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January  2018, 23(1): 275-281. doi: 10.3934/dcdsb.2018019

Solutions to resonant boundary value problem with boundary conditions involving Riemann-Stieltjes integrals

Igor Kossowski , and  Katarzyna Szymańska-Dębowska

Institute of Mathematics, Lódź University of Technology, 90-924 Lódź, ul. Wólczańska 215, Poland

* Corresponding author: Igor Kossowski

Received  October 2016 Published  January 2018

We study the nonlinear boundary value problem consisting of a system of second order differential equations and boundary conditions involving a Riemann-Stieltjes integrals. Our proofs are based on the generalized Miranda Theorem.

Keywords: Nonlocal boundary value problem, boundary value problem at resonance, the Miranda theorem.
Mathematics Subject Classification: Primary: 34B10; Secondary: 34B15, 47H11.
Citation: Igor Kossowski, Katarzyna Szymańska-Dębowska. Solutions to resonant boundary value problem with boundary conditions involving Riemann-Stieltjes integrals. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 275-281. doi: 10.3934/dcdsb.2018019
References:
[1]

C. Bai and J. Fang, Existence of positive solutions for three-point boundary value problems at resonance, J. Math. Anal. Appl., 291 (2004), 538-549. doi: 10.1016/j.jmaa.2003.11.014. Google Scholar

[2]

H. Ben-El-Mechaiekh and W. Kryszewski, Equilibria of set-valued maps on nonconvex domains, Trans. Amer. Math. Soc., 349 (1997), 4159-4179. doi: 10.1090/S0002-9947-97-01836-9. Google Scholar

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J. T. Ding and B. Z. Guo, Blow-up and global existence for nonlinear parabolic equations with Neumann boundary conditions, Comput. Math. Appl., 60 (2010), 670-679. doi: 10.1016/j.camwa.2010.05.015. Google Scholar

[4]

W. Feng, On an M-point boundary value problem, Nonlinear Anal., 30 (1997), 5369-5374. doi: 10.1016/S0362-546X(97)00360-X. Google Scholar

[5]

D. Franco, G. Infante and M. Zima, Second order nonlocal boundary value problems at resonance, Math. Nachr., 284 (2011), 875-884. doi: 10.1002/mana.200810841. Google Scholar

[6]

L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings Mathematics and its Applications, 495. Kluwer Academic Publishers, Dordrecht, 1999. Google Scholar

[7]

A. Granas and M. Frigon, Topological Methods in Differential Equations and Inclusions Kluwer Academic Publishers, 1995. doi: 10.1007/978-94-011-0339-8. Google Scholar

[8]

C. P. Gupta, A generalized multi-point boundary value problem for second order ordinary differential equations, Appl.Math. Comput., 89 (1998), 133-146. doi: 10.1016/S0096-3003(97)81653-0. Google Scholar

[9]

X. Han, Positive solutions for a three-point boundary value problem at resonance, J. Math. Anal. Appl., 336 (2007), 556-568. doi: 10.1016/j.jmaa.2007.02.069. Google Scholar

[10]

G. Infante and J. R. L. Webb, Positive solutions of some nonlocal boundary value problems, Abstr. Appl. Anal., 18 (2003), 1047-1060. doi: 10.1155/S1085337503301034. Google Scholar

[11]

L. C. Piccinnini, G. Stampacchia and G. Vidossich, Ordinary Differential Equations in $\mathbb{R}^n$ Translated from the Italian by A. LoBello. Applied Mathematical Sciences, 39. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5188-0. Google Scholar

[12]

P. Souplet and F. B. Weissler, Self-similar subsolutions and blowup for nonlinear parabolic equations, J. Math. Anal. Appl., 212 (1997), 60-74. doi: 10.1006/jmaa.1997.5452. Google Scholar

[13]

E. H. Spanier, Algebraic Topology Corrected reprint of the 1966 original. Springer-Verlag, New York, [1995?]. Google Scholar

[14]

K. Szymańska-Dębowska, On a generalization of the Miranda Theorem and its application to boundary value problems, J. Differential Equations, 258 (2015), 2686-2700. doi: 10.1016/j.jde.2014.12.022. Google Scholar

[15]

J. R. L. Webb, Optimal constants in a nonlocal boundary value problem, Nonlinear Anal., 63 (2005), 672-685. doi: 10.1016/j.na.2005.02.055. Google Scholar

[16]

J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: A unified approach, J. London Math. Soc.(2), 74 (2006), 673-693. doi: 10.1112/S0024610706023179. Google Scholar

[17]

J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems involving integral conditions, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 45-67. doi: 10.1007/s00030-007-4067-7. Google Scholar

