We describe a Lohner-type algorithm for rigorous integration of dissipative PDEs. Using it for the Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions we give a computer assisted proof the existence of multiple periodic orbits.

We study the existence of positive solutions for perturbed impulsive integral equations. Our setting is quite general and covers a wide class of impulsive boundary value problems. We also study other cases that can be treated in a similar manner. The main ingredient in our theory is the classical fixed point index theory for compact maps.

With the linear growth of the nonlinearity and a new compactness condition involving the asymptotic behavior of its potential at infinity, we establish the existence and multiplicity results of nontrivial solutions for semilinear elliptic Dirichlet problems. The nonlinearity may cross multiple eigenvalues.

In this paper we consider the existence and uniqueness of positive solutions of the multi-point boundary value problem \begin{gather} - (\phi_p(u')' + (a + g(x,u,u'))\phi_p(u) = 0 , \quad\text{a.e. on $(-1,1)$}, \tag{1} \\ u(\pm 1) = \sum^{m^\pm}_{i=1}\alpha^\pm_i u(\eta^\pm_i) , \tag{2} \end{gather} where $p> 1$, $\phi_p(s) := |s|^{p-2} s$, $s \in \mathbb R$, $m^\pm \ge 1$ are integers, and $$ \eta_i^\pm \in (-1,1),\quad \alpha_i^\pm > 0,\quad i = 1,\dots,m^\pm, \quad \sum^{m^\pm}_{i=1} \alpha_i^\pm < 1 . $$ Also, $a \in L^1(-1,1)$, and $g \colon [-1,1] \times \mathbb R^2 \to \mathbb R$ is Carathéodory, with \begin{equation} g(x,0,0) = 0, \quad x \in [-1,1]. \tag{3} \end{equation} Our criteria for existence of positive solutions of (1), (2) will be expressed in terms of the asymptotic behaviour of $g(x,s,t)$, as $s \to \infty$, and the principal eigenvalues of the multi-point boundary value problem consisting of the equation \begin{equation} -\phi_p (u')' + a \phi_p (u) = \lambda \phi_p (u) , \quad \text{on $(-1,1)$}, \tag{4} \end{equation}

In this paper we study when the minimal number of roots of the so-called convenient maps from two-dimensional CW complexes into closed surfaces is zero. We present several necessary and sufficient conditions for such a map to be root free. Among these conditions we have the existence of specific liftings for the homomorphism induced by the map on the fundamental groups, existence of the so-called mutation of a specific homomorphism also induced by the map, and existence of particular solutions of specific systems of equations on free groups over specific subgroups.

We give a definition of the spectral flow for paths of bounded essentially hyperbolic operators on a Banach space. The spectral flow induces a group homomorphism on the fundamental group of every connected component of the space of essentially hyperbolic operators. We prove that this homomorphism completes the exact homotopy sequence of a Serre fibration. This allows us to characterise its kernel and image and to produce examples of spaces where it is not injective or not surjective, unlike what happens for Hilbert spaces. For a large class of paths, namely the essentially splitting, the spectral flow of $ A $ coincides with $ -{\rm ind}(F_A) $, the Fredholm index of the differential operator $ F_A (u) = u' - A u $.

We prove, by asymptotic center techniques and some inequalities in Banach spaces, that if $E$ is $p$-uniformly convex Banach space, $C$ is a nonempty bounded closed convex subset of $E$, and $T\colon C\rightarrow C$ has lipschitzian iterates (with some restrictions), then the set of fixed-points is not only connected but even a retract of $C$. The results presented in this paper improve and extend some results in [J. Górnicki, A remark on fixed point theorems for lipschitzian mappings, J. Math. Anal. Appl. 183 (1994), 495–508], [J. Górnicki, The methods of Hilbert spaces and structure of the fixed-point set of lipschitzian mapping, Fixed Point Theory and Applications, Hindawi Publ. Corporation, 2009, Article ID 586487].