Applied Mathematics and Mechanics

, Volume 22, Issue 4, pp 495–500

# Positive Solutions of Boundary Value Problems for Second-Order Singular Nonlinear Differential Equations

• Ren-gui Li
• Li-shan Liu
Article

## Abstract

New existence results are presented for the singular second-order nonlinear boundary value problems u″ + g(t)f(u) = 0, 0 < t < 1, αu(0) − βu′ (0) = 0, γu(1) + δu′(1) = 0 under the conditions$$0 \leqslant f_0^ + < M_1 ,m_1 < f_\infty ^ - \leqslant \infty {\text{ }}or{\text{ 0}} \leqslant f_\infty ^ + < M_1 ,m_1 < f_0^ - \leqslant \infty$$, where$$f_0^ + = \overline {\lim } _{u \to 0} f\left( u \right)/u,f_\infty ^ - = \underline {\lim } _{u \to \infty } f\left( u \right)/u,f_0^ - = \underline {\lim } _{u \to 0} f\left( u \right)/u,f_\infty ^ + = \overline {\lim } _{u \to \infty } f\left( u \right)/u,$$g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.

second-order singular boundary value problems positive solutions cone fixed point

## References

1. [1]
Erbe L H, WANG Hai-yan. On the existence of positive solutions of ordinary differential equations[J]. Proc Amer Math Soc,1994,120(3):743-748.Google Scholar
2. [2]
MA Ru-yun. Positive solutions of singular second-order boundary value problem[J]. Acta Math Sinica,1998,41(6):1225-1230. (in Chinese)Google Scholar
3. [3]
Amann H. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces[J]. SIAM Rev,1976,18(4):620-709.Google Scholar
4. [4]
GUO Da-jun. Nonlinear Funcational Analysis [M]. Jinan: Shandong Science and Technology Publishing House, 1985. (in Chinese)Google Scholar