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Biagio Ricceri

Publications and source records attributed to Biagio Ricceri.

At least 55 records · Page 3Linked to original sources
arXiv2013

A characterization related to a two-point boundary value problem

In this short note, we establish the following result: Let $f:[0,+\infty[\to [0,+\infty[$, $α:[0,1]\to ]0,+\infty[$ be two continuous functions, with $f(0)=0$. Assume that, for some $a>0$, the function $ξ\to {{\int_0^ξf(t)dt}\over {ξ^2}}$ is non-increasing in $]0,a]$. Then, the following assertions are equivalent: $(i)$ for each $b>0$, the function $ξ\to {{\int_0^ξf(t)dt}\over {ξ^2}}$ is not constant in $]0,b]$ ; $(ii)$ for each $r>0$, there exists an open interval $I\subseteq ]0,+\infty[$ such that, for every $λ\in I$, the problem $$\cases {-u''=λα(t)f(u) & in $[0,1]$\cr & \cr u>0 & in $]0,1[$\cr & \cr u(0)=u(1)=0\cr}$$ has a solution $u$ satisfying $$\int_0^1|u'(t)|^2dt<r\ .$$

math.CA↗
arXiv2013

A note on spherical maxima sharing the same Lagrange multiplier

In this paper, we establish a general result on spherical maxima sharing the same Lagrange multiplier of which the following is a particular consequence: Let $X$ be a real Hilbert space. For each $r>0$, let $S_r=\{x\in X : \|x\|^2=r\}$. Let $J:X\to {\bf R}$ be a sequentially weakly upper semicontinuous functional which is Gâteaux differentiable in $X\setminus \{0\}$. Assume that $$\limsup_{x\to 0}{{J(x)}\over {\|x\|^2}}=+\infty\ .$$ Then, for each $ρ>0$, there exists an open interval $I\subseteq ]0,+\infty[$ and an increasing function $φ:I\to ]0,ρ[$ such that, for each $λ\in I$, one has $$\emptyset\neq \left \{x\in S_{φ(λ)} : J(x)=\sup_{S_{φ(λ)}}J\right\}\subseteq \{x\in X : x=λJ'(x)\}\ .$$

math.OC↗
arXiv2013

Revisiting a theorem on multifunctions of one real variable

In this paper, we intend to revisit Theorem 2 of [3] formulating it in a way that, weakening the hypotheses and, at the same time, highlighting the richer conclusion allowed by the proof, it can potentially be applicable to a broader range of different situations. Samples of such applications are also given.

math.FA↗
arXiv2012

A strict minimax inequality criterion and some of its consequences

In this paper, we point out a very flexible scheme within which a strict minimax inequality occurs. We then show the fruitfulness of this approach presenting a series of various consequences. Here is one of them: Let $Y$ be a finite-dimensional real Hilbert space, $J:Y\to {\bf R}$ a $C^1$ function with locally Lipschitzian derivative, and $φ:Y\to [0,+\infty[$ a $C^1$ convex function with locally Lipschitzian derivative at 0 and $φ^{-1}(0)=\{0\}$. Then, for each $x_0\in Y$ for wich $J'(x_0)\neq 0$, there exists $δ>0$ such that, for each $r\in ]0,δ[$, the restriction of $J$ to $B(x_0,r)$ has a unique global minimum $u_r$ which satisfies $$J(u_r)\leq J(x)-φ(x-u_r)$$ for all $x\in B(x_0,r)$, where $B(x_0,r)=\{x\in Y: \|x-x_0\|\leq r\}\ .$

math.OC↗
arXiv2012

Another four critical points theorem

In this paper, making use of Theorem 2 of [5], we establish a new four critical points theorem which can be regarded as a companion to Theorem 1 of [4]. We also present an application to the Dirichlet problem for a class of quasilinear elliptic equations.

math.AP↗
arXiv2011

A structure theorem on non-homogeneous linear equations in Hilbert spaces

A very particular by-product of the result announced in the title reads as follows: Let $(X,<\cdot,\cdot>)$ be a real Hilbert space, $T:X\to X$ a compact and symmetric linear operator, and $z\in X$ such that the equation $T(x)-\|T\|x=z$ has no solution in $X$. For each $r>0$, set $γ(r)=\sup_{x\in S_r}J(x)$, where $J(x)=< T(x)-2z,x>$ and $S_r=\{x\in X:\|x\|^2=r\}$. Then, the function $γ$ is $C^1$, increasing and strictly concave in $]0,+\infty[$, with $γ'(]0,+\infty[)=]\|T\|,+\infty[$; moreover, for each $r>0$, the problem of maximizing $J$ over $S_r$ is well-posed, and one has $$T(\hat x_r)-γ'(r)\hat x_r=z$$ where $\hat x_r$ is the only global maximum of $J_{|S_r}$.\par

math.FA↗
arXiv2009

On an elliptic Kirchhoff-type problem depending on two parameters

In this paper, we consider the Dirichlet problem associated to an elliptic Kirchhoff-type equation depending on two parameters. Under rather general and natural assumptions, we prove that, for certain values of the parameters, the problem has at least three solutions.

math.AP↗
arXiv2004

Covering dimension and nonlinear equations

Theorem: Let X and Y be two Banach spaces, Phi: X to Y a continuous, linear, surjective operator, and Psi: X to Y a completely continuous operator with bounded range. Then, one has dim{x in X : Phi(x)=Psi(x)} >= dim(Phi^{-1}(0)). Here dim denotes the covering dimension.

math.FA↗
arXiv2004

Some research perspectives in nonlinear functional analysis

The object of this lecture is to propose a series of conjectures and problems in different fields of analysis. They have been formulated with the aim of introducing some innovative methods in the study of classical topics, as open mappings, fixed points, critical points, global minima, control theory.

math.FA↗
arXiv2004

A general variational principle and some of its applications

In this paper, given a reflexive real Banach space X and two sequentially weakly lower semicontinuous functionals Phi, Psi on X with Psi strongly continuous and coercive, we are mainly interested in the existence of infinitely many local minima of the functional capital Phi + r Psi for each sufficiently real r.

math.OC↗