Abstract

The classical Hartman's Theorem in [18] for the solvability of the vector Dirichlet problem will be generalized and extended in several directions. We will consider its multivalued versions for Marchaud and upper-Caratheodory right-hand sides with only certain amount of compactness in Banach spaces. Advanced topological methods are combined with a bound sets technique. Besides the existence, the localization of solutions can be obtained in this way.

Highlights

  • IntroductionOne of the most classical results in the theory of boundary value problems (b.v.p.) is Hartman’s Theorem

  • One of the most classical results in the theory of boundary value problems (b.v.p.) is Hartman’s Theorem.Theorem 1.1. ([18, Theorem 1], cf. [19]) Let us consider the vector Dirichlet problem x(t) = f (t, x(t), x (t)), t ∈ [0, T ], (1)x(T ) = x(0) = 0, where f : [0, T ] × Rn × Rn → Rn is a continuous function

  • In [3] and [5], we proved in detail the following three results, stated here in the form of propositions, which deal with the existence and the localization of a solution of a multivalued Dirichlet problem

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Summary

Introduction

One of the most classical results in the theory of boundary value problems (b.v.p.) is Hartman’s Theorem. Hartman-type conditions, Dirichlet problem, abstract spaces, multivalued operators, topological methods, bound sets technique. It was already published in 1960, as far as we know, it has not yet been improved, but only extended in many ways After this introduction, we recall some elements of functional and multivalued analyses, jointly with the definition of a measure of non-compactness and condensing maps. A function β : P (E) → N is called a measure of non-compactness (m.n.c.) in E if β(co Ω) = β(Ω), for all Ω ∈ P (E), where co Ω denotes the closed convex hull of Ω. [8, Theorem 2.2] Let E be a Banach space and K ⊂ E a nonempty, open, convex, bounded set such that 0 ∈ K.

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