[18]

J. R. L. Webb, Existence of positive solutions for a thermostat model, Nonlinear Anal. RWA, 13 (2012), 923-938. doi: 10.1016/j.nonrwa.2011.08.027. Google Scholar

[19]

J. R. L. Webb and M. Zima, Multiple positive solutions of resonant and non-resonant nonlocal boundary value problems, Nonlinear Anal., 71 (2009), 1369-1378. doi: 10.1016/j.na.2008.12.010. Google Scholar

show all references

References:
[1]

C. Bai and J. Fang, Existence of positive solutions for three-point boundary value problems at resonance, J. Math. Anal. Appl., 291 (2004), 538-549. doi: 10.1016/j.jmaa.2003.11.014. Google Scholar

[2]

H. Ben-El-Mechaiekh and W. Kryszewski, Equilibria of set-valued maps on nonconvex domains, Trans. Amer. Math. Soc., 349 (1997), 4159-4179. doi: 10.1090/S0002-9947-97-01836-9. Google Scholar

[3]

J. T. Ding and B. Z. Guo, Blow-up and global existence for nonlinear parabolic equations with Neumann boundary conditions, Comput. Math. Appl., 60 (2010), 670-679. doi: 10.1016/j.camwa.2010.05.015. Google Scholar

[4]

W. Feng, On an M-point boundary value problem, Nonlinear Anal., 30 (1997), 5369-5374. doi: 10.1016/S0362-546X(97)00360-X. Google Scholar

[5]

D. Franco, G. Infante and M. Zima, Second order nonlocal boundary value problems at resonance, Math. Nachr., 284 (2011), 875-884. doi: 10.1002/mana.200810841. Google Scholar

[6]

L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings Mathematics and its Applications, 495. Kluwer Academic Publishers, Dordrecht, 1999. Google Scholar

[7]

A. Granas and M. Frigon, Topological Methods in Differential Equations and Inclusions Kluwer Academic Publishers, 1995. doi: 10.1007/978-94-011-0339-8. Google Scholar

[8]

C. P. Gupta, A generalized multi-point boundary value problem for second order ordinary differential equations, Appl.Math. Comput., 89 (1998), 133-146. doi: 10.1016/S0096-3003(97)81653-0. Google Scholar

[9]

X. Han, Positive solutions for a three-point boundary value problem at resonance, J. Math. Anal. Appl., 336 (2007), 556-568. doi: 10.1016/j.jmaa.2007.02.069. Google Scholar

[10]

G. Infante and J. R. L. Webb, Positive solutions of some nonlocal boundary value problems, Abstr. Appl. Anal., 18 (2003), 1047-1060. doi: 10.1155/S1085337503301034. Google Scholar

[11]

L. C. Piccinnini, G. Stampacchia and G. Vidossich, Ordinary Differential Equations in $\mathbb{R}^n$ Translated from the Italian by A. LoBello. Applied Mathematical Sciences, 39. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5188-0. Google Scholar

[12]

P. Souplet and F. B. Weissler, Self-similar subsolutions and blowup for nonlinear parabolic equations, J. Math. Anal. Appl., 212 (1997), 60-74. doi: 10.1006/jmaa.1997.5452. Google Scholar

[13]

E. H. Spanier, Algebraic Topology Corrected reprint of the 1966 original. Springer-Verlag, New York, [1995?]. Google Scholar

[14]

K. Szymańska-Dębowska, On a generalization of the Miranda Theorem and its application to boundary value problems, J. Differential Equations, 258 (2015), 2686-2700. doi: 10.1016/j.jde.2014.12.022. Google Scholar

[15]

J. R. L. Webb, Optimal constants in a nonlocal boundary value problem, Nonlinear Anal., 63 (2005), 672-685. doi: 10.1016/j.na.2005.02.055. Google Scholar

[16]

J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: A unified approach, J. London Math. Soc.(2), 74 (2006), 673-693. doi: 10.1112/S0024610706023179. Google Scholar

[17]

J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems involving integral conditions, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 45-67. doi: 10.1007/s00030-007-4067-7. Google Scholar

[18]

J. R. L. Webb, Existence of positive solutions for a thermostat model, Nonlinear Anal. RWA, 13 (2012), 923-938. doi: 10.1016/j.nonrwa.2011.08.027. Google Scholar

[19]

J. R. L. Webb and M. Zima, Multiple positive solutions of resonant and non-resonant nonlocal boundary value problems, Nonlinear Anal., 71 (2009), 1369-1378. doi: 10.1016/j.na.2008.12.010. Google Scholar

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