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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
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##### References:
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