Let $T$ be a measurable space, $X$ a Banach space while $Y$ a Banach lattice. We consider the class of "upper separated" set-valued functions $F\colon T\times X \rightarrow 2^{Y}$ and investigate the problem of the existence of Carathéodory type selection, that is, measurable in the first variable and order-convex in the second variable.

In this paper, we are mainly concerned with the existence and multiplicity of positive solutions for the following second order periodic boundary value problem involving impulsive effects $$ \begin{cases} -u''+\rho^2u=f(t,u), & t\in J',\\ -\Delta u'|_{t=t_k}=I_k(u(t_k)), & k=1,\ldots,m,\\ u(0)-u(2\pi)=0,\quad u'(0)-u'(2\pi)=0. \end{cases} $$ Here $J'=J\setminus \{t_1,\ldots, t_m\}$, $f\in C(J\times \mathbb{R}^+, \mathbb{R}^+)$, $I_k\in C( \mathbb{R}^+, \mathbb{R}^+)$, where $ \mathbb{R}^+=[0,\infty)$, $J=[0,2\pi]$. The proof of our main results relies on the fixed point theorem on cones. The paper extends some previous results and reports some new results about impulsive differential equations.

We give some generalizations of Krasnosel'skiĭ's fixed point theorem in cones, replacing norms with functionals. We will apply these theorems to obtain at least one positive solution for the boundary value problems for second-order differential equations. Two positive solution results are also obtained.

We study the existence of heteroclinics connecting the two equilibria $\pm 1$ of the third order differential equation $$u'''=f(u)+p(t)u'$$ where $f$ is a continuous function such that $f(u)(u^2-1)> 0$ if $u\neq\pm 1$ and $p$ is a bounded non negative function. Uniqueness is also addressed.

By some new properties of Stepanov-like weighted pseudo almost automorphic functions established by Chang, Zhang and N'Guérékata recently, we shall deal with weighted pseudo almost automorphic solutions to the nonautonomous semilinear evolution equations with a constant delay: $u'(t)=A(t)u(t)+f(t,u(t-h))$, $ t\in\mathbb{R}$ in a Banach space $\mathbb{X}$, where $A(t)$, $t\in\mathbb{R}$ generates an exponentially stable evolution family $\{U(t,s)\}$ and $f \colon\mathbb{R}\times\mathbb{X} \rightarrow \mathbb{X}$ is a ${\rm S}^{p}$-weighted pseudo almost automorphic function satisfying some suitable conditions. We obtain our main results by the Leray-Shauder Alternative theorem.

Suppose that $S$ is a left amenable semitopological semigroup. We prove that if $\mathcal{S}=\{ T_{t}:t\in S\} $ is a uniformly $k$-Lipschitzian semigroup on a bounded closed and convex subset $C$ of a Hilbert space and $k< \sqrt{2}$, then the set of fixed points of $\mathcal{S}$ is a Hölder continuous retract of $C$. This gives a qualitative complement to the Ishihara-Takahashi fixed point existence theorem.

In this paper, we establish several fundamental properties in impulsive semidynamical systems. First, we formulate a counterpart of the continuous dependence on the initial conditions for impulsive dynamical systems, and also establish some equivalent properties. Second, we present several theorems similar to the Poincaré-Bendixson theorem for two-dimensional impulsive systems, i.e. if the omega limit set of a bounded infinite trajectory (with an infinite number of impulses) contains no rest points, then there exists an almost recurrent orbit in the limit set. Further, if the omega limit set contains an interior point, then it is a chaotic set; otherwise, if the limit set contains no interior points, then the limit set contains a periodic orbit or a Cantor-type minimal set in which each orbit is almost recurrent.

We study the following problem introduced by J. Hadamard in 1906: to find sufficient conditions for a local diffeomorphism of an Euclidean space to be a global diffeomorphism. J. Hadamard introduced a celebrated integral condition which is a sufficient condition for the bijectivity of a local diffeomorphism. In this paper we improve the classical result of Hadamard giving a new sufficient condition for a $C^2$ mapping to be bijective.

