Using topological degree techniques, we state and prove new sufficient conditions for the existence of a solution of the Neumann boundary value problem $$ (|x'|^{p-2} x')' +f(t, x)+ h(t, x) =0, \quad x'(0) = x'(1)=0, $$ when $h$ is bounded, $f$ satisfies a one-sided growth condition, $f + h$ some sign condition, and the solutions of some associated homogeneous problem are not oscillatory. A generalization of Lyapunov inequality is proved for a $p$-Laplacian equation. Similar results are given for the periodic problem.

The existence of a fixed point for maps of the form Identity + Contraction acting on $\mathbb R^2$ is established under quite general conditions. A counterexample is given in $\mathbb R^3$.

The object here is to study the following question in the homotopy theory of configuration spaces of a general manifold $M$: When is the fibration $\mathbb F_{k+1}(M)\rightarrow\mathbb F_r(M)$, $r< k+1$, fiber homotopically trivial? The answer to this question for the special cases when $M$ is a sphere or euclidean space is given in [E. Fadell and S. Husseini, Geometry and Topology of Configuration Spaces, Springer, New York, 2001]. The key to the solution of the problem for compact manifolds $M$ is the study of an associated question for the punctured manifold $M-q$, where $q$ is a point of $M$. The fact that $M-q$ admits a nonzero vector field plays a crucial role. Also required are investigations into the Lie algebra $\pi_*(\mathbb F_{k+1}(M))$, with special attention to the punctured case $\pi_*(\mathbb F_k(M-q))$. This includes the so-called Yang-Baxter equations in homotopy, taking into account the homotopy group elements of $M$ itself as well as the classical braid elements.

We are concerned with multiplicity results for solutions of some reversed variational inequalities, in which the inequality is opposite with respect to the classical inequalities introduced by Lions and Stampacchia. The inequalities we study arise from a family (P$_\omega$) of elliptic problems of the fourth order when $\omega$ tends to $\infty$. We use two basic tools: the $\nabla$-theorems and a theorem about the multiplicity of "asymptotically critical" points. In the last section some open problems are listed.

We present two types of perturbations with reverse effects on some scalar fully nonlinear second order elliptic differential operators: on the other hand, first order perturbations which destroy the global solvability of the Dirichlet problem, in smooth bounded domains of $\mathbb R^n$; on the other hand, an integral perturbation which restore the local solvability, on compact connected manifolds without boundary.

The homotopy perturbation method proposed by the present author is further improved in this paper, which is proved to be effective and convenient to solving nonlinear equations.

In this paper, existence and uniqueness of generalized solutions of some first order Hamilton Jacobi equations are proved. This task is accomplished by showing that the value function for a certain problem of the calculus of variations is the unique solution of the PDE. This can be viewed as a representation formula of the solution.

The paper provides topological characterization for solution sets of differential inclusions with (not necessarily smooth) functional constraints in Banach spaces. The corresponding compactness and tangency conditions for the right hand-side are expressed in terms of the measure of noncompactness and the Clarke generalized gradient, respectively. The consequences of the obtained result generalize the known theorems about the structure of viable solution set for differential inclusions.

We establish an existence result for a class of quasilinear elliptic boundary value problems with jumping nonlinearities using variational arguments. First we calculate certain homotopy groups of sublevel sets of the asymptotic part of the variational functional. Then we use these groups to show that the full functional admits a linking geometry and hence a min-max critical point.

We prove two Hardy-Sobolev type inequalities in ${\mathcal D}^{1,2}({\mathbb R}^N)$, resp. in $H^1_0(\Omega)$, where $\Omega$ is a bounded domain in ${\mathbb R}^N$, $N\geq 3$. The framework involves the singular potential $\vert x\vert ^{-a}$, with $a\in (0,1)$. Our paper extends previous results established by Bianchi and Egnell [A Note on the Sobolev inequality, J. Funct. Anal. 100 (1991), 18–24], resp. by Brezis and Lieb [Inequalities with remainder terms, J. Funct. Anal. 62 (1985), 73–86], corresponding to the case $a=0$.

A reaction-diffusion equation on a family of three dimensional thin domains, collapsing onto a two dimensional subspace, is considered. In [The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Differential Equations 173 (2001), 271–320] it was proved that, as the thickness of the domains tends to zero, the solutions of the equations converge in a strong sense to the solutions of an abstract semilinear parabolic equation living in a closed subspace of $H^1$. Also, existence and upper semicontinuity of the attractors was proved. In this work, for a specific class of domains, the limit problem is completely characterized as a system of two-dimensional reaction-diffusion equations, coupled by mean of compatibility and balance boundary conditions.

We shall prove the existence of minimizers of the following functional $f(u)=\int_{0}^{T}L(x,u(x),u'(x))dx$ without convexity assumption. As a consequence of this result and the duality described in [A. Nowakowski, Metody wariacyjne dla nieliniowych problem\'ow Dirichleta (Chapter 6), Wydawnictwo Naukowo Techniczne, Warszawa, 1994] we derive the existence of solutions for the Dirichlet problem for a certain differential inclusion being a generalization of the Euler-Lagrange equation of the functional $f$.