Positive solutions for a $2n$th-order boundary value problem involving all derivatives of odd orders
Keywords
Positive solution, integro-differential equation, fixed point index, a priori estimate, symmetric positive solutionAbstract
We are concerned with the existence, multiplicity and uniqueness of positive solutions for the $2n$-order boundary value problem $$ \cases (-1)^nu^{(2n)}=f(t,u,u',-u''',\ldots, \\ (-1)^{i-1}u^{(2i-1)},\ldots, (-1)^{n-1}u^{(2n-1)}), \\ u^{(2i)}(0)=u^{(2i+1)}(1)=0, \quad i=0,\ldots,n-1. \endcases $$ where $n\geq 2$ and $f\in C([0,1]\times \mathbb{R}_+^{n+1},\mathbb{R}_+)$ $(\mathbb{R}_+:=[0,\infty))$ depends on $u$ and all derivatives of odd orders. Our main hypotheses on $f$ are formulated in terms of the linear function $g(x):=x_1+2\sum_{i=2}^{n+1}x_i$. We use fixed point index theory to establish our main results, based on a priori estimates achieved by utilizing some integral identities and an integral inequality. Finally, we apply our main results to establish the existence, multiplicity and uniqueness of positive symmetric solutions for a Lidostone problem involving an open question posed by P. W. Eloe in 2000.Downloads
Published
2011-04-23
How to Cite
1.
YANG, Zhilin and O’REGAN, Donal. Positive solutions for a $2n$th-order boundary value problem involving all derivatives of odd orders. Topological Methods in Nonlinear Analysis. Online. 23 April 2011. Vol. 37, no. 1, pp. 87 - 101. [Accessed 25 April 2024].
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