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Biagio Ricceri

Publications and source records attributed to Biagio Ricceri.

At least 37 records · Page 2Linked to original sources
arXiv2019

Another multiplicity result for the periodic solutions of certain systems

In this paper, we deal with a problem of the type $$\cases{(ϕ(u'))'=\nabla_xF(t,u) & in $[0,T]$\cr & \cr u(0)=u(T)\ , \hskip 3pt u'(0)=u'(T)\ ,\cr}$$ where, in particular, $ϕ$ is a homeomorphism from an open ball of ${\bf R}^n$ onto ${\bf R}^n$. Using the theory developed by Brezis and Mawhin in [1] jointly with our minimax theorem proved in [3], we obtain a general multiplicity result, under assumptions of qualitative nature only. Three remarkable corollaries are also presented.

math.CA↗
arXiv2019

Applying twice a minimax theorem

Here is one of the results obtained in this paper: Let $X, Y$ be two convex sets each in a real vector space, let $J:X\times Y\to {\bf R}$ be convex and without global minima in $X$ and concave in $Y$, and let $Φ:X\to {\bf R}$ be strictly convex. Also, assume that, for some topology on $X$, $Φ$ is lower semicontinuous and, for each $y\in Y$ and $λ>0$, $J(\cdot,y)$ is lower semicontinuous and $J(\cdot,y)+λΦ(\cdot)$ is inf-compact. Then, for each $r\in ]\inf_XΦ,\sup_XΦ[$ and for each closed set $S\subseteq X$ satisfying $$Φ^{-1}(r)\subseteq S\subseteq Φ^{-1}(]-\infty,r])\ ,$$ one has $$\sup_Y\inf_SJ=\inf_S\sup_YJ\ .$$

math.OC↗
arXiv2019

Minimax theorems in a fully non-convex setting

In this paper, we establish two minimax theorems for functions $f:X\times I\to {\bf R}$, where $I$ is a real interval, without assuming that $f(x,\cdot)$ is quasi-concave. Also, some related applications are presented.

math.OC↗
arXiv2017

Kirchhoff-type problems involving subcritical and superlinear nonlinearities satisfying no further condition

In this note, we deal with a problem of the type $$\cases {-h\left ( \int_Ω|\nabla u(x)|^2dx\right ) Δu=f(u) & in $Ω$\cr & \cr u_{|\partialΩ}=0\ .\cr}$$ As an application of a new general multiplicity result, we establish the existence of at least three solutions, two of which are global minima of the related energy functional. The only condition assumed on $f$ is that it be subcritical and superlinear: no condition on the behaviour of $f$ at $0$ is requested.

math.AP↗
arXiv2017

Multiple periodic solutions of Lagrangian systems of relativistic oscillators

Let $B_L$ the open ball in ${\bf R}^n$ centered at $0$, of radius $L$, and let $ϕ$ be a homeomorphism from $B_L$ onto ${\bf R}^n$ such that $ϕ(0)=0$ and $ϕ=\nablaΦ$, where the function $Φ:\bar {B_L}\to ]-\infty,0]$ is continuous and strictly convex in $\bar {B_L}$, and of class $C^1$ in $B_L$. Moreover, let $F:[0,T]\times {\bf R}^n\to {\bf R}$ be a function which is measurable in $[0,T]$, of class $C^1$ in ${\bf R}^n$ and such that $\nabla_xF$ satisfies the $L^1$-Carathéodory conditions. Set $$K=\{u\in Lip([0,T],{\bf R}^n) : |u'(t)|\leq L\ for\ a.e.\ t\in [0,T] , u(0)=u(T)\}\ ,$$ and define the functional $I:K\to {\bf R}$ by $$I(u)=\int_0^T(Φ(u'(t))+F(t,u(t)))dt$$ for all $u\in K$. In [1], Brezis and Mawhin proved that any global minimum of $I$ in $K$ is a solution of the problem $$\cases{(ϕ(u'))'=\nabla_xF(t,u) & in $[0,T]$\cr & \cr u(0)=u(T)\ , u'(0)=u'(T)\ .\cr}$$ In the present paper, we provide a set of conditions under which the functional $I$ has at least two global minima in $K$. This seems to be the first result of this kind. The main tool of our proof is the well-posedness result obtained in [3].

math.CA↗
arXiv2017

Four conjectures in Nonlinear Analysis

In this chapter, I formulate four challenging conjectures in Nonlinear Analysis. More precisely: a conjecture on the Monge-Ampère equation; a conjecture on an eigenvalue problem; a conjecture on a non-local problem; a conjecture on disconnectedness versus infinitely many solutions.

