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Fixed point approach to some two-point boundary value problems for differential inclusions on manifolds. (English) Zbl 1153.58004

Let \(M\) be a finite-dimensional complete Riemannian manifold and \(TM\) be its tangent bundle with the natural projection \(\pi: TM\to M\). Let \(F: R\times TM\to TM\) be a set-valued map such that for any point \((m,X)\in TM\) the relation \(\pi F(t,m,X)= \pi(m, X)= m\) holds.
The authors investigate the differential inclusion of the form
\[ {D\over dt}\dot m(t)\in F(t,m(t), \dot m(t)),\tag{1} \]
where \({D\over dt}\) is the covariant derivative of Levi-CivitĂ  connection. They deal with the two-point boundary value problem for inclusion (1). The main result of the paper is that if the following more restrictive estimate \[ \| F(t,m,X)\|\leq a(t,m)\| X\|^2 \] holds and conditions of the authors [Abstr. Appl. Anal. 2006, Article ID 30395, 9 p. (2006; Zbl 1140.34320)] are satisfied, a solution of the two-point boundary value problem for (1) exists on arbitrary finite time interval.

MSC:

58C06 Set-valued and function-space-valued mappings on manifolds
58C30 Fixed-point theorems on manifolds
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics

Citations:

Zbl 1140.34320
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