# American Institute of Mathematical Sciences

2015, 2015(special): 841-850. doi: 10.3934/proc.2015.0841

## Solvability of higher-order BVPs in the half-line with unbounded nonlinearities

 1 Departamento de Matemática. Universidade de Évora, Centro de Investigação em Matemática e Aplicaçoes da U.E. (CIMA-UE), Rua Romão Ramalho, 59. 7000-671 Évora 2 Centro de Investigação em Matemática e Aplicações (CIMA-UE), Portugal

Received  July 2014 Revised  April 2015 Published  November 2015

This work presents sufficient conditions for the existence of unbounded solutions of a Sturm-Liouville type boundary value problem on the half-line. One-sided Nagumo condition plays a special role because it allows an asymmetric unbounded behavior on the nonlinearity. The arguments are based on fixed point theory and lower and upper solutions method. An example is given to show the applicability of our results.
Citation: Feliz Minhós, Hugo Carrasco. Solvability of higher-order BVPs in the half-line with unbounded nonlinearities. Conference Publications, 2015, 2015 (special) : 841-850. doi: 10.3934/proc.2015.0841
##### References:
 [1] R.P. Agarwal and D. O'Regan, Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publisher, Glasgow 2001. Google Scholar [2] R.P. Agarwal and D. O'Regan, Non-linear boundary value problems on the semi-infinite interval: an upper and lower solution approach, Mathematika 49, no. 1-2, (2002) 129-140. Google Scholar [3] C. Bai and C. Li, Unbounded upper and lower solution method for third-order boundary-value problems on the half-line Electronic Journal of Differential Equations, 119 (2009), 1-12. Google Scholar [4] F. Bernis and L.A. Peletier, Two problems from draining flows involving third-order ordinary differential equations SIAM J. Math. Anal., 27 (1996), 515-527. Google Scholar [5] A. Cabada, F. Minhós and A. I. Santos, Solvability for a third order discontinuous fully equation with functional boundary conditions J. Math. Anal. Appl., 322 (2006), 735-748. Google Scholar [6] P.W. Eloe, E. R. Kaufmann and C. C. Tisdell, Multiple solutions of a boundary value problem on an unbounded domain, Dynamic Systems and Applications, 15 (2006), 53-63. Google Scholar [7] J. Fialho and F. Minhós, Existence and location results for hinged beams with unbounded nonlinearities, Nonlinear Anal., 71 (2009), 1519-1525. Google Scholar [8] J. Graef, L. Kong and F. Minhós, Higher order boundary value problems with Î|-Laplacian and functional boundary conditions , Computers and Mathematics with Applications, 61 (2011), 236-249. Google Scholar [9] M. Greguš, Third Order Linear Differential Equations, Mathematics and its Applications, Reidel Publishing Co., Dordrecht, 1987. Google Scholar [10] M. R. Grossinho, F. Minhós and A. I. Santos, A note on a class of problems for a higher order fully nonlinear equation under one sided Nagumo type condition, Nonlinear Anal., 70 (2009), 4027-4038. Google Scholar [11] H. Lian, P. Wang and W. Ge, Unbounded upper and lower solutions method for Sturm-Liouville boundary value problem on infinite intervals, Nonlinear Anal., 70 (2009), 2627-2633. Google Scholar [12] H. Lian and J. Zhao, Existence of Unbounded Solutions for a Third-Order Boundary Value Problem on Infinite Intervals, Discrete Dynamics in Nature and Society, 2012, Article ID 357697, 14 pages. Google Scholar [13] F. Minhós, Location results: an under used tool in higher order boundary value problems, International Conference on Boundary Value Problems: Mathematical Models in Engineering, Biology and Medicine, American Institute of Physics Conference Proceedings, 1124 (2009), 244-253. Google Scholar [14] F. Minhós, T. Gyulov and A. I. Santos, Lower and upper solutions for a fully nonlinear beam equations, Nonlinear Anal., 71 (2009), 281-292. Google Scholar [15] W. C. Troy, Solutions of third-order differential equations relevant to draining and coating flows, SIAM J. Math. Anal., 24 (1993), 155-171. Google Scholar [16] E.O. Tuck and L.W. Schwartz, A boundary value problem from draining and coating flows involving a third-order differential equation relevant to draining and coating flows, SIAM Rev., 32 (1990), 453-469. Google Scholar [17] B. Yan, D. O'Regan and R.P. Agarwal, Unbounded solutions for singular boundary value problems on the semi-infinite interval: Upper and lower solutions and multiplicity, J.Comput.Appl.Math., 197 (2006), 365-386. Google Scholar

