearity, the existence of one or two solutions is established. Our approach is based on a local minimum the...
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| Published in | Mathematical methods in the applied sciences Vol. 39; no. 5; pp. 1060 - 1068 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Freiburg
Blackwell Publishing Ltd
01.04.2016
Wiley Subscription Services, Inc |
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| Online Access | Get full text |
| ISSN | 0170-4214 1099-1476 |
| DOI | 10.1002/mma.3545 |
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| Abstract | We study the existence and multiplicity of solutions for a nonlinear boundary value problem subject to perturbations of impulsive terms. Under suitable assumptions on the potential of the nonlinearity, the existence of one or two solutions is established. Our approach is based on a local minimum theorem and a two non‐zero critical points for differentiable functionals. Copyright © 2015 John Wiley & Sons, Ltd. |
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| AbstractList | We study the existence and multiplicity of solutions for a nonlinear boundary value problem subject to perturbations of impulsive terms. Under suitable assumptions on the potential of the nonlinearity, the existence of one or two solutions is established. Our approach is based on a local minimum theorem and a two non‐zero critical points for differentiable functionals. Copyright © 2015 John Wiley & Sons, Ltd. We study the existence and multiplicity of solutions for a nonlinear boundary value problem subject to perturbations of impulsive terms. Under suitable assumptions on the potential of the nonlinearity, the existence of one or two solutions is established. Our approach is based on a local minimum theorem and a two non-zero critical points for differentiable functionals. |
| Author | D'Aguì, Giuseppina Di Bella, Beatrice Tersian, Stepan |
| Author_xml | – sequence: 1 givenname: Giuseppina surname: D'Aguì fullname: D'Aguì, Giuseppina email: Correspondence to: Giuseppina D'Aguì, Department of Civil, Computer, Construction, Environmental Engineering and Applied Mathematics, University of Messina, 98166 Messina, Italy., dagui@unime.it organization: Department of Civil, Computer, Construction, Environmental Engineering and Applied Mathematics, University of Messina, 98166 Messina, Italy – sequence: 2 givenname: Beatrice surname: Di Bella fullname: Di Bella, Beatrice organization: Department of Mathematics and Computer Science, University of Messina, 98166 Messina, Italy – sequence: 3 givenname: Stepan surname: Tersian fullname: Tersian, Stepan organization: Department of Mathematics, University of Ruse, 7017 Ruse, Bulgaria |
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| Cites_doi | 10.1016/j.jmaa.2006.09.021 10.2478/s13540-014-0196-y 10.1016/j.nonrwa.2007.10.022 10.1016/j.na.2011.12.003 10.1155/9789775945501 10.1515/anona-2012-0003 10.1016/j.jmaa.2004.08.034 10.1002/mma.2777 10.1142/0906 |
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| References | Bonanno G, Rodriguez-Lopez R, Tersian S. Existence of solutions to boundary value problem for impulsive fractional differential equations. Fractional Calculus and Applied Analysis 2014; 17/3:717-744. Benchohra M, Henderson J, Ntouyas N. Impulsive Differential Equations and Inclusions. Hindawi Publ. Corp.: New York, NY, USA, 2006. Cabada A, Tersian S. Existence and multiplicity of solutions to boundary value problems for fourth-order impulsive differential equations. Boundary Value Problems 2014; 105:1-12. Lakshmikantham V, Bainov DD, Simeonov PS. Theory of Impulsive Differential Equations. World Scientific: Singapore, 1989. Drábek P, Langerová M. Quasilinear boundary value problem with impulses: variational approach to resonance problem. Bound. Value Probl 2014; 64:1-14. Mawhin J. Topological Degree and Boundary Value Problems for Nonlinear Differential Equations, Topological methods for ordinary differential equations, Lecture Notes in Math., vol. 1537. Springer: Berlin, 1993, 74-142. Qian D, Li X. Periodic solutions for