Rodrigo López Pouso

Fixed point theorems in ordered abstract spaces

We extend some fixed point theorems in L-spaces, obtaining extensions of the Banach fixed point theorem to partially ordered sets.

Existence and approximation of solutions for first-order discontinuous difference equations with nonlinear global conditions in the presence of lower and upper solutions☆

Abstract This paper is devoted to the study of the existence of solutions of first-order difference equations verifying nonlinear conditions that involve the global behavior of the solution. We prove that the existence of lower and upper solutions warrants the existence of such solutions lying in the sector formed by the mentioned functions. We also can prove that some classical results for differential equations are not true in general for this case.

Examples of the nonexistence of a solution in the presence of upper and lower solutions

Standard results for boundary value problems involving second-order ordinary differential equations ensure that the existence of a well-ordered pair of lower and upper solutions together with a Nagumo condition imply existence of a solution. In this note we introduce some examples which show that existence is not guaranteed if no Nagumo condition is satisfied.

Approximate solutions to a new class of nonlinear diffusion problems

In this paper it is considered a new class of nonlinear differential equations that arises in the study of some diffusion processes. The authors present an existence and uniqueness result by constructing a sequence of approximate solutions. Theoretical and numerical aspects are considered.

Ordinary differential equations and systems with time-dependent discontinuity sets

In this paper we prove new existence results for nonautonomous systems of first order ordinary differential equations under weak conditions on the nonlinear part. Discontinuities with respect to the unknown are allowed to occur over general classes of time-dependent sets which are assumed to satisfy a kind of inverse viability condition.

NONRESONANCE CONDITIONS AND EXTREMAL SOLUTIONS FOR FIRST-ORDER IMPULSIVE PROBLEMS UNDER WEAK ASSUMPTIONS

In this work we shall study the existence of extremal solutions for an impulsive problem with functional-boundary conditions and weak regularity assumptions, not only on the right-hand side of the equation and on the functions that define the boundary conditions, but also on the impulse functions, which will be required to be nondecreasing, but not continuous as well, as is customary in the literature. Moreover, in order to prove one of our results we shall study a general impulsive linear problem, giving a complete characterisation of resonance for it.

On first-order ordinary differential equations with nonnegative right-hand sides

Existence of extremal solutions for scalar initial value problems is investigated. We shall concentrate upon nonlinearities having constant sign, which lead to new existence results.

First-Order Singular and Discontinuous Differential Equations

We use subfunctions and superfunctions to derive sufficient conditions for the existence of extremal solutions to initial value problems for ordinary differential equations with discontinuous and singular nonlinearities.

Necessary and Sufficient Conditions for Existence and Uniqueness of Solutions of Second-Order Autonomous Differential Equations

The main result ensures that the scalar problem , , has a nonconstant locally solution if and only if there exists a nontrivial interval such that , , for almost all and Necessary and sufficient conditions for local and global uniqueness and for existence of periodic solutions are also established.

Does Lipschitz with Respect to x Imply Uniqueness for the Differential Equation y' = f(x,y)?

Of course, the answer to the question posed in the title is no, in general, but, surprisingly enough, yes in a significant situation to be detailed later. This paper aims to bring to the attention of the widest possible mathematical audience an elementary result, based on the theorem of differentiation of inverse functions, which extends the applicability of uniqueness theorems. Loosely speaking, it transforms every uniqueness theorem into an alternative version of it with the roles of the dependent and the independent variables interchanged in the assumptions. A remarkable example of such a theorem is the version of Lipschitz’s theorem alluded to in the first paragraph, but there are at least as many new possibilities as old uniqueness theorems.