J.M. Soriano

Global minimum point of a convex function

In this paper we prove the existence of an unique global minimum point of a convex function under some smoothness conditions. Our proof permits us to calculate numerically such a minimum point utilizing a constructive homotopy method.

On the existence of zero points

We prove the existence of at least one zero of a continuous function under one of two smoothness conditions. The proof is constructive: numerical methods can be used to follow a solution curve with a computer. A global implicit function theorem is given.

Extremum points of a convex function

Abstract We prove the existence of a global minimum point of a convex function under some smoothness conditions. The proof is indeed constructive; numerical methods can be used to follow a solution curve with a computer.

On the number of zeros of a mapping

We prove the existence of the exact number of zeros of a mapping under smoothness conditions. The proof is constructive: numerical methods can be used to follow a solution curve with a computer. Weakening the conditions, we have the existence of at least one zero.

Existence of zeros for bounded perturbations of proper mappings

Sufficient conditions, asserting the existence of at least one zero of bounded perturbations of proper mappings, are given. The conditions are easily tested. The proof of this theorem is constructive, showing the existence of a solution curve that can be followed by numerical methods.

Mappings sharing a value on finite-dimensional spaces

Sufficient conditions are given to assert that differentiable mappings and differentiable proper mappings between R^n and R^m share a value. The proof of the result is constructive and is based upon continuation methods.

Fredholm and compact mappings sharing a value

Sufficient conditions are given to assert that two differentiable mappings between Banach spaces have common values. The proof is essentially based upon continuation methods.

On the existence of Zero points of a continuous function

Abstract A sufficient condition is given to assert that a continuous mapping between Rm and Rn has a zero. The constructive proof of the result is based upon continuation methods and supplies the existence of a path leading to the zero point.

A compactness condition

Sufficient conditions are given to assert that a differentiable mapping between K^m and K^n, where K=R or K=C, has a zero. The constructive proof of the result is based upon continuation methods and supplies the existence of a path leading to the zero point.

STABLE AND UNSTABLE STATIONARY TRAJECTORIES

The stability of stationary trajectories of a particular autonomous system is studied. The proof of the result is based upon continuation methods.