×

zbMATH — the first resource for mathematics

Positive solutions of four-point boundary value problems for higher-order \(p\)-Laplacian operator. (English) Zbl 1127.34013
This paper deals with a quasi-linear equation of the form
\[ (\phi_p(u^{(n-1)}(t)))'+g(t) f(u(t),u'(t),\dots, u^{(n-2)}(t))=0, \; t\in (0,1), \]
where \(\phi_p(s)=| s| ^{p-2},\, p>1\) is the \(p\)-Laplacian operator and \(\phi_q=\phi_p^{-1}\), being \(1/p + 1/q =1.\) The multipoint boundary condition attached to the equation is of the form
\[ u^{(i)}(0)=0\text{ for }0\leq i \leq n-3, \]
\[ \alpha \phi_p(u^{(n-2)}(0))-\beta \phi_p(u^{(n-1)}(\xi))=0\text{ for }n \geq 3, \]
\[ \gamma \phi_p(u^{(n-2)}(1))+\delta \phi_p(u^{(n-1)}(\eta))=0\text{ for }n \geq 3, \]
where \(\xi,\eta \in (0,1)\) and \(\alpha,\gamma >0, \, \beta, \delta \geq 0\). It is assumed that \(g\) is a non-negative function defined on the open interval \((0,1)\) such that \(0<\int_0^1 g(t)\, dt < +\infty\); the function \(f\) is regular and non-negative. Various technical sufficient conditions are given which guarantee the existence of at least one positive solution to the given boundary value problem. These conditions involve, among others, the quantities
\[ f_0=\lim_{u_{n-1}\to 0}\max_{0\leq u_1\leq u_2 \leq \dots \leq u_{n-2}\leq (1/\theta)u_{n-1}} {{f(u_1,u_2,\dots, u_{n-1})}\over{(u_{n-1})^{p-1}}} \]
and
\[ f_\infty=\lim_{u_{n-1}\to \infty}\max_{0\leq u_1\leq u_2 \leq \dots \leq u_{n-2}\leq (1/\theta)u_{n-1}} {{f(u_1,u_2,\dots, u_{n-1})}\over{(u_{n-1})^{p-1}}}, \]
where \(\theta\in(0,1/2)\) is a suitable constant. The proof is performed in the framework of fixed point index theory in cones. For related results we refer, together with the papers listed in the bibliography, to the recent work by S. Hua, Y. Zhou [Positive solutions of four-point boundary-value problems for higher-order with \(p\)-Laplacian operator, Electron. J. Differential Equations, 5, 1–14 (2007)].

MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ma, R., Positive solutions for a nonlinear three-point boundary value problems, Electron. J. differential equations, 34, 1-8, (1999)
[2] Ma, R.; Castaneda, N., Existence of solutions for nonlinear m-point boundary value problems, J. math. anal. appl., 256, 556-567, (2001) · Zbl 0988.34009
[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)
[5] Liu, B., Positive solutions for a nonlinear three-point boundary value problems, Appl. math. comput., 132, 11-28, (2002) · Zbl 1032.34020
[6] Wang, J.Y., The existence of positive solutions for the one-Laplacian, Proc. amer. math. soc., 125, 8, 2275-2283, (1997) · Zbl 0884.34032
[7] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cone, (1988), Academic Press San Diego · Zbl 0661.47045
[8] Guo, D.; Lakshmikantham, V.; Liu, X., Nonlinear integral equations in abstract spaces, (1996), Kluwer Academic Dordrecht · Zbl 0866.45004
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.