We study the existence of positive periodic solutions to nonlinear elliptic and parabolic equations with oblique and dynamical boundary conditions and non-local terms. The results are obtained through fixed point theory, topological degree methods and properties of related linear elliptic problems with natural boundary conditions and possibly non-symmetric principal part. As immediate consequences, we also obtain estimates on the principal eigenvalue for non-symmetric elliptic eigenvalue problems.

In this paper we study existence, uniqueness and data dependence for the solutions of some nonlinear fractional differential equations in Banach spaces. By means of topological degree method for condensing maps via a singular Gronwall inequality with mixed type integral terms, many new results are obtained.

In this paper we investigate the problem of existence and multiplicity of solutions for a non-homogeneous fourth order Yamabe type equation. We exhibit a family of solutions concentrating at two points, provided the domain contains one hole and we give a multiplicity result if the domain has multiple holes. Also we prove a multiplicity result for vanishing positive solutions in a general domain.

Fixed point theorems of Krasnosel'skiĭ type are obtained for the localization of positive solutions in a set defined by means of the norm and of a semi-norm. In applications to elliptic boundary value problems, the semi-norm comes from the Moser-Harnack inequality for nonnegative superharmonic functions whose use is crucial for the estimations from below. The paper complements and gives a fixed point alternative approach to our similar results recently established in the frame of critical point theory. It also provides a new method for discussing the existence and multiplicity of positive solutions to elliptic boundary value problems.

Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear reaction diffusion equation $$ \begin{alignat}{2} u_t+\beta(x)u-\Delta u&=f(x,u),&\quad &(t,x)\in[0,+\infty[\times\Omega,\\ u&=0,&\quad &(t,x)\in[0,+\infty[\times\partial\Omega \end{alignat} $$ in $H^1_0(\Omega)$ has finite Hausdorff dimension. Here $\Omega$ is an arbitrary, possibly unbounded, domain in $\mathbb R^3$ and $f(x,u)$ is a nonlinearity of subcritical growth. The nonlinearity $f(x,u)$ needs not to satisfy any dissipativeness assumption and the invariant subset $\mathcal I$ needs not to be an attractor. If $\Omega$ is regular, $f(x,u)$ is dissipative and $\mathcal I$ is the global attractor, we give an explicit bound on the Hausdorff dimension of $\mathcal I$ in terms of the structure parameter of the equation.

We present a random topological degree effectively applicable mainly to periodic problems for random differential inclusions. These problems can be transformed to the existence problems of random fixed points or periodic orbits of the associated Poincaré translation operators. The solvability can be so guaranteed either directly by means of nontrivial topological invariants (random degree, index of a random direct potential) or via a randomization scheme using deterministic results which are ``periodicity stable'' under implemented parameter values.

In this paper, by using the topological degree theory for multivalued maps, we develop the method of guiding functions to deal with the problem of global structure of periodic solutions for functional differential inclusions with infinite delay. As example we consider the global structure of periodic solutions of feedback control systems with infinite delay.

We show that near any given minimizing sequence of paths for the mountain pass lemma, there exists a critical point whose polarization is also a critical point. This is motivated by the fact that if any polarization of a critical point is also a critical point and the Euler-Lagrange equation is a second-order semi-linear elliptic problem, T. Bartsch, T. Weth and M. Willem (J. Anal. Math., 2005) have proved that the critical point is axially symmetric.

We present some abstract theorems on the existence of selections and graph-approximations of set-valued mappings with convex values in the equivariant setting, i.e. maps commuting with the action of a compact group. Some known results of the Michael, Browder and Cellina type are generalized to this context. The equivariant measurable as well as Carathéodory selection/approximation problems are also studied.

We prove a generalization of the individual ergodic theorem of the information theory and we apply it to give a new proof of the Shereshevsky inequality connecting the metric entropy and Lyapunov exponents of dynamical systems generated by cellular automata.

In 1970, Schirmer discussed about topological properties of the fixed point set of multifunctions ([Properties of the fixed point set of contractive multifunctions, Canad. Math. Bull. 13 (1970), 169-173]). Later, some authors continued this study by providing different conditions ([M.C. Alicu and O. Mark, Some properties of the fixed points set for multifunctions, Studia Univ. Babes-Bolyai Math. 25 (1980), 77-79] and [B. Ricceri, Une propriete topologique de l'ensemble des points fixes d'une contraction multivoque a valeurs convexes, Atti. Acc. Lincei Rend. 81 (1987), 283-286]). Recently, Sintamarian proved results on absolute retractivity of the common fixed points set of two multivalued operators ([A topological property of the common fixed points set of two multivalued operators, Nonlinear Anal. 70 (2009), 452-456] and [A topological property of the common fixed points set of two multivalued operators satisfying a Latif-Beg type condition, Fixed Point Theory 9 (2008), 561-573]). We shall present some results on absolute retractivity of the common fixed points set of two multifunctions by using different conditions.