Claudio A. Gallegos

University of Chile · Departamento de Matemáticas

We present a new Gronwall inequality for Stieltjes integrals, which improves numerous existing results, and has a simple proof based on the quotient rule for Stieltjes integrals. As an application, we obtain uniqueness theorems for measure differential equations and nabla dynamic equations. Finally, we revisit the topic from the perspective of Stie...

This paper is devoted to study the existence of solutions for a class of measure differential equations –abbreviated, MDEs– with a general nonlocal condition. Specifically, by a general nonlocal condition we understand a nonlocal condition modeled in terms of a multivalued map. We distinguish two cases, problems formulated in finite and infinite di...

In this paper, we are interested in investigating notions of stability for generalized linear differential equations (GLDEs). Initially, we propose and revisit several definitions of stability and provide a complete characterization of them in terms of upper bounds and asymptotic behaviour of the transition matrix. In addition, we illustrate our st...

We unveil instances where nonautonomous linear systems manifest distinct nonuniform \(\mu \)-dichotomy spectra despite admitting nonuniform \((\mu , \varepsilon )\)-kinematic similarity. Exploring the theoretical foundations of this lack of invariance, we discern the pivotal influence of the parameters involved in the property of nonuniform \(\mu \...

This paper is concerned with a version of the Lebesgue dominated convergence theorem (DCT) which has been stated for the Kurzweil–Stieltjes integral of real functions. Our objective in this work is to analyze the extension of this result to include vector functions with values in Banach spaces. We establish that the mentioned convergence theorem fo...

In this paper, we provide a necessary and sufficient condition ensuring the property of exponential dichotomy for periodic linear systems of generalized differential equations. This condition allows us to revisit a recent result of admissibility, obtaining an alternative formulation with a particular simplicity.

This paper is devoted to study the qualitative properties of hybrid measure differential equations (HMDEs, for short). We establish several results on the existence of global solutions, including the existence of regulated, continuous, differentiable and S-asymptotically $\omega$-periodic solutions. Furthermore, we present a result on continuous de...

In this paper, we provide a necessary and sufficient condition ensuring the property of exponential dichotomy for periodic linear systems of generalized differential equations. This condition allow us to revisit a recent result of admissibility, obtaining an alternative formulation with a particular simplicity.

In this work, we introduce the measure functional differential equations (MFDEs) with infinite time-dependent delay, and we study the correspondence between the solutions of these equations and the solutions of the generalized ordinary differential equations (GODEs, for short) in Banach spaces. Using the theory of GODEs , we obtain results concerni...

In this paper, we are interested in investigating stability results for generalized ordinary differential equations (generalized ODEs in short), and their applications to measure differential equations and dynamic equations on time scales. First, we establish stability, asymptotic and exponential stability for the trivial solution of generalized OD...

In this work we are concerned with the existence of fixed points for multivalued maps defined on Banach spaces. Using the Banach spaces scale concept, we establish the existence of a fixed point of a multivalued map in a vector subspace where the map is only locally Lipschitz continuous. We apply our results to the existence of mild solutions and a...

In this paper, we investigate the asymptotic behavior for measure differential equations and dynamic equations on time scales via generalized ordinary differential equations (generalized ODEs, for short). At first, we establish new results that guarantee the existence of unbounded solutions for generalized ODEs, and after that, using the known corr...