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In this paper, we present some uniqueness results for systems of ordinary differential equations. All of them are linked by a weak transversality condition at the initial condition, which generalizes those in the previous literature. Several examples are also provided to illustrate our results. [ABSTRACT FROM AUTHOR]

We introduce a new definition of topological degree for a meaningful class of operators which need not be continuous. Subsequently, we derive a number of fixed point theorems for such operators. As an application, we deduce a new existence result for first-order ODEs with discontinuous nonlinearities. [ABSTRACT FROM AUTHOR]

We prove an existence result for systems of differential inclusions driven by multivalued mappings which need not assume closed or convex values everywhere, and need not be semicontinuous everywhere. Moreover, we consider differentiation with respect to a nondecreasing function, thus covering discrete, continuous and impulsive problems under a unique formulation. We emphasize that our existence result appears to be new even when the derivator is the identity, i.e. when derivatives are considered in the usual sense. We also apply our existence theorem for inclusions to derive a new existence result for discontinuous Stieltjes differential equations. Examples are given to illustrate the main results. [ABSTRACT FROM AUTHOR]

Stieltjes differential equations, which contain equations with impulses and equations on time scales as particular cases, simply consist in replacing usual derivatives by derivatives with respect to a nondecreasing function. In this paper we prove new results for the existence of extremal solutions for discontinuous Stieltjes differential equations. In particular, we prove that the pointwise infimum of upper solutions of a Stieltjes differential equation is the minimal solution under certain hypotheses. These results can be adapted to the context of both difference equations and impulsive differential equations. [ABSTRACT FROM AUTHOR]

We present a novel application of Fourier analysis for solving PDEs which is much faster than the usual separation of variables method and, moreover, it implies uniqueness of the obtained solution at the same time. [ABSTRACT FROM AUTHOR]

Stieltjes differential equations, which contain equations with impulses and equations on time scales as particular cases, simply consist on replacing usual derivatives by derivatives with respect to a nondecreasing function. In this paper we prove new existence results for functional and discontinuous Stieltjes differential equations and we show that such general results have real world applications. Specifically, we show that Stieltjes differential equations are specially suitable to study populations which exhibit dormant states and/or very short (impulsive) periods of reproduction. In particular, we construct two mathematical models for the evolution of a silkworm population. Our first model can be explicitly solved, as it consists on a linear Stieltjes equation. Our second model, more realistic, is nonlinear, discontinuous and functional, and we deduce the existence of solutions by means of a result proven in this paper. [ABSTRACT FROM AUTHOR]

We present a teaching approach to uniform continuity on unbounded intervals which, hopefully, may help to meet the following pedagogical objectives: To provide students with efficient and simple criteria to decide whether a continuous function is also uniformly continuous; To provide students with skill to recognize graphically significant classes of both uniformly and nonuniformly continuous functions. Assembling some well-known facts and refining the resulting statement, we establish a useful asymptotic coincidence test for the uniform continuity on unbounded intervals. That test is the core of the present note and yields an easily applicable technique. In particular, one of its immediate consequences is the elementary fact that continuity and existence of horizontal or oblique asymptotes imply uniform continuity. [ABSTRACT FROM AUTHOR]

We present new results on existence and uniqueness of absolutely continuous solutions for systems of discontinuous ordinary differential equations. Our existence result complements an earlier theorem by Bressan and Shen. Basically, we show that a global transversality condition assumed by Bressan and Shen need only be imposed on the sets where the nonlinear part is discontinuous. Our proof, completely different to the one given by Bressan and Shen, uses Krasovskij solutions as a first step. We illustrate the applicability of our result with several examples not covered by the previous literature. The second part of this paper concerns uniqueness. Specifically, we prove uniqueness of solutions for discontinuous systems of differential equations with piecewise Lipschitz continuous nonlinearities and assuming localized Bressan–Shen transversality conditions on the boundaries between different Lipschitz continuity domains. Our uniqueness result appears to be new even in the classical case of continuous nonlinearities. [ABSTRACT FROM AUTHOR]

We establish the existence of positive solutions for systems of second–order differential equations with discontinuous nonlinear terms. To this aim, we give a multivalued vector version of Krasnosel'skiĭ's fixed point theorem in cones which we apply to a regularization of the discontinuous integral operator associated to the differential system. We include several examples to illustrate our theory. [ABSTRACT FROM AUTHOR]

This is a review paper on recent results for different types of generalized ordinary differential equations. Its scope ranges from discontinuous equations to equations on time scales. We also discuss their relation with inclusion and highlight the use of generalized integration to unify many of them under one single formulation. [ABSTRACT FROM AUTHOR]