General existence principles for Stieltjes differential equations with applications to mathematical biology |
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Authors: | Rodrigo López Pouso Ignacio Márquez Albés |
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Affiliation: | Departamento de Análise Matemática, Estatística e Optimización, Universidade de Santiago de Compostela, 15782, Faculty of Mathematics, Campus Vida, Santiago de Compostela, Spain |
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Abstract: | 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. |
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Keywords: | 34A36 34K05 34K05 Lower solution Ordinary differential equations Impulsive differential equations Dynamic equations Time scales |
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