Pablo Amster

Existence of positive T-periodic solutions of a generalized Nicholson’s blowflies model with a nonlinear harvesting term☆

Abstract We give sufficient and necessary conditions for the existence of at least one positive T -periodic solution for a generalized Nicholson’s blowflies model with a nonlinear harvesting term. Our results extend those of the previous work Li and Du (2008) [1] .

The prescribed mean curvature equation for nonparametric surfaces

We study the prescribed mean curvature equation for nonparametric surfaces, obtaining existence and uniqueness results in the Sobolev space W2,p. We also prove that under appropriate conditions the set of surfaces of mean curvature H is a connected subset of W2,p. Moreover, we obtain existence results for a boundary value problem which generalizes the one-dimensional periodic problem for the mean curvature equation.

Solutions to a stationary nonlinear Black-Scholes type equation

We study by topological methods a nonlinear differential equation generalizing the Black–Scholes formula for an option pricing model with stochastic volatility. We prove the existence of at least a solution of the stationary Dirichlet problem applying an upper and lower solutions method. Moreover, we construct a solution by an iterative procedure.

Stationary solutions for two nonlinear Black-Scholes type equations

We study by topological methods two different problems arising in the Black-Scholes model for option pricing. More specifically, we consider a nonlinear differential equation which generalizes the Black-Scholes formula when the volatility is assumed to be stochastic. On the other hand, we study a model with transaction costs.

Topological methods in the study of boundary value problems

Introduction.- Shooting type methods.- The Banach Fixed Point Theorem.- Schauders Theorem and applications.- Topological degree: an introduction.- Applications.- Basic facts on metric and normed spaces.- Brief review of ODEs.- Hints and Solutions to Selected Exercises.

Periodic solutions of systems with singularities of repulsive type

Abstract Motivated by the classical Coulomb central motion model, we study the existence of T-periodic solutions for the nonlinear second order system of singular ordinary differential equations u′′ + g(u) = p(t). Using topological degree methods, we prove that when the nonlinearity g : ℝN\{0} →ℝN is continuous, repulsive at the origin and bounded at infinity, if an appropriate Nirenberg type condition holds then either the problem has a classical solution, or else there exists a family of solutions of perturbed problems that converge uniformly and weakly in H1 to some limit function u. Furthermore, under appropriate conditions we prove that u is a classical solution.

Existence and regularity of weak solutions to the prescribed mean curvature equation for a nonparametric surface

It is known that for the parametric Plateau’s problem, weak solutions can be obtained as critical points of a functional (see [2, 6, 7, 8, 10, 11]). The nonparametric case has been studied for H = H(x,y) (and generally H = H(x1, . . . ,xn) for hypersurfaces in Rn+1) by Gilbarg, Trudinger, Simon, and Serrin, among other authors. It has been proved [5] that there exists a solution for any smooth boundary data if the mean curvature H ′ of ∂ satisfies H ′ ( x1, . . . ,xn )≥ n n−1 ∣∣H (x1, . . . ,xn)∣∣ (1.3)

Stability of Hahnfeldt angiogenesis models with time lags

Abstract Mathematical models of angiogenesis, pioneered by Hahnfeldt, are under study. To enrich the dynamics of three models, we introduced biologically motivated time-varying delays. All models under study belong to a special class of nonlinear nonautonomous delay differential systems with non-Lipschitz nonlinearities. Explicit conditions for the existence of positive global solutions and the equilibria solutions were obtained. Based on a notion of an M -matrix, new results are presented for the global stability of the system and were used to prove local stability of one model. For a local stability of a second model, the recent result for a Lienard-type second-order differential equation with delays was used. It was shown that models with delays produce a complex and nontrivial dynamics. Some open problems are presented for further studies.

On a generalization of Lazer-Leach conditions for a system of second order ODE's

We study the existence of periodic solutions for a nonlinear second order system of ordinary differential equations. Assuming suitable Lazer-Leach type conditions, we prove the existence of at least one solution applying topological degree methods.

On existence of periodic solutions for Kepler type problems

We prove existence and multiplicity of periodic motions for the forced 2-body problem under conditions of topological character. In the different cases, the lower bounds obtained for the number of solutions are related to the winding number of a curve in the plane, the homology of a space in