Flow-invariant sets and critical point theory
Jingxian Sun Shouchuan Hu
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.
keywords: connected set. Invariant set of flow critical point forward saturated solution
Positive solutions of a fourth-order boundary value problem involving derivatives of all orders
Zhilin Yang Jingxian Sun
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.
keywords: integro-differential equation positive solution a priori estimate symmetric positive solution. fixed point index Fourth-order boundary value problem

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