# Singular $\phi $-Laplacian third-order BVPs with derivative dependance

## Smaïl Djebali and Ouiza Saifi

Address:

Department of Mathematics, Faculty of Sciences, Al Imam Mohammad Ibn Saud Islamic University, P.O. Box 90950, Riyadh 11623, Saudi Arabia

Laboratoire “Théorie du point Fixe et Applications”, E.N.S., P.O. Box 92. Kouba, 16006 Algiers, Algeria, Department of Economics, Faculty of Economic and Management Sciences, Algiers University 3, Algeria

E-mail:

djebali@hotmail.com

saifi_kouba@yahoo.fr

Abstract: This work is devoted to the existence of solutions for a class of singular third-order boundary value problem associated with a $\phi $-Laplacian operator and posed on the positive half-line; the nonlinearity also depends on the first derivative. The upper and lower solution method combined with the fixed point theory guarantee the existence of positive solutions when the nonlinearity is monotonic with respect to its arguments and may have a space singularity; however no Nagumo type condition is assumed. An example of application illustrates the applicability of the existence result.

AMSclassification: primary 34B15; secondary 34B18, 34B40, 47H10.

Keywords: third order, half-line, $\phi$-Laplacian, singular problem, positive solution, derivative dependance, upper and lower solution.

DOI: 10.5817/AM2016-1-35