Boundary value problems for semi-continuous delayed differential inclusions on Riemannian manifolds

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Abstract

The existence of solutions to nonlinear second order boundary value problems for differential inclusions with time lags is investigated. The set-valued mappings generating the differential inclusions are semi-continuous with compact and convex values. It turns out that solutions exist for small time intervals and arbitrary time lags in the state variable. The proof relies on mapping degree methods.

Introduction

We consider a delayed second order differential inclusion (x(t),Ddtẋ(t))Φ(x(th),x(t),ẋ(t)) on a connected Riemannian manifold M, where Ddt denotes the covariant derivative given by the Levi-Cevita connection. The set-valued mapping Φ:M×TMCC(TM) 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 h0 in the state variable are considered as well. Then the boundary values become x(t)=x0for t[h,0],x(T)=x1, for given states x0,x1M and terminal time T>0.

Notation

As for the notation, we write C(X) for the family of non-empty compact subsets of a metric space X. For subsets A,BX we make use of the Hausdorff semi-distance distH(A,B)supaAdist(a,B)[0,]. For a normed space X and AC(X) we set as usual AmaxaAa. We also write CC(X) for the family of non-empty compact and convex subsets of a normed space X.

Assumption 1.1

The set-valued mapping Φ:M×TMCC(TM) satisfies the following growth condition. For any zM, (x,y)TM, there is a compact, convex set F(z,x,y)TxM with Φ(z,x,y)=(x,F(z,x,y)). There are constants c0 and α[0,2) such that F(z,x,y)xc(1+yxα) for all zM(x,y)TM.

By this assumption we can write the second order differential inclusion (1) in the usual form Ddtẋ(t)F(x(th),x(t),ẋ(t)). The Riemannian metric induces a canonic metric on the tangent bundle TM. Thus, we recall the definition of semi-continuous set-valued mappings on metric spaces.

Definition 1.2

Let X, Y be metric spaces. A set-valued mapping F:XC(Y) is upper semi-continuous, if for all xX and all ϵ>0 there is a δ>0 such that distH(F(z),F(x))ϵ for all zB(δ;x). A set-valued mapping F:XC(Y) is lower semi-continuous, if for all xX and all ϵ>0 there is a δ>0 such that distH(F(x),F(z))ϵ for all zB(δ;x). A set-valued mapping F:XC(Y) 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 Φ:M×TMCC(TM) satisfyAssumption 1.1. Then for any x0M , x1M{x0} , that are not conjugate along a certain geodesics, there is a T0(0,) such that for any delay h0 and any time T(0,T0]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 Tx0M of the initial value. This simplification works for small times T>0 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 T>0 is viewed as a homotopy parameter. Certain transformations show that the degenerate time T=0 represents the force free situation of a geodesic connecting the boundary values on M. 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 Φ. 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 Φ:M×TMCC(TM) satisfyAssumption 1.1. For any ϵ>0 ,

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