Cubic differential systems in real commutative two-dimensional algebras are classified up to orbital topological equivalence via the solubility of polynomial equations in algebras. As a by-product, existence of bounded solutions in such systems is studied via complex structures in the algebras. Application to the existence of periodic solutions to $n$-dimensional differential systems "cubic at infinity" is given.

The aim of this paper is to extend the definition of geodesics to conical manifolds, defined as submanifolds of ${\mathbb R}^n$ with a finite number of singularities. We look for an approach suitable both for the local geodesic problem and for the calculus of variation in the large. We give a definition which links the local solutions of the Cauchy problem (1.1) with variational geodesics, i.e. critical points of the energy functional. We prove a deformation lemma (Theorem 2.2) which leads us to extend the Lusternik-Schnirelmann theory to conical manifolds, and to estimate the number of geodesics (Theorem 3.4 and Corollary 3.5). In Section 4, we provide some applications in which conical manifolds arise naturally: in particular, we focus on the brachistochrone problem for a frictionless particle moving in $S^n$ or in ${\mathbb R}^n$ in the presence of a potential $U(x)$ unbounded from below. We conclude with an appendix in which the main results are presented in a general framework.

In this paper, we study positive periodic solutions to singular second order differential systems. It is proved that such a problem has at least two positive periodic solutions. The proof relies on a nonlinear alternative of Leray-Schauder type and on Krasnosel'skiĭ fixed point theorem on compression and expansion of cones.

We study the existence of non-collision periodic solutions with prescribed energy for the following singular Hamiltonian systems: $$ \begin{cases} \ddot q+\nabla V(q)=0, \\ \displaystyle \frac{1}{2}|\dot q|^2+V(q)=H. \end{cases} $$ In particular for the potential $V(q)\sim -1/{\rm dist} (q,D)^\alpha$, where the singular set $D$ is a non-empty compact subset of $\mathbb R^N$, we prove the existence of a non-collision periodic solution for all $H> 0$ and $\alpha\in (0,2)$.

Solutions to generalizations of the Volterra and Hammerstein integral inclusions are found by using the fixed point theorems of Covitz-Nadler and Bohnenblust-Karlin. Several illustrative examples are presented. Some conditions are given which also allow Lipschitz solutions to be obtained.

In this paper we deal with the Peano phenomenon for general initial-boundary value problems of quasilinear parabolic equations with arbitrary even order space derivatives.

The nonlinearity is assumed to be a continuous or continuously Fréchet differentiable function. Using a method of transformation to an operator equation and employing the theory of proper, Fredholm (linear and nonlinear) and Nemitskiĭ operators, we study the existence of solution of the given problem and qualitative and quantitative structure of its solution and bifurcation sets. These results can be applied to the different technical and natural science models.

In this paper we prove existence results for first and second order impulsive neutral functional differential inclusions under the mixed Lipschitz and Carathéodory conditions.

Using degree theory for countably condensing maps due to Väth, we give an invariance of domain theorem for countably condensing vector fields. The key tool is Borsuk's theorem for odd countably condensing maps.

In the last two decades, many algebras of generalized functions have been constructed, particularly the so-called generalized Sobolev algebras. Our goal is to study the latter and some of their main properties. In this framework, we pose and solve a nonlinear degenerated Dirichlet problem with irregular data such as Dirac generalized functions.

Some random coincidence point theorems are proved. The results of Benavides et al. [Random fixed points of set-valued operators, Proc. Amer. Math. Soc. 124 (1996), 831–838], Itoh [Random fixed point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl. 67 (1979), 261–273], Shahzad and Latif [A random coincidence point theorem, J. Math. Anal. Appl. 245 (2000), 633–638], Tan and Yuan [Random fixed point theorems and approximation, Stochastic Anal. Appl. 15 (1997), 103–123] and Xu [Some random fixed point theorems for condensing and nonexpansive operators, Proc. Amer. Math. Soc. 110 (1990), 495–500] are either extended or improved.