## Journals

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DCDS

In this paper, we study the relationship between flow-invariant sets for
an vector field $-f'(x)$ in a Banach space, and the critical
points of the functional $f(x)$. The Mountain-Pass
Lemma, for functionals defined on a Banach
space, is generalized to a more general setting where the domain of the
functional $f$ can
be any flow-invariant set for $-f'(x)$. Furthermore, the
intuitive approach taken in the proofs provides a new
technique in proving multiple critical points.

CPAA

This paper is mainly concerned with the existence, multiplicity and
uniqueness of positive solutions for the fourth-order boundary
value problem
\begin{eqnarray*}
u^{(4)}=f(t,u,u^\prime,-u^{\prime\prime},-u^{\prime\prime\prime}),\\
u(0)=u^\prime(1)=u^{\prime\prime}(0)=u^{\prime\prime\prime}(1)=0,
\end{eqnarray*}
where $f\in C([0,1]\times\mathbb R_+^4,\mathbb R_+)(\mathbb
R_+:=[0,\infty))$.
Based on a priori estimates achieved by utilizing some integral
identities and inequalities, we use fixed point index theory to
prove the existence, multiplicity and uniqueness of positive
solutions for the above problem. Finally, as a byproduct, our main
results are applied to establish the existence, multiplicity and
uniqueness of symmetric positive solutions for the fourth order
Lidstone problem.

## Year of publication

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