We study a Dirichlet problem for an elliptic equation of resonant type involving a general nonlocal term. Using a result of Ricceri, we prove that the solution set for such equation has a positive topological dimension, and contains a nondegenerate connected component. In particular, the solution set has the cardinality of the continuum.

In this paper we present a new approach for solving the problem of the existence of closed trajectories for autonomous differential equations without the uniqueness property. To this aim, we are using a special class of set-valued maps, called weighted carriers or weighted maps.

We prove the existence of at least two geometrically different periodic solutions with winding number $N$ for the forced relativistic pendulum. The instability of a solution is also proved. The proof is topological and based on the version of the Poincaré-Birkhoff theorem by Franks. Moreover, with some restriction on the parameters, we prove the existence of twist dynamics.

The aim of this paper is to prove the existence result of viable solutions for the differential inclusion $$ \dot{x}(t) \in F(x(t)),\qquad x(t) \in K\quad \text{on } [0,T], $$ where $F$ is an upper semicontinuous set-valued map with compact values.

We consider the stability of positive solutions to semilinear elliptic systems under a new general sublinear condition and its variants. Using the stability result and bifurcation theory, we prove the existence and uniqueness of positive solution and obtain the precise global bifurcation diagram of the system being a single monotone solution curve.

In this paper, we obtain the existence of infinitely many classical solutions to the multi-point boundary value system $$ \begin{cases} -(\phi_{p_i}(u'_{i}))'=\lambda F_{u_{i}}(x,u_{1},\ldots,u_{n}),\qquad t\in (0,1),\\ \displaystyle \\ u_{i}(0)=\sum_{j=1}^m a_ju_i(x_j),\quad u_{i}(1)=\sum_{j=1}^m b_ju_i(x_j), \end{cases} \quad i=1,\ldots,n. $$ Our analysis is based on critical point theory.

In this paper, the existence and continuation of solutions for the Cauchy initial value problem of fractional functional differential equations in an arbitrary Banach space is discussed under hypotheses based on Carathéodory condition and the measure of noncompactness. In addition, an example is given to show that the criteria on existence of solutions for the initial value problem of fractional differential equations in finite-dimensional spaces may not be true in infinite-dimensional cases.

The aim of this paper is to explore the possibilities of Conley index techniques in the study of heteroclinic connections between finite and infinite invariant sets. For this, we remind the reader of the Poincaré compactification: this transformation allows to project a $n$-dimensional vector space $X$ on the $n$-dimensional unit hemisphere of $X\times \mathbb{R}$ and infinity on its $(n-1)$-dimensional equator called the sphere at infinity. Under a normalizability condition, vector fields on $X$ are mapped to vector fields on the Poincaré hemisphere whose associated flows leave the equator invariant. The dynamics on the equator reflects the dynamics at infinity, but are now finite and may be studied by Conley index techniques. Furthermore, we observe that some non-isolated behavior may occur around the equator, and introduce the concept of an invariant set at infinity of isolated invariant dynamical complement. Through the construction of an extended phase space together with an extended flow, we are able to adapt the Conley index techniques and prove the existence of connections to such non-isolated invariant sets.

In this paper, we consider the non-autonomous $p$-Laplacian equation with a dynamic boundary condition. The existence and structure of a compact uniform attractor in $W^{1,p}(\Omega)\times W^{1-1/p,p}(\Gamma)$ are established for the case of time-dependent internal force $h(t)$. While the nonlinearity $f$ and the boundary nonlinearity $g$ are dissipative for large values without restriction on the growth order of the polynomial.

On étudie en détail les espaces des lacets des espaces de configuration de $3$ points dans $\mathbb{R}^n$ et dans la sphere $S^n$. Notre approche consiste a établir tout d'abord un résultat de formalité de ces espaces, et ensuite d'utiliser l'homologie de Hochschild pour calculer l'homologie de l'espace des lacets. Dans le cas de $\mathbb{R}^n$, nous retrouvons facilement et de façon plus conceptuelle un résultat de scindement homologique établi par Fadell et Husseini.

The main theorem of this article provides a necessary and sufficient condition for a pair of maps from a two-complex into a one-complex (a graph) can be homotoped to be coincidence free. As a consequence of it, we prove that a pair of maps from a two-complex into the circle can be homotoped to be coincidence free if and only if the two maps are homotopic. We also obtain an alternative proof for the known result that every pair of maps from a graph into the bouquet of a circle and an interval can be homotoped to be coincidence free. As applications of the main theorem, we characterize completely when a pair of maps from the bi-dimensional torus into the bouquet of a circle and an interval can be homotoped to be coincidence free, and we prove that every pair of maps from the Klein bottle into such a bouquet can be homotoped to be coincidence free.

We consider the following superlinear Kirchhoff type nonlocal problem: $$ \begin{cases} \displaystyle -\bigg(a+b\int_\Omega |\nabla u|^2dx\bigg)\Delta u =\lambda f(x,u) & \text{in } \Omega,\ a > 0, \ b > 0, \ \lambda > 0, \\ u=0 &\text{on } \partial\Omega. \end{cases} $$ Here, $f(x,u)$ does not satisfy the usual superlinear condition, that is, for some $\theta > 0$, $$ 0\leq F(x,u)\overset{\triangle}{=} \int_0^u f(x,s)ds \leq \frac1{2+\theta}f(x,u)u, \quad \text{for all } (x,u)\in \Omega \times \mathbb{R}^+ $$ or the following variant $$ 0\leq F(x,u) \overset{\triangle}{=} \int_0^u f(x,s)ds \leq \frac1{4+\theta}f(x,u)u, \quad \text{for all } (x,u)\in \Omega \times \mathbb{R}^+ $$ which is quiet important and natural. But this superlinear condition is very restrictive eliminating many nonlinearities. The aim of this paper is to discuss how to use the mountain pass theorem to show the existence of non-trivial solution to the present problem when we lose the above superlinear condition. To achieve the result, we first consider the existence of a solution for almost every positive parameter $\lambda$ by varying the parameter $\lambda$. Then, it is considered the continuation of the solutions.

In this paper, we consider initial value problems for a class of nonlinear impulsive fractional differential equations involving the Caputo fractional derivative in a Banach space. We give a natural formula of the solution and some related existence results by applying Mönch's fixed point theorem and the technique of measures of noncompactness.