ordinary differential equations with sublinear impulsive effects. Journal on Mathematical Analysis 2005; 303:288-303. Chen H, He Z. Variational approach to some damped Dirichlet problems with impulses. Math. Methods Appl. Sci 2013; 36/18:2564-2575. Bonanno G, Di Bella B, Henderson J. Existence of solutions to second-order boundary-value problems with small pertubations of impulses. Electronic Journal of Differential Equations 2013; 126:1-14. Chen J, Tisdell CC, Yuan R. On the solvability of periodic boundary value problems with impulse. Journal on Mathematical Analysis 2007; 331:902-912. Nieto JJ, O'Regan D. Variational approach to impulsive differential equations. Nonlinear Analysis, Real World Applications 2009; 70:680-690. Bonanno G. Relations between the mountain pass theorem and local minima. Advances in Nonlinear Analysis 2012; 1:205-220. Bonanno G. A critical point theorem via the Ekeland variational principle. Nonlinear Analysis 2012; 75:2992-3007. Bonanno G, Di Bella B, Henderson J. Infinitely many solutions for a boundary value problem with impulsive effects. Boundary Value Problems 2013; 278. 2014; 17/3 2014; 105 2012; 1 2009; 70 2007; 331 2013; 126 2005; 303 2006 2013; 278 1993 2013; 36/18 2012; 75 1989 2014; 64 e_1_2_5_15_1 e_1_2_5_14_1 e_1_2_5_16_1 e_1_2_5_11_1 e_1_2_5_6_1 e_1_2_5_13_1 e_1_2_5_4_1 e_1_2_5_3_1 e_1_2_5_2_1 Mawhin J (e_1_2_5_5_1) 1993 Bonanno G (e_1_2_5_7_1) 2013; 126 Bonanno G (e_1_2_5_8_1) 2013; 278 Drábek P (e_1_2_5_12_1) 2014; 64 Bonanno G (e_1_2_5_9_1) 2014; 17 Cabada A (e_1_2_5_10_1) 2014; 105 |
| References_xml | – reference: Bonanno G, Di Bella B, Henderson J. Infinitely many solutions for a boundary value problem with impulsive effects. Boundary Value Problems 2013; 278. – reference: Bonanno G. A critical point theorem via the Ekeland variational principle. Nonlinear Analysis 2012; 75:2992-3007. – reference: Bonanno G, Di Bella B, Henderson J. Existence of solutions to second-order boundary-value problems with small pertubations of impulses. Electronic Journal of Differential Equations 2013; 126:1-14. – reference: Bonanno G, Rodriguez-Lopez R, Tersian S. Existence of solutions to boundary value problem for impulsive fractional differential equations. Fractional Calculus and Applied Analysis 2014; 17/3:717-744. – reference: Bonanno G. Relations between the mountain pass theorem and local minima. Advances in Nonlinear Analysis 2012; 1:205-220. – reference: Cabada A, Tersian S. Existence and multiplicity of solutions to boundary value problems for fourth-order impulsive differential equations. Boundary Value Problems 2014; 105:1-12. – reference: Qian D, Li X. Periodic solutions for ordinary differential equations with sublinear impulsive effects. Journal on Mathematical Analysis 2005; 303:288-303. – reference: Nieto JJ, O'Regan D. Variational approach to impulsive differential equations. Nonlinear Analysis, Real World Applications 2009; 70:680-690. – reference: Mawhin J. Topological Degree and Boundary Value Problems for Nonlinear Differential Equations, Topological methods for ordinary differential equations, Lecture Notes in Math., vol. 1537. Springer: Berlin, 1993, 74-142. – reference: Chen H, He Z. Variational approach to some damped Dirichlet problems with impulses. Math. Methods Appl. Sci 2013; 36/18:2564-2575. – reference: Lakshmikantham V, Bainov DD, Simeonov PS. Theory of Impulsive Differential Equations. World Scientific: Singapore, 1989. – reference: Drábek P, Langerová M. Quasilinear boundary value problem with impulses: variational approach to resonance problem. Bound. Value Probl 2014; 64:1-14. – reference: Benchohra M, Henderson J, Ntouyas N. Impulsive Differential Equations and Inclusions. Hindawi Publ. Corp.: New York, NY, USA, 2006. – reference: Chen J, Tisdell CC, Yuan R. On the solvability of periodic boundary value problems with impulse. Journal on Mathematical Analysis 2007; 331:902-912. – start-page: 74 year: 1993 end-page: 142 – volume: 303 start-page: 288 year: 2005 end-page: 303 article-title: Periodic solutions for ordinary differential equations with sublinear impulsive effects