Existence of solutions for higher order \(\phi-\)Laplacian BVPs on the half-line using a one-sided Nagumo condition with nonordered upper and lower solutions





In this paper, we consider the following \((n+1)\)st order bvp on the half line with a \(\phi-\)Laplacian operator \[ \begin{cases} (\phi(u^{(n)}))'(t) = f(t,u(t),\ldots,u^{(n)}(t)), & \text{a.e.},\, t\in [0,+\infty), \\ n \in \mathbb{N}\setminus\{0\}, \\  \\ u^{(i)}(0) = A_{i}, \, i=0,\ldots,n-2, \\ u^{(n-1)}(0) + au^{(n)}(0) = B, \\ u^{(n)}(+\infty) = C. \end{cases} \]

The existence of solutions is obtained by applying Schaefer's fixed point theorem under a one-sided Nagumo condition with nonordered lower and upper solutions method where \(f\) is a \(L^{1}\)-Carathéodory function.


Boundary value problem , One-sided Nagumo condition , Lower and upper solutions , A priori estimates

Mathematics Subject Classification:

34B10 , 34B15 , 34B40
  • Pages: 173–193
  • Date Published: 2023-07-19
  • Vol. 25 No. 2 (2023)

R. P. Agarwal and D. O’Regan, Infinite Interval Problems for Differential, Difference and Integral Equations. Glasgow, Scotland: Kluwer Academic Publisher, 2001.

S. E. Ariaku, E. C. Mbah, C. C. Asogwa and P. U. Nwokoro, “Lower and upper solutions of first order non-linear ordinary differential equations”, IJSES, vol. 3, no. 11, pp. 59–61, 2019.

C. Bai and C. Li, “Unbounded upper and lower solution method for third order boundary value problems on the half line”, Electron. J. Differential Equations, vol. 2009, no. 119, pp. 1–12, 2009.

A. Cabada, J. A. Cid, and L. Sánchez, “Positivity and lower and upper solutions for fourth order boundary value problems”, Nonlinear Anal., vol. 67, no. 5, pp. 1599–1612, 2007.

A. Cabada and N. Dimitrov, “Existence of solutions of nth-order nonlinear difference equations with general boundary conditions”, Acta Math. Sci. Ser. B (Engl. Ed.), vol. 40, no. 1, pp. 226– 236, 2020.

A. Cabada and L. Saavedra, “Existence of solutions for nth-order nonlinear differential boundary value problems by means of fixed point theorems”, Nonlinear Anal. Real World Appl., vol. 42, pp. 180–206, 2018.

H. Carrasco and F. Minhós, ”Existence of solutions to infinite elastic beam equations with unbounded nonlinearities”, Electron. J. Differential Equations, vol. 2017, no. 192, pp. 1–11, 2017.

J. R. Graef, L. Kong and F. Minhós, “Higher order boundary value problems with φ−Laplacian and functional boundary conditions”, Comput. Math. Appl., vol. 61, no. 2, pp. 236–249, 2011.

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., vol. 70, no. 11, pp. 4027–4038, 2009.

R. Koplatadze, G. Kvinikadze and I. P. Stavroulakis, “Properties A and B of nth order linear differential equations with deviating argument”, Georgian Math. J., vol. 6, no. 6, pp. 553–566, 1999.

H. Lian and J. Zhao, “Existence of unbounded solutions for a third-order boundary value problem on infinite intervals”, Discrete Dyn. Nat. Soc., vol. 2012, 2012.

F. Minhós and H. Carrasco, “Solvability of higher-order BVPs in the half-line with unbounded nonlinearities”, Discrete Contin. Dyn. Syst., vol. 2015, pp. 841–850, 2015.

D. R. Smart; Fixed point theorems. London-New York, England-USA: Cambridge University Press, 1974.

A. Zerki, K. Bachouche and K. Ait-Mahiout, “Existence solutions for third order φ−Laplacian bvps on the half-line”, Mediterr. J. Math., vol. 19, no. 6, Art. ID 261, 2022.

Q. Zhang, D. Jiang, S. Weng and H. Gao, “Upper and lower solutions for a second-order three-point singular boundary-value problem”, Electron. J. Differential Equations, vol. 2009, Art. ID 115, 2009.

X. Zhang and L. Liu, “Positive solutions of fourth-order four-point boundary value problems with p-Laplacian operator”, J. Math. Anal. Appl., vol. 336, no. 2, pp. 1414–1423, 2007.


Download data is not yet available.



How to Cite

A. Zerki, K. Bachouche, and K. Ait-Mahiout, “Existence of solutions for higher order \(\phi-\)Laplacian BVPs on the half-line using a one-sided Nagumo condition with nonordered upper and lower solutions”, CUBO, pp. 173–193, Jul. 2023.