We study properties of an operator $S$ which assigns to compact subsets of $[0,1]$ their centers of distances. We consider its continuity points and its upper semicontinuity points as well as orbits and fixed points of this operator. We also compute centers of distances of some classic sets. Using properties of operator $S$ we show that the family of achievement sets is of the first category in the space of compact subsets of $[0,1]$.

In this article we consider a homogeneous eigenvalue problem ruled by the fractional $g$-Laplacian operator whose Euler-Lagrange equation is obtained by minimization of a quotient involving Luxemburg norms. We prove existence of an infinite sequence of variational eigenvalues and study its behavior as the fractional parameter $s\uparrow 1$ among other stability results.

A bi-spatial random attractor is obtained for the stochastic FitzHugh-Nagumo systems on unbounded thin domains when the initial space is $L^2\times L^2$ and the terminate space is $L^p\times L^2$. Furthermore, we establish the upper semi-continuity of attractors under the $p$-norm when a family of $(n+1)$-dimensional thin domains degenerates into an $n$-dimensional unbounded domain.

In this paper, we prove sharp blow-up and global existence results for a time fractional diffusion-wave equation with a nonlinear memory term in a bounded domain, where the fractional derivative in time is taken in the sense of the Caputo type. Moreover, we also give a result for nonexistence of global solutions to a wave equation with a nonlinear memory term in a bounded domain. The proof of blow-up results is based on the eigenfunction method and the asymptotic properties of solutions for an ordinary fractional differential inequality.

The aim of this paper is to study the multiplicity and concentration of positive solutions to the $(p,q)$ Kirchhoff-type problems involving a positive potential and a continuous nonlinearity with critical growth at infinity. Applying penalization techniques, truncation methods and the Lusternik-Schnirelmann theory, we investigate a relationship between the number of positive solutions and the topology of the set where the potential $V$ attains its minimum values.

In this paper, we study the existence and nonexistence of solutions to the following Choquard-type equation\begin{equation*}-\Delta u+(V+\lambda)u=(I_\alpha*F(u))f(u)\quad\text{in } \mathbb{R}^N,\end{equation*}having prescribed mass $\int_{\mathbb{R}^N}u^2=a$, where $\lambda\in\mathbb{R}$ will arise as a Lagrange multiplier, $N\geq 3$, $\alpha\in(0,N)$, $I_\alpha$ is Riesz potential. Under suitable assumptions on the potential function $V$ and the nonlinear term $f$, $a_0\in[0,\infty)$ exists such that the above equation has a positive ground state normalized solution if $a\in(a_0,\infty)$ and one has no ground state normalized solution if $a\in(0,a_0)$ when $a_0> 0$ by comparison arguments. Moreover, we obtain sufficient conditions for $a_0=0$.

In this study, we compute some examples of sectional category secat$(f)$ and sectional number sec$(f)$ for continuous maps $f$ related to finite spaces. Moreover, we introduce an invariant secat$_k(f)$ for a map $f$ between finite spaces using the $k$-th barycentric subdivision and show the equality secat$_k(f)=$ secat$(\mathcal{B}(f))$ for sufficiently large $k$, where $\mathcal{B}(f)$ is the induced map on the associated polyhedra.

We present a general scheme to find variationally characterized critical points of a functional $I\colon H \to \mathbb{R}$ on a Hilbert space $H$ with hypothesis where the usual Nehari method is not directly applicable. These critical points arise as minima of $I$ over a suitable subset of the associated Nehari set and are obtained with the aid of fibering methods. Moreover, we derive a comparison result with mountain pass critical values. The abstract results will be applied to classes of logarithmic Choquard and nonlinear Schrödinger equations.

In this paper we establish, in a separable Hilbert space, a result asserting the existence of absolutely continuous solutions for a system made up of a first-order differential inclusion governed by time and state-dependent maximal monotone operators; and an ordinary differential equation. From this result, we derive existence of absolutely continuous solutions to a second-order differential inclusion governed by time and state-dependent maximal monotone operators.

