### Nontrivial solutions of fourth-order singular boundary value problems with sign-changing nonlinear terms

#### Abstract

In this paper, the fourth-order singular boundary value problem (BVP)

$$

\aligned &u^{(4)}(t)=h(t)f(u(t)),\ \ t\in(0,1),\\

&u(0)=u(1)=u'(0)=u'(1)=0\endaligned

$$

is considered under some conditions concerning the first

characteristic value corresponding to the relevant

linear operator, where $h$ is allowed to be singular at both $t=0$

and $t=1$. In particular, $f\colon (-\infty,\infty)\rightarrow

(-\infty,\infty)$ may be a sign-changing and unbounded function

from below, and it is not also necessary to exist a control

function for $f$ from below. The existence results of nontrivial

solutions and positive-negative solutions are given by

the topological degree theory and the fixed point index theory, respectively.

$$

\aligned &u^{(4)}(t)=h(t)f(u(t)),\ \ t\in(0,1),\\

&u(0)=u(1)=u'(0)=u'(1)=0\endaligned

$$

is considered under some conditions concerning the first

characteristic value corresponding to the relevant

linear operator, where $h$ is allowed to be singular at both $t=0$

and $t=1$. In particular, $f\colon (-\infty,\infty)\rightarrow

(-\infty,\infty)$ may be a sign-changing and unbounded function

from below, and it is not also necessary to exist a control

function for $f$ from below. The existence results of nontrivial

solutions and positive-negative solutions are given by

the topological degree theory and the fixed point index theory, respectively.

#### Keywords

Nontrivial solution; topological degree; fixed point index

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