In this paper, the existence of periodic traveling-wave solutions for a nonlinear Schrödinger system is established using the topological degree theory for positive operators. The method guarantees existence of periodic solutions in a parameter region in the period and phase speed plane.

We prove the existence of at least one $T$-periodic solution $(T> 0)$ for differential equations of the form $$ \left(\frac{u'(t)}{\sqrt{1-{u'}^2(t)}}\right)' =f(u(t))+h(t),\quad \text{in } (0,T), $$ where $f$ is a continuous function defined on $\mathbb{R}$ that satisfies a strong resonance condition, $h$ is continuous and with zero mean value. Our method uses variational techniques for nonsmooth functionals.

The homotopy Conley index along heteroclinic solutions of certain parabolic evolution equations is zero under appropriate assumptions. This result implies that the so-called connecting homomorphism associated with a heteroclinic solution is an isomorphism. Hence, using $\mathbb{Z}$-coefficients it can be viewed as either $1$ or $-1$ - depending on the choice of generators for the homology Conley index. We develop a method to choose such generators, and compute the connecting homomorphism relative to these generators.

Reaction-diffusion systems of equations with integral terms are studied. Essential spectrum of the corresponding linear operators is determined and the Fredholm property is studied. Properness of nonlinear operators is proved and topological degree is constructed.

We use Lyapunov type functions to give new conditions under which a homeomorphism of a compact metric space has the shadowing property. These conditions are applied to establish the topological stability of some homeomorphisms with nonhyperbolic behavior.

Under weaker conditions on parameters, we prove strong convergence theorems of Halpern type iteration schemes for sequences of nonexpansive mappings in ${\rm CAT}(0)$ spaces. Since there is no assumption of the AKTT-condition imposed on the involved mappings, the results improve those of the authors with related researches.

In this paper we complete the study of the following non-local fractional equation involving critical nonlinearities $$ \begin{cases} (-\Delta)^s u - {\lambda}u = |u|^{2^* -2}u \; \mathrm{in} \; \Omega, \\ u=0 \; \mathrm{in} \; \mathbb{R}^n \setminus \Omega, \end{cases} $$ started in the recent papers [13], [17]-[19]. Here $s\in (0,1)$ is a fixed parameter, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda$ is a positive constant, $2^*=2n/(n-2s)$ is the fractional critical Sobolev exponent and $\Omega$ is an open bounded subset of $\mathbb R^n$, $n> 2s$, with Lipschitz boundary. Aim of this paper is to study this critical problem in the special case when $n\not=4s$ and $\lambda$ is an eigenvalue of the operator $(-\Delta)^s$ with homogeneous Dirichlet boundary datum. In this setting we prove that this problem admits a non-trivial solution, so that with the results obtained in [13], [17]-[19], we are able to show that this critical problem admits a nontrivial solution provided

$\bullet$ $n> 4s$ and $\lambda> 0$,

$\bullet$ $n=4s$ and $\lambda> 0$ is different from the eigenvalues of $(-\Delta)^s$,

$\bullet$ $2s< n< 4s$ and $\lambda> 0$ is sufficiently large.

In this way we extend completely the famous result of Brezis and Nirenberg (see [4], [5], [9], [23]) for the critical Laplace equation to the non-local setting of the fractional Laplace equation.

Let $f \in C((0,1)\times (0,\infty),\mathbb{R})$ and $n \in \mathbb{N}$ with $n \geq 2$ such that for each $u \in (0,\infty)$, $r\mapsto r^{2-2n}f(r,u)\colon (0,1)\rightarrow \mathbb{R}$ is nonincreasing and let $D=\{x=(x_1,x_2)\in\mathbb{R}^2: |x|< 1\}$. We show that each positive solution of $$ \Delta u + f(|x|,u) =0 \quad\text{in $D$,} \qquad u=0 \quad\text{on $\partial D$} $$ which satisfies $u(r,\theta)= u(r,\theta+2\pi/n)$ by the polar coordinates is radially symmetric and $u_r(|x|)< 0$ for each $r=|x| \in (0,1)$. We apply our result to the Hénon equation.