## Positive solutions of fourth-order singular three point eigenvalue problems.(English)Zbl 1126.34321

Summary: By establishing a new comparison theorem and constructing upper and lower solutions, some sufficient conditions of existence of positive solutions for the following singular fourth order three point eigenvalue problem $\begin{cases} u^{(4)} (t)=\lambda f(t,u),\;t\in(0,1),\\ u(0)=\alpha u(\eta),\;u(1)=0,\\ u''(0)=\beta u''(\eta),\;u''(1)=0,\end{cases}$ are established due to Schauder’s fixed point theorem for $$\lambda$$ large enough, where $$\alpha,\beta,\eta\in(0,1)$$ are constants, $$f$$ can be singular at $$t=0$$ and/or 1, $$u=0$$. Moreover, some peculiar cases are discussed and some further results are obtained.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B24 Sturm-Liouville theory

### Keywords:

upper and lower solutions; maximal principle
Full Text:

### References:

 [1] Agarwal, R. P.; O’Regan, D.; Wong, P. J.Y., Positive Solutions of Differential, Difference, and Integral Equations (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0923.39002 [2] Avery, R. I., A generalization of the Leggett-Williams fixed point theorem, Math. Sci. Res. Hot-line, 2, 9-14 (1998) · Zbl 0965.47038 [3] Avery, R. I.; Peterson, A. C., Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl., 42, 313-322 (2001) · Zbl 1005.47051 [4] Rachxunkova, I., Upper and lower solutions and topological degree, J. Math. Anal. Appl., 234, 311-327 (1999) · Zbl 1086.34017 [5] Guo, Y. P.; Ge, W. G., Positive solutions for three-point boundary value problems with dependence on the first order derivative, J. Math. Anal. Appl., 290, 291-301 (2004) · Zbl 1054.34025 [6] Ma, R. Y., Positive solutions for second order three-point boundary value problem, Appl. Math. Lett., 14, 1-5 (2001) · Zbl 0989.34009 [7] Gupta, C. P., Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. Math. Anal. Appl., 168, 540-551 (1992) · Zbl 0763.34009 [8] He, X. M.; Ge, W. G., Triple positive solutions for second-order three-point boundary value problems, J. Math. Anal. Appl., 268, 256-265 (2002) · Zbl 1043.34015 [9] Il’in, V. A.; Moiseev, E. I., Non-local boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, Differential Equations, 23, 803-810 (1987) · Zbl 0668.34025 [10] Krasnosel’skii, M. A., Positive Solutions of Operator Equations (1964), Noordhoff: Noordhoff Gronignen · Zbl 0121.10604 [11] Leggett, R. W.; Williams, L. R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28, 673-688 (1979) · Zbl 0421.47033 [12] Ma, Y. R., Multiplicity of positive solutions for second-order three-point boundary value problems, Comput. Math. Appl., 40, 193-204 (2000) · Zbl 0958.34019 [13] Graef, J. R.; Qian, C.; Yang, B., A three point boundary value problem for nonlinear fourth order differential equations, J. Math. Anal. Appl., 287, 217-233 (2003) · Zbl 1054.34038 [14] Gupta, C. P., A sharper condition for the solvability of a three-point second order boundary value problem, J. Math. Anal. Appl., 205, 579-586 (1997) · Zbl 0874.34014 [15] Z.L. Wei, C.C. Pang, The method of lower and upper solutions for fourth order singular $$m$$-point boundary value problems, J. Math. Anal. Appl., in press.; Z.L. Wei, C.C. Pang, The method of lower and upper solutions for fourth order singular $$m$$-point boundary value problems, J. Math. Anal. Appl., in press. · Zbl 1112.34010
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.