Harmonic solutions of periodic Carathéodory perturbations of autonomous ODE's on manifolds
The purpose of this paper is to extend some known results to the specific case of Carathéodory perturbations of autonomous vector fields. The author studies the differential equation \[ dx/dt= g(x)+\lambda f(t, x),\quad \lambda\geq 0,\tag{1} \] on a boundary-less differentiable manifold \(M\subset \mathbb{R}^k\). \(g\) and \(f\) are tangent vector fields, \(g(x)\) is continuous, while \(f(t)\) is \(T\)-periodic and satisfies the Carathéodory conditions (Lebesgue measurability in \(x\), a.e. continuity in \(t\), and on any compact set \(K\) there exists an \(L^1_T\) function \(\gamma_K(t)\) that is \(T\)-periodic and \(|f|<\gamma_K(t)\)). The author investigates properties of \(T\)-pairs \((\lambda, x)\), where \(x\) (dependent on \(\lambda\)) is \(T\)-periodic. The set \(X\) of all \(T\)-pairs is considered as a subspace of the Banach space of \(T\)-periodic continuous functions with the max norm. An easy lemma shows that any bounded subset of \(X\) is totally bounded. Thus closed and bounded subsets of \(X\) are compact. If only local boundedness is assumed then such subsets of \(X\) are locally compact. \(C_T(M)\) denotes the metric \(M\)-subspace of the Banach space of all \(T\)-periodic continuous functions in \(\mathbb{R}^k\). The author needs the following definition. Let the Cauchy problem be defined by equation (1) with the original condition \(x(0)= p\). Suppose that there exists a \(T\)-periodic solution. Then \(p\) is called a starting point for (1). The author proves the following theorem: Let \(\Omega\) be an open set of \([0,\infty)\times C_T(M)\), and assume that the Brouwer degree \(\deg(g,\Omega\times M)\) is well defined and is not zero. Then \(\Omega\) contains a connected set \(\Gamma\) of nontrivial \(T\)-pairs to (1) whose closure in \([0,\infty)\times C_T(M)\) meets \(g^{-1}(0)\cap \Omega\) and is not contained in any compact subset of \(\Omega\). In particular, if \(\Omega= [0,\infty)\times C_T(M)\), then \(\Gamma\) is unbounded. Additional results contain some consequences of the main result, such as the existence of a \(T\)-periodic solution to system (1). An examples of the applications includes the forced motion of a simple pendulum.
scientific article; zbMATH DE number 1462158