We construct a general framework that generates classes of multilinear operators between Banach spaces which encompasses, as particular cases, the several classes of summing type multilinear operators that have been studied individually in the literature. Summing operators by blocks in the isotropic and anisotropic cases are taken into account. The classes we create are shown to be Banach ideals of multilinear operators and applications to coherence and coincidence theorems are provided.

This paper deals with the Gromov-Hausdorff stability for systems generated of reaction-diffusion equations whose diffusion coefficients are simultaneously large in a bounded smooth domain. The appropriated framework is presented to establish the conjugation between the attractors by means of $\varepsilon$-isometries.

In this paper, we study the existence and asymptotic behavior to a diffusion system which is non-local in time. As consequence of our theorems we deduce new results for the fractional-in-time Keller-Segel model. Our approach is intimately related with the Sonine kernels.

In this paper, we are concerned with a generalized evolution dynamical system, called fractional differential variational-hemivariational inequality (FDVHVI, for short), which is composed of a nonlinear fractional evolution inclusion and a time-dependent mixed variational-hemivariational inequality in the framework of Banach spaces. The objective of this paper is fourfold. The first one is to investigate the nonemptiness as well as the compactness of the mild solutions set to the FDVHVI. The second aim is to study the optimal control problems described by the FDVHVI. The third goal is to establish the well-posedness results of the FDVHVI, including the existence, uniqueness, and stability. Furthermore, the sensitivity analysis of a perturbed problem associated to the FDVHVI with respect to the initial state and the two parameters is also obtained. Finally, a comprehensive fractional model is given to illustrate the validity of our main results.

We study some critical elliptic problems involving the difference of two nonlocal operators, or the difference of a local operator and a nonlocal operator. The main result is the existence of two nontrivial weak solutions, one with negative energy and the other with positive energy, for all sufficiently small values of a parameter. The proof is based on an abstract result recently obtained in [22].

This paper concerns with a class of double phase problem without Ambrosetti-Rabinowitz-type growth condition. Under reasonable hypotheses, we establish the existence of infinitely many solutions by using the variant fountain theorems due to Zou [41].

We consider the minimizers of $L^{2}$-critical inhomogeneous variational problems with a spatially decaying nonlinear term in an open bounded domain $\Omega$ of $\mathbb{R}^{N}$ which contains $0$. We prove that there is a threshold $a^{*}> 0$ such that minimizers exist for $0< a< a^{*}$ and the minimizer does not exist for any $a> a^{*}$. In contrast to the homogeneous case, we show that both the existence and nonexistence of minimizers may occur at the threshold $a^*$ depending on the value of $V(0)$, where $V(x)$ denotes the trapping potential. Moreover, under some suitable assumptions on $V(x)$, based on a detailed analysis on the concentration behavior of minimizers as $a\nearrow a^*$, we prove local uniqueness of minimizers when $a$ is close enough to $a^*$.

This paper studies the existence and multiplicity of normalized solutions to the lower critical Choquard equation with a $L^2$-subcritical local perturbation and kinds of bounded potentials\begin{equation*}\begin{cases}-\Delta u+V(x)u\\\qquad =\lambda u+\big(I_{\alpha}\ast|u|^{({N+\alpha})/{N}}\big)|u|^{({N+\alpha})/{N}-2}u+\mu|u|^{q-2}u& \text{in } \mathbb{R}^N, \\\displaystyle\int_{\mathbb{R}^N}|u|^2dx=a^2,\end{cases}\end{equation*}where $N\geq 1$, $\mu, a> 0$, $2< q< 2+{4}/{N}$, $\alpha\in (0,N)$, $I_{\alpha}$ is the Riesz potential, $V(x)$ is a bounded potential and $\lambda\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier.

We give formal normal forms for parabolic logarithmic transseries $f=z+\dots $, with respect to parabolic logarithmic normalizations. Normalizations are given algorithmically, using fixed point theorems, as limits of Picard's sequences in appropriate complete metric spaces, in contrast to transfinite term-by-term eliminations described in former works. Furthermore, we give the explicit formula for the residual coefficient in the normal form and show that, in the larger logarithmic class, we can even eliminate the residual term from the normal form.