Abstract
We consider the initial value problem for a functional differential inclusion with a Volterra multivalued mapping that is not necessarily decomposable inL1n[a,b]. The concept of the decomposable hull of a set is introduced. Using this concept, we define a generalized solution of such a problem and study its properties. We have proven that standard results on local existence and continuation of a generalized solution remain true. The question on the estimation of a generalized solution with respect to a given absolutely continuous function is studied. The density principle is proven for the generalized solutions. Asymptotic properties of the set of generalized approximate solutions are studied.
Highlights
During the last years, mathematicians have been intensively studying see 1, 2 perturbed inclusions that are generated by the algebraic sum of the values of two multivalued mappings, one of which is decomposable
We consider a functional differential inclusion with a Volterra-Tikhonov type in the sequel Volterra type multivalued mapping and we prove that for such an inclusion, the theorem on existence and continuation of a local generalized solution holds true
We introduce the concept of a generalized solution of a functional differential inclusion with a right-hand side which is not necessarily decomposable
Summary
Mathematicians have been intensively studying see 1, 2 perturbed inclusions that are generated by the algebraic sum of the values of two multivalued mappings, one of which is decomposable. Do certain mathematical models of sophisticated multicomponent systems of automatic control see 14 , where, due to the failure of some devices, objects are controlled by different control laws different right-hand sides with the diverse sets of the control admissible values This means that the object’s control law consists of a set of the controlling subsystems. We consider a functional differential inclusion with a Volterra-Tikhonov type in the sequel Volterra type multivalued mapping and we prove that for such an inclusion, the theorem on existence and continuation of a local generalized solution holds true This justifies one of the requirements, which were formulated in the monograph of Filippov 4 for generalized solutions of differential equations with discontinuous right-hand sides. The reason for that is that a multivalued mapping that determines a generalized solution the definition is given below may not be closed in the weak topology of Ln1 a, b , as this mapping is not necessarily convex-valued
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