Existence, non-existence and multiplicity results for a third order eigenvalue three-point boundary value problem

Volume 10, Issue 10, pp 5445--5463 http://dx.doi.org/10.22436/jnsa.010.10.28

Publication Date: October 28, 2017 Submission Date: September 02, 2016

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Authors

Alberto Cabada - Instituto de Matemáticas, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain. Lucía López-Somoza - Instituto de Matemáticas, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain. Feliz Minhós - Departamento de Matematica, Escola de Ciencias e Tecnologia, Universidade de Évora, Évora, Portugal. - Centro de Investigação em Matemática e Aplicações (CIMA), Instituto de Investigação e Formação Avançada, Universidade de Évora, Évora, Portugal.

Abstract

This paper provides sufficient conditions to guarantee the existence, non-existence and multiplicity of solutions for a third order eigenvalue fully differential equation, coupled with three point boundary value conditions. Although the change of sign, some bounds for the second derivative of the Green's function are obtained, which allow to define a different kind of cone that, as far as we know, has not been previously used in the literature. The main arguments are based on the fixed point index theory for bounded and unbounded sets. Some examples are presented in order to show that the different existence theorems proved are not comparable.

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ISRP Style

Alberto Cabada, Lucía López-Somoza, Feliz Minhós, Existence, non-existence and multiplicity results for a third order eigenvalue three-point boundary value problem, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 10, 5445--5463

AMA Style

Cabada Alberto, López-Somoza Lucía, Minhós Feliz, Existence, non-existence and multiplicity results for a third order eigenvalue three-point boundary value problem. J. Nonlinear Sci. Appl. (2017); 10(10):5445--5463

Chicago/Turabian Style

Cabada, Alberto, López-Somoza, Lucía, Minhós, Feliz. "Existence, non-existence and multiplicity results for a third order eigenvalue three-point boundary value problem." Journal of Nonlinear Sciences and Applications, 10, no. 10 (2017): 5445--5463

Keywords

Nonlinear boundary value problems
parameter dependence
multipoint boundary value problems
Green functions
degree theory
fixed points in cones

MSC

34B08
34B10
34B15
34B18
34B27

References

[1] H. Amann , Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM. Rev., 18 (1976), 620–709.

[2] A. Cabada, G. Infante, F. A. F. Tojo, Nontrivial solutions of Hammerstein integral equations with reflections, Bound. Value Probl., 2013 (2013), 22 pages.

[3] A. Cabada, F. Minhós, A. I. Santos, Solvability for a third order discontinuous fully equation with nonlinear functional boundary conditions, J. Math. Anal. Appl., 322 (2006), 735–748.

[4] J. Graef, L. Kong, F. Minhós, Generalized Hammerstein equations and applications, Results Math., 72 (2017), 369–383.

[5] A. Granas, J. Dugundji, Fixed Point Theory, Springer, New York (2003)
- Google Scholar

[6] M. Greguš, Third Order Linear Differential Equations, Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht (1987)

[7] D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York (1988)
- MathSciNet

[8] L. Guo, J. Sun, Y. Zhao, Existence of positive solutions for nonlinear third-order three-point boundary value problems, Nonlinear Anal., 68 (2008), 3151–3158.

[9] G. Infante, P. Pietramala, A third order boundary value problem subject to nonlinear boundary conditions, Math. Bohem., 135 (2010), 113–121.

[10] G. Infante, P. Pietramala, F. A. F. Tojo, Nontrivial solutions of local and nonlocal Neumann boundary-value problems, Proc. Royal Soc. Edinburgh, 146 (2016), 337–369.
- View Article
- Google Scholar

[11] G. Infante, J. R. L. Webb , Three point boundary value problems with solutions that change sign, J. Integral Equations Appl., 15 (2003), 37–57.

[12] Y. Liu, Z. Weiguo, L. Xiping, S. Chunfang, C. Hua , Positive solutions for a nonlinear third order multipoint boundary value problem, Pac. J. Math., 249 (2011), 177–188.

[13] F. Minhós, R. Sousa , On the solvability of third-order three point systems of differential equations with dependence on the first derivative, Bull. Braz. Math. Soc., 48 (2017), 485–503.