An existence and qualitative result for discontinuous implicit differential equations

Paolo Cubiotti


Let T > 0 and YRn. Given a function f:[0,T] × Rn ×  Y → R, we consider the Cauchy problem f(t, u, u′) = 0 in [0, T], u(0) = ξ. We prove an existence and qualitative result for the generalized solutions of the above problem. In particular, our result does not require the continuity of f with respect to the first two variables. As a matter of fact, a function f(t, x, y) satisfying our assumptions could be discontinuous (with respect to x) even at all points xRn. We also study the dependence of the solution set ST(ξ) from the initial point ξRn. In particular, we prove that, under our assumptions, the multifunction ST admits a multivalued selection Φ which is upper semicontinuous with nonempty compact acyclic values.


Implicit discontinuous differential equations; Cauchy problem; Generalized solutions; Selections; Differential inclusions

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Copyright (c) 2018 Paolo Cubiotti

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