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Optimal control problem for a viscoelastic beam and its galerkin approximation
Solutions to resonant boundary value problem with boundary conditions involving Riemann-Stieltjes integrals
Institute of Mathematics, Lódź University of Technology, 90-924 Lódź, ul. Wólczańska 215, Poland |
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.
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. |
[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. |
[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. |
[4] |
W. Feng,
On an M-point boundary value problem, Nonlinear Anal., 30 (1997), 5369-5374.
doi: 10.1016/S0362-546X(97)00360-X. |
[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. |
[6] |
L. Górniewicz,
Topological Fixed Point Theory of Multivalued Mappings Mathematics and its Applications, 495. Kluwer Academic Publishers, Dordrecht, 1999. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
E. H. Spanier,
Algebraic Topology Corrected reprint of the 1966 original. Springer-Verlag, New York, [1995?]. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[4] |
W. Feng,
On an M-point boundary value problem, Nonlinear Anal., 30 (1997), 5369-5374.
doi: 10.1016/S0362-546X(97)00360-X. |
[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. |
[6] |
L. Górniewicz,
Topological Fixed Point Theory of Multivalued Mappings Mathematics and its Applications, 495. Kluwer Academic Publishers, Dordrecht, 1999. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
E. H. Spanier,
Algebraic Topology Corrected reprint of the 1966 original. Springer-Verlag, New York, [1995?]. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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