### Positive solutions for a $2n$th-order boundary value problem involving all derivatives of odd orders

#### Abstract

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.

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.

#### Keywords

Positive solution; integro-differential equation; fixed point index; a priori estimate; symmetric positive solution

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