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##### References:
 [1] R.P. Agarwal and D. O'Regan, Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publisher, Glasgow 2001. Google Scholar [2] R.P. Agarwal and D. O'Regan, Non-linear boundary value problems on the semi-infinite interval: an upper and lower solution approach, Mathematika 49, no. 1-2, (2002) 129-140. Google Scholar [3] C. Bai and C. Li, Unbounded upper and lower solution method for third-order boundary-value problems on the half-line Electronic Journal of Differential Equations, 119 (2009), 1-12. Google Scholar [4] F. Bernis and L.A. Peletier, Two problems from draining flows involving third-order ordinary differential equations SIAM J. Math. Anal., 27 (1996), 515-527. Google Scholar [5] A. Cabada, F. Minhós and A. I. Santos, Solvability for a third order discontinuous fully equation with functional boundary conditions J. Math. Anal. Appl., 322 (2006), 735-748. Google Scholar [6] P.W. Eloe, E. R. Kaufmann and C. C. Tisdell, Multiple solutions of a boundary value problem on an unbounded domain, Dynamic Systems and Applications, 15 (2006), 53-63. Google Scholar [7] J. Fialho and F. Minhós, Existence and location results for hinged beams with unbounded nonlinearities, Nonlinear Anal., 71 (2009), 1519-1525. Google Scholar [8] J. Graef, L. Kong and F. Minhós, Higher order boundary value problems with Î|-Laplacian and functional boundary conditions , Computers and Mathematics with Applications, 61 (2011), 236-249. Google Scholar [9] M. Greguš, Third Order Linear Differential Equations, Mathematics and its Applications, Reidel Publishing Co., Dordrecht, 1987. Google Scholar [10] M. R. Grossinho, F. Minhós and A. I. Santos, A note on a class of problems for a higher order fully nonlinear equation under one sided Nagumo type condition, Nonlinear Anal., 70 (2009), 4027-4038. Google Scholar [11] H. Lian, P. Wang and W. Ge, Unbounded upper and lower solutions method for Sturm-Liouville boundary value problem on infinite intervals, Nonlinear Anal., 70 (2009), 2627-2633. Google Scholar [12] H. Lian and J. Zhao, Existence of Unbounded Solutions for a Third-Order Boundary Value Problem on Infinite Intervals, Discrete Dynamics in Nature and Society, 2012, Article ID 357697, 14 pages. Google Scholar [13] F. Minhós, Location results: an under used tool in higher order boundary value problems, International Conference on Boundary Value Problems: Mathematical Models in Engineering, Biology and Medicine, American Institute of Physics Conference Proceedings, 1124 (2009), 244-253. Google Scholar [14] F. Minhós, T. Gyulov and A. I. Santos, Lower and upper solutions for a fully nonlinear beam equations, Nonlinear Anal., 71 (2009), 281-292. Google Scholar [15] W. C. Troy, Solutions of third-order differential equations relevant to draining and coating flows, SIAM J. Math. Anal., 24 (1993), 155-171. Google Scholar [16] E.O. Tuck and L.W. Schwartz, A boundary value problem from draining and coating flows involving a third-order differential equation relevant to draining and coating flows, SIAM Rev., 32 (1990), 453-469. Google Scholar [17] B. Yan, D. O'Regan and R.P. Agarwal, Unbounded solutions for singular boundary value problems on the semi-infinite interval: Upper and lower solutions and multiplicity, J.Comput.Appl.Math., 197 (2006), 365-386. Google Scholar
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