On maximum principles in the class of oscillating functions
The paper discusses a maximum principle technique that is applicable to solutions of ordinary differential equations that might oscillate near the boundary, and thus covering an area left untouched by classical theory. The main result is Theorem 2.1: Assume that (i) \(\alpha <\beta \) are given numbers; (ii) \(a,b,c:(\alpha ,\beta )\rightarrow \mathbb{R}\) are given functions such that \(c\neq 0\) a.e., \(a/c\in W_{loc}^{1,1}(\alpha ,\beta )\), \(b/c\), \(h\in L_{\text{loc}}^1(\alpha ,\beta )\), where \(h=(a/c)^{\prime }-2(b/c)\); (iii) \(M\) is the set of functions \(u\) such that \(au^{\prime \prime }+bu^{\prime }+cu=0\) a.e. in \((\alpha ,\beta )\), \(u\in \) \(W_{\text{loc}}^{2,1}(\alpha ,\beta )\cap C[\alpha ,\beta ]\); (iv) the following conditions for \(s\in \{\alpha ,\beta \}\) are considered: \(c_1\)) \((a/c)(x)>0\) for \(x\) close to \(s\); \(c_2\)) \((a/c)(x)<0\) for \(x\) close to \(s\); \(c_3\)) lim\(_{x\rightarrow s}(a/c)(x)=0\) and \(M\subseteq \{u:\limsup_{y\rightarrow s}| u^{\prime }(y)| <+\infty \}\); \(c_4\)) \(\limsup_{x\rightarrow s}| a/c| (x)<+\infty \) and \(M\subseteq \{u:\lim_{y\rightarrow s}u^{\prime }(y)=0\}\). Then we have: (1) if \(h(x)\leq 0\) a.e. in \((\alpha ,\beta )\) and \(\alpha \) satisfies one of the conditions \(c_2),c_3),c_4)\), then the elements of \(M\) attain their maximum in \(\{\alpha ,\beta \}\). If additionally \(\beta \) satisfies one of the conditions \(c_1)\), \(c_3)\), \(c_4)\), then the elements of \(M\) attain their maximum in \(\alpha \). (2) if \(h(x)\geq 0\) a.e. in \((\alpha ,\beta )\) and \(\beta \) satisfies one of the conditions \(c_2)\), \(c_3)\), \(c_4)\), then the elements of \(M\) attain their maximum in \(\{\alpha ,\beta \}\). If additionally \(\alpha \) satisfies one of the conditions \(c_1)\), \(c_3)\), \(c_4)\), then the elements of \(M\) attain their maximum in \(\beta \). A result in the case when \(h\) changes its sign in \((\alpha ,\beta )\) is obtained in Corollary 2.1. The question of the dimension of \(M\) is addressed in Corollary 3.5. The results are applied to estimate the \(L^\infty \) norm of special functions and the conclusions of this investigation have intimate connections to the specialized literature. The paper is extremely interesting, easy to read and of upmost interest for anyone concerned with maximum principles for ODEs.
scientific article; zbMATH DE number 2173760