math.AP↗
arXiv2017

On a minimax theorem: an improvement, a new proof and an overview of its applications

Theorem 1 of [14], a minimax result for functions $f:X\times Y\to {\bf R}$, where $Y$ is a real interval, was partially extended to the case where $Y$ is a convex set in a Hausdorff topological vector space ([15], Theorem 3.2). In doing that, a key tool was a partial extension of the same result to the case where $Y$ is a convex set in ${\bf R}^n$ ([7], Theorem 4.2). In the present paper, we first obtain a full extension of the result in [14] by means of a new proof fully based on the use of the result itself via an inductive argument. Then, we present an overview of the various and numerous applications of these results.

math.FA↗
arXiv2016

On the infimum of certain functionals

In this note, in particular, we establish the following result: Let $X$ be a real Banach space, $φ\in X^*\setminus \{0\}$ and $ψ:X\to {\bf R}$ a Lipschitzian functional with Lipschitz constant equal to $φ\|_X^{*}$. Then, we have $$\max\left\{\inf_{x\in X}(φ(x)+ψ(x)),\inf_{x\in X}(φ(x)-ψ(x))\right\}=\inf_{x\in X}(φ(x)+|ψ(x)|)$$ and $$\liminf_{\|x\|\to +\infty}(φ(x)+|ψ(x)|)=\inf_{x\in X}(φ(x)+|ψ(x)|)\ .$$

math.FA↗
arXiv2015

Miscellaneous applications of certain minimax theorems. I

Here is one of the results of this paper (with the convention ${{1}\over {0}}=+\infty$): Let $X$ be a real Hilbert space and let $J:X\to {\bf R}$ be a $C^1$ functional, with compact derivative, such that $$α^*:=\max\left \{0,\limsup_{\|x\|\to +\infty}{{J(x)}\over {\|x\|^2}}\right \}<β^*:=\sup_{x\in X\setminus \{0\}}{{J(x)}\over {\|x\|^2}}<+\infty\ .$$ Then, for every $λ\in \left ]{{1}\over {2β^*}}, {{1}\over {2α^*}}\right [$ and for every convex set $C\subseteq X$ dense in $X$, there exists $\tilde y\in C$ such that the equation $$x=λJ'(x)+\tilde y$$ has at least three solutions, two of which are global minima of the functional $x\to {{1}\over {2}}\|x\|^2-λJ(x)-\langle x,\tilde y\rangle$ .

math.FA↗
arXiv2015

A minimax theorem in infinite-dimensional topological vector spaces

In this paper, we obtain a minimax theorem by means of which, in turn, we prove the following result: Let $E$ be an infinite-dimensional reflexive real Banach space, $T:E\to E$ a non-zero compact linear operator, $φ:E\to {\bf R}$ a lower semicontinuous, convex and coercive functional, $I\subset {\bf R}$ a compact interval, with $0\in I$, $ψ:I\to {\bf R}$ a lower semicontinuous convex function. Then, for each $r>φ(0)$, one has $$\sup_{x\in X}\inf_{λ\in I}(φ(T(x)-λx)+ψ(λ))=r+ψ(0)\ ,$$ where $$X=\{x\in E : φ(T(x))\leq r\}\ .$$

math.FA↗
arXiv2015

Convex hull-like property and supported images of open sets

In this note, as a particular case of a more general result, we obtain the following theorem: Let $Ω\subseteq {\bf R}^n$ be a non-empty bounded open set and let $f:\overline Ω\to {\bf R}^n$ be a continuous function which is $C^1$ in $Ω$. Then, at least one of the following assertions holds: $(a)$ $f(Ω)\subseteq \hbox {conv}(f(\partial Ω))\ .$ $(b)$ There exists a non-empty open set $X\subseteq Ω$, with $\overline {X}\subseteq Ω$, satisfying the following property: for every continuous function $g:Ω\to {\bf R}^n$ which is $C^1$ in $X$, there exists $\tildeλ>0$ such that, for each $λ>\tildeλ$, the Jacobian determinant of the function $g+λf$ vanishes at some point of $X$. As a consequence, if $n=2$ and $h:Ω\to {\bf R}$ is a non-negative function, for each $u\in C^2(Ω)\cap C^1(\overline Ω)$ satisfying in $Ω$ the Monge-Ampère equation $$u_{xx}u_{yy}-u_{xy}^2=h\ ,$$ one has $$\nabla u(Ω)\subseteq \hbox {conv}(\nabla u(\partialΩ))\ .$$

