Nonlocal difference equations with sign-changing coefficients
We consider second-order difference equations of the form −A∑j=1Nαj(utj)qΔ2u(n)=λf(n,u(n+1))subject to the Dirichlet boundary conditions u(0)=0=u(b+2). We demonstrate that using a nonstandard cone and associated open set can allow one to deduce the existence of at least one positive solution even in...
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| Published in | Applied mathematics letters Vol. 106; p. 106371 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Ltd
01.08.2020
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| Online Access | Get full text |
| ISSN | 0893-9659 1873-5452 |
| DOI | 10.1016/j.aml.2020.106371 |
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| Abstract | We consider second-order difference equations of the form −A∑j=1Nαj(utj)qΔ2u(n)=λf(n,u(n+1))subject to the Dirichlet boundary conditions u(0)=0=u(b+2). We demonstrate that using a nonstandard cone and associated open set can allow one to deduce the existence of at least one positive solution even in the case where the function A may change sign. Jensen’s inequality plays an important role in our analysis. |
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| AbstractList | We consider second-order difference equations of the form −A∑j=1Nαj(utj)qΔ2u(n)=λf(n,u(n+1))subject to the Dirichlet boundary conditions u(0)=0=u(b+2). We demonstrate that using a nonstandard cone and associated open set can allow one to deduce the existence of at least one positive solution even in the case where the function A may change sign. Jensen’s inequality plays an important role in our analysis. |
| ArticleNumber | 106371 |
| Author | Lyons, Benjamin Goodrich, Christopher S. |
| Author_xml | – sequence: 1 givenname: Christopher S. surname: Goodrich fullname: Goodrich, Christopher S. email: c.goodrich@unsw.edu.au organization: School of Mathematics and Statistics, UNSW Australia, Sydney, NSW 2052, Australia – sequence: 2 givenname: Benjamin surname: Lyons fullname: Lyons, Benjamin email: 2021178@creightonprep.org organization: Department of Mathematics, Creighton Preparatory School, Omaha, NE 68114, USA |
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| Cites_doi | 10.1103/RevModPhys.63.129 10.1016/j.jde.2017.09.011 10.1016/S0362-546X(01)00478-3 10.1016/S0895-7177(00)00154-0 10.1007/s11117-016-0427-z 10.1080/10236198.2015.1125896 10.1007/s10231-018-0738-8 10.1080/10236198.2019.1639684 10.1016/j.anihpc.2004.12.001 10.1016/j.aml.2007.02.019 10.1016/S0362-546X(96)00165-4 10.1080/17476933.2015.1064404 10.1080/10236190008808220 10.1016/j.jmaa.2016.04.023 10.1016/S0034-4877(99)80005-6 10.1016/j.na.2004.08.010 10.1103/PhysRevE.49.3771 |
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| Keywords | Jensen’s inequality Sign-changing coefficient Nonlocal difference equation Coercivity Positive solution |
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| References | Goodrich (b1) 2020 Bohner, Peterson (b10) 2001 Goodrich (b22) 2017; 21 Aly (b2) 1994; 49 Goodrich (b17) 2018; 264 Biler, Krzywicki, Nadzieja (b4) 1998; 42 Dahal, Goodrich (b19) 2019; 25 Erbe, Peterson (b12) 2000; 6 Erbe, Peterson (b11) 2000; 32 Ma, Zhong (b15) 2011; 15 do Ó, Lorca, Sánchez, Ubilla (b6) 2016; 61 Graef, Kong, Wang (b14) 2008; 21 Bavaud (b3) 1991; 63 Stańczy (b8) 2001; 47 Yan, Wang (b9) 2016; 442 Goodrich (b20) 2016; 22 Goodrich, Peterson (b13) 2015 Corrêa (b21) 2004; 59 Goodrich (b18) 2018; 197 Webb (b16) 2009; 13 Esposito, Grossi, Pistoia (b7) 2005; 22 Biler, Nadzieja (b5) 1997; 30 Guo, Lakshmikantham (b23) 1988 Bohner (10.1016/j.aml.2020.106371_b10) 2001 Goodrich (10.1016/j.aml.2020.106371_b20) 2016; 22 Dahal (10.1016/j.aml.2020.106371_b19) 2019; 25 Guo (10.1016/j.aml.2020.106371_b23) 1988 do Ó (10.1016/j.aml.2020.106371_b6) 2016; 61 Goodrich (10.1016/j.aml.2020.106371_b22) 2017; 21 Aly (10.1016/j.aml.2020.106371_b2) 1994; 49 Corrêa (10.1016/j.aml.2020.106371_b21) 2004; 59 Goodrich (10.1016/j.aml.2020.106371_b17) 2018; 264 Biler (10.1016/j.aml.2020.106371_b5) 1997; 30 Yan (10.1016/j.aml.2020.106371_b9) 2016; 442 Erbe (10.1016/j.aml.2020.106371_b11) 2000; 32 Graef (10.1016/j.aml.2020.106371_b14) 2008; 21 Esposito (10.1016/j.aml.2020.106371_b7) 2005; 22 Webb (10.1016/j.aml.2020.106371_b16) 2009; 13 Goodrich (10.1016/j.aml.2020.106371_b18) 2018; 197 Biler (10.1016/j.aml.2020.106371_b4) 1998; 42 Stańczy (10.1016/j.aml.2020.106371_b8) 2001; 47 Erbe (10.1016/j.aml.2020.106371_b12) 2000; 6 Ma (10.1016/j.aml.2020.106371_b15) 2011; 15 Goodrich (10.1016/j.aml.2020.106371_b1) 2020 Goodrich (10.1016/j.aml.2020.106371_b13) 2015 Bavaud (10.1016/j.aml.2020.106371_b3) 1991; 63 |
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