On the Solvability of Third-Order Three Point Systems of Differential Equations with Dependence on the First Derivative

  • Feliz MinhósEmail author
  • Robert de Sousa


This paper presents sufficient conditions for the solvability of the third order three point boundary value problem
$$\begin{aligned} \left\{ \begin{array}{c} -u^{\prime \prime \prime }(t)=f(t,\,v(t),\,v^{\prime }(t)) \\ -v^{\prime \prime \prime }(t)=h(t,\,u(t),\,u^{\prime }(t)) \\ u(0)=u^{\prime }(0)=0,u^{\prime }(1)=\alpha u^{\prime }(\eta ) \\ v(0)=v^{\prime }(0)=0,v^{\prime }(1)=\alpha v^{\prime }(\eta ). \end{array} \right. \end{aligned}$$
The arguments apply Green’s function associated to the linear problem and the Guo–Krasnosel’skiĭ theorem of compression-expansion cones. The dependence on the first derivatives is overcome by the construction of an adequate cone and suitable conditions of superlinearity/sublinearity near 0 and \(+\infty \). Last section contains an example to illustrate the applicability of the theorem.


Coupled systems Green functions Guo–Krasnosel’skiĭ fixed-point in cones Positive solution 

Mathematics Subject Classification

34B15 34B18 34B27 34L30 


  1. Asif, N.A., Khan, R.A.: Positive solutions to singular system with four-point coupled boundary conditions. J. Math. Anal. Appl. 386, 848–861 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bernis, F., Peletier, L.A.: Two problems from draining flows involving third-order ordinary differential equations. SIAM J. Math. Anal. 27(2), 515–527 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Cui, Y., Sun, J.: On existence of positive solutions of coupled integral boundary value problems for a nonlinear singular superlinear differential system. Electron. J. Qual. Theory Differ. Equ. 41, 1–13 (2012)MathSciNetCrossRefGoogle Scholar
  4. Danziger, L., Elmergreen, G.: The thyroid-pituitary homeostatic mechanism. Bull. Math. Biophys. 18, 1–13 (1956)CrossRefGoogle Scholar
  5. Guo, D., Lakshmikantham, V.: Nonlinear problems in abstract cones. Academic Press, Cambridge (1988)zbMATHGoogle Scholar
  6. Henderson, J., Luca, R.: Boundary value problems for systems of differential. Difference and fractional equations, positive solutions, Elsevier (2015)Google Scholar
  7. Henderson, J., Luca, R.: Positive solutions for systems of nonlinear second-order multipoint boundary value problems. Math. Methods Appl. Sci. 37, 2502–2516 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Infante, G., Minhós, F., Pietramala, P.: Non-negative solutions of systems of ODEs with coupled boundary conditions. Commun. Nonlinear Sci. Numer. Simulat. 17, 4952–4960 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Infante, G., Pietramala, P.: Nonnegative solutions for a system of impulsive BVPs with nonlinear nonlocal BCs. Nonlinear Anal. Model. Control 19(3), 413–431 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Jankowski, T.: Nonnegative solutions to nonlocal boundary value problems for systems of second-order differential equations dependent on the first-order derivatives. Nonlinear Anal. 87, 83–101 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Kang, P., Wei, Z.: Existence of positive solutions for systems of bending elastic beam equations. Electron. J. Differ. Equ. 19 (2012)Google Scholar
  12. Lee, E.K., Lee, Y.H.: Multiple positive solutions of a singular Emden–Fowler type problem for second-order impulsive differential systems, Bound. Value Probl., Art. ID 212980, p. 22 (2011)Google Scholar
  13. Li, Y., Guo, Y., Li, G.: Existence of positive solutions for systems of nonlinear third-order differential equations. Commun. Nonlinear Sci. Numer. Simulat. 14, 3792–3797 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Li-Jun, G., Jian-Ping, S., Ya-Hong, Z.: Existence of positive solutions for nonlinear third-order three-point boundary value problems. Nonlinear Anal. 68, 3151–3158 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Liu, X., Chen, H., Lü, Y.: Explicit solutions of the generalized KdV equations with higher order nonlinearity. Appl. Math. Comput. 171, 315–319 (2005)MathSciNetzbMATHGoogle Scholar
  16. Liu, L., Kanga, P., Wub, Y., Wiwatanapataphee, B.: Positive solutions of singular boundary value problems for systems of nonlinear fourth order differential equations. Nonlinear Anal. 68, 485–498 (2008)MathSciNetCrossRefGoogle Scholar
  17. Stakgold, I., Holst, M.: Green’s functions and Boundary Value Problems. John Wiley and Sons, 3rd ed., New Jersey (2011)Google Scholar
  18. Tuck, E.O., Schwartz, L.W.: A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows. SIAM Rev. 32(3), 453–469 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2016

Authors and Affiliations

  1. 1.Departamento de Matemática, Escola de Ciências e Tecnologia, Centro de Investigação em Matemática e Aplicações (CIMA), Instituto de Investigação e Formação AvançadaUniversidade de ÉvoraÉvoraPortugal
  2. 2.Faculdade de Ciências e Tecnologia, Núcleo de Matemática e Aplicações (NUMAT)Universidade de Cabo Verde, Campus de PalmarejoPraiaCabo Verde

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