Solvability of a boundary value problem with the nonlinearity satisfying a sign condition
Let f: [0,\(\pi\) ]\(\times R\to R\) be a given function satisfying Caratheodory's conditions and the sign condition f(x,u)u\(\geq 0\) for \(x\in [0,\pi]\), \(u\in R\). Let h: [0,\(\pi\) ]\(\to R\) be a given function in \(L^ 1[0,\pi]\) such that \(\int^{\pi}_{0}h(x) \sin x dx=0.\) This paper is concerned with the existence of a solution for the boundary value problem \[ (1)\quad -u''(x)-u(x)+f(x,u(x))=h(x),\quad u(0)=u(\pi)=0, \] and the boundary value problem \[ (2)\quad u''(x)+u(x)+f(x,u(x))=h(x),\quad u(0)=u(\pi)=0, \] when f additionally satisfies \(\limsup_{| u| \to \infty}(f(x,u)/u)=\beta <3\). The boundary value problem (2) was earlier studied by \textit{J. Mawhin}, \textit{J. R. Ward} and \textit{M. Willem} [Proc. Am. Math. Soc. 93, 667-674 (1985; Zbl 0559.34014)] using variational methods when f(x,u) is non-decreasing in u and \(h(x)=0\). The methods of this paper use Leray-Schauder continuation theorem and estimates involving the use of Wirtinger type inequalities.
scientific article; zbMATH DE number 4040329
scientific article