Nonlinear Analysis: Theory, Methods & Applications
Boundary value problems for semi-continuous delayed differential inclusions on Riemannian manifolds
Introduction
We consider a delayed second order differential inclusion on a connected Riemannian manifold , where denotes the covariant derivative given by the Levi-Cevita connection. The set-valued mapping is assumed to be semi-continuous, and hence incorporates the case of differential equations generated by discontinuous vector fields. Problems of this type frequently arise from the presence of friction in mechanical systems. In order to include the application of delayed feedback controls, time lags in the state variable are considered as well. Then the boundary values become for given states and terminal time . Notation As for the notation, we write for the family of non-empty compact subsets of a metric space . For subsets we make use of the Hausdorff semi-distance . For a normed space and we set as usual . We also write for the family of non-empty compact and convex subsets of a normed space . Assumption 1.1 The set-valued mapping satisfies the following growth condition. For any , , there is a compact, convex set with There are constants and such that for all .
By this assumption we can write the second order differential inclusion (1) in the usual form The Riemannian metric induces a canonic metric on the tangent bundle . Thus, we recall the definition of semi-continuous set-valued mappings on metric spaces. Definition 1.2 Let , be metric spaces. A set-valued mapping is upper semi-continuous, if for all and all there is a such that for all . A set-valued mapping is lower semi-continuous, if for all and all there is a such that for all . A set-valued mapping is semi-continuous, if it is upper or lower semi-continuous.
The following theorem is the main result of the paper. Theorem 1.3 Let the semi-continuous set-valued mapping satisfyAssumption 1.1. Then for any , , that are not conjugate along a certain geodesics, there is a such that for any delay and any time the boundary value problem(1), (2)has a solution.
There is a rich literature on nonlinear boundary value problems for differential inclusions in linear spaces. Many results can already be found in [4]. In contrast, second order boundary value problems on Riemannian manifolds are considered in a few papers only; see [8], [9], [10]. An introduction to this field can be found in [7]. The technique presented in this work relies on an application of the Bressan–Colombo theorem along with Schauder’s fixed point theorem. To this end, the problem needs to be formulated within the tangent space of the initial value. This simplification works for small times only. Most results are proved for uniformly bounded forces. In the recent paper [10] the framework is extended to less than quadratically growing (w.r.t. the velocity variable) forces.
The method of proof in the present paper is based on mapping degree arguments, such as those presented in [6]. The time horizon is viewed as a homotopy parameter. Certain transformations show that the degenerate time represents the force free situation of a geodesic connecting the boundary values on . Although the method of proof is completely different to the one used in [10], the supposition of less than quadratically growing forces is needed in the present paper as well.
Section snippets
Proof of Theorem 1.3
In this section we prove Theorem 1.3 for the case of upper semi-continuous right-hand sides . Here, the set-valued situation can be reduced to a single-valued situation via approximate selections. The lower semi-continuous case does not require a separate proof, since by Michael’s selection theorem (see e.g. Lemma 2.1 in [5] for a strong version in Banach spaces) there is a continuous selection. Lemma 2.1 Let the upper semi-continuous set-valued mapping satisfyAssumption 1.1. For any ,
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