We introduce the notion of invariant subspaces for multilinear operators from which the invariant subspace problems for multilinear and polynomial operators arise. We prove that polynomial operators acting in a finite dimensional complex space and even polynomial operators acting in a finite dimensional real space have eigenvalues. These results enable us to prove that polynomial and multilinear operators acting in a finite dimensional complex space, even polynomial and even multilinear operators acting in a finite dimensional real space have nontrivial invariant subspaces. Furthermore, we prove that odd polynomial operators acting in a finite dimensional real space either have eigenvalues or are homotopic to scalar operators; we then use this result to prove that odd polynomial and odd multilinear operators acting in a finite dimensional real space may or may not have invariant subspaces.

In this paper, we study the Fučík spectrum of a linear operator in a general setting. We illustrate the influence of various aspects on the structure of the Fučík spectrum. Mainly, we describe the inadmissible areas where the Fučík spectrum of a given operator cannot be located.

We study the existence and multiplicity of positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation $$ \begin{cases} \displaystyle -{\rm div}\bigg( \frac{\nabla v} {\sqrt{1 - |\nabla v|^2}}\bigg)= f(|x|,v) &\quad \text{in } B_R, \\ v=0 & \quad \text{on } \partial B_R, \end{cases} $$ where $B_R$ is a ball in $\mathbb{R}^N$ ($N\ge 2$). According to the behaviour of $f=f(r,s)$ near $s=0$, we prove the existence of either one, two or three positive solutions. All results are obtained by reduction to an equivalent non-singular one-dimensional problem, to which variational methods can be applied in a standard way.

This paper is about boundary value problems of the form \begin{equation*} \begin{cases} -{\rm div} [\nabla \Phi(\nabla u)] = f(x,u) &\mbox{in } \Omega, \\ u=0 &\mbox{on } \partial\Omega, \end{cases} \end{equation*} where $\Phi$ is a convex function of $\xi\in \mathbb{R}^N$, rather than a function of the norm $|\xi|$. The problem is formulated appropriately in an anisotropic Orlicz-Sobolev space associated with $\Phi$. We study the existence of solutions and some other properties of the above problem and its corresponding variational inequality in such space.

In this paper, we firstly present the existence of nondecreasing solutions of a fractional quadratic integral equations involving Erdélyi-Kober singular kernels for three provided parameters $\alpha \in (1/2\},1)$, $\beta\in (0,1]$ and $\gamma\in [\beta(1-\alpha)-1,\infty)$. Moreover, we prove this restriction on $\alpha$ and $\beta$ can not be improved. Secondly, we obtain the uniqueness and nonuniqueness of the monotonic solutions by utilizing a weakly singular integral inequality and putting $\gamma\in [1/2-\alpha,\infty)$. Finally, two numerical examples are given to illustrate the obtained results.

In this paper, we consider the existence and multiplicity of solutions to the elliptic equation with resonance. We classify the linear elliptic equation and obtain some new conditions on the existence and multiplicity for asymptotically linear elliptic equation by using critical point theory.

In this work we deal with existence and uniqueness of almost automorphic solutions for abstract semilinear differential equations using a mix of fixed point theory and extrapolation spaces theory. We apply our abstract results in the framework of transmission problems for the Bernoulli-Euler plate equation and heat conduction theory.

This paper concerns results about minimal, recurrent and almost periodic motions in impulsive semidynamical systems. In the first part, we investigate general properties of minimal sets. In the sequel, we study some relations among minimal, recurrent and almost periodic motions. Some important results from the classical dynamical systems theory are generalized to the impulsive case, as Birkhoff's theorem for instance.

In the present paper we mainly consider the second order evolution inclusion with proximal normal cone: \begin{equation} \begin{cases} -\ddot{x}(t)\in N_{K(t)}(\dot{x}(t))+F(t,x(t),\dot{x}(t)), \quad \text{a.e.}\\ \dot x(t)\in K(t),\\ x(0)=x_0,\quad\dot x(0)=u_0, \end{cases} \tag{$*$} \end{equation} where $t\in I=[0,T]$, $E$ is a separable reflexive Banach space, $K(t)$ a ball compact and $r$-prox-regular subset of $E$, $N_{K(t)}(\,\cdot\,)$ the proximal normal cone of $K(t)$ and $F$ an u.s.c. set-valued mapping with nonempty closed convex values. First, we prove the existence of solutions of $(*)$. After, we give an other existence result of $(*)$ when $K(t)$ is replaced by $K(x(t))$.

This paper is devoted to the asymptotic behavior of the semi-linear strongly damped wave equation with forcing term only belongs to $H^{-1}$. Some refined decompositions of the solution have been presented, which allow to remove the quasi-monotone condition $f'(s)> -k$. The asymptotic regularity and existence of a finite-dimensional exponential attractor are established under the usual assumptions.

In this paper, we make application of some three-critical points results to establish the existence of at least three solutions for a boundary value problem for the quasilinear second order differential equation on a compact interval $[a,b]\subset\mathbb{R}$, $$ \begin{cases} -u''=(\lambda f(x,u)+g(x,u))h(x,u') &\text{in } (a,b),\\ u(a)=u(b)=0, \end{cases} $$ under appropriate hypotheses. We exhibit the existence of at least three (weak) solutions.

Given a map $\Phi$ defined on bounded subsets of the (base) metric space $X$ and with bounded sets as its values, one can follow the orbits $A,\Phi(A),\Phi^2(A),\ldots$, of nonempty, closed, and bounded sets $A$ in $X$. This is the system $(\Phi, X)$. On the other hand, the same orbits can be viewed as trajectories of points in the hyperspace $X^\sharp$ of nonempty, closed, and bounded subsets of $X$. This is the system $(\Phi, X^\sharp)$. We study the existence and properties of global attractors for both $(\Phi, X)$ and $(\Phi, X^\sharp)$. We give very basic conditions on $\Phi$, stated at the level of the base space $X$, that are necessary and sufficient for the existence of a global attractor for $(\Phi, X)$. Continuity is not among those conditions, but if $\Phi$ is continuous in a certain sense then the attractor and the $\omega$-limit sets are $\Phi$-invariant. If $(\Phi, X)$ has a global attractor, then $(\Phi, X^\sharp)$ has a global attractor as well. Every point of the global attractor of $(\Phi, X^\sharp)$ is a compact set in $X$, and the union of all the points of that attractor is the global attractor of $(\Phi, X)$.

This paper studies existence results for boundary value problems of nonlinear fractional hybrid differential inclusions of quadratic type, by using a fixed point theorem of Dhage [B.C. Dhage, Existence results for neutral functional differential inclusions in Banach algebras, Nonlinear Anal. 64 (2006), 1290-1306].

In this paper, the existence and localization result will be proven for multivalued vector Dirichlet problem with an upper-Carathéodory right-hand side by using bound sets approach. Since Scorza-Dragoni type technique will be furthermore applied, the conditions for bounding functions can be required directly on the boundaries of bound sets and not at some vicinity of them.

We consider the semilinear homogeneous Dirichlet boundary value problem for the second order equation on a finite interval with nonlinear impulses in the derivative at prescribed points. We introduce Landesman-Lazer type necessary and sufficient conditions for resonance problems and generalize the Fredholm alternative results for linear operators. An interaction between nonlinear restoring force and nonlinear impulses is presented.

In this paper we survey a set of Brouwer fixed point theorem equivalents. These equivalents are divided into four loops related to (1) the Borsuk retraction theorem, (2) the Himmelberg fixed point theorem, (3) the Gale lemma and (4) the Nash equilibrium theorem.