×

Topological conditions for the existence of bounded solutions of quasihomogeneous systems. (Russian) Zbl 0585.34030

Consider the systems \[ (1)\quad \dot x=P(x,t),\quad (2)\quad \dot x=P(x,t)+X(x,t),\quad (x,t)\in {\mathbb{R}}^ n\times {\mathbb{R}}, \] where \(P\in C^ 1\) \(({\mathbb{R}}^ n\times {\mathbb{R}})\), \(P(\lambda x,t)=\lambda^ mP(x,t)\) for all \((x,t)\in {\mathbb{R}}^ n\times {\mathbb{R}}\) and \(\lambda\geq 0\), \(m>1\); P is bounded and uniformly continuous in \(\{(x,t):\| x\| <1\}\); \(X\in C^ 0({\mathbb{R}}^ n\times {\mathbb{R}})\), satisfying \(\lim_{\| x\| \to \infty}\| x\|^{-m}\| X(x,t)\| \rightrightarrows 0.\) Let \(V\in C^ 1({\mathbb{R}}^ n\times {\mathbb{R}})\), satisfying the conditions \(V(\lambda x,t)=\lambda^{\alpha}V(x,t),\) for all \((x,t)\in {\mathbb{R}}^ n\times {\mathbb{R}}\), \(\lambda\geq 0\), \(\alpha >1\); \[ (\partial V/\partial x)(x,t)\cdot P(x,t)\geq 1;\quad | V(x,t)| +| (\partial V/\partial t)(x,t)| +\| (\partial V/\partial x)(x,t)\| \leq M, \] for \(\| x\| =1\), \(t\in {\mathbb{R}}\). Define \(\hat S_+=\{(x,t):\| x\| =1\), V(x,t)\(\leq 0\}\), \(\hat S_- =\{(x,t):\| x\| =1\), V(x,t)\(\geq 0\}\), and \(S_{\pm}(t_ 0)=\hat S_{\pm}\cap \{t=t_ 0\}.\)
By applying the principle of Wazewski the following theorem is proved: Theorem I. Suppose that either \(S_+(0)\) or \(S_-(0)\), associated with system (1), is not contractible, then system (2) has at least one bounded solution.
Reviewer: B.Li

MSC:

34C11 Growth and boundedness of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: EuDML