math.AP↗
arXiv2014

Energy functionals of Kirchhoff-type problems having multiple global minima

In this paper, using the theory developed in [8], we obtain some results of a totally new type about a class of non-local problems. Here is a sample: Let $Ω\subset {\bf R}^n$ be a smooth bounded domain, with $n\geq 4$, let $a, b, ν\in {\bf R}$, with $a\geq 0$ and $b>0$, and let $p\in \left ] 0,{{n+2}\over {n-2}}\right [$. Then, for each $λ>0$ large enough and for each convex set $C\subseteq L^2(Ω)$ whose closure in $L^2(Ω)$ contains $H^1_0(Ω)$, there exists $v^*\in C$ such that the problem $$\cases {-\left ( a+b\int_Ω|\nabla u(x)|^2dx\right )Δu =ν|u|^{p-1}u+λ(u-v^*(x)) & in $Ω$\cr & \cr u=0 & on $\partialΩ$\cr}$$ has at least three weak solutions, two of which are global minima in $H^1_0(Ω)$ of the corresponding energy functional.

math.AP↗
arXiv2014

Singular points of non-monotone potential operators

In this paper, we establish some results about the singular points of certain non-monotone potential operators. Here is a sample: If $X$ is an infinite-dimensional reflexive real Banach space and if $T:X\to X^*$ is a non-monotone, closed, continuous potential operator such that the functional $x\to \int_0^1T(sx)(x)ds$ is sequentially weakly lower semicontinuous and $\lim_{\|x\|\to +\infty}(\int_0^1T(sx)(x)ds+φ(x))=+\infty$ for all $φ\in X^*$, then the set of all singular points of $T$ is not $σ$-compact.

math.FA↗
arXiv2014

Weights sharing the same eigenvalue

Here is the simplest particular case of our main result: let $f:{\bf R}\to {\bf R}$ be a function of class $C^1$, with $\sup_{\bf R}f'>0$, such that $$\lim_{|ξ|\to +\infty}{{f(ξ)}\over ξ}=0\ .$$ Then, for each $λ>{{π^2}\over {\sup_{\bf R}f'}}$, the set of all $u\in H^1_0(]0,1[)$ for which the problem $$\cases{-v''=λf'(u(x)) v & in $]0,1[$\cr & \cr v(0)=v(1)=0\cr}$$ has a non-zero solution is closed and not $σ$-compact in $H^1_0(]0,1[)$.

math.AP↗
arXiv2014

A property shared by continuous linear functions and holomorphic functions

In this note, we continue to highlight some applications of Theorem 1 of [3]. Here is a sample: Let $X$ be an open set in ${\bf C}^n$, $Ω$ an open convex set in ${\bf C}$ and $f, g : X\to {\bf C}$ two holomorphic functions such that $f(X)\capΩ\neq\emptyset$, $f(X)\setminusΩ\neq \emptyset$ and $g(X)\subseteq Ω$. Then, there exists a set $A$ in $[0,1]$ with the following properties:$(a)$ for each $x\in X$, there exists $λ\in A$ such that $λg(x)+(1-λ)f(x)\inΩ$\ ; $(b)$ for each finite set $B$ in $A$, there exists $u\in X$ such that $μg(u)+(1-μ)f(u)\in {\bf C}\setminusΩ$ for all $μ\in B$.

math.FA↗
arXiv2013

Integral functionals on $L^p$-spaces: infima over sub-level sets

In this paper, we establish the following result: Let $(T,{\cal F},μ)$ be a $σ$-finite measure space, let $Y$ be a reflexive real Banach space, and let $φ, ψ:Y\to {\bf R}$ be two sequentially weakly lower semicontinuous functionals such that $$\inf_{y\in Y}{\min\{φ(y),ψ(y)\}\over {1+\|y\|^p}}>-\infty$$ for some $p>0$. Moreover, assume that $φ$ has no global minima, while $φ+λψ$ is coercive and has a unique global minimum for each $λ>0$. Then, for each $γ\in L^{\infty}(T)\cap L^1(T)\setminus \{0\}$, with $γ\geq 0$, and for each $r>\inf_{Y}ψ$, if we put $$V_{γ,r}= \left \{u\in L^p(T,Y) : \int_Tγ(t)ψ(u(t))dμ\leq r\int_Tγ(t)dμ\right \}\ ,$$ we have $$\inf_{u\in V_{γ,r}} \int_Tγ(t)φ(u(t))dμ= \inf_{ψ^{-1}(r)}φ\int_Tγ(t)dμ .$$

math.OC↗