Positive solutions for nonlinear \(m\)-point boundary value problems of Dirichlet type via fixed-point index theory
This paper deals with the \(m\)-point boundary value problem \[ u''(t)+a(t)u'(t)+b(t)u(t)+g(t)f(u(t))=0, \quad t \in [0,1], \] \[ u(0)=0, u(1)=\sum^{m-2}_{i=1}\alpha_i u(\xi_i). \] It is assumed that \(a, b : [0,1] \to \mathbb R\) are continuous, with \(b(t)\leq 0 \), and \(\xi_i\in \;]0,1[\), \(\alpha_i \in \; ]0,+\infty[\) are such that \(0<\sum^{m-2}_{i=1}\alpha_i\phi_1(\xi_i)<1\), where \(\phi_1 \) is the solution of \[ u''(t)+a(t)u'(t)+b(t)u(t) =0, \quad t \in [0,1], \quad u(0)=0, u(1)=1. \] Further, \(g : [0,1] \to [0,+\infty[\) is continuous, with \(\int_0^1 g >0\), and \(f : [0,+\infty[ \; \to [0,+\infty[\) is continuous too. Under suitable conditions on the quotient \(f(u)/u\), the authors prove the existence of at least one or at least two positive solutions. The proof is based on some results on Hammerstein integral equations obtained by \textit{K. Lan} and \textit{J. R. L. Webb} [J. Differ. Equations 148, 407--421 (1998; Zbl 0909.34013)] and \textit{K. Q. Lan} [J. Lond. Math. Soc., II. Ser. 63, No.3, 690--704 (2001; Zbl 1032.34019)].
scientific article; zbMATH DE number 2073697
scientific article