A general nonlinear initial boundary value problem \begin{align} \frac{\partial u}{\partial t} - F(x,t,u,D^{1}u,\dots, D^{2m}u)&=f(x,t), \tag*{$(1)$} \\ &\hskip -30pt (x,t)\in Q_{T}\equiv \Omega\times (0,T), \\ G_{j}(x,t,u,\dots, D^{m_{j}}u)&=g_{j}(x,t), \tag*{$(2)$}\\ &\hskip-30pt (x,t)\in S_{T}\equiv \partial\Omega\times (0,T), j=\overline{1,m}, \\ u(x,0)=h(x),\quad& x\in\Omega \tag*{$(3)$} \end{align} is being considered, where $\Omega$ is a bounded open set in $\mathbb R^n$ with sufficiently smooth boundary. The problem (1)-(3) is then reduced to an operator equation $Au=0$, where the operator $A$ satisfies (S)$_+$ condition. The local and global solvability of the problem (1)-(3) are achieved via topological methods developed by the first author. Further applications involving the convergence of Galerkin approximations are also given.

In this paper we consider the exterior Neumann problem (P) involving the critical Sobolew exponent. We investigate two cases where the coefficient $a$ interferes or not with the spectrum of the Lapalce operator with the Neumann boundary conditions. In both cases we establish the existence of solutions.

In this paper we construct an example of superlinear indefinite weighted elliptic mixed boundary value problem exhibiting a mushroom shaped compact component of positive solutions emanating from the trivial solution curve at two simple eigenvalues of a related linear weighted boundary value problem. To perform such construction we have to adapt to our general setting some of the rescaling arguments of H. Amann and J. López-Gómez [Section 4, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations 146 (1998), 336–374] to get a priori bounds for the positive solutions. Then, using the theory of [H. Amann, Dual semigroups and second order linear elliptic boundary value problems, Israel J. Math. 45 (1983), 225–254], [S. Cano-Casanova, Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems, Nonlinear Anal. 49 (2002), 361–430] and [S. Cano-Casanova and J. López-Gómez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems, J. Differential Equations 178 (2002), 123–211], we give some sufficient conditions on the nonlinearity and the several potentials of our model setting so that the set of values of the parameter for which the problem possesses a positive solution is bounded. Finally, the existence of the component of positive solutions emanating from the trivial curve follows from the unilateral results of P. H. Rabinowitz ([Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513], [J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics, vol. 426, CRC Press, Boca Raton, 2001]). Monotonicity methods, re-scaling arguments, Liouville type theorems, local bifurcation and global continuation are among the main technical tools used to carry out our analysis.

We study $2\pi$-periodic solutions of $$ u''+f(t,u)=0 $$ using positively homogeneous asymptotic approximations of this equation near zero and infinity. Our main results concern the degree of $I-P$, where $P$ is the Poincaré map associated to these approximations. We indicate classes of problems, some with degree 1 and others with degree different from 1. Considering results based on first order approximations, we work out examples of equations for which the degree is the negative of any integer.

In this paper we study the existence of periodic solutions of asymptotically linear Hamiltonian systems which may not satisfy the Palais-Smale condition. By using the Conley index theory and the Galerkin approximation methods, we establish the existence of at least two nontrivial periodic solutions for the corresponding systems.

This paper deals with the existence of absolutely continuous solutions of a differential inclusion with state constraint in a separable Banach space $$ x( 0) =x_{0}, \quad x( t) \in C( t) ,\quad \dot{x}( t) \in F( t,x( t) ) $$ where $C\colon [ 0,a] \rightarrow X$ is a multifunction with closed graph $G$ and $F\colon G\rightarrow X$ is a convex compact valued multifunction which is separately measurable in $t\in[ 0,a] $ and separately upper semicontinuous in $x\in X$. Application to a non convex sweeping process is also considered.

This paper discusses the topological structure of the set of solutions for a variety of Volterra equations and inclusions. Our results rely on the existence of a maximal solution for an appropriate ordinary differential equation.

The dimension and infinitesimal groups of a Cantor dynamical system $(X,T)$ are inductive limits of sequences of homomorphisms defined by a proper Bratteli diagram of $(X,T)$. A method of selecting sequences of homomorphisms determining the dimension and the infinitesimal groups of $(X,T)$ based on non-proper Bratteli diagrams is described. The dimension and infinitesimal groups of Rudin-Shapiro, Morse and Chacon flows are computed.