Carathéodory periodic perturbations of the zero vector field on manifolds
The main result of this paper can be stated as follows. Consider the equation \[ \dot x= \lambda f(t,x), \tag{1} \] where \(\lambda\geq 0\), \(f:\mathbb{R}\times M\to \mathbb{R}^k\) is a \(T\)-periodic Carathéodory tangent vector field, and \(M\) is a boundaryless smooth manifold in \(\mathbb{R}^k\). Let \(\omega(p) ={1\over T} \int^T_0 f(t,p)dt\), \(p\in M\), let \(\Omega\) be an open subset of \([0,\infty) \times C_T(\mathbb{R},M)\), and assume that \(\deg (\omega, \Omega \cap M)\neq 0\). Then the equation (1) admits in \(\Omega\) a connected set \(\Gamma\) of nontrivial \(T\)-pairs \((\lambda,x)\) where \(\lambda>0\) and \(x\) is a solution of (1) whose closure in \(\Omega\) is noncompact and meets \(\Omega \cap M\) in the set of zeros of \(\omega\). In addition, if \(M\) is a closed submanifold of \(\mathbb{R}^k\), then \(\Gamma\) cannot be contained in a bounded and complete subset of \(\Omega\).
scientific article; zbMATH DE number 1157135