Strongly nonlinear second-order ODE's with unilateral conditions
The paper deals with the strongly nonlinear Neumann boundary value problem \[ (N)\quad-(\varphi (u'(t)))'=f(t,u(t))-q(t),\;u'(a)=u'(b)=0, \] where \([a,b] \subset \mathbb{R}\), \(\varphi\) is an odd increasing homeomorphism on \(\mathbb{R}\) and \(f:[a,b] \times \mathbb{R} \to \mathbb{R}\) is a Carathéodory function. Solutions of \((N)\) are intended in the sense that \(u \in C^ 1([a,b])\) with \(\varphi(u')\) absolutely continuous and satisfying \((N)\) for a.e. \(t \in[a,b]\). The authors introduce so-called upper and lower \(\sigma\)-conditions to \(\varphi\): for any \(\sigma>1\), \(\limsup_{s \to \infty} {\varphi(\sigma s) \over \varphi(s)} <+\infty\), and \(\liminf_{s \to \infty} {\varphi(\sigma s) \over \varphi(s)} >1\). They define a positive constant \(\wedge_ 1\) and prove existence results for \((N)\) provided e.g. (i) \(\varphi\) satisfies the upper and lower \(\sigma\)- conditions; (ii) \(f\) satisfies the sign condition \((f(t,s)-\overline q) s \geq 0\) for a.e. \(t \in(a,b)\) and \(| s |\) large, where \(\overline q\) denotes the mean value of \(q \in L^ 1(a,b)\); (iii) \(\limsup_{s \to \infty} {f(t,s) \over \varphi(s)} \leq k\), uniformly on \([a,b]\), where \(0 \leq k< \wedge_ 1\). The proofs are based on a modification of the continuation theorem by Mawhin and on the asymptotic behaviour of certain time-maps associated to the auxiliary autonomous equation \((\varphi(u'))'+g(u)=0\).
strongly nonlinear Neumann boundary value problem
scientific article; zbMATH DE number 459191