publication-title: Journal on Mathematical Analysis – volume: 70 start-page: 680 year: 2009 end-page: 690 article-title: Variational approach to impulsive differential equations publication-title: Nonlinear Analysis, Real World Applications – volume: 36/18 start-page: 2564 year: 2013 end-page: 2575 article-title: Variational approach to some damped Dirichlet problems with impulses publication-title: Math. Methods Appl. Sci – volume: 331 start-page: 902 year: 2007 end-page: 912 article-title: On the solvability of periodic boundary value problems with impulse publication-title: Journal on Mathematical Analysis – volume: 105 start-page: 1 year: 2014 end-page: 12 article-title: Existence and multiplicity of solutions to boundary value problems for fourth‐order impulsive differential equations publication-title: Boundary Value Problems – volume: 126 start-page: 1 year: 2013 end-page: 14 article-title: Existence of solutions to second‐order boundary‐value problems with small pertubations of impulses publication-title: Electronic Journal of Differential Equations – volume: 1 start-page: 205 year: 2012 end-page: 220 article-title: Relations between the mountain pass theorem and local minima publication-title: Advances in Nonlinear Analysis – year: 2006 – year: 1989 – volume: 278 year: 2013 article-title: Infinitely many solutions for a boundary value problem with impulsive effects publication-title: Boundary Value Problems – volume: 64 start-page: 1 year: 2014 end-page: 14 article-title: Quasilinear boundary value problem with impulses: variational approach to resonance problem publication-title: Bound. Value Probl – volume: 75 start-page: 2992 year: 2012 end-page: 3007 article-title: A critical point theorem via the Ekeland variational principle publication-title: Nonlinear Analysis – volume: 17/3 start-page: 717 year: 2014 end-page: 744 article-title: Existence of solutions to boundary value problem for impulsive fractional differential equations publication-title: Fractional Calculus and Applied Analysis – ident: e_1_2_5_4_1 doi: 10.1016/j.jmaa.2006.09.021 – volume: 17 start-page: 717 year: 2014 ident: e_1_2_5_9_1 article-title: Existence of solutions to boundary value problem for impulsive fractional differential equations publication-title: Fractional Calculus and Applied Analysis doi: 10.2478/s13540-014-0196-y – ident: e_1_2_5_13_1 doi: 10.1016/j.nonrwa.2007.10.022 – ident: e_1_2_5_16_1 doi: 10.1016/j.na.2011.12.003 – ident: e_1_2_5_2_1 doi: 10.1155/9789775945501 – ident: e_1_2_5_14_1 doi: 10.1515/anona-2012-0003 – ident: e_1_2_5_15_1 – start-page: 74 volume-title: Topological Degree and Boundary Value Problems for Nonlinear Differential Equations year: 1993 ident: e_1_2_5_5_1 – ident: e_1_2_5_6_1 doi: 10.1016/j.jmaa.2004.08.034 – volume: 278 year: 2013 ident: e_1_2_5_8_1 article-title: Infinitely many solutions for a boundary value problem with impulsive effects publication-title: Boundary Value Problems – volume: 64 start-page: 1 year: 2014 ident: e_1_2_5_12_1 article-title: Quasilinear boundary value problem with impulses: variational approach to resonance problem publication-title: Bound. Value Probl – volume: 105 start-page: 1 year: 2014 ident: e_1_2_5_10_1 article-title: Existence and multiplicity of solutions to boundary value problems for fourth‐order impulsive differential equations publication-title: Boundary Value Problems – ident: e_1_2_5_11_1 doi: 10.1002/mma.2777 – ident: e_1_2_5_3_1 doi: 10.1142/0906 – volume: 126 start-page: 1 year: 2013 ident: e_1_2_5_7_1 article-title: Existence of solutions to second‐order boundary‐value problems with small pertubations of impulses publication-title: Electronic Journal of Differential Equations |
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| SubjectTerms | Boundary value problems Critical point critical point theory Dirichlet boundary conditions impulsive differential equations Mathematical analysis Nonlinearity Perturbation methods Theorems |
| Title | Multiplicity results for superlinear boundary value problems with impulsive effects |
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