## Positive solutions of fourth-order nonlinear singular boundary value problems.(English)Zbl 1136.34021

Summary: We consider the existence of positive solutions for the following fourth-order singular Sturm-Liouville boundary value problem:
$\begin{cases} \frac{1}{p(t)}\,(p(t)u'''(t))'-g(t)F(t,u)=0,\quad 0<t<1,\\ \alpha_1u(0)-\beta_1u'(0)=0,\quad \gamma_1 u(1)+\delta_1u'(1)=0,\\ \alpha_2u''(0)-\beta_2\lim_{t\to 0+}p(t)u'''(t)=0,\\ \gamma_2u''(1)+\delta_2\lim_{t\to 1-}p(t)u'''(t)=0\end{cases}$
where $$g$$, $$p$$ may be singular at $$t=0$$ and/or 1. Moreover $$F(t,x)$$ may also have singularity at $$x=0.$$ Existence and multiplicity theorems of positive solutions for the fourth-order singular Sturm-Liouville boundary value problem are obtained by using the first eigenvalue of the corresponding linear problems. Our results significantly extend and improve many known results including singular and nonsingular cases.

### MSC:

 34B16 Singular nonlinear boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
Full Text:

### References:

 [1] Ma, R. Y.; Wang, H. Y., On the existence of positive solutions of fourth order ordinary differential equation, Appl. Anal., 59, 225-231 (1995) · Zbl 0841.34019 [2] Schroder, J., Fourth-order two-point boundary value problems; Estimates by two side bounds, Nonlinear Anal., 8, 107-144 (1984) · Zbl 0533.34019 [3] Webb, J. R.L., Positive solutions for some three-point boundary value problems via fixed point index, Nonlinear Anal., 47, 4319-4332 (2001) · Zbl 1042.34527 [4] Liu, B., Positive solutions for a nonlinear boundary value problems, J. Com. Math. Appl., 44, 201-217 (2002) · Zbl 1008.34014 [5] Liu, B., Positive solutions for a nonlinear three-point boundary value problems, Appl. Math. Comput., 132, 11-28 (2002) · Zbl 1032.34020 [6] Anderson, D., Multiple positive solution for a three-point boundary value problem, Math. Comput. Modeling, 27, 49-57 (1998) · Zbl 0906.34014 [7] Yao, Q., Existence of positive solutions for a third-order three-point boundary value problem with semipositone nonlinearity, J. Math. Res. Exposition, 23, 591-596 (2003) · Zbl 1050.34030 [8] Ma, R., Positive solutions for a nonlinear three-point boundary value problems, Electron. J. Differ. Equ., 34, 1-8 (1999) · Zbl 0926.34009 [9] Agarwal, R. P., On the fourth-order boundary value problems arising in beam analysis, Differential Integral Equations, 2, 91-110 (1989) · Zbl 0715.34032 [10] Yang, Y. S., Fourth order two-point boundary value problem, Proc. Amer. Math. Soc., 104, 175-180 (1988) · Zbl 0671.34016 [11] Ma, R. Y.; Zhang, J. H.; Fu, S. M., The method of lower and upper solutions for fourth-order tow-point boundary value problems, J. Math. Anal. Appl., 215, 415-422 (1997) · Zbl 0892.34009 [12] Liu, B., Positive solutions of fourth-order two point boundary value problem, Appl. Math. Comput., 148, 407-420 (2004) · Zbl 1039.34018 [13] Wong, P. J.Y.; Agarwal, R. P., Eigenvalue of Lidstone boundary value problems, Appl. Math. Comput., 104, 15-31 (1999) · Zbl 0923.39002 [14] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cone (1988), Academic Press: Academic Press Sandiego · Zbl 0661.47045 [15] Guo, D. J.; Sun, J., Nonlinear Integral Equations (1987), Shandong Science and Technology Press: Shandong Science and Technology Press Jinan, (in Chinese) [16] Liu, L. L.; Sun, Y., Positive solutions of singular boundary value problems of differential equations, Math. Acta. Scien., 25A, 4, 554-563 (2005), (in Chinese) · Zbl 1110.34308
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.