×

The existence of positive solutions for some nonlinear boundary value problems with linear mixed boundary conditions. (English) Zbl 1086.34022

The authors are interested in the study of positive solutions to the equation
\[ ( \varphi ( x^{\prime }) ) ^{\prime }+a( t) f( x( t) ) =0, \] where \(\varphi :\mathbb R\to \mathbb{R}\) is an increasing homeomorphism and \(\varphi ( 0) =0,\) \(f\in C( [ 0,\infty ) ,[ 0,\infty ) ) \) and \(a \in C( ( 0,1) ,[ 0,\infty ) ) \) and \( a( t) \) can be singular at \(t=0\) and \(t=1,\) subject to the linear mixed boundary value conditions
\[ x( 0) -\beta x^{\prime }( 0) =0,\;x( 1) +\delta x^{\prime }( 1) =0,\; \]
with \(\beta ,\;\delta \in \mathbb{R}\), \(\beta \geq 0,\;\delta \geq 0.\) The method used is an application of a fixed-point index theorem in cones. They provide several results depending essentially on the limits \( f_{0}=\lim_{x\to 0}\frac{f(x)}{\varphi (x)}\) and \(f_{\infty }=\lim_{x\to +\infty }\frac{f(x)}{\varphi (x)}.\)
Existence of positive solutions for p-Laplacian operators is not new, see for instance [L. H. Erbe and H. Wang, Proc. Am. Math. Soc. 120, 743–748 (1994; Zbl 0802.34018)], [J. Wang, Proc. Am. Math. Soc. 125, 2275–2283 (1997; Zbl 0884.34032)], [W. Sun and W. Ge, Acta. Math. Sin. 44, 577–580 (2001; Zbl 1024.34016)], [H. Chen, X. Liu, Y. Guo and W. Ge, Ann. Diff. Equations 19, 256–260 (2003; Zbl 1050.34021)], [B. Li, Acta Math. Sci., Ser. A, Chin. Ed. 23, 257–264 (2003; Zbl 1051.34015)], [Y. Guo and W. Ge, J. Math. Anal. Appl. 286, 491–508 (2003; Zbl 1045.34005)].
The original novelty is that the results obtained concern the cases where \(\varphi :\mathbb R\to \mathbb{R}\) is an increasing homeomorphism. An interesting example closes the paper.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Erbe, L.H.; Wang, H., On the existence of positive solutions of ordinary differential equations, Proc. amer. math. soc., 120, 743-748, (1994) · Zbl 0802.34018
[2] Erbe, L.H., Multiple positive solutions of some boundary value problems, J. math. anal. appl., 184, 640-648, (1994) · Zbl 0805.34021
[3] Wang, J.Y., The existence of positive solution for the one dimensional p-Laplacian, Proc. amer. math. soc., 125, 2275-2283, (1997) · Zbl 0884.34032
[4] Sun, W.; Ge, W., The existence of positive solutions for a class of nonlinear boundary value problem, Acta math. sinica, 4, 577-580, (2001) · Zbl 1024.34016
[5] Chen, H.; Liu, X.; Guo, Y.; Ge, W., Multiple positive solutions of boundary value problems for quasilinear differential equations, Ann. differential equations, 19, 256-260, (2003) · Zbl 1050.34021
[6] Li, B., The existence of positive solutions for singular boundary value problems with p-Laplacian operators, Math. acta sci. ser. A, 23, 257-264, (2003) · Zbl 1051.34015
[7] Yang, X., Positive solutions for nonlinear singular boundary value problems, Appl. math. comput., 130, 225-234, (2002) · Zbl 1030.34021
[8] Agarwal, R.P., Positive solutions for nonlinear singular boundary value problems, Appl. math. lett., 12, 115-120, (1999) · Zbl 0934.34015
[9] Guo, Y.; Ge, W., Three positive solutions for the one-dimensional p-Laplacian, J. math. anal. appl., 286, 491-508, (2003) · Zbl 1045.34005
[10] Guo, D., Nonlinear functional analysis, (1985), Shangdong Science and Technology Press Jinan
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.